1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
3793
3794
3795
3796
3797
3798
3799
3800
3801
3802
3803
3804
3805
3806
3807
3808
3809
3810
3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
3824
3825
3826
3827
3828
3829
3830
3831
3832
3833
3834
3835
3836
3837
3838
3839
3840
3841
3842
3843
3844
3845
3846
3847
3848
3849
3850
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880
3881
3882
3883
3884
3885
3886
3887
3888
3889
3890
3891
3892
3893
3894
3895
3896
3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
3954
3955
3956
3957
3958
3959
3960
3961
3962
3963
3964
3965
3966
3967
3968
3969
3970
3971
3972
3973
3974
3975
3976
3977
3978
3979
3980
3981
3982
3983
3984
3985
3986
3987
3988
3989
3990
3991
3992
3993
3994
3995
3996
3997
3998
3999
4000
4001
4002
4003
4004
4005
4006
4007
4008
4009
4010
4011
4012
4013
4014
4015
4016
4017
4018
4019
4020
4021
4022
4023
4024
4025
4026
4027
4028
4029
4030
4031
4032
4033
4034
4035
4036
4037
4038
4039
4040
4041
4042
4043
4044
4045
4046
4047
4048
4049
4050
4051
4052
4053
4054
4055
4056
4057
4058
4059
4060
4061
4062
4063
4064
4065
4066
4067
4068
4069
4070
4071
4072
4073
4074
4075
4076
4077
4078
4079
4080
4081
4082
4083
4084
4085
4086
4087
4088
4089
4090
4091
4092
4093
4094
4095
4096
4097
4098
4099
4100
4101
4102
4103
4104
4105
4106
4107
4108
4109
4110
4111
4112
4113
4114
4115
4116
4117
4118
4119
4120
4121
4122
4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
4137
4138
4139
4140
4141
4142
4143
4144
4145
4146
4147
4148
4149
4150
4151
4152
4153
4154
4155
4156
4157
4158
4159
4160
4161
4162
4163
4164
4165
4166
4167
4168
4169
4170
4171
4172
4173
4174
4175
4176
4177
4178
4179
4180
4181
4182
4183
4184
4185
4186
4187
4188
4189
4190
4191
4192
4193
4194
4195
4196
4197
4198
4199
4200
4201
4202
4203
4204
4205
4206
4207
4208
4209
4210
4211
4212
4213
4214
4215
4216
4217
4218
4219
4220
4221
4222
4223
4224
4225
4226
4227
4228
4229
4230
4231
4232
4233
4234
4235
4236
4237
4238
4239
4240
4241
4242
4243
4244
4245
4246
4247
4248
4249
4250
4251
4252
4253
4254
4255
4256
4257
4258
4259
4260
4261
4262
4263
4264
4265
4266
4267
4268
4269
4270
4271
4272
4273
4274
4275
4276
4277
4278
4279
4280
4281
4282
4283
4284
4285
4286
4287
4288
4289
4290
4291
4292
4293
4294
4295
4296
4297
4298
4299
4300
4301
4302
4303
4304
4305
4306
4307
4308
4309
4310
4311
4312
4313
4314
4315
4316
4317
4318
4319
4320
4321
4322
4323
4324
4325
4326
4327
4328
4329
4330
4331
4332
4333
4334
4335
4336
4337
4338
4339
4340
4341
4342
4343
4344
4345
4346
4347
4348
4349
4350
4351
4352
4353
4354
4355
4356
4357
4358
4359
4360
4361
4362
4363
4364
4365
4366
4367
4368
4369
4370
4371
4372
4373
4374
4375
4376
4377
4378
4379
4380
4381
4382
4383
4384
4385
4386
4387
4388
4389
4390
4391
4392
4393
4394
4395
4396
4397
4398
4399
4400
4401
4402
4403
4404
4405
4406
4407
4408
4409
4410
4411
4412
4413
4414
4415
4416
4417
4418
4419
4420
4421
4422
4423
4424
4425
4426
4427
4428
4429
4430
4431
4432
4433
4434
4435
4436
4437
4438
4439
4440
4441
4442
4443
4444
4445
4446
4447
4448
4449
4450
4451
4452
4453
4454
4455
4456
4457
4458
4459
4460
4461
4462
4463
4464
4465
4466
4467
4468
4469
4470
4471
4472
4473
4474
4475
4476
4477
4478
4479
4480
4481
4482
4483
4484
4485
4486
4487
4488
4489
4490
4491
4492
4493
4494
4495
4496
4497
4498
4499
4500
4501
4502
4503
4504
4505
4506
4507
4508
4509
4510
4511
4512
4513
4514
4515
4516
4517
4518
4519
4520
4521
4522
4523
4524
4525
4526
4527
4528
4529
4530
4531
4532
4533
4534
4535
4536
4537
4538
4539
4540
4541
4542
4543
4544
4545
4546
4547
4548
4549
4550
4551
4552
4553
4554
4555
4556
4557
4558
4559
4560
4561
4562
4563
4564
4565
4566
4567
4568
4569
4570
4571
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
4591
4592
4593
4594
4595
4596
4597
4598
4599
4600
4601
4602
4603
4604
4605
4606
4607
4608
4609
4610
4611
4612
4613
4614
4615
4616
4617
4618
4619
4620
4621
4622
4623
4624
4625
4626
4627
4628
4629
4630
4631
4632
4633
4634
4635
4636
4637
4638
4639
4640
4641
4642
4643
4644
4645
4646
4647
4648
4649
4650
4651
4652
4653
4654
4655
4656
4657
4658
4659
4660
4661
4662
4663
4664
4665
4666
4667
4668
4669
4670
4671
4672
4673
4674
4675
4676
4677
4678
4679
4680
4681
4682
4683
4684
4685
4686
4687
4688
4689
4690
4691
4692
4693
4694
4695
4696
4697
4698
4699
4700
4701
4702
4703
4704
4705
4706
4707
4708
4709
4710
4711
4712
4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
4739
4740
4741
4742
4743
4744
4745
4746
4747
4748
4749
4750
4751
4752
4753
4754
4755
4756
4757
4758
4759
4760
4761
4762
4763
4764
4765
4766
4767
4768
4769
4770
4771
4772
4773
4774
4775
4776
4777
4778
4779
4780
4781
4782
4783
4784
4785
4786
4787
4788
4789
4790
4791
4792
4793
4794
4795
4796
4797
4798
4799
4800
4801
4802
4803
4804
4805
4806
4807
4808
4809
4810
4811
4812
4813
4814
4815
4816
4817
4818
4819
4820
4821
4822
4823
4824
4825
4826
4827
4828
4829
4830
4831
4832
4833
4834
4835
4836
4837
4838
4839
4840
4841
4842
4843
4844
4845
4846
4847
4848
4849
4850
4851
4852
4853
4854
4855
4856
4857
4858
4859
4860
4861
4862
4863
4864
4865
4866
4867
4868
4869
4870
4871
4872
4873
4874
4875
4876
4877
4878
4879
4880
4881
4882
4883
4884
4885
4886
4887
4888
4889
4890
4891
4892
4893
4894
4895
4896
4897
4898
4899
4900
4901
4902
4903
4904
4905
4906
4907
4908
4909
4910
4911
4912
4913
4914
4915
4916
4917
4918
4919
4920
4921
4922
4923
4924
4925
4926
4927
4928
4929
4930
4931
4932
4933
4934
4935
4936
4937
4938
4939
4940
4941
4942
4943
4944
4945
4946
4947
4948
4949
4950
4951
4952
4953
4954
4955
4956
4957
4958
4959
4960
4961
4962
4963
4964
4965
4966
4967
4968
4969
4970
4971
4972
4973
4974
4975
4976
4977
4978
4979
4980
4981
4982
4983
4984
4985
4986
4987
4988
4989
4990
4991
4992
4993
4994
4995
4996
4997
4998
4999
5000
5001
5002
5003
5004
5005
5006
5007
5008
5009
5010
5011
5012
5013
5014
5015
5016
5017
5018
5019
5020
5021
5022
5023
5024
5025
5026
5027
5028
5029
5030
5031
5032
5033
5034
5035
5036
5037
5038
5039
5040
5041
5042
5043
5044
5045
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
5059
5060
5061
5062
5063
5064
5065
5066
5067
5068
5069
5070
5071
5072
5073
5074
5075
5076
5077
5078
5079
5080
5081
5082
5083
5084
5085
5086
5087
5088
5089
5090
5091
5092
5093
5094
5095
5096
5097
5098
5099
5100
5101
5102
5103
5104
5105
5106
5107
5108
5109
5110
5111
5112
5113
5114
5115
5116
5117
5118
5119
5120
5121
5122
5123
5124
5125
5126
5127
5128
5129
5130
5131
5132
5133
5134
5135
5136
5137
5138
5139
5140
5141
5142
5143
5144
5145
5146
5147
5148
5149
5150
5151
5152
5153
5154
5155
5156
5157
5158
5159
5160
5161
5162
5163
5164
5165
5166
5167
5168
5169
5170
5171
5172
5173
5174
5175
5176
5177
5178
5179
5180
5181
5182
5183
5184
5185
5186
5187
5188
5189
5190
5191
5192
5193
5194
5195
5196
5197
5198
5199
5200
5201
5202
5203
5204
5205
5206
5207
5208
5209
5210
5211
5212
5213
5214
5215
5216
5217
5218
5219
5220
5221
5222
5223
5224
5225
5226
5227
5228
5229
5230
5231
5232
5233
5234
5235
5236
5237
5238
5239
5240
5241
5242
5243
5244
5245
5246
5247
5248
5249
5250
5251
5252
5253
5254
5255
5256
5257
5258
5259
5260
5261
5262
5263
5264
5265
5266
5267
5268
5269
5270
5271
5272
5273
5274
5275
5276
5277
5278
5279
5280
5281
5282
5283
5284
5285
5286
5287
5288
5289
5290
5291
5292
5293
5294
5295
5296
5297
5298
5299
5300
5301
5302
5303
5304
5305
5306
5307
5308
5309
5310
5311
5312
5313
5314
5315
5316
5317
5318
5319
5320
5321
5322
5323
5324
5325
5326
5327
5328
5329
5330
5331
5332
5333
5334
5335
5336
5337
5338
5339
5340
5341
5342
5343
5344
5345
5346
5347
5348
5349
5350
5351
5352
5353
5354
5355
5356
5357
5358
5359
5360
5361
5362
5363
5364
5365
5366
5367
5368
5369
5370
5371
5372
5373
5374
5375
5376
5377
5378
5379
5380
5381
5382
5383
5384
5385
5386
5387
5388
5389
5390
5391
5392
5393
5394
5395
5396
5397
5398
5399
5400
5401
5402
5403
5404
5405
5406
5407
5408
5409
5410
5411
5412
5413
5414
5415
5416
5417
5418
5419
5420
5421
5422
5423
5424
5425
5426
5427
5428
5429
5430
5431
5432
5433
5434
5435
5436
5437
5438
5439
5440
5441
5442
5443
5444
5445
5446
5447
5448
5449
5450
5451
5452
5453
5454
5455
5456
5457
5458
5459
5460
5461
5462
5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
5475
5476
5477
5478
5479
5480
5481
5482
5483
5484
5485
5486
5487
5488
5489
5490
5491
5492
5493
5494
5495
5496
5497
5498
5499
5500
5501
5502
5503
5504
5505
5506
5507
5508
5509
5510
5511
5512
5513
5514
5515
5516
5517
5518
5519
5520
5521
5522
5523
5524
5525
5526
5527
5528
5529
5530
5531
5532
5533
5534
5535
5536
5537
5538
5539
5540
5541
5542
5543
5544
5545
5546
5547
5548
5549
5550
5551
5552
5553
5554
5555
5556
5557
5558
5559
5560
5561
5562
5563
5564
5565
5566
5567
5568
5569
5570
5571
5572
5573
5574
5575
5576
5577
5578
5579
5580
5581
5582
5583
5584
5585
5586
5587
5588
5589
5590
5591
5592
5593
5594
5595
5596
5597
5598
5599
5600
5601
5602
5603
5604
5605
5606
5607
5608
5609
5610
5611
5612
5613
5614
5615
5616
5617
5618
5619
5620
5621
5622
5623
5624
5625
5626
5627
5628
5629
5630
5631
5632
5633
5634
5635
5636
5637
5638
5639
5640
5641
5642
5643
5644
5645
5646
5647
5648
5649
5650
5651
5652
5653
5654
5655
5656
5657
5658
5659
5660
5661
5662
5663
5664
5665
5666
5667
5668
5669
5670
5671
5672
5673
5674
5675
5676
5677
5678
5679
5680
5681
5682
5683
5684
5685
5686
5687
5688
5689
5690
5691
5692
5693
5694
5695
5696
5697
5698
5699
5700
5701
5702
5703
5704
5705
5706
5707
5708
5709
5710
5711
5712
5713
5714
5715
5716
5717
5718
5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
5729
5730
5731
5732
5733
5734
5735
5736
5737
5738
5739
5740
5741
5742
5743
5744
5745
5746
5747
5748
5749
5750
5751
5752
5753
5754
5755
5756
5757
5758
5759
5760
5761
5762
5763
5764
5765
5766
5767
5768
5769
5770
5771
5772
5773
5774
5775
5776
5777
5778
5779
5780
5781
5782
5783
5784
5785
5786
5787
5788
5789
5790
5791
5792
5793
5794
5795
5796
5797
5798
5799
5800
5801
5802
5803
5804
5805
5806
5807
5808
5809
5810
5811
5812
5813
5814
5815
5816
5817
5818
5819
5820
5821
5822
5823
5824
5825
5826
5827
5828
5829
5830
5831
5832
5833
5834
5835
5836
5837
5838
5839
5840
5841
5842
5843
5844
5845
5846
5847
5848
5849
5850
5851
5852
5853
5854
5855
5856
5857
5858
5859
5860
5861
5862
5863
5864
5865
5866
5867
5868
5869
5870
5871
5872
5873
5874
5875
5876
5877
5878
5879
5880
5881
5882
5883
5884
5885
5886
5887
5888
5889
5890
5891
5892
5893
5894
5895
5896
5897
5898
5899
5900
5901
5902
5903
5904
5905
5906
5907
5908
5909
5910
5911
5912
5913
5914
5915
5916
5917
5918
5919
5920
5921
5922
5923
5924
5925
5926
5927
5928
5929
5930
5931
5932
5933
5934
5935
5936
5937
5938
5939
5940
5941
5942
5943
5944
5945
5946
5947
5948
5949
5950
5951
5952
5953
5954
5955
5956
5957
5958
5959
5960
5961
5962
5963
5964
5965
5966
5967
5968
5969
5970
5971
5972
5973
5974
5975
5976
5977
5978
5979
5980
5981
5982
5983
5984
5985
5986
5987
5988
5989
5990
5991
5992
5993
5994
5995
5996
5997
5998
5999
6000
6001
6002
6003
6004
6005
6006
6007
6008
6009
6010
6011
6012
6013
6014
6015
6016
6017
6018
6019
6020
6021
6022
6023
6024
6025
6026
6027
6028
6029
6030
6031
6032
6033
6034
6035
6036
6037
6038
6039
6040
6041
6042
6043
6044
6045
6046
6047
6048
6049
6050
6051
6052
6053
6054
6055
6056
6057
6058
6059
6060
6061
6062
6063
6064
6065
6066
6067
6068
6069
6070
6071
6072
6073
6074
6075
6076
6077
6078
6079
6080
6081
6082
6083
6084
6085
6086
6087
6088
6089
6090
6091
6092
6093
6094
6095
6096
6097
6098
6099
6100
6101
6102
6103
6104
6105
6106
6107
6108
6109
6110
6111
6112
6113
6114
6115
6116
6117
6118
6119
6120
6121
6122
6123
6124
6125
6126
6127
6128
6129
6130
6131
6132
6133
6134
6135
6136
6137
6138
6139
6140
6141
6142
6143
6144
6145
6146
6147
6148
6149
6150
6151
6152
6153
6154
6155
6156
6157
6158
6159
6160
6161
6162
6163
6164
6165
6166
6167
6168
6169
6170
6171
6172
6173
6174
6175
6176
6177
6178
6179
6180
6181
6182
6183
6184
6185
6186
6187
6188
6189
6190
6191
6192
6193
6194
6195
6196
6197
6198
6199
6200
6201
6202
6203
6204
6205
6206
6207
6208
6209
6210
6211
6212
6213
6214
6215
6216
6217
6218
6219
6220
6221
6222
6223
6224
6225
6226
6227
6228
6229
6230
6231
6232
6233
6234
6235
6236
6237
6238
6239
6240
6241
6242
6243
6244
6245
6246
6247
6248
6249
6250
6251
6252
6253
6254
6255
6256
6257
6258
6259
6260
6261
6262
6263
6264
6265
6266
6267
6268
6269
6270
6271
6272
6273
6274
6275
6276
6277
6278
6279
6280
6281
6282
6283
6284
|
<!DOCTYPE html PUBLIC '-//W3C//DTD XHTML 1.0 Transitional//EN' 'http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd'><html xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>OWL 2 Web Ontology Language RDF-Based Semantics</title>
<meta content="text/html; charset=utf-8" http-equiv="Content-Type" />
<style type="text/css">
.editsection { display: none; }
</style>
<link href="owl.css" rel="stylesheet" type="text/css" />
<link href="http://www.w3.org/StyleSheets/TR/W3C-REC" rel="stylesheet" type="text/css" />
<script src="http://www.w3.org/2007/OWL/toggles.js" type="text/javascript"></script>
</head>
<body>
<div class="head">
<a href="http://www.w3.org/"><img alt="W3C" height="48" src="../../../Icons/w3c_home" width="72" /></a><h1 id="title" style="clear:both">OWL 2 Web Ontology Language <br /><span id="short-title">RDF-Based Semantics</span></h1>
<h2 id="W3C-doctype">W3C Recommendation 27 October 2009</h2>
<!-- no inplace warning -->
<dl>
<dt>This version:</dt>
<dd><a href="index.html" id="this-version-url">http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/</a></dd>
<dt>Latest version (series 2):</dt>
<dd><a href="http://www.w3.org/TR/owl2-rdf-based-semantics/">http://www.w3.org/TR/owl2-rdf-based-semantics/</a></dd>
<dt>Latest Recommendation:</dt>
<dd><a href="http://www.w3.org/TR/owl-rdf-based-semantics">http://www.w3.org/TR/owl-rdf-based-semantics</a></dd>
<dt>Previous version:</dt>
<dd><a href="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/">http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/</a> (<a href="http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/diff-from-20090922">color-coded diff</a>)</dd>
</dl>
<dl><dt>Editors:</dt><dd><a href="http://www.fzi.de/michael.schneider">Michael Schneider</a>, FZI Research Center for Information Technology</dd>
<dt>Contributors: (in alphabetical order)</dt><dd><a href="http://semanticweb.org/wiki/Jeremy_J._Carroll">Jeremy Carroll</a>, HP (now at TopQuadrant)</dd>
<dd><a href="http://www.w3.org/People/Ivan/">Ivan Herman</a>, W3C/ERCIM</dd>
<dd><a href="http://ect.bell-labs.com/who/pfps/">Peter F. Patel-Schneider</a>, Bell Labs Research, Alcatel-Lucent</dd>
</dl>
<p>Please refer to the <a href="http://www.w3.org/2007/OWL/errata"><strong>errata</strong></a> for this document, which may include some normative corrections.</p>
<p>This document is also available in these non-normative formats: <a href="http://www.w3.org/2009/pdf/REC-owl2-rdf-based-semantics-20091027.pdf">PDF version</a>.</p>
<p>See also <a href="http://www.w3.org/2007/OWL/translation/owl2-rdf-based-semantics">translations</a>.</p>
<p class="copyright"><a href="http://www.w3.org/Consortium/Legal/ipr-notice#Copyright">Copyright</a> © 2009 <a href="http://www.w3.org/"><acronym title="World Wide Web Consortium">W3C</acronym></a><sup>®</sup> (<a href="http://www.csail.mit.edu/"><acronym title="Massachusetts Institute of Technology">MIT</acronym></a>, <a href="http://www.ercim.org/"><acronym title="European Research Consortium for Informatics and Mathematics">ERCIM</acronym></a>, <a href="http://www.keio.ac.jp/">Keio</a>), All Rights Reserved. W3C <a href="http://www.w3.org/Consortium/Legal/ipr-notice#Legal_Disclaimer">liability</a>, <a href="http://www.w3.org/Consortium/Legal/ipr-notice#W3C_Trademarks">trademark</a> and <a href="http://www.w3.org/Consortium/Legal/copyright-documents">document use</a> rules apply.</p>
</div>
<hr />
<h2><a id="abstract" name="abstract">Abstract</a></h2>
<div>
<div><p>The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally defined meaning. OWL 2 ontologies provide classes, properties, individuals, and data values and are stored as Semantic Web documents. OWL 2 ontologies can be used along with information written in RDF, and OWL 2 ontologies themselves are primarily exchanged as RDF documents. The OWL 2 <a href="../REC-owl2-overview-20091027/index.html" title="Document Overview">Document Overview</a> describes the overall state of OWL 2, and should be read before other OWL 2 documents.</p><p>This document defines the RDF-compatible model-theoretic semantics of OWL 2.</p></div>
</div>
<h2 class="no-toc no-num">
<a id="status" name="status">Status of this Document</a>
</h2>
<h4 class="no-toc no-num" id="may-be">May Be Superseded</h4>
<p><em>This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the <a href="http://www.w3.org/TR/">W3C technical reports index</a> at http://www.w3.org/TR/.</em></p>
<!-- no eventStatusExtra -->
<!-- no statusExtra -->
<div>
<h4 class="no-toc no-num" id="sotd-xml-dep">XML Schema Datatypes Dependency</h4>
<p>OWL 2 is defined to use datatypes defined in the <a href="http://www.w3.org/TR/xmlschema-2/">XML Schema Definition Language (XSD)</a>. As of this writing, the latest W3C Recommendation for XSD is version 1.0, with <a href="http://www.w3.org/TR/xmlschema11-1/">version 1.1</a> progressing toward Recommendation. OWL 2 has been designed to take advantage of the new datatypes and clearer explanations available in XSD 1.1, but for now those advantages are being partially put on hold. Specifically, until XSD 1.1 becomes a W3C Recommendation, the elements of OWL 2 which are based on it should be considered <em>optional</em>, as detailed in <a href="../REC-owl2-conformance-20091027/index.html#XML_Schema_Datatypes">Conformance, section 2.3</a>. Upon the publication of XSD 1.1 as a W3C Recommendation, those elements cease to be optional and are to be considered required as otherwise specified.</p>
<p>We suggest that for now developers and users follow the <a href="http://www.w3.org/TR/2009/CR-xmlschema11-1-20090430/">XSD 1.1 Candidate Recommendation</a>. Based on discussions between the Schema and OWL Working Groups, we do not expect any implementation changes will be necessary as XSD 1.1 advances to Recommendation.</p>
</div>
<h4 class="no-toc no-num" id="status-changes">Summary of Changes</h4>
<div>There have been no <a href="http://www.w3.org/2005/10/Process-20051014/tr#substantive-change">substantive</a> changes since the <a href="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/">previous version</a>. For details on the minor changes see the <a href="index.html#changelog">change log</a> and <a href="http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/diff-from-20090922">color-coded diff</a>.</div>
<h4 class="no-toc no-num" id="please">Please Send Comments</h4><p>Please send any comments to <a class="mailto" href="mailto:public-owl-comments@w3.org">public-owl-comments@w3.org</a>
(<a class="http" href="http://lists.w3.org/Archives/Public/public-owl-comments/">public
archive</a>). Although work on this document by the <a href="http://www.w3.org/2007/OWL/">OWL Working Group</a> is complete, comments may be addressed in the <a href="http://www.w3.org/2007/OWL/errata">errata</a> or in future revisions. Open discussion among developers is welcome at <a class="mailto" href="mailto:public-owl-dev@w3.org">public-owl-dev@w3.org</a> (<a class="http" href="http://lists.w3.org/Archives/Public/public-owl-dev/">public archive</a>).</p>
<h4 class="no-toc no-num" id="endorsement">Endorsed By W3C</h4>
<p><em>This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.</em></p>
<h4 class="no-toc no-num" id="patents">Patents</h4>
<p><em>This document was produced by a group operating under the <a href="http://www.w3.org/Consortium/Patent-Policy-20040205/">5 February 2004 W3C Patent Policy</a>. W3C maintains a <a href="http://www.w3.org/2004/01/pp-impl/41712/status" rel="disclosure">public list of any patent disclosures</a> made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent.</em></p>
<hr title="Separator After Status Section" />
<table class="toc" id="toc" summary="Contents"><tr><td><div id="toctitle"><h2>Table of Contents</h2></div>
<ul>
<li class="toclevel-1"><a href="index.html#Introduction_.28Informative.29"><span class="tocnumber">1</span> <span class="toctext">Introduction (Informative)</span></a></li>
<li class="toclevel-1"><a href="index.html#Ontologies"><span class="tocnumber">2</span> <span class="toctext">Ontologies</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Syntax"><span class="tocnumber">2.1</span> <span class="toctext">Syntax</span></a></li>
<li class="toclevel-2"><a href="index.html#Content_of_Ontologies_.28Informative.29"><span class="tocnumber">2.2</span> <span class="toctext">Content of Ontologies (Informative)</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Vocabulary"><span class="tocnumber">3</span> <span class="toctext">Vocabulary</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Standard_Prefixes"><span class="tocnumber">3.1</span> <span class="toctext">Standard Prefixes</span></a></li>
<li class="toclevel-2"><a href="index.html#Vocabulary_Terms"><span class="tocnumber">3.2</span> <span class="toctext">Vocabulary Terms</span></a></li>
<li class="toclevel-2"><a href="index.html#Datatype_Names"><span class="tocnumber">3.3</span> <span class="toctext">Datatype Names</span></a></li>
<li class="toclevel-2"><a href="index.html#Facet_Names"><span class="tocnumber">3.4</span> <span class="toctext">Facet Names</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Interpretations"><span class="tocnumber">4</span> <span class="toctext">Interpretations</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Datatype_Maps"><span class="tocnumber">4.1</span> <span class="toctext">Datatype Maps</span></a></li>
<li class="toclevel-2"><a href="index.html#Vocabulary_Interpretations"><span class="tocnumber">4.2</span> <span class="toctext">Vocabulary Interpretations</span></a></li>
<li class="toclevel-2"><a href="index.html#Satisfaction.2C_Consistency_and_Entailment"><span class="tocnumber">4.3</span> <span class="toctext">Satisfaction, Consistency and Entailment</span></a></li>
<li class="toclevel-2"><a href="index.html#Parts_of_the_Universe"><span class="tocnumber">4.4</span> <span class="toctext">Parts of the Universe</span></a></li>
<li class="toclevel-2"><a href="index.html#Class_Extensions"><span class="tocnumber">4.5</span> <span class="toctext">Class Extensions</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Semantic_Conditions"><span class="tocnumber">5</span> <span class="toctext">Semantic Conditions</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_the_Parts_of_the_Universe"><span class="tocnumber">5.1</span> <span class="toctext">Semantic Conditions for the Parts of the Universe</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes"><span class="tocnumber">5.2</span> <span class="toctext">Semantic Conditions for the Vocabulary Classes</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties"><span class="tocnumber">5.3</span> <span class="toctext">Semantic Conditions for the Vocabulary Properties</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Boolean_Connectives"><span class="tocnumber">5.4</span> <span class="toctext">Semantic Conditions for Boolean Connectives</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Enumerations"><span class="tocnumber">5.5</span> <span class="toctext">Semantic Conditions for Enumerations</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Property_Restrictions"><span class="tocnumber">5.6</span> <span class="toctext">Semantic Conditions for Property Restrictions</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Datatype_Restrictions"><span class="tocnumber">5.7</span> <span class="toctext">Semantic Conditions for Datatype Restrictions</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_the_RDFS_Vocabulary"><span class="tocnumber">5.8</span> <span class="toctext">Semantic Conditions for the RDFS Vocabulary</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Equivalence_and_Disjointness"><span class="tocnumber">5.9</span> <span class="toctext">Semantic Conditions for Equivalence and Disjointness</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_N-ary_Disjointness"><span class="tocnumber">5.10</span> <span class="toctext">Semantic Conditions for N-ary Disjointness</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Sub_Property_Chains"><span class="tocnumber">5.11</span> <span class="toctext">Semantic Conditions for Sub Property Chains</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Inverse_Properties"><span class="tocnumber">5.12</span> <span class="toctext">Semantic Conditions for Inverse Properties</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Property_Characteristics"><span class="tocnumber">5.13</span> <span class="toctext">Semantic Conditions for Property Characteristics</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Keys"><span class="tocnumber">5.14</span> <span class="toctext">Semantic Conditions for Keys</span></a></li>
<li class="toclevel-2"><a href="index.html#Semantic_Conditions_for_Negative_Property_Assertions"><span class="tocnumber">5.15</span> <span class="toctext">Semantic Conditions for Negative Property Assertions</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Appendix:_Axiomatic_Triples_.28Informative.29"><span class="tocnumber">6</span> <span class="toctext">Appendix: Axiomatic Triples (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Axiomatic_Triples_in_RDF"><span class="tocnumber">6.1</span> <span class="toctext">Axiomatic Triples in RDF</span></a></li>
<li class="toclevel-2"><a href="index.html#Axiomatic_Triples_for_the_Vocabulary_Classes"><span class="tocnumber">6.2</span> <span class="toctext">Axiomatic Triples for the Vocabulary Classes</span></a></li>
<li class="toclevel-2"><a href="index.html#Axiomatic_Triples_for_the_Vocabulary_Properties"><span class="tocnumber">6.3</span> <span class="toctext">Axiomatic Triples for the Vocabulary Properties</span></a></li>
<li class="toclevel-2"><a href="index.html#A_Set_of_Axiomatic_Triples"><span class="tocnumber">6.4</span> <span class="toctext">A Set of Axiomatic Triples</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Appendix:_Relationship_to_the_Direct_Semantics_.28Informative.29"><span class="tocnumber">7</span> <span class="toctext">Appendix: Relationship to the Direct Semantics (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Example_on_Semantic_Differences"><span class="tocnumber">7.1</span> <span class="toctext">Example on Semantic Differences</span></a></li>
<li class="toclevel-2"><a href="index.html#Correspondence_Theorem"><span class="tocnumber">7.2</span> <span class="toctext">Correspondence Theorem</span></a></li>
<li class="toclevel-2"><a href="index.html#Proof_for_the_Correspondence_Theorem"><span class="tocnumber">7.3</span> <span class="toctext">Proof for the Correspondence Theorem</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Appendix:_Comprehension_Conditions_.28Informative.29"><span class="tocnumber">8</span> <span class="toctext">Appendix: Comprehension Conditions (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Comprehension_Conditions_for_Sequences"><span class="tocnumber">8.1</span> <span class="toctext">Comprehension Conditions for Sequences</span></a></li>
<li class="toclevel-2"><a href="index.html#Comprehension_Conditions_for_Boolean_Connectives"><span class="tocnumber">8.2</span> <span class="toctext">Comprehension Conditions for Boolean Connectives</span></a></li>
<li class="toclevel-2"><a href="index.html#Comprehension_Conditions_for_Enumerations"><span class="tocnumber">8.3</span> <span class="toctext">Comprehension Conditions for Enumerations</span></a></li>
<li class="toclevel-2"><a href="index.html#Comprehension_Conditions_for_Property_Restrictions"><span class="tocnumber">8.4</span> <span class="toctext">Comprehension Conditions for Property Restrictions</span></a></li>
<li class="toclevel-2"><a href="index.html#Comprehension_Conditions_for_Datatype_Restrictions"><span class="tocnumber">8.5</span> <span class="toctext">Comprehension Conditions for Datatype Restrictions</span></a></li>
<li class="toclevel-2"><a href="index.html#Comprehension_Conditions_for_Inverse_Properties"><span class="tocnumber">8.6</span> <span class="toctext">Comprehension Conditions for Inverse Properties</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Appendix:_Changes_from_OWL_1_.28Informative.29"><span class="tocnumber">9</span> <span class="toctext">Appendix: Changes from OWL 1 (Informative)</span></a></li>
<li class="toclevel-1"><a href="index.html#Appendix:_Change_Log_.28Informative.29"><span class="tocnumber">10</span> <span class="toctext">Appendix: Change Log (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Changes_Since_Proposed_Recommendation"><span class="tocnumber">10.1</span> <span class="toctext">Changes Since Proposed Recommendation</span></a></li>
<li class="toclevel-2"><a href="index.html#Changes_Since_Candidate_Recommendation"><span class="tocnumber">10.2</span> <span class="toctext">Changes Since Candidate Recommendation</span></a></li>
<li class="toclevel-2"><a href="index.html#Changes_Since_Last_Call"><span class="tocnumber">10.3</span> <span class="toctext">Changes Since Last Call</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="index.html#Acknowledgments"><span class="tocnumber">11</span> <span class="toctext">Acknowledgments</span></a></li>
<li class="toclevel-1"><a href="index.html#References"><span class="tocnumber">12</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2"><a href="index.html#Normative_References"><span class="tocnumber">12.1</span> <span class="toctext">Normative References</span></a></li>
<li class="toclevel-2"><a href="index.html#Nonnormative_References"><span class="tocnumber">12.2</span> <span class="toctext">Nonnormative References</span></a></li>
</ul>
</li>
</ul>
</td></tr></table><script type="text/javascript"> if (window.showTocToggle) { var tocShowText = "show"; var tocHideText = "hide"; showTocToggle(); } </script>
<p><br />
</p>
<a name="Introduction_.28Informative.29"></a><h2> <span class="mw-headline">1 Introduction (Informative) </span></h2>
<div id="topic-intro-purpose"></div>
<p>This document defines the RDF-compatible model-theoretic semantics of OWL 2,
referred to as the <i>"OWL 2 RDF-Based Semantics"</i>.
The OWL 2 RDF-Based Semantics gives a formal meaning
to every <a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-Graph-syntax" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax"><i>RDF graph</i></a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]
and is fully compatible with the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/"><i>RDF Semantics specification</i></a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
The specification provided here
is the successor to
the original <a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html"><i>OWL 1 RDF-Compatible Semantics</i> specification</a>
[<cite><a href="index.html#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>].
</p>
<div id="topic-intro-rdfcompatible"></div>
<p>Technically,
the OWL 2 RDF-Based Semantics
is defined as a
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DefSemanticExtension" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension"><i>semantic extension</i></a>
of
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#dtype_interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#dtype_interp">"D-Entailment"</a>
(RDFS with datatype support),
as specified in the <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
In other words,
the meaning given to an RDF graph by the OWL 2 RDF-Based Semantics
includes the meaning provided by the semantics of RDFS with datatypes,
and additional meaning is specified for all the language constructs of OWL 2,
such as Boolean connectives,
sub property chains
and qualified cardinality restrictions
(see the <a href="../REC-owl2-syntax-20091027/index.html" title="Syntax">OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for further information
on all the language constructs of OWL 2).
The definition of the semantics for the extra constructs
follows the design principles
as applied to the RDF Semantics.
</p>
<div id="topic-intro-documentcontent"></div>
<p>The content of this document is not meant to be self-contained
but builds on top of the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics document</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
by adding those aspects
that are specific to OWL 2.
Hence,
the complete definition of the OWL 2 RDF-Based Semantics
is given by
the <i>combination</i> of both
the RDF Semantics document
and the document at hand.
In particular,
the terminology used in the RDF Semantics
is reused here
except for cases
where a conflict exists with the rest of the OWL 2 specification.
</p>
<div id="topic-intro-outline"></div>
<p>The remainder of this section
provides an overview
of some of the distinguishing features
of the OWL 2 RDF-Based Semantics
and outlines the document's structure and content.
</p>
<div id="topic-intro-ontologies"></div>
<p>In <a href="index.html#Ontologies" title="">Section 2</a>,
the <i>syntax</i>
over which the OWL 2 RDF-Based Semantics is defined
is the set of all
<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-Graph-syntax" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax"><i>RDF graphs</i></a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>].
The OWL 2 RDF-Based Semantics
provides a precise formal meaning
for every RDF graph.
The language
that is determined
by RDF graphs
being interpreted using the OWL 2 RDF-Based Semantics
is called
<i>"OWL 2 Full"</i>.
In this document,
RDF graphs are also called
<i>"OWL 2 Full ontologies"</i>,
or simply <i>"ontologies"</i>,
unless there is risk of confusion.
</p>
<div id="topic-intro-vocabulary"></div>
<p>The OWL 2 RDF-Based Semantics
interprets the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defRDFV" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defRDFV">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defRDFSV" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defRDFSV">RDFS <i>vocabularies</i></a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
and the <i>OWL 2 RDF-Based vocabulary</i>
together with an extended set of <i>datatypes</i>
and their constraining <i>facets</i>
(see <a href="index.html#Vocabulary" title="">Section 3</a>).
</p>
<div id="topic-intro-interpretation"></div>
<p><i>OWL 2 RDF-Based interpretations</i>
(<a href="index.html#Interpretations" title="">Section 4</a>)
are defined on a <i>universe</i>
(see <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp">Section 1.3 of the RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
for an overview of
the basic intuition of model-theoretic semantics).
The universe is divided into <i>parts</i>,
namely <i>individuals</i>, <i>classes</i>, and <i>properties</i>,
which are identified with their RDF counterparts
(see <a href="index.html#fig-partshierarchy" title="">Figure 1</a>).
The part of individuals equals the whole universe.
This means
that all classes and properties are also
individuals in their own right.
Further,
every name interpreted by an OWL 2 RDF-Based interpretation
denotes an individual.
</p>
<div id="topic-intro-subparts"></div>
<p>The three basic parts are divided into further parts as follows.
The part of individuals subsumes the part of <i>data values</i>,
which comprises the denotations of all literals.
Also subsumed by the individuals is the part of <i>ontologies</i>.
The part of classes subsumes the part of <i>datatypes</i>,
which are classes
consisting entirely of data values.
Finally,
the part of properties subsumes the parts of
<i>object properties</i>,
<i>data properties</i>,
<i>ontology properties</i>
and <i>annotation properties</i>.
The part of object properties equals the whole part of properties,
and therefore all other kinds of properties are also object properties.
</p>
<div id="topic-intro-annotations"></div>
<p>For <i>annotations properties</i>
note that annotations are not "semantic-free"
under the OWL 2 RDF-Based Semantics.
Just like every other triple or set of triples occurring in an RDF graph,
an annotation is assigned a truth value by any given OWL 2 RDF-Based interpretation.
Hence,
although annotations are meant to be "semantically weak",
i.e., their formal meaning does not significantly exceed
that originating from the RDF Semantics specification,
adding an annotation
may still change the meaning of an ontology.
A similar discussion holds for statements
that are built from <i>ontology properties</i>,
such as <span class="name">owl:imports</span>,
which are used to define relationships between two ontologies.
</p>
<div id="topic-intro-extensions"></div>
<p>Every class represents a specific set of individuals,
called the <i>class extension</i> of the class:
an individual <i>a</i> is an instance of a class <i>C</i>,
if <i>a</i> is a member of the class extension ICEXT(<i>C</i>).
Since a class is itself an individual under the OWL 2 RDF-Based Semantics,
classes are distinguished from their respective class extensions.
This distinction allows,
for example,
that a class may be an instance of itself
by being a member of its own class extension.
Also,
two classes may be equivalent
by sharing the same class extension,
although being different individuals,
e.g., they do not need to share the same properties.
Similarly,
every property has an associated <i>property extension</i>
that consists of pairs of individuals:
an individual <i>a<sub>1</sub></i>
has a relationship to an individual <i>a<sub>2</sub></i>
with respect to a property <i>p</i>
if the pair
( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> )
is a member of the property extension IEXT(<i>p</i>).
Again, properties are distinguished from their property extensions.
In general,
if there are no further constraints,
an arbitrary extension may be associated with
a given class or property,
and two interpretations may associate
distinct extensions
with the same class or property.
</p>
<div id="topic-intro-roleplay"></div>
<p>Individuals may <i>play different "roles"</i>.
For example,
an individual can be
both a data property and an annotation property,
since the different parts of the universe
of an OWL 2 RDF-Based interpretation
are not required to be mutually disjoint,
or an individual can be
both a class and a property
by associating
both a class extension and a property extension
with it.
In the latter case
there will be no specific relationship
between the class extension and the property extension
of such an individual
without further constraints.
For example,
the same individual
can have an empty class extension
while having a nonempty property extension.
</p>
<div id="topic-intro-conditions"></div>
<p>The main part of the OWL 2 RDF-Based Semantics is <a href="index.html#Semantic_Conditions" title="">Section 5</a>,
which specifies
a formal meaning for all the OWL 2 language constructs
by means of the
<i>OWL 2 RDF-Based semantic conditions</i>.
These semantic conditions extend all the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">semantic conditions given in the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
The OWL 2 RDF-Based semantic conditions effectively determine
which sets of RDF triples are assigned a specific meaning
and what this meaning is.
For example,
semantic conditions exist
that allow one to interpret the triple
"<i>C</i> <span class="name">owl:disjointWith</span> <i>D</i>"
to mean that the denotations of the IRIs
<i>C</i> and <i>D</i>
have disjoint class extensions.
</p>
<div id="topic-intro-localization"></div>
<p>There is usually no need to provide <i>localizing information</i>
(e.g., by means of "typing triples")
for the IRIs occurring in an ontology.
As for the RDF Semantics,
the OWL 2 RDF-Based semantic conditions have been designed
to ensure that the denotation of any IRI
will be in the appropriate part of the universe.
For example,
the RDF triple
"<i>C</i> <span class="name">owl:disjointWith</span> <i>D</i>"
is sufficient to deduce that
the denotations of the IRIs
<i>C</i> and <i>D</i>
are actually <i>classes</i>.
It is not necessary to explicitly add additional typing triples
"<i>C</i> <span class="name">rdf:type rdfs:Class</span>"
and
"<i>D</i> <span class="name">rdf:type rdfs:Class</span>"
to the ontology.
</p>
<div id="topic-intro-axiomatic"></div>
<p>In the RDF Semantics,
this kind of "automatic localization"
was to some extent achieved by so called
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFS_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples"><i>"axiomatic triples"</i></a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
such as
"<span class="name">rdf:type rdf:type rdf:Property</span>"
or
"<span class="name">rdf:type rdfs:domain rdfs:Resource</span>".
However,
there is no explicit normative collection
of additional axiomatic triples
for the OWL 2 RDF-Based Semantics;
instead,
the specific axiomatic aspects of the OWL 2 RDF-Based Semantics
are determined by a subset of the OWL 2 RDF-Based semantic conditions.
<a href="index.html#Appendix:_Axiomatic_Triples_.28Informative.29" title="">Section 6</a>
discusses axiomatic triples in general
and provides an example set of axiomatic triples
that is compatible with the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-intro-correspondence"></div>
<p><a href="index.html#Appendix:_Relationship_to_the_Direct_Semantics_.28Informative.29" title="">Section 7</a> compares
the OWL 2 RDF-Based Semantics
with the <a href="../REC-owl2-direct-semantics-20091027/index.html" title="Direct Semantics"><i>OWL 2 Direct Semantics</i></a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
While
the OWL 2 RDF-Based Semantics is based on the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
the OWL 2 Direct Semantics
is a <i>description logic</i> style semantics.
Several fundamental differences
exist between the two semantics,
but
there is also a strong relationship
basically stating that the OWL 2 RDF-Based Semantics
is able to reflect all logical conclusions
of the OWL 2 Direct Semantics.
This means that the OWL 2 Direct Semantics
can
in a sense
be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.
The precise relationship is given by the
<a href="index.html#Correspondence_Theorem" title=""><i>OWL 2 correspondence theorem</i></a>.
</p>
<div id="topic-intro-changes"></div>
<p>Significant effort has been spent
in keeping the design of the OWL 2 RDF-Based Semantics
as close as possible
to that of the original specification of the
<a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html"><i>OWL 1 RDF-Compatible Semantics</i></a>
[<cite><a href="index.html#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>].
While this aim was achieved to a large degree,
the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects.
In most cases,
this is because of serious technical problems
that would have arisen
from a conservative
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DefSemanticExtension" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension">semantic extension</a>.
One important change is that
while so called <i>"comprehension conditions"</i>
for the OWL 2 RDF-Based Semantics
(see <a href="index.html#Appendix:_Comprehension_Conditions_.28Informative.29" title="">Section 8</a>)
still exist,
these are <i>not</i> part of the
normative set of semantic conditions anymore.
The OWL 2 RDF-Based Semantics also corrects several errors of OWL 1.
A list of differences between the two languages is given in
<a href="index.html#Appendix:_Changes_from_OWL_1_.28Informative.29" title="">Section 9</a>.
</p>
<div id="topic-intro-rfc2119"></div>
<p>The italicized keywords <em class="RFC2119" title="MUST in RFC 2119 context">MUST</em>, <em class="RFC2119" title="MUST NOT in RFC 2119 context">MUST NOT</em>, <em class="RFC2119" title="SHOULD in RFC 2119 context">SHOULD</em>, <em class="RFC2119" title="SHOULD NOT in RFC 2119 context">SHOULD NOT</em>, and <em class="RFC2119" title="MAY in RFC 2119 context">MAY</em> are used to specify normative features of OWL 2 documents and tools, and are interpreted as specified in RFC 2119 [<cite><a href="index.html#ref-rfc-2119" title="">RFC 2119</a></cite>].
</p>
<div class="image left" id="fig-partshierarchy">
<p><img alt="Parts Hierarchy of the OWL 2 RDF-Based Semantics. Each node is labeled with a class IRI that represents a part of the universe of an OWL 2 RDF-based interpretation. Arrows point from parts to their super parts." border="0" height="318" src="Owl2RdfBasedSemanticsPartsHierarchy.png" width="600" /><br />
<span class="caption">Figure 1: Parts Hierarchy of the OWL 2 RDF-Based Semantics</span><br />
Each <i>node</i> is labeled with a class IRI
that represents a part of the universe
of an OWL 2 RDF-based interpretation.
<i>Arrows</i> point from parts to their super parts.
</p>
</div>
<a name="Ontologies"></a><h2> <span class="mw-headline">2 Ontologies </span></h2>
<p>This section determines the <i>syntax</i>
for the OWL 2 RDF-Based Semantics,
and gives an overview on typical <i>content of ontologies</i>
for ontology management tasks.
</p>
<a name="Syntax"></a><h3> <span class="mw-headline">2.1 Syntax </span></h3>
<p><span id="topic-ont-rdfgraph"></span>
</p><p>Following <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#graphsyntax" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#graphsyntax">Sections 0.2 and 0.3 of the RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
the OWL 2 RDF-Based Semantics
is defined on every <i><b>RDF graph</b></i>
(<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-rdf-graph" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-rdf-graph">Section 6.2 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]),
i.e. on every set of <i><b>RDF triples</b></i>
(<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-triples" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples">Section 6.1 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]).
</p><p><span id="topic-ont-iri"></span>
</p><p>In accordance with the rest of the OWL 2 specification
(see <a href="../REC-owl2-syntax-20091027/index.html#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
this document
uses an extended notion of an RDF graph
by allowing the RDF triples in an RDF graph
to contain arbitrary <i><b>IRIs</b></i>
("Internationalized Resource Identifiers")
according to <a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt">RFC 3987</a>
[<cite><a href="index.html#ref-rfc-3987" title="">RFC 3987</a></cite>].
In contrast,
the <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#urisandlit" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#urisandlit">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
is defined on RDF graphs containing
<a class="external text" href="http://www.ietf.org/rfc/rfc2396.txt" title="http://www.ietf.org/rfc/rfc2396.txt"><i>URIs</i></a>
[<cite><a href="index.html#ref-rfc-2396" title="">RFC 2396</a></cite>].
This change
is backward compatible
with the RDF specification,
since URIs are also IRIs.
</p><p><span id="topic-ont-noteiriref"></span>
</p><p><i>Terminological note:</i>
The document at hand
uses the term "IRI"
in accordance with the rest of the OWL 2 specification
(see <a href="../REC-owl2-syntax-20091027/index.html#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
whereas the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#urisandlit" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#urisandlit">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
uses the term "URI reference".
According to
<a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt">RFC 3987</a>
[<cite><a href="index.html#ref-rfc-3987" title="">RFC 3987</a></cite>],
the term "IRI"
stands for an absolute resource identifier with optional fragment,
which is what is being used throughout this document.
In contrast,
the term "IRI reference" additionally covers <i>relative</i> references,
which are never used in this document.
</p><p><span id="topic-ont-iriabbrev"></span>
</p><p><i>Convention:</i>
In this document,
IRIs are abbreviated
in the way defined by
<a href="../REC-owl2-syntax-20091027/index.html#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>],
i.e., the abbreviations consist of
a <i>prefix name</i> and a <i>local part</i>,
such as
"<span class="name">prefix:localpart</span>".
</p><p><span id="topic-ont-generalrdf"></span>
</p><p>The definition of an RDF triple
according to
<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-triples" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples">Section 6.1 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]
is restricted to cases
where the <i>subject</i> of an RDF triple is
an IRI
or a
<i>blank node</i>
(<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-blank-nodes" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-blank-nodes">Section 6.6 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]),
and where the <i>predicate</i> of an RDF triple is
an IRI.
As a consequence,
the definition does not treat cases,
where,
for example,
the subject of a triple is a <i>literal</i>
(<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-Graph-Literal" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal">Section 6.5 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]),
as in
<span class="name">"s" ex:p ex:o</span>,
or where the predicate of a triple is a blank node,
as in
<span class="name">ex:s _:p ex:o</span>.
In order to allow for interoperability
with other existing and future technologies and tools,
the document at hand
does not explicitly forbid the use of
<i><b>generalized RDF graphs</b></i> consisting of <i><b>generalized RDF triples</b></i>,
which are defined to allow for
IRIs, literals and blank nodes
to occur in the subject, predicate and object position.
Thus,
an RDF graph
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
contain generalized RDF triples,
but an implementation is not required to support generalized RDF graphs.
Note that every RDF graph consisting entirely of RDF triples according to
<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-triples" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples">Section 6.1 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>]
is also a generalized RDF graph.
</p><p><span id="topic-ont-owl2full"></span>
</p><p><i>Terminological notes:</i>
The term
<i><b>"OWL 2 Full"</b></i>
refers to the language
that is determined
by the set of all RDF graphs
being interpreted using the OWL 2 RDF-Based Semantics.
Further,
in this document
the term
<i><b>"OWL 2 Full ontology"</b></i>
(or simply <i><b>"ontology"</b></i>,
unless there is any risk of confusion)
will be used interchangeably
with the term "RDF graph".
</p>
<a name="Content_of_Ontologies_.28Informative.29"></a><h3> <span class="mw-headline">2.2 Content of Ontologies (Informative) </span></h3>
<p>While there do not exist any syntactic restrictions
on the set of RDF graphs
that can be interpreted by the OWL 2 RDF-Based Semantics,
in practice
an ontology will often contain certain kinds of constructs
that are aimed to support ontology management tasks.
Examples are
<i><b>ontology headers</b></i>
and
<i><b>ontology IRIs</b></i>,
as well as constructs that are about
<i><b>versioning</b></i>,
<i><b>importing</b></i>
and
<i><b>annotating</b></i> of ontologies,
including the concept of <i><b>incompatibility</b></i> between ontologies.
</p><p>These topics are outside the scope of this semantics specification.
<a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
deals with these topics in detail,
and can therefore be used as a guide
on how to apply these constructs in OWL 2 Full ontologies accordingly.
The mappings of all these constructs to their respective RDF encoding
are defined in
the <a href="../REC-owl2-mapping-to-rdf-20091027/index.html" title="Mapping to RDF Graphs">OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>].
</p>
<a name="Vocabulary"></a><h2> <span class="mw-headline">3 Vocabulary </span></h2>
<p>This section specifies the <i>OWL 2 RDF-Based vocabulary</i>,
and lists the names of the <i>datatypes</i> and <i>facets</i>
used under the OWL 2 RDF-Based Semantics.
</p>
<a name="Standard_Prefixes"></a><h3> <span class="mw-headline">3.1 Standard Prefixes </span></h3>
<p><a href="index.html#table-vocab-prefixes" title="">Table 3.1</a>
lists the standard prefix names
and their prefix IRIs
used in this document.
</p>
<div class="left" id="table-vocab-prefixes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.1: Standard Prefixes</span>
</caption><tr>
<th style="text-align: center">
</th><th style="text-align: center"> Prefix Name
</th><th style="text-align: center"> Prefix IRI
</th></tr>
<tr>
<td> <span id="item-vocab-prefixes-owl"></span>OWL
</td><td class="name"> owl
</td><td class="name"> http://www.w3.org/2002/07/owl#
</td></tr>
<tr>
<td> <span id="item-vocab-prefixes-rdf"></span>RDF
</td><td class="name"> rdf
</td><td class="name"> http://www.w3.org/1999/02/22-rdf-syntax-ns#
</td></tr>
<tr>
<td> <span id="item-vocab-prefixes-rdfs"></span>RDFS
</td><td class="name"> rdfs
</td><td class="name"> http://www.w3.org/2000/01/rdf-schema#
</td></tr>
<tr>
<td> <span id="item-vocab-prefixes-xsd"></span>XML Schema
</td><td class="name"> xsd
</td><td class="name"> http://www.w3.org/2001/XMLSchema#
</td></tr>
</table>
</div>
<a name="Vocabulary_Terms"></a><h3> <span class="mw-headline">3.2 Vocabulary Terms </span></h3>
<p><a href="index.html#table-vocab-owl" title="">Table 3.2</a>
lists the IRIs of the <i>OWL 2 RDF-Based vocabulary</i>,
which is the set of vocabulary terms
that are specific for the OWL 2 RDF-Based Semantics.
This vocabulary
extends the RDF and RDFS vocabularies
as specified in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">Sections 3.1</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">4.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
respectively.
<a href="index.html#table-vocab-owl" title="">Table 3.2</a>
does not mention those IRIs
that will be listed in
<a href="index.html#Datatype_Names" title="">Section 3.3</a> on datatype names
or
<a href="index.html#Facet_Names" title="">Section 3.4</a> on facet names.
</p>
<div id="topic-vocab-narydatatype"></div>
<p>Implementations are <i>not</i> required
to support the IRI <span class="name">owl:onProperties</span>,
but
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
support it
in order to realize
<i>n-ary dataranges</i> with arity ≥ 2
(see
Sections
<a href="../REC-owl2-syntax-20091027/index.html#Data_Ranges" title="Syntax">7</a>
and
<a href="../REC-owl2-syntax-20091027/index.html#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for further information).
</p>
<div id="topic-vocab-deprecatedterms"></div>
<p><b>Note:</b>
The use of the IRI <span class="name">owl:DataRange</span> has been deprecated as of OWL 2.
The IRI <span class="name">rdfs:Datatype</span>
<em class="RFC2119" title="SHOULD in RFC 2119 context">SHOULD</em>
be used instead.
</p>
<div class="left" id="table-vocab-owl">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.2: OWL 2 RDF-Based Vocabulary</span>
</caption>
<tr>
<td class="name"> owl:AllDifferent owl:AllDisjointClasses owl:AllDisjointProperties owl:allValuesFrom owl:annotatedProperty owl:annotatedSource owl:annotatedTarget owl:Annotation owl:AnnotationProperty owl:assertionProperty owl:AsymmetricProperty owl:Axiom owl:backwardCompatibleWith owl:bottomDataProperty owl:bottomObjectProperty owl:cardinality owl:Class owl:complementOf owl:DataRange owl:datatypeComplementOf owl:DatatypeProperty owl:deprecated owl:DeprecatedClass owl:DeprecatedProperty owl:differentFrom owl:disjointUnionOf owl:disjointWith owl:distinctMembers owl:equivalentClass owl:equivalentProperty owl:FunctionalProperty owl:hasKey owl:hasSelf owl:hasValue owl:imports owl:incompatibleWith owl:intersectionOf owl:InverseFunctionalProperty owl:inverseOf owl:IrreflexiveProperty owl:maxCardinality owl:maxQualifiedCardinality owl:members owl:minCardinality owl:minQualifiedCardinality owl:NamedIndividual owl:NegativePropertyAssertion owl:Nothing owl:ObjectProperty owl:onClass owl:onDataRange owl:onDatatype owl:oneOf owl:onProperty owl:onProperties owl:Ontology owl:OntologyProperty owl:priorVersion owl:propertyChainAxiom owl:propertyDisjointWith owl:qualifiedCardinality owl:ReflexiveProperty owl:Restriction owl:sameAs owl:someValuesFrom owl:sourceIndividual owl:SymmetricProperty owl:targetIndividual owl:targetValue owl:Thing owl:topDataProperty owl:topObjectProperty owl:TransitiveProperty owl:unionOf owl:versionInfo owl:versionIRI owl:withRestrictions
</td></tr>
</table>
</div>
<a name="Datatype_Names"></a><h3> <span class="mw-headline">3.3 Datatype Names </span></h3>
<p><a href="index.html#table-vocab-datatypes" title="">Table 3.3</a>
lists the IRIs of the <i>datatypes</i> used in the OWL 2 RDF-Based Semantics.
The datatype <span class="name">rdf:XMLLiteral</span> is described in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef">Section 3.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
All other datatypes are described in
<a href="../REC-owl2-syntax-20091027/index.html#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
The normative set of datatypes of the OWL 2 RDF-Based Semantics equals the set of datatypes
described in
<a href="../REC-owl2-syntax-20091027/index.html#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p>
<div class="left" id="table-vocab-datatypes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.3: Datatypes of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td class="name"> xsd:anyURI xsd:base64Binary xsd:boolean xsd:byte xsd:dateTime xsd:dateTimeStamp xsd:decimal xsd:double xsd:float xsd:hexBinary xsd:int xsd:integer xsd:language xsd:long xsd:Name xsd:NCName xsd:negativeInteger xsd:NMTOKEN xsd:nonNegativeInteger xsd:nonPositiveInteger xsd:normalizedString rdf:PlainLiteral xsd:positiveInteger owl:rational owl:real xsd:short xsd:string xsd:token xsd:unsignedByte xsd:unsignedInt xsd:unsignedLong xsd:unsignedShort rdf:XMLLiteral
</td></tr>
</table>
</div>
<a name="Facet_Names"></a><h3> <span class="mw-headline">3.4 Facet Names </span></h3>
<p><a href="index.html#table-vocab-facets" title="">Table 3.4</a>
lists the IRIs of the <i>facets</i> used in the OWL 2 RDF-Based Semantics.
Each datatype listed in <a href="index.html#Datatype_Names" title="">Section 3.3</a>
has a (possibly empty) set of constraining facets.
All facets are described in
<a href="../REC-owl2-syntax-20091027/index.html#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
in the context of their respective datatypes.
The normative set of facets of the OWL 2 RDF-Based Semantics equals the set of facets
described in
<a href="../REC-owl2-syntax-20091027/index.html#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p>
<div id="topic-vocab-facetexample"></div>
<p>In this specification,
facets are used for defining <i>datatype restrictions</i>
(see <a href="index.html#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>).
For example,
to refer to the set of all strings of length 5
one can restrict
the datatype <span class="name">xsd:string</span>
(<a href="index.html#Datatype_Names" title="">Section 3.3</a>)
by the facet <span class="name">xsd:length</span>
and the value 5.
</p>
<div class="left" id="table-vocab-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.4: Facets of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td class="name"> rdf:langRange xsd:length xsd:maxExclusive xsd:maxInclusive xsd:maxLength xsd:minExclusive xsd:minInclusive xsd:minLength xsd:pattern
</td></tr>
</table>
</div>
<a name="Interpretations"></a><h2> <span class="mw-headline">4 Interpretations </span></h2>
<p>The OWL 2 RDF-Based Semantics provides
<i>vocabulary interpretations</i> and <i>vocabulary entailment</i>
(see <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#vocabulary_entail" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#vocabulary_entail">Section 2.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>])
for the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS</a>
vocabularies
and for the
<a href="index.html#Vocabulary" title="">OWL 2 RDF-Based vocabulary</a>.
This section defines
<i>OWL 2 RDF-Based datatype maps</i>
and
<i>OWL 2 RDF-Based interpretations</i>,
and specifies what
<i>satisfaction</i> of ontologies,
<i>consistency</i> and <i>entailment</i>
means under the OWL 2 RDF-Based Semantics.
In addition,
the so called <i>"parts" of the universe</i>
of an OWL 2 RDF-Based interpretation
are defined.
</p>
<a name="Datatype_Maps"></a><h3> <span class="mw-headline">4.1 Datatype Maps </span></h3>
<div id="topic-int-rdfdatatype"></div>
<p>According to
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDatatype" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatype">Section 5.1 of the RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
a <i><b>datatype</b></i> <i>d</i> has the following components:
</p>
<ul><li> LS(<i>d</i>), the <i>lexical space</i> of <i>d</i>, which is a set of <i>lexical forms</i>;
</li><li> VS(<i>d</i>), the <i>value space</i> of <i>d</i>, which is a set of <i>data values</i>;
</li><li> L2V(<i>d</i>), the <i>lexical-to-value mapping</i> of <i>d</i>, which maps lexical forms in LS(<i>d</i>) to data values in VS(<i>d</i>).
</li></ul>
<div id="topic-int-datavalueterm"></div>
<p><i>Terminological notes:</i>
The document at hand uses the term
<i>"data value"</i>
in accordance with the rest of the OWL 2 specification
(see
<a href="../REC-owl2-syntax-20091027/index.html#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
whereas the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
uses the term
<i>"datatype value"</i> instead.
Further, the names "LS" and "VS",
which stand for the lexical space and the value space of a datatype,
respectively,
are <i>not</i> used in the RDF Semantics specification,
but have been introduced here for easier reference.
</p><p>In this document,
the basic definition of a datatype
is extended to take <i>facets</i> into account.
See <a href="index.html#Facet_Names" title="">Section 3.4</a>
for information and an example on facets.
Note that
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">Section 5.1 of the RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
explicitly permits
that semantic extensions
may impose more elaborate datatyping conditions
than those listed above.
</p>
<div id="topic-int-datatypewithfacets"></div>
<p>A <i><b>datatype with facets</b></i> <i>d</i>
is a datatype that has the following additional components:
</p>
<ul><li> FS(<i>d</i>), the <i>facet space</i> of <i>d</i>, which is a set of pairs of the form ( <i>F</i> , <i>v</i> ), where <i>F</i> is an IRI called the <i>constraining facet</i> and <i>v</i> is an arbitrary data value called the <i>constraining value</i>;
</li><li> F2V(<i>d</i>), the <i>facet-to-value mapping</i> of <i>d</i>, which maps each facet-value pair ( <i>F</i> , <i>v</i> ) in FS(<i>d</i>) to a subset of VS(<i>d</i>).
</li></ul>
<div id="topic-int-facetnature"></div>
<p>Note that
it is not further specified
what the nature of the denotation of a facet IRI is,
i.e. it is only known that a facet IRI denotes some individual.
Semantic extensions
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
impose further restrictions on the denotations of facets.
In fact,
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>
will define additional restrictions on facets.
</p>
<div id="topic-int-facetvalueothertype"></div>
<p>Also note
that for a datatype <i>d</i>
and a facet-value pair ( <i>F</i> , <i>v</i> ) in FS(<i>d</i>)
the value <i>v</i>
is not required
to be included in the value space VS(<i>d</i>) of <i>d</i> itself.
For example,
the datatype <span class="name">xsd:string</span>
(<a href="index.html#Datatype_Names" title="">Section 3.3</a>)
has the facet <span class="name">xsd:length</span>
(<a href="index.html#Facet_Names" title="">Section 3.4</a>),
which takes nonnegative integers as its constraining values
rather than strings.
</p>
<div id="topic-int-facetdatatypeassumption"></div>
<p>In this document,
it will always be assumed from now on that
any datatype <i>d</i> is a datatype with facets.
If the facet space FS(<i>d</i>) of a datatype <i>d</i>
has not been explicitly defined,
or if it is not derived from another datatype's facet space
according to some well defined condition,
then FS(<i>d</i>) is the empty set.
Unless there is any risk of confusion,
the term <i>"datatype"</i>
will always refer to a datatype with facets.
</p>
<div id="topic-int-rdfdatatypemap"></div>
<p><a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDatatypeMap" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatypeMap">Section 5.1 of the RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
further
defines a <i><b>datatype map</b></i> <i>D</i> to be
a set of name-datatype pairs
( <i>u</i> , <i>d</i> )
consisting of an IRI <i>u</i> and a datatype <i>d</i>,
such that no IRI appears twice in the set.
As a consequence of what has been said before,
in this document
every datatype map <i>D</i> will entirely consist of datatypes with facets.
</p><p>The following definition specifies what an <i>OWL 2 RDF-Based datatype map</i> is.
</p>
<div id="def-owldatatypemap">
<p><b>Definition 4.1 (OWL 2 RDF-Based Datatype Map):</b>
A datatype map <i>D</i>
is an <i>OWL 2 RDF-Based datatype map</i>,
if and only if
for every datatype name <i>u</i> listed in <a href="index.html#Datatype_Names" title="">Section 3.3</a>
and its respective set of constraining facets (<a href="index.html#Facet_Names" title="">Section 3.4</a>)
there is
a name-datatype pair ( <i>u</i>, <i>d</i> ) in <i>D</i>
with the specified
lexical space LS(<i>d</i>),
value space VS(<i>d</i>),
lexical-to-value mapping L2V(<i>d</i>),
facet space FS(<i>d</i>) and
facet-to-value mapping F2V(<i>d</i>).
</p>
</div>
<div id="topic-int-facetclosedowlmap"></div>
<p>Note that <a href="index.html#def-owldatatypemap" title="">Definition 4.1</a>
does not prevent <i>additional</i> datatypes
to be in an OWL 2 RDF-Based datatype map.
For the special case of
an OWL 2 RDF-Based datatype map <i>D</i>
that exclusively contains the datatypes listed in
<a href="index.html#Datatype_Names" title="">Section 3.3</a>,
it is ensured that
there are datatypes available for all the facet values,
i.e.,
for every name-datatype pair ( <i>u</i> , <i>d</i> ) in <i>D</i>
and for every facet-value pair
( <i>F</i> , <i>v</i> )
in FS(<i>d</i>)
there exists a name-datatype pair ( <i>u<sup>*</sup></i> , <i>d<sup>*</sup></i> ) in <i>D</i>
such that <i>v</i> is in VS(<i>d<sup>*</sup></i>).
</p>
<a name="Vocabulary_Interpretations"></a><h3> <span class="mw-headline">4.2 Vocabulary Interpretations </span></h3>
<div id="topic-int-rdfinterpretation"></div>
<p>From the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
let <i>V</i> be a set of literals and IRIs
containing the RDF and RDFS vocabularies,
and let <i>D</i> be a datatype map according to
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDatatypeMap" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatypeMap">Section 5.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
(and accordingly <a href="index.html#Datatype_Maps" title="">Section 4.1</a>).
A <i><b>D-interpretation</b></i> <i>I</i> of <i>V</i> with respect to <i>D</i> is a tuple
</p>
<div class="indent">
<p><i>I</i> = ( IR , IP , IEXT , IS , IL , LV ) .
</p>
</div>
<p>IR is the <i>universe</i> of <i>I</i>,
i.e., a nonempty set
that contains at least
the denotations of literals and IRIs in <i>V</i>.
IP is a subset of IR,
the <i>properties</i> of <i>I</i>.
LV,
the <i>data values</i> of <i>I</i>,
is a subset of IR
that contains at least the set of plain literals
(see <a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-Graph-Literal" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal">Section 6.5 of RDF Concepts</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>])
in <i>V</i>,
and
the value spaces of each datatype of <i>D</i>.
IEXT is used to associate properties with their <i>property extension</i>,
and is a mapping from IP to the powerset of IR × IR.
IS is a mapping from <i>IRIs</i> in <i>V</i> to their denotations in IR.
In particular,
IS(<i>u</i>) = <i>d</i>
for any name-datatype pair ( <i>u</i> , <i>d</i> ) in <i>D</i>.
IL is a mapping from <i>typed literals</i>
"<i>s</i>"<span class="name">^^</span><i>u</i>
in <i>V</i>
to their denotations in IR,
where IL("<i>s</i>"<span class="name">^^</span><i>u</i>) = L2V(<i>d</i>)(<i>s</i>),
provided that <i>d</i> is a datatype of <i>D</i>,
IS(<i>u</i>) = <i>d</i>, and
<i>s</i> is in the lexical space LS(<i>d</i>);
otherwise
IL("<i>s</i>"<span class="name">^^</span><i>u</i>)
is not in LV.
</p>
<div id="topic-int-ifunction"></div>
<p><i>Convention:</i>
Following the practice introduced in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">Section 1.4 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
for a given interpretation <i>I</i> of a vocabulary <i>V</i>
the notation
"<i>I</i>(<i>x</i>)"
will be used
instead of "IL(<i>x</i>)" and "IS(<i>x</i>)"
for the typed literals and IRIs <i>x</i> in <i>V</i>,
respectively.
</p>
<div id="topic-int-rdfconditions"></div>
<p>As detailed in the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
a D-interpretation has to meet all the semantic conditions
for <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">ground graphs</a>
and <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">blank nodes</a>,
those for <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF interpretations</a> and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS interpretations</a>,
and the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">"general semantic conditions for datatypes"</a>.
</p><p>In this document,
the basic definition of a D-interpretation
is extended to take <i>facets</i> into account.
</p>
<div id="topic-int-facetinterpretation"></div>
<p>A <i><b>D-interpretation with facets</b></i> <i>I</i>
is a D-interpretation for a datatype map <i>D</i>
consisting entirely of datatypes with facets
(<a href="index.html#Datatype_Maps" title="">Section 4.1</a>),
where <i>I</i> meets the following additional semantic conditions:
for each name-datatype pair ( <i>u</i> , <i>d</i> ) in <i>D</i>
and each facet-value pair ( <i>F</i> , <i>v</i> ) in the facet space FS(<i>d</i>)
</p>
<ul><li> <i>F</i> is in the vocabulary <i>V</i> of <i>I</i>;
</li><li> a name-datatype pair ( <i>u<sup>*</sup></i> , <i>d<sup>*</sup></i> ) exists in <i>D</i>, such that <i>v</i> is in the value space VS(<i>d<sup>*</sup></i>).
</li></ul>
<div id="topic-int-facetinterpretassumption"></div>
<p>In this document,
it will always be assumed from now on that
any D-interpretation <i>I</i>
is a D-interpretation with facets.
Unless there is any risk of confusion,
the term <i>"D-interpretation"</i>
will always refer to a D-interpretation with facets.
</p><p>The following definition specifies what an <i>OWL 2 RDF-Based interpretation</i> is.
</p>
<div id="def-owlinterpretation">
<p><b>Definition 4.2 (OWL 2 RDF-Based Interpretation):</b>
Let <i>D</i> be an OWL 2 RDF-Based datatype map,
and let <i>V</i> be a vocabulary
that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names
listed in <a href="index.html#Vocabulary" title="">Section 3</a>.
An <i>OWL 2 RDF-Based interpretation</i>,
<i>I</i> = ( IR , IP , IEXT , IS , IL , LV ),
of <i>V</i> with respect to <i>D</i>
is a D-interpretation of <i>V</i> with respect to <i>D</i>
that meets all the extra semantic conditions
given in <a href="index.html#Semantic_Conditions" title="">Section 5</a>.
</p>
</div>
<a name="Satisfaction.2C_Consistency_and_Entailment"></a><h3> <span class="mw-headline">4.3 Satisfaction, Consistency and Entailment </span></h3>
<p>The following definitions specify
what it means for an RDF graph to be <i>satisfied</i>
by a given OWL 2 RDF-Based interpretation,
to be <i>consistent</i>
under the OWL 2 RDF-Based Semantics,
and to <i>entail</i> another RDF graph.
</p>
<div id="topic-int-rdfsatisfaction"></div>
<p>The notion of <i>satisfaction</i> under the OWL 2 RDF-Based Semantics
is based on the notion of satisfaction for
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">D-interpretations</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defsatis" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defsatis">Simple interpretations</a>,
as defined in the RDF Semantics
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
In essence,
in order to satisfy an RDF graph,
an interpretation <i>I</i> has to satisfy all the triples in the graph,
i.e.,
for a triple "<i>s p o</i>"
it is necessary that the relationship
( <i>I</i>(<i>s</i>) , <i>I</i>(<i>o</i>) ) ∈ IEXT(<i>I</i>(<i>p</i>))
holds
(special treatment exists for blank nodes,
as detailed in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">Section 1.5 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]).
In other words,
the given graph has to be compatible with
the specific form of the IEXT mapping of <i>I</i>.
The distinguishing aspect of <i>OWL 2 RDF-Based satisfaction</i> is
that an interpretation <i>I</i> needs to meet
all the OWL 2 RDF-Based semantic conditions
(see <a href="index.html#Semantic_Conditions" title="">Section 5</a>),
which have a constraining effect
on the possible forms an IEXT mapping can have.
</p>
<div id="def-owlsatisfaction">
<p><b>Definition 4.3 (OWL 2 RDF-Based Satisfaction):</b>
Let <i>G</i> be an RDF graph,
let <i>D</i> be an OWL 2 RDF-Based datatype map,
let <i>V</i> be a vocabulary
that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in <a href="index.html#Vocabulary" title="">Section 3</a>,
and let <i>I</i> be a D-interpretation of <i>V</i> with respect to <i>D</i>.
<i>I</i> <i>OWL 2 RDF-Based satisfies</i> <i>G</i> with respect to <i>V</i> and <i>D</i>
if and only if
<i>I</i> is an OWL 2 RDF-Based interpretation of <i>V</i> with respect to <i>D</i>
that
satisfies <i>G</i>
as a <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">D-interpretation</a> of <i>V</i> with respect to <i>D</i>
according to the RDF Semantics
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p>
</div>
<div id="def-owlconsistency">
<p><b>Definition 4.4 (OWL 2 RDF-Based Consistency):</b>
Let <i>S</i> be a collection of RDF graphs,
and let <i>D</i> be an OWL 2 RDF-Based datatype map.
<i>S</i> is <i>OWL 2 RDF-Based consistent</i> with respect to <i>D</i>
if and only if
there is some OWL 2 RDF-Based interpretation <i>I</i> with respect to <i>D</i>
of some vocabulary <i>V</i>
that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in <a href="index.html#Vocabulary" title="">Section 3</a>,
such that <i>I</i> OWL 2 RDF-Based satisfies all the RDF graphs in <i>S</i>
with respect to <i>V</i> and <i>D</i>.
</p>
</div>
<div id="def-owlentailment">
<p><b>Definition 4.5 (OWL 2 RDF-Based Entailment):</b>
Let <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i> be collections of RDF graphs,
and let <i>D</i> be an OWL 2 RDF-Based datatype map.
<i>S<sub>1</sub></i> <i>OWL 2 RDF-Based entails</i> <i>S<sub>2</sub></i> with respect to <i>D</i>
if and only if
for every OWL 2 RDF-Based interpretation <i>I</i> with respect to <i>D</i>
of any vocabulary <i>V</i> that includes
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in <a href="index.html#Vocabulary" title="">Section 3</a>
the following holds:
If <i>I</i>
OWL 2 RDF-Based satisfies all the RDF graphs in <i>S<sub>1</sub></i>
with respect to <i>V</i> and <i>D</i>,
then <i>I</i>
OWL 2 RDF-Based satisfies all the RDF graphs in <i>S<sub>2</sub></i>
with respect to <i>V</i> and <i>D</i>.
</p>
</div>
<a name="Parts_of_the_Universe"></a><h3> <span class="mw-headline">4.4 Parts of the Universe </span></h3>
<p><a href="index.html#table-int-parts" title="">Table 4.1</a>
defines the <i>"parts"</i> of the universe
of a given OWL 2 RDF-Based interpretation <i>I</i>.
</p><p>The second column tells the <i>name</i> of the part.
The third column gives a <i>definition</i> of the part
in terms of the mapping IEXT of <i>I</i>,
and by referring to a particular term
of the RDF, RDFS or OWL 2 RDF-Based vocabulary.
</p>
<div id="topic-int-partsdefexample"></div>
<p>As an example,
the part of all datatypes is named "IDC",
and it is defined as the set of all individuals <i>x</i>
for which the relationship
"( <i>x</i> , <i>I</i>(<span class="name">rdfs:Datatype</span>) )
∈
IEXT(<i>I</i>(<span class="name">rdf:type</span>))"
holds.
According to the semantics of <span class="name">rdf:type</span>,
as defined in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfssemcond1" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1">Section 4.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
this means that the name "IDC"
denotes the class extension
(see <a href="index.html#Class_Extensions" title="">Section 4.5</a>)
of <i>I</i>(<span class="name">rdfs:Datatype</span>).
</p>
<div class="left" id="table-int-parts">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 4.1: Parts of the Universe</span>
</caption>
<tr>
<th style="text-align: center">
</th><th style="text-align: center"> Name of<br />Part <i>S</i>
</th><th style="text-align: center"> Definition of <i>S</i> as<br />{ <i>x</i> ∈ IR | ( <i>x</i> , <i>I</i>(<i>E</i>) ) ∈ IEXT(<i>I</i>(<span class="name">rdf:type</span>)) }<br />where IRI <i>E</i> is
</th></tr>
<tr>
<td> <span id="item-int-parts-individuals"></span>individuals
</td><td> IR
</td><td> <span class="name">rdfs:Resource</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-datavalues"></span>data values
</td><td> LV
</td><td> <span class="name">rdfs:Literal</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-ontologies"></span>ontologies
</td><td> IX
</td><td> <span class="name">owl:Ontology</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-classes"></span>classes
</td><td> IC
</td><td> <span class="name">rdfs:Class</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-datatypes"></span>datatypes
</td><td> IDC
</td><td> <span class="name">rdfs:Datatype</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-properties"></span>properties
</td><td> IP
</td><td> <span class="name">rdf:Property</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-dataproperties"></span>data properties
</td><td> IODP
</td><td> <span class="name">owl:DatatypeProperty</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-ontologyproperties"></span>ontology properties
</td><td> IOXP
</td><td> <span class="name">owl:OntologyProperty</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-annotationproperties"></span>annotation properties
</td><td> IOAP
</td><td> <span class="name">owl:AnnotationProperty</span>
</td></tr>
</table>
</div>
<a name="Class_Extensions"></a><h3> <span class="mw-headline">4.5 Class Extensions </span></h3>
<p>The mapping ICEXT from IC to the powerset of IR,
which associates classes with their <i>class extension</i>,
is defined
for every <i>c</i> ∈ IC
as
</p>
<div class="indent">
<p>ICEXT(<i>c</i>) = { <i>x</i> ∈ IR | ( <i>x</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:type</span>)) } .
</p>
</div>
<a name="Semantic_Conditions"></a><h2> <span class="mw-headline">5 Semantic Conditions </span></h2>
<p>This section defines the semantic conditions of the OWL 2 RDF-Based Semantics.
The semantic conditions presented here
are basically only those for the specific constructs of OWL 2.
The complete set of semantic conditions for the OWL 2 RDF-Based Semantics
is the combination of the semantic conditions presented here
and the semantic conditions
for
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">Simple Entailment</a>,
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef">RDF</a>,
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">RDFS</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">D-Entailment</a>,
as specified in
the RDF Semantics specification
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p><p>All semantic conditions in this section
are defined with respect to an interpretation <i>I</i>.
<a href="index.html#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a>
specifies semantic conditions for the different parts of the universe
of the interpretation being considered
(compare <a href="index.html#Parts_of_the_Universe" title="">Section 4.4</a>).
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>
and
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>
list semantic conditions for the classes and the properties of the OWL 2 RDF-Based vocabulary.
In the rest of this section,
the OWL 2 RDF-Based semantic conditions
for the different language constructs of OWL 2
are specified.
</p>
<div id="topic-semcond-conventions"></div>
<p><b>Conventions used in this Section</b>
</p>
<div id="topic-semcond-convention-iff"></div>
<p><i>iff:</i>
Throughout this section
the term "iff" is used as a shortform for "if and only if".
</p>
<div id="topic-semcond-convention-andcomma"></div>
<p><i>Conjunctive commas:</i>
A comma
("<span class="name">,</span>")
separating two assertions in a semantic condition, as in
"<i>c</i> ∈ IC , <i>p</i> ∈ IP",
is read as a logical <i>"and"</i>.
Further,
a comma separating two variables,
as in
"<i>c</i>, <i>d</i> ∈ IC",
is used for abbreviating two comma separated assertions,
"<i>c</i> ∈ IC , <i>d</i> ∈ IC"
in this example.
</p>
<div id="topic-semcond-convention-varscope"></div>
<p><i>Unscoped variables:</i>
If no explicit scope is given for a variable "<i>x</i>",
as in "∀ <i>x</i> : …" or "{ <i>x</i> | … }",
then "<i>x</i>" is unconstrained,
which means <i>x</i> ∈ IR,
i.e. "<i>x</i>" denotes an arbitrary individual in the universe.
</p>
<div id="topic-semcond-convention-setcard"></div>
<p><i>Set cardinality:</i>
For a set <i>S</i>,
an expression of the form "#<i>S</i>" means the number of elements in <i>S</i>.
</p>
<div id="topic-semcond-convention-seq"></div>
<p><i>Sequence expressions:</i>
An expression of the form
"<i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ <i>S</i>"
means that "<i>s</i>" represents an RDF list of <i>n</i> ≥ 0
individuals <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i>,
all of them being members of the set <i>S</i>.
Precisely,
<i>s</i> = <i>I</i>(<span class="name">rdf:nil</span>) for <i>n</i> = 0;
and for <i>n</i> > 0
there exist
<i>z<sub>1</sub></i> ∈ IR , … , <i>z<sub>n</sub></i> ∈ IR,
such that
</p>
<div class="indent">
<p><i>s</i> = <i>z<sub>1</sub></i> ,<br />
<i>a<sub>1</sub></i> ∈ <i>S</i> ,
( <i>z<sub>1</sub></i> , <i>a<sub>1</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:first</span>)) ,
( <i>z<sub>1</sub></i> , <i>z<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:rest</span>)) ,<br />
… ,<br />
<i>a<sub>n</sub></i> ∈ <i>S</i>,
( <i>z<sub>n</sub></i> , <i>a<sub>n</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:first</span>)) ,
( <i>z<sub>n</sub></i> , <i>I</i>(<span class="name">rdf:nil</span>) ) ∈ IEXT(<i>I</i>(<span class="name">rdf:rest</span>)) .
</p>
</div>
<p>Note, as mentioned in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#collections" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#collections">Section 3.3.3 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
there are no semantic constraints that enforce "well-formed" sequence structures.
So, for example,
it is possible for a sequence head <i>s</i> to refer to more than one sequence.
</p>
<div id="topic-semcond-convention-setnames"></div>
<p><i>Set names:</i>
The following names are used as convenient abbreviations for certain sets:
</p>
<ul><li> ISEQ: The set of all sequences. This set equals the class extension of <span class="name">rdf:List</span>, i.e., ISEQ := ICEXT(<i>I</i>(<span class="name">rdf:List</span>)).
</li><li> INNI: The set of all nonnegative integers. This set equals the value space of the datatype <span class="name">xsd:nonNegativeInteger</span>, i.e., INNI := ICEXT(<i>I</i>(<span class="name">xsd:nonNegativeInteger</span>)), but is also subsumed by the value spaces of other numerical datatypes, such as <span class="name">xsd:integer</span>.
</li></ul>
<div id="topic-semcond-conditionform"></div>
<p><b>Notes on the Form of Semantic Conditions (Informative)</b>
</p>
<div id="topic-semcond-conditionform-correspondence"></div>
<p>One design goal of OWL 2
was to ensure an appropriate degree of alignment
between the OWL 2 RDF-Based Semantics and the
<a href="../REC-owl2-direct-semantics-20091027/index.html" title="Direct Semantics">OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
under the different constraints the two semantics have to meet.
The way this semantic alignment is described
is via the <i>OWL 2 correspondence theorem</i>
in <a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>.
For this theorem to hold,
the semantic conditions
that treat the RDF encoding
of OWL 2 <i>axioms</i>
(compare <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Axioms" title="Mapping to RDF Graphs">Section 3.2.5 of the OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
and
<a href="../REC-owl2-syntax-20091027/index.html#Axioms" title="Syntax">Section 9 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
such as
<a href="index.html#Semantic_Conditions_for_Inverse_Properties" title="">inverse property axioms</a>,
must have the form of "iff" ("if-and-only-if") conditions.
This means that these semantic conditions
completely determine the semantics
of the encoding of these constructs.
On the other hand,
the RDF encoding
of OWL 2 <i>expressions</i>
(compare <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Expressions" title="Mapping to RDF Graphs">Section 3.2.4 of the OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
and
<a href="../REC-owl2-syntax-20091027/index.html#Property_Expressions" title="Syntax">Sections 6 – 8 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
such as
<a href="index.html#Semantic_Conditions_for_Property_Restrictions" title="">property restrictions</a>,
are treated by "if-then" conditions.
These weaker semantic conditions for expressions
are sufficient for the correspondence theorem to hold,
so there is no necessity to define stronger "iff" conditions under the OWL 2 RDF-Based Semantics
for these language constructs.
</p>
<div id="topic-semcond-conditionform-axiomexpressions"></div>
<p>Special cases are
the semantic conditions for
<a href="index.html#Semantic_Conditions_for_Boolean_Connectives" title="">Boolean connectives</a>
of classes
and for
<a href="index.html#Semantic_Conditions_for_Enumerations" title="">enumerations</a>.
These language constructs build OWL 2 expressions.
But for backward compatibility reasons
there is also RDF encoding of <i>axioms</i>
based on the vocabulary for these language constructs
(see Table 18 in <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Axioms" title="Mapping to RDF Graphs">Section 3.2.5 of the OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
For example, an RDF expression of the form
</p>
<div class="indent">
<p><span class="name">ex:c<sub>1</sub> owl:unionOf ( ex:c<sub>2</sub> ex:c<sub>3</sub> ) .</span>
</p>
</div>
<p>is mapped by the reverse RDF mapping
to an OWL 2 axiom
that states the equivalence of the class denoted by
<span class="name">ex:c<sub>1</sub></span>
with the union of the classes denoted by
<span class="name">ex:c<sub>2</sub></span>
and
<span class="name">ex:c<sub>3</sub></span>.
In order to ensure that the
<a href="index.html#Correspondence_Theorem" title="">correspondence theorem</a>
holds,
and in accordance with the original
<a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html#5.2" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#5.2">OWL 1 RDF-Compatible Semantics specification</a>
[<cite><a href="index.html#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>],
the semantic conditions for the mentioned language constructs are therefore
"iff" conditions.
</p>
<div id="topic-semcond-conditionform-multitripleaxiom"></div>
<p>Further,
special treatment exists for OWL 2 axioms
that have a <i>multi-triple representation</i> in RDF,
where the different triples share a common <i>"root node"</i>,
such as the blank node
"<span class="name">_:x</span>"
in the following example:
</p>
<div class="indent">
<p><span class="name">_:x rdf:type owl:AllDisjointClasses .</span><br />
<span class="name">_:x owl:members ( ex:c<sub>1</sub> ex:c<sub>2</sub> ) .</span>
</p>
</div>
<p>In essence,
the semantic conditions for the encoding of these language constructs
are "iff" conditions,
as usual for axioms.
However,
in order to cope with the specific syntactic aspect of a "root node",
the "iff" conditions of these language constructs have been split into two "if-then" conditions,
where the "if-then" condition representing the right-to-left direction
contains an additional premise
having the form
"∃ <i>z</i> ∈ IR".
The purpose of this premise is to ensure the existence of an individual
that is needed to satisfy the root node
under the OWL 2 RDF-Based semantics.
The language constructs in question are
<i>n-ary disjointness axioms</i>
in <a href="index.html#Semantic_Conditions_for_N-ary_Disjointness" title="">Section 5.10</a>,
and
<i>negative property assertions</i>
in <a href="index.html#Semantic_Conditions_for_Negative_Property_Assertions" title="">Section 5.15</a>.
</p>
<div id="topic-semcond-conditionform-deducedrhs"></div>
<p>The "if-then" semantic conditions in this section
sometimes do not explicitly list all typing statements in their consequent
that one might expect.
For example,
the semantic condition for
<span class="name">owl:someValuesFrom</span> restrictions in
<a href="index.html#Semantic_Conditions_for_Property_Restrictions" title="">Section 5.6</a>
does not list the statement
"<i>x</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>))"
on its right hand side.
Consequences are generally not mentioned,
if they can already be deduced by other means.
Often,
these redundant consequences follow from the
semantic conditions for
<i>vocabulary classes</i> and <i>vocabulary properties</i>
in
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>
and
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>,
respectively,
occasionally in connection with the semantic conditions
for the <i>parts of the universe</i>
in
<a href="index.html#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a>.
In the example above,
the omitted consequence can be obtained
from the third column of the entry for
<span class="name">owl:someValuesFrom</span>
in the table in
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>,
which determines that
IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>))
⊆
ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IC.
</p>
<a name="Semantic_Conditions_for_the_Parts_of_the_Universe"></a><h3> <span class="mw-headline">5.1 Semantic Conditions for the Parts of the Universe </span></h3>
<p><a href="index.html#table-semcond-parts" title="">Table 5.1</a>
lists the semantic conditions
for the parts of the universe
of the OWL 2 RDF-Based interpretation being considered.
Additional semantic conditions affecting these parts
are given in <a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>.
</p><p>The first column tells the <i>name</i> of the part,
as defined in
<a href="index.html#Parts_of_the_Universe" title="">Section 4.4</a>.
The second column defines
certain <i>conditions</i> on the part.
In most cases,
the column specifies for the part
by which other part it is subsumed,
and thus the position of the part
in the "parts hierarchy" of the universe
is narrowed down.
The third column provides further
<i>information about the instances</i>
of those parts
that consist of classes or properties.
In general,
if the part consists of classes,
then for the class extensions of the member classes
is specified by which part of the universe they are subsumed.
If the part consists of properties,
then the domains and ranges of the member properties are determined.
</p>
<div class="left" id="table-semcond-parts">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.1: Semantic Conditions for the Parts of the Universe</span>
</caption>
<tr>
<th style="text-align: center"> Name of<br />Part <i>S</i>
</th><th style="text-align: center"> Conditions on <i>S</i>
</th><th style="text-align: center"> Conditions on<br />Instances <i>x</i> of <i>S</i>
</th></tr>
<tr>
<td> <span id="item-semcond-parts-individuals"></span>IR
</td><td> <i>S</i> ≠ ∅
</td><td>
</td></tr>
<tr>
<td> <span id="item-semcond-parts-datavalues"></span>LV
</td><td> <i>S</i> ⊆ IR
</td><td>
</td></tr>
<tr>
<td> <span id="item-semcond-parts-ontologies"></span>IX
</td><td> <i>S</i> ⊆ IR
</td><td>
</td></tr>
<tr>
<td> <span id="item-semcond-parts-classes"></span>IC
</td><td> <i>S</i> ⊆ IR
</td><td> ICEXT(<i>x</i>) ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-parts-datatypes"></span>IDC
</td><td> <i>S</i> ⊆ IC
</td><td> ICEXT(<i>x</i>) ⊆ LV
</td></tr>
<tr>
<td> <span id="item-semcond-parts-properties"></span>IP
</td><td> <i>S</i> ⊆ IR
</td><td> IEXT(<i>x</i>) ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-parts-dataproperties"></span>IODP
</td><td> <i>S</i> ⊆ IP
</td><td> IEXT(<i>x</i>) ⊆ IR × LV
</td></tr>
<tr>
<td> <span id="item-semcond-parts-ontologyproperties"></span>IOXP
</td><td> <i>S</i> ⊆ IP
</td><td> IEXT(<i>x</i>) ⊆ IX × IX
</td></tr>
<tr>
<td> <span id="item-semcond-parts-annotationproperties"></span>IOAP
</td><td> <i>S</i> ⊆ IP
</td><td> IEXT(<i>x</i>) ⊆ IR × IR
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_the_Vocabulary_Classes"></a><h3> <span class="mw-headline">5.2 Semantic Conditions for the Vocabulary Classes </span></h3>
<p><a href="index.html#table-semcond-classes" title="">Table 5.2</a>
lists the semantic conditions for the classes
that have IRIs in the OWL 2 RDF-Based vocabulary.
In addition,
the table contains all those classes
with IRIs in the RDF and RDFS vocabularies
that represent
parts of the universe
of the OWL 2 RDF-Based interpretation being considered
(<a href="index.html#Parts_of_the_Universe" title="">Section 4.4</a>).
The semantic conditions for the remaining classes
with names in the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS</a> vocabularies
can be found in the RDF Semantics specification
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p><p>The first column tells the <i>IRI</i> of the class.
The second column defines
of what particular <i>kind</i> a class is,
i.e. whether it is a general class (a member of the part IC)
or a datatype (a member of IDC).
The third column specifies
for the class extension of the class
by which part of the universe
(<a href="index.html#Parts_of_the_Universe" title="">Section 4.4</a>)
it is <i>subsumed</i>:
from an entry of the form
"ICEXT(<i>I</i>(<i>C</i>)) ⊆ <i>S</i>",
for a class IRI <i>C</i>
and a set <i>S</i>,
and given an RDF triple of the form
"<i>u</i> <span class="name">rdf:type</span> <i>C</i>",
one can deduce
that the relationship
"<i>I</i>(<i>u</i>) ∈ <i>S</i>"
holds.
Note that some entries are of the form
"ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>",
which means that the class extension is exactly specified to be that set.
See <a href="index.html#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a>
for further semantic conditions
on those classes that represent <i>parts</i>.
</p>
<div id="topic-semcond-classes-datatypes"></div>
<p>Not included in this table are the <i>datatypes</i> of the OWL 2 RDF-Based Semantics
with IRIs listed in <a href="index.html#Datatype_Names" title="">Section 3.3</a>.
For each such datatype IRI <i>E</i>,
the following semantic conditions hold
(as a consequence of
the fact that <i>E</i> is a member of the datatype map
of every OWL 2 RDF-Based interpretation
according to
<a href="index.html#def-owlinterpretation" title="">Definition 4.2</a>,
and by the "general semantic conditions for datatypes"
listed in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">Section 5.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]):
</p>
<ul><li> <i>I</i>(<i>E</i>) ∈ IDC
</li><li> ICEXT(<i>I</i>(<i>E</i>)) ⊆ LV
</li></ul>
<div class="left" id="table-semcond-classes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.2: Semantic Conditions for the Vocabulary Classes</span>
</caption>
<tr>
<th style="text-align: center"> IRI <i>E</i>
</th><th style="text-align: center"> <i>I</i>(<i>E</i>)
</th><th style="text-align: center"> ICEXT(<i>I</i>(<i>E</i>))
</th></tr>
<tr>
<td> <span id="item-semcond-classes-alldifferent"></span><span class="name">owl:AllDifferent</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-alldisjointclasses"></span><span class="name">owl:AllDisjointClasses</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-alldisjointproperties"></span><span class="name">owl:AllDisjointProperties</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-annotation"></span><span class="name">owl:Annotation</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-annotationproperty"></span><span class="name">owl:AnnotationProperty</span>
</td><td> ∈ IC
</td><td> = IOAP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-asymmetricproperty"></span><span class="name">owl:AsymmetricProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-axiom"></span><span class="name">owl:Axiom</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-rdfsclass"></span><span class="name">rdfs:Class</span>
</td><td> ∈ IC
</td><td> = IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-owlclass"></span><span class="name">owl:Class</span>
</td><td> ∈ IC
</td><td> = IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-datarange"></span><span class="name">owl:DataRange</span>
</td><td> ∈ IC
</td><td> = IDC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-datatype"></span><span class="name">rdfs:Datatype</span>
</td><td> ∈ IC
</td><td> = IDC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-dataproperty"></span><span class="name">owl:DatatypeProperty</span>
</td><td> ∈ IC
</td><td> = IODP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-deprecatedclass"></span><span class="name">owl:DeprecatedClass</span>
</td><td> ∈ IC
</td><td> ⊆ IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-deprecatedproperty"></span><span class="name">owl:DeprecatedProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-functionalproperty"></span><span class="name">owl:FunctionalProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-inversefunctionalproperty"></span><span class="name">owl:InverseFunctionalProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-irreflexiveproperty"></span><span class="name">owl:IrreflexiveProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-literal"></span><span class="name">rdfs:Literal</span>
</td><td> ∈ IDC
</td><td> = LV
</td></tr>
<tr>
<td> <span id="item-semcond-classes-namedindividual"></span><span class="name">owl:NamedIndividual</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-negativepropertyassertion"></span><span class="name">owl:NegativePropertyAssertion</span>
</td><td> ∈ IC
</td><td> ⊆ IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-nothing"></span><span class="name">owl:Nothing</span>
</td><td> ∈ IC
</td><td> = ∅
</td></tr>
<tr>
<td> <span id="item-semcond-classes-objectproperty"></span><span class="name">owl:ObjectProperty</span>
</td><td> ∈ IC
</td><td> = IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-ontology"></span><span class="name">owl:Ontology</span>
</td><td> ∈ IC
</td><td> = IX
</td></tr>
<tr>
<td> <span id="item-semcond-classes-ontologyproperty"></span><span class="name">owl:OntologyProperty</span>
</td><td> ∈ IC
</td><td> = IOXP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-property"></span><span class="name">rdf:Property</span>
</td><td> ∈ IC
</td><td> = IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-reflexiveproperty"></span><span class="name">owl:ReflexiveProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-resource"></span><span class="name">rdfs:Resource</span>
</td><td> ∈ IC
</td><td> = IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-restriction"></span><span class="name">owl:Restriction</span>
</td><td> ∈ IC
</td><td> ⊆ IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-symmetricproperty"></span><span class="name">owl:SymmetricProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-thing"></span><span class="name">owl:Thing</span>
</td><td> ∈ IC
</td><td> = IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-transitiveproperty"></span><span class="name">owl:TransitiveProperty</span>
</td><td> ∈ IC
</td><td> ⊆ IP
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_the_Vocabulary_Properties"></a><h3> <span class="mw-headline">5.3 Semantic Conditions for the Vocabulary Properties </span></h3>
<p><a href="index.html#table-semcond-properties" title="">Table 5.3</a>
lists the semantic conditions for the properties
that have IRIs in the OWL 2 RDF-Based vocabulary.
In addition,
the table contains all those properties
with IRIs in the RDFS vocabulary
that are specified to be annotation properties
under the OWL 2 RDF-Based Semantics.
The semantic conditions for the remaining properties
with names in the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS</a>
vocabularies
can be found in the RDF Semantics specification
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p><p>The first column tells the <i>IRI</i> of the property.
The second column defines
of what particular <i>kind</i> a property is,
i.e. whether it is a general property (a member of the part IP),
a datatype property (a member of IODP),
an ontology property (a member of IOXP) or
an annotation property (a member of IOAP).
The third column specifies
the <i>domain and range</i> of the property:
from an entry of the form
"IEXT(<i>I</i>(<i>p</i>)) ⊆ <i>S<sub>1</sub></i> × <i>S<sub>2</sub></i>",
for a property IRI <i>p</i>
and sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>,
and given an RDF triple
"<i>s</i> <i>p</i> <i>o</i>",
one can deduce the relationships
"<i>I</i>(<i>s</i>) ∈ <i>S<sub>1</sub></i>"
and
"<i>I</i>(<i>o</i>) ∈ <i>S<sub>2</sub></i>".
Note that some entries are of the form
"IEXT(<i>I</i>(<i>p</i>)) = <i>S<sub>1</sub></i> × <i>S<sub>2</sub></i>",
which means that the property extension is exactly specified
to be the Cartesian product of the two sets.
</p>
<div id="topic-semcond-properties-facets"></div>
<p>Not included in this table are the <i>facets</i> of the OWL 2 RDF-Based Semantics
with IRIs
listed in <a href="index.html#Facet_Names" title="">Section 3.4</a>,
which are used to specify datatype restrictions
(see <a href="index.html#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>).
For each such facet IRI <i>E</i>,
the following semantic conditions
<i>extend</i>
the basic semantics specification
that has been given for
<i>datatypes with facets</i>
in <a href="index.html#Datatype_Maps" title="">Section 4.1</a>:
</p>
<ul><li> <i>I</i>(<i>E</i>) ∈ IODP
</li><li> IEXT(<i>I</i>(<i>E</i>)) ⊆ IR × LV
</li></ul>
<div id="topic-semcond-properties-narydatatype"></div>
<p>Implementations are <i>not</i> required
to support the semantic condition for
<span class="name">owl:onProperties</span>,
but
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
support it
in order to realize
<i>n-ary dataranges</i> with arity ≥ 2
(see
Sections
<a href="../REC-owl2-syntax-20091027/index.html#Data_Ranges" title="Syntax">7</a>
and
<a href="../REC-owl2-syntax-20091027/index.html#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for further information).
</p>
<div id="topic-semcond-properties-informativenotes"></div>
<p><i>Informative notes:</i>
</p><p><span class="name">owl:topObjectProperty</span>
relates every two individuals in the universe with each other.
Likewise, <span class="name">owl:topDataProperty</span>
relates every individual with every data value.
Further,
<span class="name">owl:bottomObjectProperty</span>
and
<span class="name">owl:bottomDataProperty</span>
stand both for the <i>empty</i> relationship.
</p><p>The ranges of the properties
<span class="name">owl:deprecated</span> and <span class="name">owl:hasSelf</span>
are not restricted in any form,
and, in particular,
they are not restricted to Boolean values.
The actual object values of these properties
do not have any intended meaning,
but could as well have been defined to be of any other value.
Therefore, the semantics given here are of a form
that the values can be arbitrarily chosen
without leading to any nontrivial semantic conclusions.
It is, however, recommended to still use an object literal of the form
<span class="name">"true"^^xsd:boolean</span>
in ontologies,
in order to not get in conflict
with the required usage of these properties
in scenarios that ask for applying the reverse RDF mapping
(compare Table 13 in
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Expressions" title="Mapping to RDF Graphs">Section 3.2.4 of the OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
for <span class="name">owl:hasSelf</span>,
and
<a href="../REC-owl2-syntax-20091027/index.html#Annotation_Properties" title="Syntax">Section 5.5 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for <span class="name">owl:deprecated</span>).
</p><p>The range of the property
<span class="name">owl:annotatedProperty</span>
is unrestricted,
i.e. it is <i>not</i> specified as the set of properties.
Annotations are meant to be "semantically weak",
i.e. their formal meaning should not significantly exceed
that originating from the RDF Semantics specification.
</p><p>Several properties,
such as <span class="name">owl:priorVersion</span>,
have been specified as both ontology properties and annotation properties,
in order to be in line with both
the original
<a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html#5.2" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#5.2">OWL 1 RDF-Compatible Semantics specification</a>
[<cite><a href="index.html#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>]
and
the rest of the OWL 2 specification
(see <a href="../REC-owl2-syntax-20091027/index.html#Annotation_Properties" title="Syntax">Section 5.5 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
</p>
<div class="left" id="table-semcond-properties">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.3: Semantic Conditions for the Vocabulary Properties</span>
</caption>
<tr>
<th style="text-align: center"> IRI <i>E</i>
</th><th style="text-align: center"> <i>I</i>(<i>E</i>)
</th><th style="text-align: center"> IEXT(<i>I</i>(<i>E</i>))
</th></tr>
<tr>
<td> <span id="item-semcond-properties-allvaluesfrom"></span><span class="name">owl:allValuesFrom</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-annotatedproperty"></span><span class="name">owl:annotatedProperty</span>
</td><td> ∈ IP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-annotatedsource"></span><span class="name">owl:annotatedSource</span>
</td><td> ∈ IP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-annotatedtarget"></span><span class="name">owl:annotatedTarget</span>
</td><td> ∈ IP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-assertionproperty"></span><span class="name">owl:assertionProperty</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) × IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-backwardcompatiblewith"></span><span class="name">owl:backwardCompatibleWith</span>
</td><td> ∈ IOXP ,<br />∈ IOAP
</td><td> ⊆ IX × IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-bottomdataproperty"></span><span class="name">owl:bottomDataProperty</span>
</td><td> ∈ IODP
</td><td> = ∅
</td></tr>
<tr>
<td> <span id="item-semcond-properties-bottomobjectproperty"></span><span class="name">owl:bottomObjectProperty</span>
</td><td> ∈ IP
</td><td> = ∅
</td></tr>
<tr>
<td> <span id="item-semcond-properties-cardinality"></span><span class="name">owl:cardinality</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-comment"></span><span class="name">rdfs:comment</span>
</td><td> ∈ IOAP
</td><td> ⊆ IR × LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-complementof"></span><span class="name">owl:complementOf</span>
</td><td> ∈ IP
</td><td> ⊆ IC × IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-datatypecomplementof"></span><span class="name">owl:datatypeComplementOf</span>
</td><td> ∈ IP
</td><td> ⊆ IDC × IDC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-deprecated"></span><span class="name">owl:deprecated</span>
</td><td> ∈ IOAP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-differentfrom"></span><span class="name">owl:differentFrom</span>
</td><td> ∈ IP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-disjointunionof"></span><span class="name">owl:disjointUnionOf</span>
</td><td> ∈ IP
</td><td> ⊆ IC × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-disjointwith"></span><span class="name">owl:disjointWith</span>
</td><td> ∈ IP
</td><td> ⊆ IC × IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-distinctmembers"></span><span class="name">owl:distinctMembers</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-equivalentclass"></span><span class="name">owl:equivalentClass</span>
</td><td> ∈ IP
</td><td> ⊆ IC × IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-equivalentproperty"></span><span class="name">owl:equivalentProperty</span>
</td><td> ∈ IP
</td><td> ⊆ IP × IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-haskey"></span><span class="name">owl:hasKey</span>
</td><td> ∈ IP
</td><td> ⊆ IC × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-hasself"></span><span class="name">owl:hasSelf</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-hasvalue"></span><span class="name">owl:hasValue</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-imports"></span><span class="name">owl:imports</span>
</td><td> ∈ IOXP
</td><td> ⊆ IX × IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-incompatiblewith"></span><span class="name">owl:incompatibleWith</span>
</td><td> ∈ IOXP ,<br />∈ IOAP
</td><td> ⊆ IX × IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-intersectionof"></span><span class="name">owl:intersectionOf</span>
</td><td> ∈ IP
</td><td> ⊆ IC × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-inverseof"></span><span class="name">owl:inverseOf</span>
</td><td> ∈ IP
</td><td> ⊆ IP × IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-isdefinedby"></span><span class="name">rdfs:isDefinedBy</span>
</td><td> ∈ IOAP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-label"></span><span class="name">rdfs:label</span>
</td><td> ∈ IOAP
</td><td> ⊆ IR × LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-maxcardinality"></span><span class="name">owl:maxCardinality</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-maxqualifiedcardinality"></span><span class="name">owl:maxQualifiedCardinality</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-members"></span><span class="name">owl:members</span>
</td><td> ∈ IP
</td><td> ⊆ IR × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-mincardinality"></span><span class="name">owl:minCardinality</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-minqualifiedcardinality"></span><span class="name">owl:minQualifiedCardinality</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-onclass"></span><span class="name">owl:onClass</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-ondatarange"></span><span class="name">owl:onDataRange</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IDC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-ondatatype"></span><span class="name">owl:onDatatype</span>
</td><td> ∈ IP
</td><td> ⊆ IDC × IDC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-oneof"></span><span class="name">owl:oneOf</span>
</td><td> ∈ IP
</td><td> ⊆ IC × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-onproperty"></span><span class="name">owl:onProperty</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-onproperties"></span><span class="name">owl:onProperties</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-priorversion"></span><span class="name">owl:priorVersion</span>
</td><td> ∈ IOXP ,<br />∈ IOAP
</td><td> ⊆ IX × IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-propertychainaxiom"></span><span class="name">owl:propertyChainAxiom</span>
</td><td> ∈ IP
</td><td> ⊆ IP × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-propertydisjointwith"></span><span class="name">owl:propertyDisjointWith</span>
</td><td> ∈ IP
</td><td> ⊆ IP × IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-qualifiedcardinality"></span><span class="name">owl:qualifiedCardinality</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-sameas"></span><span class="name">owl:sameAs</span>
</td><td> ∈ IP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-seealso"></span><span class="name">rdfs:seeAlso</span>
</td><td> ∈ IOAP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-somevaluesfrom"></span><span class="name">owl:someValuesFrom</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) × IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-sourceindividual"></span><span class="name">owl:sourceIndividual</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-targetindividual"></span><span class="name">owl:targetIndividual</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-targetvalue"></span><span class="name">owl:targetValue</span>
</td><td> ∈ IP
</td><td> ⊆ ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) × LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-topdataproperty"></span><span class="name">owl:topDataProperty</span>
</td><td> ∈ IODP
</td><td> = IR × LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-topobjectproperty"></span><span class="name">owl:topObjectProperty</span>
</td><td> ∈ IP
</td><td> = IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-unionof"></span><span class="name">owl:unionOf</span>
</td><td> ∈ IP
</td><td> ⊆ IC × ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-versioninfo"></span><span class="name">owl:versionInfo</span>
</td><td> ∈ IOAP
</td><td> ⊆ IR × IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-versioniri"></span><span class="name">owl:versionIRI</span>
</td><td> ∈ IOXP
</td><td> ⊆ IX × IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-withrestrictions"></span><span class="name">owl:withRestrictions</span>
</td><td> ∈ IP
</td><td> ⊆ IDC × ISEQ
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Boolean_Connectives"></a><h3> <span class="mw-headline">5.4 Semantic Conditions for Boolean Connectives </span></h3>
<p><a href="index.html#table-semcond-booleans" title="">Table 5.4</a>
lists the semantic conditions for Boolean connectives,
including
intersections, unions and complements
of classes and datatypes.
An intersection or a union of a collection of datatypes
or a complement of a datatype
is itself a datatype.
While a complement of a class is created w.r.t. the whole universe,
a datatype complement is created for a datatype w.r.t. the set of data values only.
</p>
<div id="topic-semcond-booleans-informativenotes"></div>
<p><i>Informative notes:</i>
Of the three pairs of semantic conditions in the table
every first is an "iff" condition,
since the corresponding OWL 2 language constructs
are both
class expressions and axioms.
In contrast,
the semantic condition on datatype complements
is an "if-then" condition,
since it only corresponds to a datarange expression.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
For the remaining semantic conditions
that treat the cases of intersections and unions of datatypes
it is sufficient to have "if-then" conditions,
since stronger "iff" conditions would be redundant
due to the more general "iff" conditions
that already exist for classes.
Note that the datatype related semantic conditions
do not apply to empty sets,
but one can still receive a datatype from an empty set
by explicitly asserting the resulting class
to be an instance of class <span class="name">rdfs:Datatype</span>.
</p>
<div class="left" id="table-semcond-booleans">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.4: Semantic Conditions for Boolean Connectives</span>
</caption>
<tr>
<th colspan="4" style="text-align: center"> <span id="item-semcond-booleans-intersectionof-main"></span>if <i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IR then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:intersectionOf</span>))
</td><th colspan="2" style="text-align: center"> iff
</th><td> <i>z</i> , <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC ,<br />ICEXT(<i>z</i>) = ICEXT(<i>c<sub>1</sub></i>) ∩ … ∩ ICEXT(<i>c<sub>n</sub></i>)
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-booleans-intersectionof-data"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> <i>s</i> sequence of <i>d<sub>1</sub></i> , … , <i>d<sub>n</sub></i> ∈ IDC , <i>n</i> ≥ 1 ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:intersectionOf</span>))
</td><td colspan="2"> <i>z</i> ∈ IDC
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="4" style="text-align: center"> <span id="item-semcond-booleans-unionof-main"></span>if <i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IR then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:unionOf</span>))
</td><th colspan="2" style="text-align: center"> iff
</th><td> <i>z</i> , <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC ,<br />ICEXT(<i>z</i>) = ICEXT(<i>c<sub>1</sub></i>) ∪ … ∪ ICEXT(<i>c<sub>n</sub></i>)
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-booleans-unionof-data"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> <i>s</i> sequence of <i>d<sub>1</sub></i> , … , <i>d<sub>n</sub></i> ∈ IDC , <i>n</i> ≥ 1 ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:unionOf</span>))
</td><td colspan="2"> <i>z</i> ∈ IDC
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<td> <span id="item-semcond-booleans-complementof"></span>( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:complementOf</span>))
</td><th colspan="2" style="text-align: center"> iff
</th><td> <i>z</i> , <i>c</i> ∈ IC ,<br />ICEXT(<i>z</i>) = IR \ ICEXT(<i>c</i>)
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-booleans-datatypecomplementof"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> ( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:datatypeComplementOf</span>))
</td><td colspan="2"> ICEXT(<i>z</i>) = LV \ ICEXT(<i>d</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Enumerations"></a><h3> <span class="mw-headline">5.5 Semantic Conditions for Enumerations </span></h3>
<p><a href="index.html#table-semcond-enums" title="">Table 5.5</a>
lists the semantic conditions for enumerations,
i.e. classes that consist of an explicitly given finite set of instances.
In particular, an enumeration entirely consisting of data values is a datatype.
</p>
<div id="topic-semcond-enums-informativenotes"></div>
<p><i>Informative notes:</i>
The first semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is both a class expression and an axiom.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
For the remaining semantic condition
that treats the case of enumerations of data values
it is sufficient to have an "if-then" condition,
since a stronger "iff" condition would be redundant
due to the more general "iff" condition
that already exists for individuals.
Note that the data value related semantic condition
does not apply to empty sets,
but one can still receive a datatype from an empty set
by explicitly asserting the resulting class
to be an instance of class <span class="name">rdfs:Datatype</span>.
</p>
<div class="left" id="table-semcond-enums">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.5: Semantic Conditions for Enumerations</span>
</caption>
<tr>
<th colspan="4" style="text-align: center"> <span id="item-semcond-enums-main"></span>if <i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:oneOf</span>))
</td><th colspan="2" rowspan="1" style="text-align: center"> iff
</th><td> <i>z</i> ∈ IC ,<br />ICEXT(<i>z</i>) = { <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> }
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-enums-data"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> <i>s</i> sequence of <i>v<sub>1</sub></i> , … , <i>v<sub>n</sub></i> ∈ LV , <i>n</i> ≥ 1 ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:oneOf</span>))
</td><td colspan="2"> <i>z</i> ∈ IDC
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Property_Restrictions"></a><h3> <span class="mw-headline">5.6 Semantic Conditions for Property Restrictions </span></h3>
<p><a href="index.html#table-semcond-restrictions" title="">Table 5.6</a>
lists the semantic conditions for property restrictions.
</p><p><i>Value restrictions</i> require that
some or all of the values of a certain property
must be instances of a given class or data range,
or that the property has a specifically defined value.
By placing a <i>self restriction</i> on some given property
one only considers those individuals
that are reflexively related to themselves via this property.
<i>Cardinality restrictions</i> determine
how often a certain property is allowed
to be applied to a given individual.
<i>Qualified cardinality restrictions</i>
are more specific than cardinality restrictions
in that they determine the quantity of a property application
with respect to a particular class or data range
from which the property values are taken.
</p>
<div id="topic-semcond-restrictions-narydatatype"></div>
<p>Implementations are <i>not</i> required
to support the semantic conditions for
<span class="name">owl:onProperties</span>,
but
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
support them
in order to realize
<i>n-ary dataranges</i> with arity ≥ 2
(see
Sections
<a href="../REC-owl2-syntax-20091027/index.html#Data_Ranges" title="Syntax">7</a>
and
<a href="../REC-owl2-syntax-20091027/index.html#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for further information).
</p>
<div id="topic-semcond-restrictions-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "if-then" conditions,
since the corresponding OWL 2 language constructs
are class expressions.
The "if-then" conditions generally only list those consequences
on their right hand side
that are specific for the respective condition,
i.e. consequences that do not already follow by other means.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
Note that the semantic condition for <i>self restrictions</i>
does not constrain the right hand side of
a <span class="name">owl:hasSelf</span> assertion
to be the Boolean value <span class="name">"true"^^xsd:boolean</span>.
See <a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a> for an explanation.
</p>
<div class="left" id="table-semcond-restrictions">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.6: Semantic Conditions for Property Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <span id="item-semcond-restrictions-somevaluesfrom"></span>( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | ∃ <i>y</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-somevaluesfrom-nary"></span><i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IR , <i>n</i> ≥ 1 ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td><td> <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | ∃ <i>y<sub>1</sub></i> , … , <i>y<sub>n</sub></i> : ( <i>x</i> , <i>y<sub>k</sub></i> ) ∈ IEXT(<i>p<sub>k</sub></i>) for each 1 ≤ <i>k</i> ≤ <i>n</i> and ( <i>y<sub>1</sub></i> , … , <i>y<sub>n</sub></i> ) ∈ ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-allvaluesfrom"></span>( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | ∀ <i>y</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) implies <i>y</i> ∈ ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-allvaluesfrom-nary"></span><i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IR , <i>n</i> ≥ 1 ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td><td> <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | ∀ <i>y<sub>1</sub></i> , … , <i>y<sub>n</sub></i> : ( <i>x</i> , <i>y<sub>k</sub></i> ) ∈ IEXT(<i>p<sub>k</sub></i>) for each 1 ≤ <i>k</i> ≤ <i>n</i> implies ( <i>y<sub>1</sub></i> , … , <i>y<sub>n</sub></i> ) ∈ ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-hasvalue"></span>( <i>z</i> , <i>a</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:hasValue</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | ( <i>x</i> , <i>a</i> ) ∈ IEXT(<i>p</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-hasself"></span>( <i>z</i> , <i>v</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:hasSelf</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | ( <i>x</i> , <i>x</i> ) ∈ IEXT(<i>p</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-mincardinality"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:minCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) } ≥ <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-maxcardinality"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:maxCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) } ≤ <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-cardinality"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:cardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) } = <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-minqualifiedcardinality"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onClass</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>c</i>) } ≥ <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-minqualifiedcardinality-data"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>))
</td><td> <i>p</i> ∈ IODP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> ∈ LV | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>d</i>) } ≥ <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-maxqualifiedcardinality"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onClass</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>c</i>) } ≤ <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-maxqualifiedcardinality-data"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>))
</td><td> <i>p</i> ∈ IODP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> ∈ LV | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>d</i>) } ≤ <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-qualifiedcardinality"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onClass</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>c</i>) } = <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-qualifiedcardinality-data"></span>( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>))
</td><td> <i>p</i> ∈ IODP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> ∈ LV | ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and <i>y</i> ∈ ICEXT(<i>d</i>) } = <i>n</i> }
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Datatype_Restrictions"></a><h3> <span class="mw-headline">5.7 Semantic Conditions for Datatype Restrictions </span></h3>
<p><a href="index.html#table-semcond-facets" title="">Table 5.7</a>
lists the semantic conditions for datatype restrictions,
which are used to define sub datatypes of existing datatypes
by restricting the original datatype
by means of a set of facet-value pairs.
See <a href="index.html#Facet_Names" title="">Section 3.4</a>
for information and an example on constraining facets.
</p>
<div id="topic-semcond-facets-empty"></div>
<p>Certain special cases exist:
If no facet-value pair is applied to a given datatype,
then the resulting datatype will be equivalent to the original datatype.
Further,
if a facet-value pair is applied to a datatype
without being a member of the datatype's facet space,
then the ontology cannot be satisfied
and will therefore be inconsistent.
In particular,
a datatype restriction with one or more specified facet-value pairs
will result in an inconsistent ontology,
if applied to a datatype with an empty facet space.
</p>
<div id="topic-semcond-facets-facetspace"></div>
<p>The set <b>IFS</b>
is defined by
IFS(<i>d</i>) := { ( <i>I</i>(<i>F</i>) , <i>v</i> ) | ( <i>F</i> , <i>v</i> ) ∈ FS(<i>d</i>) } ,
where
<i>d</i> is a datatype,
<i>F</i> is the IRI of a constraining facet,
and <i>v</i> is a constraining value of the facet.
This set corresponds to the facet space FS(<i>d</i>),
as defined in <a href="index.html#Datatype_Maps" title="">Section 4.1</a>,
but rather consists of
pairs of the <i>denotation</i> of a facet and a value.
</p>
<div id="topic-semcond-facets-facetmapping"></div>
<p>The mapping <b>IF2V</b>
is defined by
IF2V(<i>d</i>)(( <i>I</i>(<i>F</i>) , <i>v</i> )) := F2V(<i>d</i>)(( <i>F</i> , <i>v</i> )) ,
where
<i>d</i> is a datatype,
<i>F</i> is the IRI of a constraining facet,
and <i>v</i> is a constraining value of the facet.
This mapping corresponds to the facet-to-value mapping F2V(<i>d</i>),
as defined in <a href="index.html#Datatype_Maps" title="">Section 4.1</a>,
resulting in the same subsets of the value space VS(<i>d</i>),
but rather applies to
pairs of the <i>denotation</i> of a facet and a value.
</p>
<div id="topic-semcond-facets-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "if-then" condition,
since the corresponding OWL 2 language construct
is a datarange expression.
The "if-then" condition only lists those consequences
on its right hand side
that are specific for the condition,
i.e. consequences that do not already follow by other means.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.7: Semantic Conditions for Datatype Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-semcond-facets"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>z</i><sub>1</sub><i> , … , </i>z<i><sub>n</sub></i> ∈ IR ,<br /><i>f<sub>1</sub></i> , … , <i>f<sub>n</sub></i> ∈ IP ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDatatype</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:withRestrictions</span>)) ,<br />( <i>z<sub>1</sub></i> , <i>v<sub>1</sub></i> ) ∈ IEXT(<i>f<sub>1</sub></i>) , … , ( <i>z<sub>n</sub></i> , <i>v<sub>n</sub></i> ) ∈ IEXT(<i>f<sub>n</sub></i>)
</td><td> <i>f<sub>1</sub></i> , … , <i>f<sub>n</sub></i> ∈ IODP ,<br /><i>v<sub>1</sub></i> , … , <i>v<sub>n</sub></i> ∈ LV ,<br />( <i>f<sub>1</sub></i> , <i>v<sub>1</sub></i> ) , … , ( <i>f<sub>n</sub></i> , <i>v<sub>n</sub></i> ) ∈ IFS(<i>d</i>) ,<br />ICEXT(<i>z</i>) = ICEXT(<i>d</i>) ∩ IF2V(<i>d</i>)(( <i>f<sub>1</sub></i> , <i>v<sub>1</sub></i> )) ∩ … ∩ IF2V(<i>d</i>)(( <i>f<sub>n</sub></i> , <i>v<sub>n</sub></i> ))
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_the_RDFS_Vocabulary"></a><h3> <span class="mw-headline">5.8 Semantic Conditions for the RDFS Vocabulary </span></h3>
<p><a href="index.html#table-semcond-rdfs" title="">Table 5.8</a>
<i>extends</i> the RDFS semantic conditions
for subclass axioms, subproperty axioms, domain axioms and range axioms.
The semantic conditions provided here are "iff" conditions,
while the original semantic conditions,
as specified in
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">Section 4.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
are weaker "if-then" conditions.
Only the additional semantic conditions are given here
and the other conditions of RDF and RDFS
are retained.
</p>
<div id="topic-semcond-rdfs-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-rdfs">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.8: Semantic Conditions for the RDFS Vocabulary</span>
</caption>
<tr>
<td> <span id="item-semcond-rdfs-subclassof"></span>( <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdfs:subClassOf</span>))
</td><th rowspan="4" style="text-align: center"> iff
</th><td> <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ∈ IC ,<br />ICEXT(<i>c<sub>1</sub></i>) ⊆ ICEXT(<i>c<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-rdfs-subpropertyof"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdfs:subPropertyOf</span>))
</td><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ∈ IP ,<br />IEXT(<i>p<sub>1</sub></i>) ⊆ IEXT(<i>p<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-rdfs-domain"></span>( <i>p</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">rdfs:domain</span>))
</td><td> <i>p</i> ∈ IP , <i>c</i> ∈ IC ,<br />∀ <i>x</i> , <i>y</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) implies <i>x</i> ∈ ICEXT(<i>c</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-rdfs-range"></span>( <i>p</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">rdfs:range</span>))
</td><td> <i>p</i> ∈ IP , <i>c</i> ∈ IC ,<br />∀ <i>x</i> , <i>y</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) implies <i>y</i> ∈ ICEXT(<i>c</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Equivalence_and_Disjointness"></a><h3> <span class="mw-headline">5.9 Semantic Conditions for Equivalence and Disjointness </span></h3>
<p><a href="index.html#table-semcond-eqdis" title="">Table 5.9</a>
lists the semantic conditions for specifying
that two individuals are equal or different from each other,
and that either two classes or two properties
are equivalent or disjoint with each other,
respectively.
The
property <span class="name">owl:equivalentClass</span>
is also used to formulate <i>datatype definitions</i>
(see <a href="../REC-owl2-syntax-20091027/index.html#Datatype_Definitions" title="Syntax">Section 9.4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for information about datatype definitions).
In addition,
the table treats disjoint union axioms.
</p>
<div id="topic-semcond-eqdis-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-eqdis">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.9: Semantic Conditions for Equivalence and Disjointness</span>
</caption>
<tr>
<td> <span id="item-semcond-eqdis-sameas"></span>( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:sameAs</span>))
</td><th rowspan="6" style="text-align: center"> iff
</th><td> <i>a<sub>1</sub></i> = <i>a<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-differentfrom"></span>( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:differentFrom</span>))
</td><td> <i>a<sub>1</sub></i> ≠ <i>a<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-equivalentclass"></span>( <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:equivalentClass</span>))
</td><td> <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ∈ IC ,<br />ICEXT(<i>c<sub>1</sub></i>) = ICEXT(<i>c<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-disjointwith"></span>( <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:disjointWith</span>))
</td><td> <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ∈ IC ,<br />ICEXT(<i>c<sub>1</sub></i>) ∩ ICEXT(<i>c<sub>2</sub></i>) = ∅
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-equivalentproperty"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:equivalentProperty</span>))
</td><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ∈ IP ,<br />IEXT(<i>p<sub>1</sub></i>) = IEXT(<i>p<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-propertydisjointwith"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:propertyDisjointWith</span>))
</td><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ∈ IP ,<br />IEXT(<i>p<sub>1</sub></i>) ∩ IEXT(<i>p<sub>2</sub></i>) = ∅
</td></tr>
<tr>
<td colspan="3">
</td></tr>
<tr>
<th colspan="3" style="text-align: center"> <span id="item-semcond-eqdis-disjointunionof"></span>if <i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IR then
</th></tr>
<tr>
<td> ( <i>c</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:disjointUnionOf</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>c</i> , <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC ,<br />ICEXT(<i>c</i>) = ICEXT(<i>c<sub>1</sub></i>) ∪ … ∪ ICEXT(<i>c<sub>n</sub></i>) ,<br />ICEXT(<i>c<sub>j</sub></i>) ∩ ICEXT(<i>c<sub>k</sub></i>) = ∅ for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_N-ary_Disjointness"></a><h3> <span class="mw-headline">5.10 Semantic Conditions for N-ary Disjointness </span></h3>
<p><a href="index.html#table-semcond-ndis" title="">Table 5.10</a>
lists the semantic conditions for specifying
n-ary diversity and disjointness axioms,
i.e. that several given individuals
are mutually different from each other,
and that several given classes or properties
are mutually disjoint with each other,
respectively.
</p>
<div id="topic-semcond-ndis-variants"></div>
<p>Note that there are two alternative ways
to specify <span class="name">owl:AllDifferent</span> axioms,
by using either the property
<span class="name">owl:members</span>
that is used for all other constructs, too,
or by applying the legacy property
<span class="name">owl:distinctMembers</span>.
Both variants have an equivalent formal meaning.
</p>
<div id="topic-semcond-ndis-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic conditions essentially represent "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
However,
there are actually <i>two</i> semantic conditions for each language construct
due to the multi-triple RDF encoding of these language constructs.
The "if-then" conditions only list those consequences
on their right hand side
that are specific for the respective condition,
i.e. consequences that do not already follow by other means.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-ndis">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.10: Semantic Conditions for N-ary Disjointness</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-members-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR ,<br /><i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td><td> <i>a<sub>j</sub></i> ≠ <i>a<sub>k</sub></i> for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-members-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR ,<br /><i>a<sub>j</sub></i> ≠ <i>a<sub>k</sub></i> for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td><td> <i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-distinctmembers-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR ,<br /><i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:distinctMembers</span>))
</td><td> <i>a<sub>j</sub></i> ≠ <i>a<sub>k</sub></i> for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-distinctmembers-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR ,<br /><i>a<sub>j</sub></i> ≠ <i>a<sub>k</sub></i> for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td><td> <i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:distinctMembers</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointclasses-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IR ,<br /><i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDisjointClasses</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td><td> <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC ,<br />ICEXT(<i>c<sub>j</sub></i>) ∩ ICEXT(<i>c<sub>k</sub></i>) = ∅ for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointclasses-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC ,<br />ICEXT(<i>c<sub>j</sub></i>) ∩ ICEXT(<i>c<sub>k</sub></i>) = ∅ for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td><td> <i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDisjointClasses</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointproperties-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IR ,<br /><i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDisjointProperties</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td><td> <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP ,<br />IEXT(<i>p<sub>j</sub></i>) ∩ IEXT(<i>p<sub>k</sub></i>) = ∅ for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointproperties-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP ,<br />IEXT(<i>p<sub>j</sub></i>) ∩ IEXT(<i>p<sub>k</sub></i>) = ∅ for each 1 ≤ <i>j</i> ≤ <i>n</i> and each 1 ≤ <i>k</i> ≤ <i>n</i> such that <i>j</i> ≠ <i>k</i>
</td><td> <i>z</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AllDisjointProperties</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Sub_Property_Chains"></a><h3> <span class="mw-headline">5.11 Semantic Conditions for Sub Property Chains </span></h3>
<p><a href="index.html#table-semcond-chains" title="">Table 5.11</a>
lists the semantic conditions for sub property chains,
which allow for specifying complex property subsumption axioms.
</p><p>As an example,
one can define a sub property chain axiom
that specifies
the chain consisting of the property extensions
of properties
<span class="name">ex:hasFather</span>
and
<span class="name">ex:hasBrother</span>
to be a sub relation of
the extension of the property
<span class="name">ex:hasUncle</span>.
</p>
<div id="topic-semcond-chains-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
The semantics has been specified in a way
such that a sub property chain axiom can be satisfied
without requiring the existence of a property
that has the property chain as its property extension.
</p>
<div class="left" id="table-semcond-chains">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.11: Semantic Conditions for Sub Property Chains</span>
</caption>
<tr>
<th colspan="3" style="text-align: center"> <span id="item-semcond-chains"></span>if <i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IR then
</th></tr>
<tr>
<td> ( <i>p</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:propertyChainAxiom</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>p</i> ∈ IP ,<br /><i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP ,<br />∀ <i>y<sub>0</sub></i> , … , <i>y<sub>n</sub></i> : ( <i>y<sub>0</sub></i> , <i>y<sub>1</sub></i> ) ∈ IEXT(<i>p<sub>1</sub></i>) and … and ( <i>y<sub>n-1</sub></i> , <i>y<sub>n</sub></i> ) ∈ IEXT(<i>p<sub>n</sub></i>) implies ( <i>y<sub>0</sub></i> , <i>y<sub>n</sub></i> ) ∈ IEXT(<i>p</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Inverse_Properties"></a><h3> <span class="mw-headline">5.12 Semantic Conditions for Inverse Properties </span></h3>
<p><a href="index.html#table-semcond-inverses" title="">Table 5.12</a>
lists the semantic conditions for inverse property axioms.
The inverse of a given property
is the corresponding property
with subject and object swapped
for each property assertion built from it.
</p>
<div id="topic-semcond-inverses-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-inverses">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.12: Semantic Conditions for Inverse Properties</span>
</caption>
<tr>
<td> <span id="item-semcond-inverses"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:inverseOf</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ∈ IP ,<br />IEXT(<i>p<sub>1</sub></i>) = { ( <i>x</i> , <i>y</i> ) | ( <i>y</i> , <i>x</i> ) ∈ IEXT(<i>p<sub>2</sub></i>) }
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Property_Characteristics"></a><h3> <span class="mw-headline">5.13 Semantic Conditions for Property Characteristics </span></h3>
<p><a href="index.html#table-semcond-characteristics" title="">Table 5.13</a>
lists the semantic conditions for property characteristics.
</p><p>If a property is <i>functional</i>,
then at most one distinct value can be assigned
to any given individual
via this property.
An <i>inverse functional</i> property can be regarded as a "key" property,
i.e. no two different individuals
can be assigned the same value
via this property.
A <i>reflexive</i> property relates every individual in the universe to itself,
whereas an <i>irreflexive</i> property does not relate any individual with itself.
If two individuals are related by a <i>symmetric</i> property,
then this property also relates them reversely,
while this is never the case for an <i>asymmetric</i> property.
A <i>transitive</i> property
that relates an individual <i>a</i> with an individual <i>b</i>,
and the latter with an individual <i>c</i>,
also relates <i>a</i> with <i>c</i>.
</p>
<div id="topic-semcond-characteristics-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-characteristics">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.13: Semantic Conditions for Property Characteristics</span>
</caption>
<tr>
<td> <span id="item-semcond-characteristics-functionalproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:FunctionalProperty</span>))
</td><th rowspan="7" style="text-align: center"> iff
</th><td> <i>p</i> ∈ IP ,<br />∀ <i>x</i> , <i>y<sub>1</sub></i> , <i>y<sub>2</sub></i> : ( <i>x</i> , <i>y<sub>1</sub></i> ) ∈ IEXT(<i>p</i>) and ( <i>x</i> , <i>y<sub>2</sub></i> ) ∈ IEXT(<i>p</i>) implies <i>y<sub>1</sub></i> = <i>y<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-inversefunctionalproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:InverseFunctionalProperty</span>))
</td><td> <i>p</i> ∈ IP ,<br />∀ <i>x<sub>1</sub></i> , <i>x<sub>2</sub></i> , <i>y</i> : ( <i>x<sub>1</sub></i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and ( <i>x<sub>2</sub></i> , <i>y</i> ) ∈ IEXT(<i>p</i>) implies <i>x<sub>1</sub></i> = <i>x<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-reflexiveproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:ReflexiveProperty</span>))
</td><td> <i>p</i> ∈ IP ,<br />∀ <i>x</i> : ( <i>x</i> , <i>x</i> ) ∈ IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-irreflexiveproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:IrreflexiveProperty</span>))
</td><td> <i>p</i> ∈ IP ,<br />∀ <i>x</i> : ( <i>x</i> , <i>x</i> ) ∉ IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-symmetricproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:SymmetricProperty</span>))
</td><td> <i>p</i> ∈ IP ,<br />∀ <i>x</i> , <i>y</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) implies ( <i>y</i> , <i>x</i> ) ∈ IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-asymmetricproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:AsymmetricProperty</span>))
</td><td> <i>p</i> ∈ IP ,<br />∀ <i>x</i> , <i>y</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) implies ( <i>y</i> , <i>x</i> ) ∉ IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-transitiveproperty"></span><i>p</i> ∈ ICEXT(<i>I</i>(<span class="name">owl:TransitiveProperty</span>))
</td><td> <i>p</i> ∈ IP ,<br />∀ <i>x</i> , <i>y</i> , <i>z</i> : ( <i>x</i> , <i>y</i> ) ∈ IEXT(<i>p</i>) and ( <i>y</i> , <i>z</i> ) ∈ IEXT(<i>p</i>) implies ( <i>x</i> , <i>z</i> ) ∈ IEXT(<i>p</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Keys"></a><h3> <span class="mw-headline">5.14 Semantic Conditions for Keys </span></h3>
<p><a href="index.html#table-semcond-keys" title="">Table 5.14</a>
lists the semantic conditions for Keys.
</p><p>Keys provide an alternative to inverse functional properties
(see <a href="index.html#Semantic_Conditions_for_Property_Characteristics" title="">Section 5.13</a>).
They allow for defining a property as a key local to a given class:
the specified property
will have the features of a key
only for individuals being instances of the class,
and no assumption is made
about individuals
for which membership of the class cannot be entailed.
Further,
it is possible to define "compound keys",
i.e. several properties can be combined into a single key
applicable to composite values.
Note that
keys are not functional by default
under the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-semcond-keys-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-keys">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.14: Semantic Conditions for Keys</span>
</caption>
<tr>
<th colspan="3" style="text-align: center"> <span id="item-semcond-keys"></span>if <i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IR then
</th></tr>
<tr>
<td> ( <i>c</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:hasKey</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>c</i> ∈ IC ,<br /><i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP ,<br />∀ <i>x</i> , <i>y</i> , <i>z<sub>1</sub></i> , … , <i>z<sub>n</sub></i> :<br /> if <i>x</i> ∈ ICEXT(<i>c</i>) and <i>y</i> ∈ ICEXT(<i>c</i>) and<br /> ( <i>x</i> , <i>z<sub>k</sub></i> ) ∈ IEXT(<i>p<sub>k</sub></i>) and ( <i>y</i> , <i>z<sub>k</sub></i> ) ∈ IEXT(<i>p<sub>k</sub></i>) for each 1 ≤ <i>k</i> ≤ <i>n</i><br /> then <i>x</i> = <i>y</i>
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Negative_Property_Assertions"></a><h3> <span class="mw-headline">5.15 Semantic Conditions for Negative Property Assertions </span></h3>
<p><a href="index.html#table-semcond-negassert" title="">Table 5.15</a>
lists the semantic conditions for negative property assertions.
They allow to state that
two given individuals are <i>not</i> related by a given property.
</p>
<div id="topic-semcond-negassert-datavariant"></div>
<p>The second form based on <span class="name">owl:targetValue</span>
is more specific than the first form based on <span class="name">owl:targetIndividual</span>
in that the second form is restricted
to the case of negative <i>data</i> property assertions.
Note that the second form
will coerce the target value of a negative property assertion
into a data value,
due to the range defined for the property
<span class="name">owl:targetValue</span>
in
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>.
</p>
<div id="topic-semcond-negassert-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic conditions essentially represent "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
However,
there are actually <i>two</i> semantic conditions for each language construct,
due to the multi-triple RDF encoding of these language constructs.
The "if-then" conditions only list those consequences
on their right hand side
that are specific for the respective condition,
i.e. consequences that do not already follow by other means.
See the
<a href="index.html#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-negassert">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.15: Semantic Conditions for Negative Property Assertions</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-object-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>a<sub>1</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>a<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:targetIndividual</span>))
</td><td> ( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) ∉ IEXT(<i>p</i>)
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-object-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>a<sub>1</sub></i> ∈ IR ,<br /><i>p</i> ∈ IP ,<br /><i>a<sub>2</sub></i> ∈ IR ,<br />( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) ∉ IEXT(<i>p</i>)
</td><td> ( <i>z</i> , <i>a<sub>1</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>a<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:targetIndividual</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-data-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>a</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>v</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:targetValue</span>))
</td><td> <i>p</i> ∈ IODP ,<br />( <i>a</i> , <i>v</i> ) ∉ IEXT(<i>p</i>)
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-data-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>a</i> ∈ IR ,<br /><i>p</i> ∈ IODP ,<br /><i>v</i> ∈ LV ,<br />( <i>a</i> , <i>v</i> ) ∉ IEXT(<i>p</i>)
</td><td> ( <i>z</i> , <i>a</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>v</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:targetValue</span>))
</td></tr>
</table>
</div>
<a name="Appendix:_Axiomatic_Triples_.28Informative.29"></a><h2> <span class="mw-headline">6 Appendix: Axiomatic Triples (Informative) </span></h2>
<p>The RDF Semantics specification
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
defines so called <i>"axiomatic triples"</i>
as part of the semantics of
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDF_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDF_axiomatic_triples">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFS_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples">RDFS</a>.
Unlike the RDF Semantics,
the OWL 2 RDF-Based Semantics does not normatively specify any axiomatic triples,
since one cannot expect to find a set of RDF triples
that fully captures all "axiomatic aspects"
of the OWL 2 RDF-Based Semantics.
Furthermore,
axiomatic triples for the OWL 2 RDF-Based Semantics could,
in principle,
contain arbitrarily complex class expressions,
e.g. the union of several classes,
and by this it becomes nonobvious
which of several possible nonequivalent sets of axiomatic triples
should be selected.
However,
the OWL 2 RDF-Based Semantics includes many semantic conditions
that can in a sense be regarded as being "axiomatic",
and thus can be considered a replacement for the missing axiomatic triples.
After an overview on axiomatic triples for RDF and RDFS
in <a href="index.html#Axiomatic_Triples_in_RDF" title="">Section 6.1</a>,
Sections <a href="index.html#Axiomatic_Triples_for_the_Vocabulary_Classes" title="">6.2</a>
and
<a href="index.html#Axiomatic_Triples_for_the_Vocabulary_Properties" title="">6.3</a>
will discuss how the "axiomatic" semantic conditions
of the OWL 2 RDF-Based Semantics
relate to axiomatic triples.
Based on this discussion,
an explicit example set of axiomatic triples
that is compatible with the OWL 2 RDF-Based Semantics
will be provided in
<a href="index.html#A_Set_of_Axiomatic_Triples" title="">Section 6.4</a>.
</p>
<a name="Axiomatic_Triples_in_RDF"></a><h3> <span class="mw-headline">6.1 Axiomatic Triples in RDF </span></h3>
<p>In
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDF_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDF_axiomatic_triples">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#RDFS_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples">RDFS</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
axiomatic triples are used
to provide basic meaning
for all the vocabulary terms
of the two languages.
This formal meaning is independent of any given RDF graph,
and it even holds for vocabulary terms,
which do not occur in a graph
that is interpreted by an RDF or RDFS interpretation.
As a consequence,
all the axiomatic triples of RDF and RDFS
are entailed by the <i>empty</i> graph,
when being interpreted under the semantics of RDF or RDFS,
respectively.
</p>
<div id="topic-axiomatic-rdf-examples"></div>
<p>Examples of RDF and RDFS axiomatic triples are:
</p>
<div class="indent">
<p>(1) <span class="name">rdf:type rdf:type rdf:Property .</span><br />
(2) <span class="name">rdf:type rdfs:domain rdfs:Resource .</span><br />
(3) <span class="name">rdf:type rdfs:range rdfs:Class .</span><br />
(4) <span class="name">rdfs:Datatype rdfs:subClassOf rdfs:Class .</span><br />
(5) <span class="name">rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .</span>
</p>
</div>
<p>As shown by these examples,
axiomatic triples are typically used by the RDF Semantics specification
to determine the part of the universe
to which the denotation of a vocabulary term belongs (1).
In the case of a property,
the domain (2) and range (3) is specified as well.
Also, in some cases,
hierarchical relationships
between classes (4) or properties (5) of the vocabulary
are determined.
</p>
<div id="topic-axiomatic-rdf-semcond"></div>
<p>Under the OWL 2 RDF-Based Semantics,
all the axiomatic triples of RDF and RDFS
could, in principle, be replaced by
"axiomatic" semantic conditions
that have neither premises nor bound variables.
By applying the <i>RDFS semantic conditions</i>
given in <a href="index.html#Semantic_Conditions_for_the_RDFS_Vocabulary" title="">Section 5.8</a>,
the example axiomatic triples (1) – (5)
can be equivalently restated as:
</p>
<div class="indent">
<p><i>I</i>(<span class="name">rdf:type</span>) ∈ ICEXT(<i>I</i>(<span class="name">rdf:Property</span>)) ,<br />
IEXT(<i>I</i>(<span class="name">rdf:type</span>)) ⊆ ICEXT(<i>I</i>(<span class="name">rdfs:Resource</span>)) × ICEXT(<i>I</i>(<span class="name">rdfs:Class</span>)) ,<br />
ICEXT(<i>I</i>(<span class="name">rdfs:Datatype</span>)) ⊆ ICEXT(<i>I</i>(<span class="name">rdfs:Class</span>)) ,<br />
IEXT(<i>I</i>(<span class="name">rdfs:isDefinedBy</span>)) ⊆ IEXT(<i>I</i>(<span class="name">rdfs:seeAlso</span>)) .
</p>
</div>
<div id="topic-axiomatic-rdf-simple"></div>
<p>All the axiomatic triples of RDF and RDFS
can be considered <i>"simple"</i>
in the sense that
they have in their object position
only single terms
from the RDF and RDFS vocabularies,
and no complex class or property expressions
appear there.
</p>
<a name="Axiomatic_Triples_for_the_Vocabulary_Classes"></a><h3> <span class="mw-headline">6.2 Axiomatic Triples for the Vocabulary Classes </span></h3>
<p>The semantic conditions for <i>vocabulary classes</i>
in <a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>
can be considered as corresponding to
a set of axiomatic triples
for the classes in the vocabulary of the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-axiomatic-classes-secondcoldef"></div>
<p>First,
for each IRI <i>E</i>
occurring in the first column of <a href="index.html#table-semcond-classes" title="">Table 5.2</a>,
if the <i>second</i> column contains an entry
of the form
"<i>I</i>(<i>E</i>) ∈ <i>S</i>"
for some set <i>S</i>,
then this entry corresponds to an RDF triple of the form
"<i>E</i> <span class="name">rdf:type</span> <i>C</i>",
where <i>C</i> is the IRI of a vocabulary class with ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>.
In the table, <i>S</i> will always be either
the <a href="index.html#Parts_of_the_Universe" title="">part</a> IC of all classes,
or some sub part of IC.
Hence, in a corresponding RDF triple the IRI <i>C</i> will be
one of
"<span class="name">rdfs:Class</span>",
"<span class="name">owl:Class</span>"
(<i>S</i>=IC in both cases)
or "<span class="name">rdfs:Datatype</span>" (<i>S</i>=IDC).
</p>
<div id="topic-axiomatic-classes-secondcolexample"></div>
<p>For example,
for the IRI "<span class="name">owl:FunctionalProperty</span>",
the semantic condition
</p>
<div class="indent">
<p><i>I</i>(<span class="name">owl:FunctionalProperty</span>) ∈ IC
</p>
</div>
<p>has the corresponding axiomatic triple
</p>
<div class="indent">
<p><span class="name">owl:FunctionalProperty rdf:type rdfs:Class .</span>
</p>
</div>
<div id="topic-axiomatic-classes-thirdcoldef"></div>
<p>Further,
for each IRI <i>E</i> in the first column of the table,
if the <i>third</i> column contains an entry
of the form
"ICEXT(<i>I</i>(<i>E</i>)) ⊆ <i>S</i>"
(or "ICEXT(<i>I</i>(<i>E</i>)) = <i>S</i>")
for some set <i>S</i>,
then this entry corresponds to an RDF triple of the form
"<i>E</i> <span class="name">rdfs:subClassOf</span> <i>C</i>"
(or additionally "<i>C</i> <span class="name">rdfs:subClassOf</span> <i>E</i>"),
where <i>C</i> is the IRI of a vocabulary class with ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>.
In each case,
<i>S</i> will be one of
the <a href="index.html#Parts_of_the_Universe" title="">parts of the universe</a> of <i>I</i>.
</p>
<div id="topic-axiomatic-classes-thirdcolexample"></div>
<p>For example,
the semantic condition
</p>
<div class="indent">
<p>ICEXT(<i>I</i>(<span class="name">owl:FunctionalProperty</span>)) ⊆ IP
</p>
</div>
<p>has the corresponding axiomatic triple
</p>
<div class="indent">
<p><span class="name">owl:FunctionalProperty rdfs:subClassOf rdf:Property .</span>
</p>
</div>
<div id="topic-axiomatic-classes-partstab"></div>
<p>In addition,
the semantic conditions for the
<i>parts of the universe</i>
in <a href="index.html#table-semcond-parts" title="">Table 5.1</a>
of <a href="index.html#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a>
have to be taken into account.
In particular,
if an entry in the <i>second</i> column of <a href="index.html#table-semcond-parts" title="">Table 5.1</a>
is of the form
"<i>S<sub>1</sub></i> ⊆ <i>S<sub>2</sub></i>"
for some sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>,
then this corresponds to an RDF triple
of the form
"<i>C<sub>1</sub></i> <span class="name">owl:subClassOf</span> <i>C<sub>2</sub></i>",
where
<i>C<sub>1</sub></i> and <i>C<sub>2</sub></i>
are the IRIs of vocabulary classes with
ICEXT(<i>I</i>(<i>C<sub>1</sub></i>)) = <i>S<sub>1</sub></i>
and
ICEXT(<i>I</i>(<i>C<sub>2</sub></i>)) = <i>S<sub>2</sub></i>,
respectively,
according to
<a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>.
</p>
<div id="topic-axiomatic-classes-datatypes"></div>
<p><a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>
also specifies semantic conditions
for all the <i>datatypes</i> of the OWL 2 RDF-Based Semantics,
as listed in <a href="index.html#Datatype_Names" title="">Section 3.3</a>.
For each datatype IRI <i>E</i>,
such as <i>E</i> := "<span class="name">xsd:string</span>",
for the semantic conditions
"<i>I</i>(<i>E</i>) ∈ IDC"
and
"ICEXT(<i>I</i>(<i>E</i>)) ⊆ LV"
the corresponding axiomatic triples are of the form
</p>
<div class="indent">
<p><i>E</i> <span class="name">rdf:type rdfs:Datatype .</span><br />
<i>E</i> <span class="name">rdfs:subClassOf rdfs:Literal .</span>
</p>
</div>
<div id="topic-axiomatic-classes-simple"></div>
<p>In analogy to
<a href="index.html#Axiomatic_Triples_in_RDF" title="">Section 6.1</a>
for the RDF axiomatic triples,
all the axiomatic triples for the vocabulary classes
(including datatypes)
can be considered <i>"simple"</i>
in the sense that
they will have in their object position
only single terms
from the RDF, RDFS and OWL 2 RDF-Based vocabularies
(<a href="index.html#Vocabulary_Terms" title="">Section 3.2</a>).
</p>
<div id="topic-axiomatic-classes-redundant"></div>
<p>Note that some of the axiomatic triples obtained in this way
already follow from the semantics of
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef">RDF</a>
and
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">RDFS</a>,
as defined in
the RDF Semantics [<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p>
<a name="Axiomatic_Triples_for_the_Vocabulary_Properties"></a><h3> <span class="mw-headline">6.3 Axiomatic Triples for the Vocabulary Properties </span></h3>
<p>The semantic conditions for <i>vocabulary properties</i>
in <a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>
can be considered as corresponding to
a set of axiomatic triples
for the properties in the vocabulary of the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-axiomatic-properties-secondcoldef"></div>
<p>First,
for each IRI <i>E</i>
occurring in the first column of <a href="index.html#table-semcond-properties" title="">Table 5.3</a>,
if the <i>second</i> column contains an entry
of the form
"<i>I</i>(<i>E</i>) ∈ <i>S</i>" for some set <i>S</i>,
then this entry corresponds to an RDF triple of the form
"<i>E</i> <span class="name">rdf:type</span> <i>C</i>",
where <i>C</i> is the IRI of a vocabulary class with ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>.
In the table,
<i>S</i> will always be either
the <a href="index.html#Parts_of_the_Universe" title="">part</a> IP of all properties,
or some sub part of IP.
Hence, in a corresponding RDF triple the IRI <i>C</i> will be
one of
"<span class="name">rdf:Property</span>",
"<span class="name">owl:ObjectProperty</span>",
(<i>S</i>=IP in both cases),
"<span class="name">owl:DatatypeProperty</span>" (<i>S</i>=IODP),
"<span class="name">owl:OntologyProperty</span>" (<i>S</i>=IOXP)
or "<span class="name">owl:AnnotationProperty</span>" (<i>S</i>=IOAP).
</p>
<div id="topic-axiomatic-properties-secondcolexample"></div>
<p>For example,
for the IRI "<span class="name">owl:disjointWith</span>",
the semantic condition
</p>
<div class="indent">
<p><i>I</i>(<span class="name">owl:disjointWith</span>) ∈ IP
</p>
</div>
<p>has the corresponding axiomatic triple
</p>
<div class="indent">
<p><span class="name">owl:disjointWith rdf:type rdf:Property .</span>
</p>
</div>
<div id="topic-axiomatic-properties-thirdcoldef"></div>
<p>Further,
for each IRI <i>E</i> in the first column of the table,
if the <i>third</i> column contains an entry
of the form
"IEXT(<i>I</i>(<i>E</i>)) ⊆ <i>S<sub>1</sub></i> × <i>S<sub>2</sub></i>"
for some sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>,
then this entry corresponds to RDF triples of the form
"<i>E</i> <span class="name">rdfs:domain</span> <i>C<sub>1</sub></i>"
and
"<i>E</i> <span class="name">rdfs:range</span> <i>C<sub>2</sub></i>",
where <i>C<sub>1</sub></i> and <i>C<sub>2</sub></i>
are the IRIs of vocabulary classes with
ICEXT(<i>I</i>(<i>C<sub>1</sub></i>)) = <i>S<sub>1</sub></i>
and
ICEXT(<i>I</i>(<i>C<sub>2</sub></i>)) = <i>S<sub>2</sub></i>,
respectively.
Note that the sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>
do <i>not always</i> correspond
to any of the <a href="index.html#Parts_of_the_Universe" title="">parts of the universe</a> of <i>I</i>.
</p>
<div id="topic-axiomatic-properties-thirdcolexample"></div>
<p>For example,
the semantic condition
</p>
<div class="indent">
<p>IEXT(<i>I</i>(<span class="name">owl:disjointWith</span>)) ⊆ IC × IC
</p>
</div>
<p>has the corresponding axiomatic triples
</p>
<div class="indent">
<p><span class="name">owl:disjointWith rdfs:domain owl:Class .</span><br />
<span class="name">owl:disjointWith rdfs:range owl:Class .</span>
</p>
</div>
<div id="topic-axiomatic-properties-thirdcolexeption"></div>
<p>Exceptions are the semantic conditions
"IEXT(<i>I</i>(<span class="name">owl:topObjectProperty</span>)) = IR × IR"
and
"IEXT(<i>I</i>(<span class="name">owl:topDataProperty</span>)) = IR × LV",
since the <i>exactly</i> specified property extensions of these properties
cannot be expressed solely by domain and range axiomatic triples.
For example,
the domain and range axiomatic triples for
<span class="name">owl:sameAs</span>
are equal to those for
<span class="name">owl:topObjectProperty</span>,
but the property extension of
<span class="name">owl:sameAs</span>
is different from the property extension of
<span class="name">owl:topObjectProperty</span>.
</p>
<div id="topic-axiomatic-properties-facets"></div>
<p><a href="index.html#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>
also specifies semantic conditions
for all the <i>facets</i> of the OWL 2 RDF-Based Semantics,
as listed in <a href="index.html#Facet_Names" title="">Section 3.4</a>.
For each facet IRI <i>E</i>,
such as <i>E</i> := "<span class="name">xsd:length</span>",
for the semantic conditions
"<i>I</i>(<i>E</i>) ∈ IODP"
and
"IEXT(<i>I</i>(<i>E</i>)) ⊆ IR × LV"
the corresponding axiomatic triples are of the form
</p>
<div class="indent">
<p><i>E</i> <span class="name">rdf:type owl:DatatypeProperty .</span><br />
<i>E</i> <span class="name">rdfs:domain rdfs:Resource .</span><br />
<i>E</i> <span class="name">rdfs:range rdfs:Literal .</span><br />
</p>
</div>
<div id="topic-axiomatic-properties-simple"></div>
<p>In analogy to
<a href="index.html#Axiomatic_Triples_in_RDF" title="">Section 6.1</a>
for the RDF axiomatic triples,
all the axiomatic triples for the vocabulary properties
(including facets)
can be considered <i>"simple"</i>
in the sense that
they will have in their object position
only single terms
from the RDF, RDFS and OWL 2 RDF-Based vocabularies
(<a href="index.html#Vocabulary_Terms" title="">Section 3.2</a>).
</p>
<a name="A_Set_of_Axiomatic_Triples"></a><h3> <span class="mw-headline">6.4 A Set of Axiomatic Triples </span></h3>
<p>This section provides a concrete example set of axiomatic triples
based on the discussion in the Sections
<a href="index.html#Axiomatic_Triples_for_the_Vocabulary_Classes" title="">6.2</a>
and
<a href="index.html#Axiomatic_Triples_for_the_Vocabulary_Properties" title="">6.3</a>.
The axiomatic triples are grouped by different tables
for the <a href="index.html#table-axiomatic-classes" title="">classes</a>
and the <a href="index.html#table-axiomatic-properties" title="">properties</a>
of the OWL 2 RDF-Based vocabulary,
for the <a href="index.html#table-axiomatic-datatypes" title="">datatypes</a>
and the <a href="index.html#table-axiomatic-facets" title="">facets</a>
of the OWL 2 RDF-Based Semantics,
and for some of the
<a href="index.html#table-axiomatic-rdfs" title="">classes and properties of the RDFS vocabulary</a>.
Note that this set of axiomatic triples
is not meant to be free of redundancy.
</p>
<div class="left" id="table-axiomatic-classes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.1: Axiomatic Triples for the Classes of the OWL 2 RDF-Based Vocabulary</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-classes-alldifferent">owl:AllDifferent rdf:type rdfs:Class .<br />owl:AllDifferent rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-alldisjointclasses">owl:AllDisjointClasses rdf:type rdfs:Class .<br />owl:AllDisjointClasses rdfs:subClassOf rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-alldisjointproperties">owl:AllDisjointProperties rdf:type rdfs:Class .<br />owl:AllDisjointProperties rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-annotation">owl:Annotation rdf:type rdfs:Class .<br />owl:Annotation rdfs:subClassOf rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-annotationproperty">owl:AnnotationProperty rdf:type rdfs:Class .<br />owl:AnnotationProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-asymmetricproperty">owl:AsymmetricProperty rdf:type rdfs:Class .<br />owl:AsymmetricProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-axiom">owl:Axiom rdf:type rdfs:Class .<br />owl:Axiom rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-class">owl:Class rdf:type rdfs:Class .<br />owl:Class rdfs:subClassOf rdfs:Class .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-datarange">owl:DataRange rdf:type rdfs:Class .<br />owl:DataRange rdfs:subClassOf rdfs:Datatype .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-datatypeproperty">owl:DatatypeProperty rdf:type rdfs:Class .<br />owl:DatatypeProperty rdfs:subClassOf rdf:Property .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-deprecatedclass">owl:DeprecatedClass rdf:type rdfs:Class .<br />owl:DeprecatedClass rdfs:subClassOf rdfs:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-deprecatedproperty">owl:DeprecatedProperty rdf:type rdfs:Class .<br />owl:DeprecatedProperty rdfs:subClassOf rdf:Property .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-functionalproperty">owl:FunctionalProperty rdf:type rdfs:Class .<br />owl:FunctionalProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-inversefunctionalproperty">owl:InverseFunctionalProperty rdf:type rdfs:Class .<br />owl:InverseFunctionalProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-irreflexiveproperty">owl:IrreflexiveProperty rdf:type rdfs:Class .<br />owl:IrreflexiveProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-namedindividual">owl:NamedIndividual rdf:type rdfs:Class .<br />owl:NamedIndividual rdfs:subClassOf owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-negativepropertyassertion">owl:NegativePropertyAssertion rdf:type rdfs:Class .<br />owl:NegativePropertyAssertion rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-nothing">owl:Nothing rdf:type owl:Class .<br />owl:Nothing rdfs:subClassOf owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-objectproperty">owl:ObjectProperty rdf:type rdfs:Class .<br />owl:ObjectProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-ontology">owl:Ontology rdf:type rdfs:Class .<br />owl:Ontology rdfs:subClassOf rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-ontologyproperty">owl:OntologyProperty rdf:type rdfs:Class .<br />owl:OntologyProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-reflexiveproperty">owl:ReflexiveProperty rdf:type rdfs:Class .<br />owl:ReflexiveProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-restriction">owl:Restriction rdf:type rdfs:Class .<br />owl:Restriction rdfs:subClassOf owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-symmetricproperty">owl:SymmetricProperty rdf:type rdfs:Class .<br />owl:SymmetricProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-thing">owl:Thing rdf:type owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-transitiveproperty">owl:TransitiveProperty rdf:type rdfs:Class .<br />owl:TransitiveProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
</table>
</div>
<div class="left" id="table-axiomatic-properties">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.2: Axiomatic Triples for the Properties of the OWL 2 RDF-Based Vocabulary</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-properties-allvaluesfrom">owl:allValuesFrom rdf:type rdf:Property .<br />owl:allValuesFrom rdfs:domain owl:Restriction .<br />owl:allValuesFrom rdfs:range rdfs:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-annotatedproperty">owl:annotatedProperty rdf:type rdf:Property .<br />owl:annotatedProperty rdfs:domain rdfs:Resource .<br />owl:annotatedProperty rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-annotatedsource">owl:annotatedSource rdf:type rdf:Property .<br />owl:annotatedSource rdfs:domain rdfs:Resource .<br />owl:annotatedSource rdfs:range rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-annotatedtarget">owl:annotatedTarget rdf:type rdf:Property .<br />owl:annotatedTarget rdfs:domain rdfs:Resource .<br />owl:annotatedTarget rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-assertionproperty">owl:assertionProperty rdf:type rdf:Property .<br />owl:assertionProperty rdfs:domain owl:NegativePropertyAssertion .<br />owl:assertionProperty rdfs:range rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-backwardcompatiblewith">owl:backwardCompatibleWith rdf:type owl:AnnotationProperty .<br />owl:backwardCompatibleWith rdf:type owl:OntologyProperty .<br />owl:backwardCompatibleWith rdfs:domain owl:Ontology .<br />owl:backwardCompatibleWith rdfs:range owl:Ontology .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-bottomdataproperty">owl:bottomDataProperty rdf:type owl:DatatypeProperty .<br />owl:bottomDataProperty rdfs:domain owl:Thing .<br />owl:bottomDataProperty rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-bottomobjectproperty">owl:bottomObjectProperty rdf:type owl:ObjectProperty .<br />owl:bottomObjectProperty rdfs:domain owl:Thing .<br />owl:bottomObjectProperty rdfs:range owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-cardinality">owl:cardinality rdf:type rdf:Property .<br />owl:cardinality rdfs:domain owl:Restriction .<br />owl:cardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-complementof">owl:complementOf rdf:type rdf:Property .<br />owl:complementOf rdfs:domain owl:Class .<br />owl:complementOf rdfs:range owl:Class .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-datatypecomplementof">owl:datatypeComplementOf rdf:type rdf:Property .<br />owl:datatypeComplementOf rdfs:domain rdfs:Datatype .<br />owl:datatypeComplementOf rdfs:range rdfs:Datatype .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-deprecated">owl:deprecated rdf:type owl:AnnotationProperty .<br />owl:deprecated rdfs:domain rdfs:Resource .<br />owl:deprecated rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-differentfrom">owl:differentFrom rdf:type rdf:Property .<br />owl:differentFrom rdfs:domain owl:Thing .<br />owl:differentFrom rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-disjointunionof">owl:disjointUnionOf rdf:type rdf:Property .<br />owl:disjointUnionOf rdfs:domain owl:Class .<br />owl:disjointUnionOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-disjointwith">owl:disjointWith rdf:type rdf:Property .<br />owl:disjointWith rdfs:domain owl:Class .<br />owl:disjointWith rdfs:range owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-distinctmembers">owl:distinctMembers rdf:type rdf:Property .<br />owl:distinctMembers rdfs:domain owl:AllDifferent .<br />owl:distinctMembers rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-equivalentclass">owl:equivalentClass rdf:type rdf:Property .<br />owl:equivalentClass rdfs:domain rdfs:Class .<br />owl:equivalentClass rdfs:range rdfs:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-equivalentproperty">owl:equivalentProperty rdf:type rdf:Property .<br />owl:equivalentProperty rdfs:domain rdf:Property .<br />owl:equivalentProperty rdfs:range rdf:Property .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-haskey">owl:hasKey rdf:type rdf:Property .<br />owl:hasKey rdfs:domain owl:Class .<br />owl:hasKey rdfs:range rdf:List .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-hasself">owl:hasSelf rdf:type rdf:Property .<br />owl:hasSelf rdfs:domain owl:Restriction .<br />owl:hasSelf rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-hasvalue">owl:hasValue rdf:type rdf:Property .<br />owl:hasValue rdfs:domain owl:Restriction .<br />owl:hasValue rdfs:range rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-imports">owl:imports rdf:type owl:OntologyProperty .<br />owl:imports rdfs:domain owl:Ontology .<br />owl:imports rdfs:range owl:Ontology .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-incompatiblewith">owl:incompatibleWith rdf:type owl:AnnotationProperty .<br />owl:incompatibleWith rdf:type owl:OntologyProperty .<br />owl:incompatibleWith rdfs:domain owl:Ontology .<br />owl:incompatibleWith rdfs:range owl:Ontology .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-intersectionof">owl:intersectionOf rdf:type rdf:Property .<br />owl:intersectionOf rdfs:domain rdfs:Class .<br />owl:intersectionOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-inverseof">owl:inverseOf rdf:type rdf:Property .<br />owl:inverseOf rdfs:domain owl:ObjectProperty .<br />owl:inverseOf rdfs:range owl:ObjectProperty .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-maxcardinality">owl:maxCardinality rdf:type rdf:Property .<br />owl:maxCardinality rdfs:domain owl:Restriction .<br />owl:maxCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-maxqualifiedcardinality">owl:maxQualifiedCardinality rdf:type rdf:Property .<br />owl:maxQualifiedCardinality rdfs:domain owl:Restriction .<br />owl:maxQualifiedCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-members">owl:members rdf:type rdf:Property .<br />owl:members rdfs:domain rdfs:Resource .<br />owl:members rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-mincardinality">owl:minCardinality rdf:type rdf:Property .<br />owl:minCardinality rdfs:domain owl:Restriction .<br />owl:minCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-minqualifiedcardinality">owl:minQualifiedCardinality rdf:type rdf:Property .<br />owl:minQualifiedCardinality rdfs:domain owl:Restriction .<br />owl:minQualifiedCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-onclass">owl:onClass rdf:type rdf:Property .<br />owl:onClass rdfs:domain owl:Restriction .<br />owl:onClass rdfs:range owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-ondatarange">owl:onDataRange rdf:type rdf:Property .<br />owl:onDataRange rdfs:domain owl:Restriction .<br />owl:onDataRange rdfs:range rdfs:Datatype .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-ondatatype">owl:onDatatype rdf:type rdf:Property .<br />owl:onDatatype rdfs:domain rdfs:Datatype .<br />owl:onDatatype rdfs:range rdfs:Datatype .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-oneof">owl:oneOf rdf:type rdf:Property .<br />owl:oneOf rdfs:domain rdfs:Class .<br />owl:oneOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-onproperty">owl:onProperty rdf:type rdf:Property .<br />owl:onProperty rdfs:domain owl:Restriction .<br />owl:onProperty rdfs:range rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-onproperties">owl:onProperties rdf:type rdf:Property .<br />owl:onProperties rdfs:domain owl:Restriction .<br />owl:onProperties rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-priorversion">owl:priorVersion rdf:type owl:AnnotationProperty .<br />owl:priorVersion rdf:type owl:OntologyProperty .<br />owl:priorVersion rdfs:domain owl:Ontology .<br />owl:priorVersion rdfs:range owl:Ontology .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-propertychainaxiom">owl:propertyChainAxiom rdf:type rdf:Property .<br />owl:propertyChainAxiom rdfs:domain owl:ObjectProperty .<br />owl:propertyChainAxiom rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-propertydisjointwith">owl:propertyDisjointWith rdf:type rdf:Property .<br />owl:propertyDisjointWith rdfs:domain rdf:Property .<br />owl:propertyDisjointWith rdfs:range rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-qualifiedcardinality">owl:qualifiedCardinality rdf:type rdf:Property .<br />owl:qualifiedCardinality rdfs:domain owl:Restriction .<br />owl:qualifiedCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-sameas">owl:sameAs rdf:type rdf:Property .<br />owl:sameAs rdfs:domain owl:Thing .<br />owl:sameAs rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-somevaluesfrom">owl:someValuesFrom rdf:type rdf:Property .<br />owl:someValuesFrom rdfs:domain owl:Restriction .<br />owl:someValuesFrom rdfs:range rdfs:Class .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-sourceindividual">owl:sourceIndividual rdf:type rdf:Property .<br />owl:sourceIndividual rdfs:domain owl:NegativePropertyAssertion .<br />owl:sourceIndividual rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-targetindividual">owl:targetIndividual rdf:type rdf:Property .<br />owl:targetIndividual rdfs:domain owl:NegativePropertyAssertion .<br />owl:targetIndividual rdfs:range owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-targetvalue">owl:targetValue rdf:type rdf:Property .<br />owl:targetValue rdfs:domain owl:NegativePropertyAssertion .<br />owl:targetValue rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-topdataproperty">owl:topDataProperty rdf:type owl:DatatypeProperty .<br />owl:topDataProperty rdfs:domain owl:Thing .<br />owl:topDataProperty rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-topobjectproperty">owl:topObjectProperty rdf:type rdf:ObjectProperty .<br />owl:topObjectProperty rdfs:domain owl:Thing .<br />owl:topObjectProperty rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-unionof">owl:unionOf rdf:type rdf:Property .<br />owl:unionOf rdfs:domain rdfs:Class .<br />owl:unionOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-versioninfo">owl:versionInfo rdf:type owl:AnnotationProperty .<br />owl:versionInfo rdfs:domain rdfs:Resource .<br />owl:versionInfo rdfs:range rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-versioniri">owl:versionIRI rdf:type owl:OntologyProperty .<br />owl:versionIRI rdfs:domain owl:Ontology .<br />owl:versionIRI rdfs:range owl:Ontology .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-withrestrictions">owl:withRestrictions rdf:type rdf:Property .<br />owl:withRestrictions rdfs:domain rdfs:Datatype .<br />owl:withRestrictions rdfs:range rdf:List .<br /></span>
</td><td>
</td></tr>
</table>
</div>
<div class="left" id="table-axiomatic-datatypes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.3: Axiomatic Triples for the Datatypes of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-anyuri">xsd:anyURI rdf:type rdfs:Datatype .<br />xsd:anyURI rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-base64binary">xsd:base64Binary rdf:type rdfs:Datatype .<br />xsd:base64Binary rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-boolean">xsd:boolean rdf:type rdfs:Datatype .<br />xsd:boolean rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-byte">xsd:byte rdf:type rdfs:Datatype .<br />xsd:byte rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-datetime">xsd:dateTime rdf:type rdfs:Datatype .<br />xsd:dateTime rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-datetimestamp">xsd:dateTimeStamp rdf:type rdfs:Datatype .<br />xsd:dateTimeStamp rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-decimal">xsd:decimal rdf:type rdfs:Datatype .<br />xsd:decimal rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-double">xsd:double rdf:type rdfs:Datatype .<br />xsd:double rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-float">xsd:float rdf:type rdfs:Datatype .<br />xsd:float rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-hexbinary">xsd:hexBinary rdf:type rdfs:Datatype .<br />xsd:hexBinary rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-int">xsd:int rdf:type rdfs:Datatype .<br />xsd:int rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-integer">xsd:integer rdf:type rdfs:Datatype .<br />xsd:integer rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-language">xsd:language rdf:type rdfs:Datatype .<br />xsd:language rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-long">xsd:long rdf:type rdfs:Datatype .<br />xsd:long rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-name">xsd:Name rdf:type rdfs:Datatype .<br />xsd:Name rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-ncname">xsd:NCName rdf:type rdfs:Datatype .<br />xsd:NCName rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-negativeinteger">xsd:negativeInteger rdf:type rdfs:Datatype .<br />xsd:negativeInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-nmtoken">xsd:NMTOKEN rdf:type rdfs:Datatype .<br />xsd:NMTOKEN rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-nonnegativeinteger">xsd:nonNegativeInteger rdf:type rdfs:Datatype .<br />xsd:nonNegativeInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-nonpositiveinteger">xsd:nonPositiveInteger rdf:type rdfs:Datatype .<br />xsd:nonPositiveInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-normalizedstring">xsd:normalizedString rdf:type rdfs:Datatype .<br />xsd:normalizedString rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-plainliteral">rdf:PlainLiteral rdf:type rdfs:Datatype .<br />rdf:PlainLiteral rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-positiveinteger">xsd:positiveInteger rdf:type rdfs:Datatype .<br />xsd:positiveInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-rational">owl:rational rdf:type rdfs:Datatype .<br />owl:rational rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-real">owl:real rdf:type rdfs:Datatype .<br />owl:real rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-short">xsd:short rdf:type rdfs:Datatype .<br />xsd:short rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-string">xsd:string rdf:type rdfs:Datatype .<br />xsd:string rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-token">xsd:token rdf:type rdfs:Datatype .<br />xsd:token rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-unsignedbyte">xsd:unsignedByte rdf:type rdfs:Datatype .<br />xsd:unsignedByte rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-unsignedint">xsd:unsignedInt rdf:type rdfs:Datatype .<br />xsd:unsignedInt rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-unsignedlong">xsd:unsignedLong rdf:type rdfs:Datatype .<br />xsd:unsignedLong rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-unsignedshort">xsd:unsignedShort rdf:type rdfs:Datatype .<br />xsd:unsignedShort rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-xmlliteral">rdf:XMLLiteral rdf:type rdfs:Datatype .<br />rdf:XMLLiteral rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td>
</td></tr>
</table>
</div>
<div class="left" id="table-axiomatic-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.4: Axiomatic Triples for the Facets of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-facets-langrange">rdf:langRange rdf:type owl:DatatypeProperty .<br />rdf:langRange rdfs:domain rdfs:Resource .<br />rdf:langRange rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-length">xsd:length rdf:type owl:DatatypeProperty .<br />xsd:length rdfs:domain rdfs:Resource .<br />xsd:length rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-maxexclusive">xsd:maxExclusive rdf:type owl:DatatypeProperty .<br />xsd:maxExclusive rdfs:domain rdfs:Resource .<br />xsd:maxExclusive rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-maxinclusive">xsd:maxInclusive rdf:type owl:DatatypeProperty .<br />xsd:maxInclusive rdfs:domain rdfs:Resource .<br />xsd:maxInclusive rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-maxlength">xsd:maxLength rdf:type owl:DatatypeProperty .<br />xsd:maxLength rdfs:domain rdfs:Resource .<br />xsd:maxLength rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-minexclusive">xsd:minExclusive rdf:type owl:DatatypeProperty .<br />xsd:minExclusive rdfs:domain rdfs:Resource .<br />xsd:minExclusive rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-mininclusive">xsd:minInclusive rdf:type owl:DatatypeProperty .<br />xsd:minInclusive rdfs:domain rdfs:Resource .<br />xsd:minInclusive rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-minlength">xsd:minLength rdf:type owl:DatatypeProperty .<br />xsd:minLength rdfs:domain rdfs:Resource .<br />xsd:minLength rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-pattern">xsd:pattern rdf:type owl:DatatypeProperty .<br />xsd:pattern rdfs:domain rdfs:Resource .<br />xsd:pattern rdfs:range rdfs:Literal .<br /></span>
</td><td>
</td></tr></table>
</div>
<div class="left" id="table-axiomatic-rdfs">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.5: Additional Axiomatic Triples for Classes and Properties of the RDFS Vocabulary</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-class">rdfs:Class rdfs:subClassOf owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-comment">rdfs:comment rdf:type owl:AnnotationProperty .<br />rdfs:comment rdfs:domain rdfs:Resource .<br />rdfs:comment rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-datatype">rdfs:Datatype rdfs:subClassOf owl:DataRange .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-isdefinedby">rdfs:isDefinedBy rdf:type owl:AnnotationProperty .<br />rdfs:isDefinedBy rdfs:domain rdfs:Resource .<br />rdfs:isDefinedBy rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-label">rdfs:label rdf:type owl:AnnotationProperty .<br />rdfs:label rdfs:domain rdfs:Resource .<br />rdfs:label rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-literal">rdfs:Literal rdf:type rdfs:Datatype .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-property">rdf:Property rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-resource">rdfs:Resource rdfs:subClassOf owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-seealso">rdfs:seeAlso rdf:type owl:AnnotationProperty .<br />rdfs:seeAlso rdfs:domain rdfs:Resource .<br />rdfs:seeAlso rdfs:range rdfs:Resource .<br /></span>
</td><td>
</td></tr>
</table>
</div>
<a name="Appendix:_Relationship_to_the_Direct_Semantics_.28Informative.29"></a><h2> <span class="mw-headline">7 Appendix: Relationship to the Direct Semantics (Informative) </span></h2>
<p>This section compares
the OWL 2 RDF-Based Semantics
with the
<a href="../REC-owl2-direct-semantics-20091027/index.html" title="Direct Semantics"><i>OWL 2 Direct Semantics</i></a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
While
the OWL 2 RDF-Based Semantics is based on the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
the OWL 2 Direct Semantics
is a <i>description logic</i> style semantics.
Several fundamental differences
exist between the two semantics,
but
there is also a strong relationship
basically stating that the OWL 2 RDF-Based Semantics is able
to reflect all logical conclusions
of the OWL 2 Direct Semantics.
This means that the OWL 2 Direct Semantics
can
in a sense
be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.
</p><p>Technically,
the comparison will be performed
by comparing the sets of <i>entailments</i>
that hold for each of the two semantics,
respectively.
The definition of an <i><b>OWL 2 RDF-Based entailment</b></i>
was given in
<a href="index.html#Satisfaction.2C_Consistency_and_Entailment" title="">Section 4.3</a>
of this document,
while the definition of an <i><b>OWL 2 Direct entailment</b></i>
is provided in
<a href="../REC-owl2-direct-semantics-20091027/index.html#Inference_Problems" title="Direct Semantics">Section 2.5 of the OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
In both cases,
entailments are defined for pairs of ontologies,
and such an ordered pair of two ontologies will be called an
<i><b>entailment query</b></i>
in this section.
</p><p>Comparing the two semantics by means of entailments
will only be meaningful
if the entailment queries
allow for applying
both
the OWL 2 RDF-Based Semantics
and the
OWL 2 Direct Semantics
to them.
In order to ensure this,
the comparison will be restricted to entailment queries,
for which the left-hand side and right-hand side ontologies
are both
<i><b>OWL 2 DL ontologies in RDF graph form</b></i>.
These are RDF graphs that,
by applying the
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs"><i>reverse RDF mapping</i></a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>],
can be transformed
into corresponding
<i><b>OWL 2 DL ontologies in Functional Syntax form</b></i>
according to the <i>functional style syntax</i> defined in the
<a href="../REC-owl2-syntax-20091027/index.html" title="Syntax">OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>],
and which must further meet
all the <i>restrictions on OWL 2 DL ontologies</i>
that are specified in
<a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
In fact,
these restrictions must be <i>mutually met</i> by
both ontologies that occur in an entailment query,
i.e.
all these restrictions need to be satisfied
as if the two ontologies would be part of a single ontology.
Any entailment query that adheres to the conditions defined here
will be called an
<i><b>OWL 2 DL entailment query</b></i>.
</p><p>Ideally,
the relationship between
the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics
would be of the form that
every OWL 2 DL entailment query
that is an OWL 2 Direct entailment
is also an OWL 2 RDF-Based entailment.
However,
this desirable relationship
cannot hold in general
due to a variety of differences
that exist between
the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
as demonstrated in
<a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>.
</p><p>Fortunately,
the problems resulting from these semantic differences
can be overcome
in a way that
for every OWL 2 DL entailment query
there is another one
for which
the desired entailment relationship indeed holds,
and the new entailment query is
semantically equivalent to the original entailment query
under the OWL 2 Direct Semantics.
This is the gist of the
<i>OWL 2 correspondence theorem</i>,
which will be presented in
<a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>.
The
<i>proof</i> of this theorem,
given in <a href="index.html#Proof_for_the_Correspondence_Theorem" title="">Section 7.3</a>,
will further demonstrate
that such a substitute OWL 2 DL entailment query
can always be algorithmically constructed
by means of simple syntactic transformations.
</p>
<a name="Example_on_Semantic_Differences"></a><h3> <span class="mw-headline">7.1 Example on Semantic Differences </span></h3>
<p>This section will show
that differences exist
between
the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics,
and it will be demonstrated
how these semantic differences
complicate a comparison
of the two semantics
in terms of entailments.
An example OWL 2 DL entailment query will be given,
which will happen to be an OWL 2 Direct entailment
without being an OWL 2 RDF-Based entailment.
The section will explain
the different reasons
and will provide a resolution
of each of them.
It will turn out
that the example entailment query
can be syntactically transformed
into another
OWL 2 DL entailment query
that is both
an OWL 2 Direct entailment
and
an OWL 2 RDF-Based entailment,
while being semantically unchanged
compared to the original entailment query
under the OWL 2 Direct Semantics.
This example will motivate
the <i>OWL 2 correspondence theorem</i>
in <a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>
and its proof
in <a href="index.html#Proof_for_the_Correspondence_Theorem" title="">Section 7.3</a>.
</p><p>The example entailment query consists of the following
pair
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
of RDF graphs:
</p>
<div id="topic-correspondence-diff-example-rdf"></div>
<div class="indent">
<p><i>G<sub>1</sub><sup>*</sup></i> :
</p>
<div class="indent">
<p>(1) <span class="name">ex:o1 rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c1 rdfs:subClassOf ex:c2 .</span>
</p>
</div>
</div>
<div class="indent">
<p><i>G<sub>2</sub><sup>*</sup></i> :
</p>
<div class="indent">
<p>(1) <span class="name">ex:o2 rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c3 rdf:type owl:Class .</span><br />
(5) <span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
(6) <span class="name">_:x rdf:type owl:Class .</span><br />
(7) <span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span><br />
(8) <span class="name">ex:c3 rdfs:label "c3" .</span>
</p>
</div>
</div>
<p>Both <i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>
are
OWL 2 DL ontologies in RDF graph form
and can therefore be mapped by the
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
to the following two OWL 2 DL ontologies in Functional Syntax form
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>):
</p>
<div id="topic-correspondence-diff-example-fs"></div>
<div class="indent">
<p>F(<i>G<sub>1</sub><sup>*</sup></i>) :
</p>
<div class="indent">
<p>(1) <span class="name">Ontology( ex:o1</span><br />
(2) <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) <span class="name">SubClassOf( ex:c1 ex:c2 )</span><br />
(5) <span class="name">)</span>
</p>
</div>
</div>
<div class="indent">
<p>F(<i>G<sub>2</sub><sup>*</sup></i>) :
</p>
<div class="indent">
<p>(1) <span class="name">Ontology( ex:o2</span><br />
(2) <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) <span class="name">Declaration( Class( ex:c3 ) )</span><br />
(5) <span class="name">SubClassOf( ex:c1 ObjectUnionOf( ex:c2 ex:c3 ) )</span><br />
(6) <span class="name">AnnotationAssertion( rdfs:label ex:c3 "c3" )</span><br />
(7) <span class="name">)</span>
</p>
</div>
</div>
<p>Note that
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
mutually meet the restrictions on OWL 2 DL ontologies
as specified in
<a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
For example,
none of the IRIs being declared as a class in F(<i>G<sub>1</sub><sup>*</sup></i>)
is declared as a datatype in F(<i>G<sub>2</sub><sup>*</sup></i>),
since this would not be allowed for an OWL 2 DL entailment query.
</p><p>It follows that
F(<i>G<sub>1</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>2</sub><sup>*</sup></i>).
To show this,
only the axioms
(4) of F(<i>G<sub>1</sub><sup>*</sup></i>)
and
(5) of F(<i>G<sub>2</sub><sup>*</sup></i>)
have to be considered.
None of the other statements in the two ontologies
are relevant for this OWL 2 Direct entailment to hold,
since they do not have a formal meaning
under the OWL 2 Direct Semantics.
However,
it turns out that the RDF graph
<i>G<sub>1</sub><sup>*</sup></i>
does <i>not</i> OWL 2 RDF-Based entail
<i>G<sub>2</sub><sup>*</sup></i>,
for reasons discussed in detail now.
</p>
<div id="topic-correspondence-diff-reason-annotations"></div>
<p><b><i>Reason 1: An Annotation in F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
The ontology F(<i>G<sub>2</sub><sup>*</sup></i>)
contains an annotation (6).
The OWL 2 Direct Semantics
does not give a formal meaning to
annotations.
In contrast,
under the OWL 2 RDF-Based Semantics
<i>every</i> RDF triple occurring in an RDF graph
has a formal meaning,
including
the corresponding annotation triple (8) in <i>G<sub>2</sub><sup>*</sup></i>.
Since this annotation triple
only occurs in <i>G<sub>2</sub><sup>*</sup></i>
but not in <i>G<sub>1</sub><sup>*</sup></i>,
there will exist OWL 2 RDF-Based interpretations
that satisfy <i>G<sub>1</sub><sup>*</sup></i>
without satisfying triple (8) of <i>G<sub>2</sub><sup>*</sup></i>.
Hence,
<i>G<sub>1</sub><sup>*</sup></i>
does <i>not</i> OWL 2 RDF-Based entail <i>G<sub>2</sub><sup>*</sup></i>.
</p><p><i><b>Resolution of Reason 1.</b></i>
The annotation triple (8) in <i>G<sub>2</sub><sup>*</sup></i>
will be removed,
which will avoid requiring
OWL 2 RDF-Based interpretations to interpret this triple.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since annotations are strictly optional in OWL 2 DL ontologies
and may therefore be omitted.
Also, this operation will not change the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since annotations do not have a formal meaning under this semantics.
</p>
<div id="topic-correspondence-diff-reason-declarations"></div>
<p><b><i>Reason 2: An Entity Declaration exclusively in F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
The ontology F(<i>G<sub>2</sub><sup>*</sup></i>)
contains an entity declaration for the class IRI
<span class="name">ex:c3</span> (4),
for which there is no corresponding entity declaration
in F(<i>G<sub>1</sub><sup>*</sup></i>).
The OWL 2 Direct Semantics does not give a formal meaning to
entity declarations,
while the OWL 2 RDF-Based Semantics gives a formal meaning
to the corresponding declaration statement (4) in <i>G<sub>2</sub><sup>*</sup></i>.
The consequences are analog to those described for reason 1.
</p><p><i><b>Resolution of Reason 2.</b></i>
The declaration statement (4) in <i>G<sub>2</sub><sup>*</sup></i>
will be copied to <i>G<sub>1</sub><sup>*</sup></i>.
An OWL 2 RDF-Based interpretation
that satisfies the modified graph <i>G<sub>1</sub><sup>*</sup></i>
will then also satisfy the declaration statement.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the copied declaration statement is not in conflict
with any of the other entity declarations
in <i>G<sub>1</sub><sup>*</sup></i>.
Also, this operation will not change the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since entity declarations do not have a formal meaning under this semantics.
</p>
<div id="topic-correspondence-diff-reason-headers"></div>
<p><b><i>Reason 3: Different Ontology IRIs in F(</i>G<sub>1</sub><sup>*</sup><i>) and F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
The ontology IRIs for the two ontologies,
given by (1) in F(<i>G<sub>1</sub><sup>*</sup></i>)
and by (1) in F(<i>G<sub>2</sub><sup>*</sup></i>),
differ from each other.
The OWL 2 Direct Semantics does not give a formal meaning to
ontology headers,
while the OWL 2 RDF-Based Semantics gives a formal meaning
to the corresponding header triples
(1) in <i>G<sub>1</sub><sup>*</sup></i>
and
(1) in <i>G<sub>2</sub><sup>*</sup></i>.
Since these header triples differ from each other,
the consequences are analog to those described for reason 1.
</p><p><i><b>Resolution of Reason 3.</b></i>
The IRI
in the subject position of the header triple (1)
in <i>G<sub>2</sub><sup>*</sup></i>
is changed into a blank node.
Due to the existential semantics of blank nodes under the OWL 2 RDF-Based Semantics
the resulting triple will then be entailed
by triple (1)
in <i>G<sub>1</sub><sup>*</sup></i>.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since an ontology IRI is optional for an OWL 2 DL ontology.
(Note, however, that it would have been an error to simply remove
triple (1) from <i>G<sub>2</sub><sup>*</sup></i>,
since an OWL 2 DL ontology is required to contain an ontology header.)
Also, this operation will not change the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since ontology headers do not have a formal meaning under this semantics.
</p>
<div id="topic-correspondence-diff-reason-expressions"></div>
<p><b><i>Reason 4: A Class Expression in F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
Axiom (5) of F(<i>G<sub>2</sub><sup>*</sup></i>)
contains a class expression
that represents the union of the two classes
denoted by
<span class="name">ex:c2</span>
and
<span class="name">ex:c3</span>.
Within <i>G<sub>2</sub><sup>*</sup></i>,
this class expression is represented
by the triples (6) and (7),
both having the blank node
"<span class="name">_:x</span>"
in their respective subject position.
The way the OWL 2 RDF-Based Semantics interprets these two triples
differs from the way
the OWL 2 Direct Semantics treats the class expression
in axiom (5) of F(<i>G<sub>2</sub><sup>*</sup></i>).
</p><p>The OWL 2 Direct Semantics treats classes as <i>sets</i>,
i.e. subsets of the universe.
Thus,
the IRIs
<span class="name">ex:c2</span>
and
<span class="name">ex:c3</span>
in F(<i>G<sub>2</sub><sup>*</sup></i>)
denote two sets,
and the class expression
in axiom (5) of F(<i>G<sub>2</sub><sup>*</sup></i>)
therefore represents the set
that consists of the union of these two sets.
</p><p>The OWL 2 RDF-Based Semantics,
on the other hand,
treats classes as <i>individuals</i>,
i.e. members of the universe.
While every class under the OWL 2 RDF-Based Semantics
represents a certain subset of the universe,
namely its class extension,
this set is actually distinguished from the class itself.
For two given classes
it is ensured under the OWL 2 RDF-Based Semantics,
just as for the OWL 2 Direct Semantics,
that the union of their class extensions will always exist
as a subset of the universe.
However,
there is no guarantee
that there will also exist
an individual in the universe
that has this set union as its class extension.
</p><p>Under the OWL 2 RDF-Based Semantics,
triple (7) of <i>G<sub>2</sub><sup>*</sup></i>
essentially claims that a class exists
being the union of two other classes.
But since
the existence of such a union class
is not ensured by <i>G<sub>1</sub><sup>*</sup></i>,
there will be OWL 2 RDF-Based interpretations
that satisfy <i>G<sub>1</sub><sup>*</sup></i>
without satisfying
triple (7) of <i>G<sub>2</sub><sup>*</sup></i>.
Hence,
<i>G<sub>1</sub><sup>*</sup></i>
does <i>not</i>
OWL 2 RDF-Based entail
<i>G<sub>2</sub><sup>*</sup></i>.
</p><p><i><b>Resolution of Reason 4.</b></i>
The triples (6) and (7) of <i>G<sub>2</sub><sup>*</sup></i>
are copied to <i>G<sub>1</sub><sup>*</sup></i>
together with the new triple
"<span class="name">_:x owl:equivalentClass _:x</span>".
In addition,
for the IRI
<span class="name">ex:c3</span>,
which only occurs in the union class expression
but not in <i>G<sub>1</sub><sup>*</sup></i>,
an entity declaration is added
to <i>G<sub>1</sub><sup>*</sup></i>
by the resolution of reason 2.
If an OWL 2 RDF-Based interpretation satisfies the modified graph <i>G<sub>1</sub><sup>*</sup></i>,
then the triples (6) and (7) of <i>G<sub>2</sub><sup>*</sup></i>
will now be satisfied.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the whole set of added triples
validly encodes an OWL 2 axiom,
and since none of the restrictions on OWL 2 DL ontologies is hurt.
Also, this operation will not change
the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since the added equivalence axiom
is a tautology under this semantics.
</p><p>Note that it would have been an error
to simply copy the
triples (6) and (7) of <i>G<sub>2</sub><sup>*</sup></i>
to <i>G<sub>1</sub><sup>*</sup></i>,
without also adding the new triple
"<span class="name">_:x owl:equivalentClass _:x</span>".
This would have produced a class expression
that has no connection to any axiom in the ontology.
An OWL 2 DL ontology is basically a set of axioms
and does not allow for the occurrence of
"dangling" class expressions.
This is the reason for actually "embedding" the class expression
in an axiom.
It would have also been wrong
to use an <i>arbitrary</i> axiom for such an embedding,
since it has to be ensured
that the formal meaning of the original ontology does not change
under the OWL 2 Direct Semantics.
However,
any <i>tautological</i> axiom
that contains the original class expression
would have been sufficient for this purpose as well.
</p>
<div id="topic-correspondence-diff-resolution-complete"></div>
<p><i><b>Complete Resolution: The Transformed Entailment Query.</b></i>
</p><p>Combining the resolutions of all the above reasons
leads to the following new pair of RDF graphs
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ):
</p>
<div id="topic-correspondence-diff-resolution-rdf"></div>
<div class="indent">
<p><i>G<sub>1</sub></i> :
</p>
<div class="indent">
<p>(1) <span class="name">ex:o1 rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c3 rdf:type owl:Class .</span><br />
(5) <span class="name">ex:c1 rdfs:subClassOf ex:c2 .</span><br />
(6) <span class="name">_:x owl:equivalentClass _:x .</span><br />
(7) <span class="name">_:x rdf:type owl:Class .</span><br />
(8) <span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<div class="indent">
<p><i>G<sub>2</sub></i> :
</p>
<div class="indent">
<p>(1) <span class="name">_:o rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c3 rdf:type owl:Class .</span><br />
(5) <span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
(6) <span class="name">_:x rdf:type owl:Class .</span><br />
(7) <span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<div id="topic-correspondence-diff-resolution-reiteration"></div>
<p>The following list reiterates the changes compared to the original RDF graphs
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>:
</p>
<ul><li> <i><b>Resolution of Reason 1 (Annotation):</b></i> Triple (8) in <i>G<sub>2</sub><sup>*</sup></i> has been removed, i.e. there is no corresponding annotation triple in <i>G<sub>2</sub></i>.
</li><li> <i><b>Resolution of Reason 2 (Entity Declaration):</b></i> Triple (4) in <i>G<sub>2</sub><sup>*</sup></i> has been copied to <i>G<sub>1</sub><sup>*</sup></i>, becoming triple (4) in <i>G<sub>1</sub></i>.
</li><li> <i><b>Resolution of Reason 3 (Ontology IRIs):</b></i> The IRI in the subject position of triple (1) in <i>G<sub>2</sub><sup>*</sup></i> has been changed into a blank node, becoming triple (1) in <i>G<sub>2</sub></i>.
</li><li> <i><b>Resolution of Reason 4 (Class Expression):</b></i> Triples (6) and (7) in <i>G<sub>2</sub><sup>*</sup></i> have been copied to <i>G<sub>1</sub><sup>*</sup></i> together with the new triple "<span class="name">_:x owl:equivalentClass _:x</span>", becoming triples (6), (7) and (8) in <i>G<sub>1</sub></i>.
</li></ul>
<p><i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are again
OWL 2 DL ontologies in RDF graph form
and can be mapped to the following
OWL 2 DL ontologies in Functional Syntax form
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>),
which again mutually meet the restrictions on OWL 2 DL ontologies:
</p>
<div id="topic-correspondence-diff-resolution-fs"></div>
<div class="indent">
<p>F(<i>G<sub>1</sub></i>) :
</p>
<div class="indent">
<p>(1) <span class="name">Ontology( ex:o1</span><br />
(2) <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) <span class="name">Declaration( Class( ex:c3 ) )</span><br />
(5) <span class="name">SubClassOf( ex:c1 ex:c2 )</span><br />
(6) <span class="name">EquivalentClasses( ObjectUnionOf( ex:c2 ex:c3 ) ObjectUnionOf( ex:c2 ex:c3 ) )</span><br />
(7) <span class="name">)</span>
</p>
</div>
</div>
<div class="indent">
<p>F(<i>G<sub>2</sub></i>) :
</p>
<div class="indent">
<p>(1) <span class="name">Ontology(</span><br />
(2) <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) <span class="name">Declaration( Class( ex:c3 ) )</span><br />
(5) <span class="name">SubClassOf( ex:c1 ObjectUnionOf( ex:c2 ex:c3 ) )</span><br />
(6) <span class="name">)</span>
</p>
</div>
</div>
<p>As said earlier,
all the applied changes
preserve the formal meaning
of the original OWL 2 DL ontologies
under the OWL 2 Direct Semantics.
Hence,
it is still the case
that
F(<i>G<sub>1</sub></i>)
OWL 2 Direct entails
F(<i>G<sub>2</sub></i>).
However,
due to the syntactic transformation
the situation has changed for the OWL 2 RDF-Based Semantics:
it is now possible to show,
by following the lines of argumentation
for the resolutions of the different reasons given above,
that <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>
as well.
</p>
<a name="Correspondence_Theorem"></a><h3> <span class="mw-headline">7.2 Correspondence Theorem </span></h3>
<p>This section presents the <i>OWL 2 correspondence theorem</i>,
which compares the semantic expressivity of
the OWL 2 RDF-Based Semantics
with that of
the OWL 2 Direct Semantics.
The theorem basically states that
the OWL 2 RDF-Based Semantics is able to reflect all the semantic conclusions
of the OWL 2 Direct Semantics,
where the notion of a "semantic conclusion"
is technically expressed in terms of an <i>entailment</i>.
</p><p>However,
as discussed in
<a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>,
there exist semantic differences
between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics,
which do not allow for stating
that <i>any</i> OWL 2 DL entailment query
that is an OWL 2 Direct entailment
will always also be an
OWL 2 RDF-Based entailment.
Nevertheless,
it can still be ensured that
any given OWL 2 DL entailment query
can be <i>substituted</i>
by another OWL 2 DL entailment query
in a way
that for the substitute entailment query
the desired relationship will really hold,
while preserving the formal meaning
compared to the original entailment query
under the OWL 2 Direct Semantics.
</p><p>In fact,
the theorem only makes the seemingly weak assertion
that such a substitute entailment query
will always <i>exist</i>.
But the actual
<i>proof for the theorem</i>
in <a href="index.html#Proof_for_the_Correspondence_Theorem" title="">Section 7.3</a>
will be more concrete
in that it will substitute each given OWL 2 DL entailment query
with a variant
that can be algorithmically constructed
by applying a set of simple syntactic transformations
to the original entailment query.
One can get an idea of how this works
from <a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>.
</p>
<div id="topic-correspondence-datatypemaps"></div>
<p><i><b>Technical Note on Corresponding Datatype Maps.</b></i>
A distinction exists
between the format of
an <i>OWL 2 RDF-Based datatype map</i>,
as defined by <a href="index.html#def-owldatatypemap" title="">Definition 4.1</a>,
and the format of an <i>OWL 2 Direct datatype map</i>,
as defined in
<a href="../REC-owl2-direct-semantics-20091027/index.html#def_datatype_map" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
It is, however, possible to translate
between
an OWL 2 RDF-Based datatype map <i>D</i>
and
the corresponding OWL 2 Direct datatype map F(<i>D</i>)
in the following way:
</p><p>For an <a href="index.html#def-owldatatypemap" title="">OWL 2 RDF-Based datatype map</a> <i>D</i>,
the <i>corresponding OWL 2 Direct datatype map</i>
F(<i>D</i>) := (
<i>N<sub>DT</sub></i> ,
<i>N<sub>LS</sub></i> ,
<i>N<sub>FS</sub></i> ,
<i>⋅ <sup>DT</sup></i> ,
<i>⋅ <sup>LS</sup></i> ,
<i>⋅ <sup>FS</sup></i>
)
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
is given by
</p>
<ul><li> <i>Datatype Names:</i> <i>N<sub>DT</sub></i> is defined as the set of all IRIs <i>u</i>, for which there is a datatype <i>d</i>, such that ( <i>u</i> , <i>d</i> ) ∈ <i>D</i>.
</li><li> <i>Lexical Space:</i> For each datatype name <i>u</i> ∈ <i>N<sub>DT</sub></i>, set <i>N<sub>LS</sub>(u)</i> := LS(<i>d</i>), where ( <i>u</i> , <i>d</i> ) ∈ <i>D</i>.
</li><li> <i>Facet Space:</i> For each datatype name <i>u</i> ∈ <i>N<sub>DT</sub></i>, set <i>N<sub>FS</sub>(u)</i> := FS(<i>d</i>), where ( <i>u</i> , <i>d</i> ) ∈ <i>D</i>.
</li><li> <i>Value Space:</i> For each datatype name <i>u</i> ∈ <i>N<sub>DT</sub></i>, set <i>(u) <sup>DT</sup></i> := VS(<i>d</i>), where ( <i>u</i> , <i>d</i> ) ∈ <i>D</i>.
</li><li> <i>Lexical-to-Value Mapping:</i> For each datatype name <i>u</i> ∈ <i>N<sub>DT</sub></i> and each lexical form <i>a</i> ∈ <i>N<sub>LS</sub>(u)</i>, set ( <i>a</i> , <i>u</i> ) <sup><i>LS</i></sup> := L2V(<i>d</i>)(<i>a</i>), where ( <i>u</i> , <i>d</i> ) ∈ <i>D</i>.
</li><li> <i>Facet-to-Value Mapping:</i> For each datatype name <i>u</i> ∈ <i>N<sub>DT</sub></i> and each facet-value pair ( <i>F</i> , <i>v</i> ) ∈ <i>N<sub>FS</sub>(u)</i>, set ( <i>F</i> , <i>v</i> ) <sup><i>FS</i></sup> := F2V(<i>d</i>)(( <i>F</i> , <i>v</i> )), where ( <i>u</i> , <i>d</i> ) ∈ <i>D</i>.
</li></ul>
<div id="thm-correspondence">
<p><b>Theorem 7.1 (OWL 2 Correspondence Theorem):</b>
</p><p>Let <i>D</i> be an OWL 2 RDF-Based datatype map
according to <a href="index.html#def-owldatatypemap" title="">Definition 4.1</a>,
with F(<i>D</i>)
being the
OWL 2 Direct datatype map
according to
<a href="../REC-owl2-direct-semantics-20091027/index.html#def_datatype_map" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
that corresponds to <i>D</i> according to the
<a href="index.html#topic-correspondence-datatypemaps" title=""><i>technical note on corresponding datatype maps</i></a>.
Let
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>
be RDF graphs
that are
OWL 2 DL ontologies in RDF graph form,
with
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
being the
OWL 2 DL ontologies in
<a href="../REC-owl2-syntax-20091027/index.html" title="Syntax">Functional Syntax</a>
form
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
that result from applying
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">the reverse RDF mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
to
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>,
respectively.
Let
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
mutually meet
the restrictions on OWL 2 DL ontologies
as specified in
<a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p><p>Then,
there exist RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
that are
OWL 2 DL ontologies in RDF graph form,
such that all the following relationships hold,
with
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
being the
OWL 2 DL ontologies in Functional Syntax form
that result from applying the reverse RDF mapping
to
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>,
respectively:
</p>
<ol><li> F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>) mutually meet the restrictions on OWL 2 DL ontologies.
</li><li> F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>1</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>1</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>1</sub></i>) with respect to <i>F(D)</i>.
</li><li> F(<i>G<sub>2</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>2</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) with respect to <i>F(D)</i>.
</li><li> If F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) with respect to <i>F(D)</i>, then <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i> with respect to <i>D</i>.
</li></ol>
</div>
<a name="Proof_for_the_Correspondence_Theorem"></a><h3> <span class="mw-headline">7.3 Proof for the Correspondence Theorem </span></h3>
<p>This is the sketch of a proof for
<a href="index.html#thm-correspondence" title=""><i>Theorem 7.1 (OWL 2 Correspondence Theorem)</i></a>
in
<a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>.
The proof sketch provides
the basic line of argumentation for showing the theorem.
However,
for complexity reasons,
some technical aspects of the theorem are only coarsely treated,
and the proof sketch also refrains
from considering the full amount of OWL 2 language constructs.
For certain steps of the proof
there are example calculations
that focus only on a small fraction of language constructs,
but which can be taken as a hint
on how a complete proof
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.
A complete proof could make use of the observation
that the definitions of the OWL 2 Direct Semantics
and the OWL 2 RDF-Based Semantics,
despite their technical differences
as outlined in <a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>,
are closely aligned with respect to the different language constructs of OWL 2.
</p><p>The proof sketch will make use of an approach
that will be called <i>"balancing"</i> throughout this section,
and which will now be introduced.
The basic idea is to substitute
the original pair of RDF graphs in an OWL 2 DL entailment query
by another entailment query
having the same semantic characteristics
under the OWL 2 Direct Semantics,
but for which the technical differences
between the two semantics specifications
have no relevant consequences
under the OWL 2 RDF-Based Semantics anymore.
A concrete example
for the application of this approach
was given in <a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>.
</p>
<div id="def-balanced">
<p><b>Definition (Balanced):</b>
A pair of RDF graphs
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is called
<i>balanced</i>,
if and only if
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are OWL 2 DL ontologies in RDF graph form,
such that all the following conditions hold,
with
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
being the
OWL 2 DL ontologies in
<a href="../REC-owl2-syntax-20091027/index.html" title="Syntax">Functional Syntax</a>
form
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
that result from applying the
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
to
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>,
respectively:
</p>
<ol><li> F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>) mutually meet the restrictions on OWL 2 DL ontologies as specified in <a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> [<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</li><li> Nodes in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>:
<ol><li> for every IRI <i>u</i> occurring in <i>G<sub>1</sub></i> or <i>G<sub>2</sub></i> that corresponds to a non-built-in entity in F(<i>G<sub>1</sub></i>) or F(<i>G<sub>2</sub></i>), respectively, the graph contains, for every entity type <i>T</i> of <i>u</i>, a declaration statement of the form "<i>u</i> <span class="name">rdf:type</span> <i>t</i>", where <i>t</i> is the vocabulary class IRI corresponding to <i>T</i> (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_the_Ontology_Header_and_Declarations" title="Mapping to RDF Graphs">Table 7 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] and <a href="../REC-owl2-syntax-20091027/index.html#Entity_Declarations_and_Typing" title="Syntax">Section 5.8 of the OWL 2 Structural Specification</a> [<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]);
</li><li> every plain or typed literal occurring in <i>G<sub>2</sub></i> also occurs in <i>G<sub>1</sub></i> (see <a href="../REC-owl2-syntax-20091027/index.html#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> [<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
</li></ol>
</li><li> <i>G<sub>2</sub></i> contains exactly one ontology header consisting of a single RDF triple of the form "<i>x</i> <span class="name">rdf:type owl:Ontology</span>", where <i>x</i> is either a blank node or, if an ontology IRI is used in <i>G<sub>1</sub></i>, may alternatively equal that ontology IRI (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_the_Ontology_Header_and_Declarations" title="Mapping to RDF Graphs">Table 4 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
</li><li> <i>G<sub>2</sub></i> does <i>not</i> contain:
<ol><li> the RDF encoding of an annotation (see Sections <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Annotations" title="Mapping to RDF Graphs">3.2.2</a> and <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Ontology_Annotations" title="Mapping to RDF Graphs">3.2.3</a>, and <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Axioms" title="Mapping to RDF Graphs">Table 17</a> in the OWL 2 RDF Mapping [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> a statement with an ontology property such as "<span class="name">owl:imports</span>";
</li><li> a deprecation statement based on "<span class="name">owl:DeprecatedClass</span>", "<span class="name">owl:DeprecatedProperty</span>" and "<span class="name">owl:deprecated</span>" (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Axioms" title="Mapping to RDF Graphs">Table 16 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> an annotation property axiom based on "<span class="name">rdfs:subClassOf</span>", "<span class="name">rdfs:domain</span>" and "<span class="name">rdfs:range</span>" (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Axioms" title="Mapping to RDF Graphs">Table 16 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
</li></ol>
</li><li> Any of the following sub graphs of <i>G<sub>2</sub></i> is also a sub graph of <i>G<sub>1</sub></i>:
<ol><li> the RDF encoding of an entity declaration (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_the_Ontology_Header_and_Declarations" title="Mapping to RDF Graphs">Table 7 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> the RDF encoding of a property expression (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Expressions" title="Mapping to RDF Graphs">Table 11 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> the RDF encoding of a class expression (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Expressions" title="Mapping to RDF Graphs">Tables 13 and 15 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> the RDF encoding of a data range expression (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Expressions" title="Mapping to RDF Graphs">Tables 12 and 14 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> an RDF sequence (see <a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">Table 3 in the OWL 2 RDF Mapping</a> [<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
</li></ol>
</li></ol>
</div>
<div id="thm-balancing">
<p><b>Balancing Lemma:</b>
An algorithm exists
that terminates on every valid input
and that has the following input/output behavior:
</p><p>The <i>valid input</i> of the algorithm
is given by
all the pairs of RDF graphs
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> ),
where
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>
are OWL 2 DL ontologies in RDF graph form,
with
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
being the
OWL 2 DL ontologies in
<a href="../REC-owl2-syntax-20091027/index.html" title="Syntax">Functional Syntax</a>
form
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
that result from applying the
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
to
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>,
respectively.
Further,
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
have to mutually meet
the restrictions on OWL 2 DL ontologies
as specified in
<a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p><p>For a valid input,
the <i>output</i> of the algorithm
is a pair of RDF graphs
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ),
where
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are OWL 2 DL ontologies in RDF graph form,
such that
for any OWL 2 RDF-Based datatype map <i>D</i>
according to <a href="index.html#def-owldatatypemap" title="">Definition 4.1</a>
all the following relationships hold,
with
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
being the
OWL 2 DL ontologies in Functional Syntax form
that result from applying the reverse RDF mapping
to
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>,
respectively,
and
with F(<i>D</i>)
being the
OWL 2 Direct datatype map
according to
<a href="../REC-owl2-direct-semantics-20091027/index.html#def_datatype_map" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
that corresponds to <i>D</i> according to the
<a href="index.html#topic-correspondence-datatypemaps" title=""><i>technical note on corresponding datatype maps</i></a>
in <a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>:
</p>
<ol><li> The pair ( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) is <a href="index.html#def-balanced" title=""><i>balanced</i></a>.
</li><li> F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>1</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>1</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>1</sub></i>) with respect to <i>F(D)</i>.
</li><li> F(<i>G<sub>2</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>2</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) with respect to <i>F(D)</i>.
</li></ol>
</div>
<div id="topic-correspondence-proof-balancing"></div>
<p><i><b>Proof for the Balancing Lemma:</b></i>
</p><p>Let the graph pair
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
be a valid input.
The resulting RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are constructed as follows,
starting from copies of
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>,
respectively.
</p><p>Since the initial versions of
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are OWL 2 DL ontologies in RDF graph form,
the <i>canonical parsing process</i> (CP)
for computing the reverse RDF mapping,
as described in
<a href="../REC-owl2-syntax-20091027/index.html#Ontologies" title="Syntax">Section 3 of the OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>],
can be applied.
Based on CP, it is possible to identify within these graphs
</p>
<ul><li> all entity types for every non-built-in IRI,
</li><li> all blank nodes that correspond to anonymous individuals, and
</li><li> all sub graphs that correspond to OWL 2 language constructs (ontology headers, declarations, expressions, axioms and annotations) as described in the <a href="../REC-owl2-syntax-20091027/index.html" title="Syntax">OWL 2 Structural Specification</a> [<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</li></ul>
<p>Based on this observation, the following steps are performed:
</p>
<ol><li> Consistently substitute all blank nodes in <i>G<sub>2</sub></i> such that <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> have no common blank nodes.
</li><li> Apply CP to <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> (without changing these graphs) to identify the entity types of the IRIs, the anonymous individuals, and the sub graphs encoding OWL 2 language constructs.
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i>: remove <i>g</i> from <i>G<sub>2</sub></i>, if <i>g</i> is the RDF encoding of
<ul><li> an <i>annotation</i>, or
</li><li> a <i>deprecation statement</i>, or
</li><li> an <i>annotation property axiom</i>.
</li></ul>
</li><li> For the sub graph <i>g</i> of <i>G<sub>2</sub></i> corresponding to the <i>ontology header</i> in F(<i>G<sub>2</sub></i>): substitute <i>g</i> in <i>G<sub>2</sub></i> by a triple of the form "<i>x</i> <span class="name">rdf:type owl:Ontology</span>", where <i>x</i> is a new blank node not yet used in <i>G<sub>2</sub></i>.
</li><li> For each non-built-in IRI <i>u</i> in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> and for each entity type <i>T</i> of <i>u</i> identified by CP: add to <i>G<sub>1</sub></i> or <i>G<sub>2</sub></i>, respectively, the RDF triple "<i>u</i> <span class="name">rdf:type</span> <i>t</i>", where <i>t</i> is the vocabulary class IRI corresponding to <i>T</i>.
</li><li> For each plain or typed literal <i>L</i> in <i>G<sub>2</sub></i>: add to <i>G<sub>1</sub></i> the RDF triple "<i>o</i> <span class="name">rdfs:comment</span> <i>L</i>", where <i>o</i> is the IRI or blank node of the ontology header triple "<i>o</i> <span class="name">rdf:type</span> <span class="name">owl:Ontology</span>" in <i>G<sub>1</sub></i>.
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of an <i>entity declaration</i>: add <i>g</i> to <i>G<sub>1</sub></i>.
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of a <i>property expression</i> with root blank node <i>x</i>: add <i>g</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>x</i> <span class="name">owl:equivalentProperty</span> <i>x</i>".
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of a <i>class expression</i> with root blank node <i>x</i>: add <i>g</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>x</i> <span class="name">owl:equivalentClass</span> <i>x</i>".
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of a <i>data range expression</i> with root blank node <i>x</i>:
<ul><li> If <i>g</i> is part of a <i>data property restriction expression</i>, then nothing needs to be done, since the comprising restriction expression is covered by the treatment of class expressions, and therefore <i>g</i> occurs in <i>G<sub>1</sub></i> as well.
</li><li> Otherwise, add a declaration triple to <i>G<sub>1</sub></i> for a new data property <i>p</i> that does not yet occur in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>. Then, the RDF encoding <i>r</i> of a <i>universal data property restriction expression</i> on property <i>p</i> is created for <i>g</i>. Let <i>r</i> have the new root blank node <i>y</i>. Add <i>r</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>y</i> <span class="name">owl:equivalentClass</span> <i>y</i>".
</li></ul>
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is an RDF sequence with root blank node <i>x</i>, which does not occur in the RDF encoding of language constructs already treated by one of the earlier steps, i.e. <i>g</i> is part of the encoding of an axiom: create the RDF encoding <i>r</i> of an enumeration class expression with a new root blank node <i>y</i> having the main RDF triple "<i>y</i> <span class="name">owl:oneOf</span> <i>x</i>". Then, add <i>r</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>y</i> <span class="name">owl:equivalentClass</span> <i>y</i>". Additionally, for every IRI <i>u</i> being a member of the RDF sequence, add to <i>G<sub>1</sub></i> a typing triple "<i>u</i> <span class="name">rdf:type owl:NamedIndividual</span>". If one of the sequence members is a blank node <i>z</i> that is the root node of some property expression or class expression <i>e</i>, then select a new IRI <i>w</i> not yet occurring in <i>G<sub>1</sub></i>, consistently replace <i>z</i> by <i>w</i> everywhere in <i>r</i>, add to <i>G<sub>1</sub></i> the triple "<i>w</i> <span class="name">owl:equivalentProperty</span> <i>z</i>" or "<i>w</i> <span class="name">owl:equivalentClass</span> <i>z</i>", respectively, and add to <i>G<sub>1</sub></i> the two triples "<i>w</i> <span class="name">rdf:type owl:NamedIndividual</span>" and "<i>w</i> <span class="name">rdf:type</span> <i>t</i>", where <i>t</i> is the vocabulary class IRI that represents the appropriate entity type of the expression <i>e</i>. No further treatment of <i>e</i> is needed, since <i>e</i> is treated by the earlier steps covering expressions.
</li></ol>
<p>In the following it is shown that all the claims of the balancing lemma hold.
</p><p><i><b>A: Existence of a Terminating Algorithm.</b></i>
An algorithm <i>exists</i>
for mapping
the input graph pair
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
to the output graph pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ),
since CP (applied in step 2)
is described in the form of an algorithm
in the
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>],
and since all other steps
can obviously be performed algorithmically.
The algorithm <i>terminates</i>,
since CP terminates
on arbitrary input graphs,
and since all other steps
can obviously be executed in finite time.
</p><p><i><b>B: The Resulting RDF Graphs are OWL 2 DL Ontologies.</b></i>
The RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are OWL 2 DL ontologies in RDF graph form
that mutually meet
the restrictions on OWL 2 DL ontologies,
since the original RDF graphs
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>
have this feature,
and since each of the steps described above
transforms a pair of RDF graphs with this feature
again into a pair of RDF graphs with this feature,
for the following reasons:
</p>
<ul><li> The consistent substitution of blank nodes in step 1 does not change the structure of an OWL 2 DL ontology.
</li><li> The application of CP in step 2 does not change the graphs.
</li><li> Annotations, deprecation statements and annotation property axioms are optional information in an OWL 2 DL ontology and can therefore be omitted in step 3.
</li><li> The ontology header of an OWL 2 DL ontology does neither require the existence of an ontology IRI nor of any ontology properties, and so the substitution of the ontology header in step 4 is a valid operation.
</li><li> If an entity has some particular entity type for which there is no explicitly given entity declaration, then the entity declaration may be added, as done in step 5.
</li><li> It is allowed to add arbitrary annotations to the ontology header of an OWL 2 DL ontology, as done in step 6.
</li><li> Entity declarations may be copied from <i>G<sub>2</sub></i> to <i>G<sub>1</sub></i> in step 7 without conflict, since the original ontologies have been assumed to mutually meet the restrictions on OWL 2 DL ontologies regarding different entity declarations for the same IRI (e.g. that one IRI must not be the name of both an object property and a data property).
</li><li> Adding to <i>G<sub>1</sub></i> an axiom that claims equivalence of some property expression (step 8) or class expression (step 9) with itself, where the expression already occurs in <i>G<sub>2</sub></i>, is an allowed operation, since the original ontologies are assumed to mutually meet the restrictions on OWL 2 DL ontologies concerning property and class expressions, and since no syntactic restrictions exist on this specific use of equivalence axioms.
</li><li> For the case of data ranges (step 10) it is sufficient to note that placing universal property restrictions on arbitrary (simple or complex) property expressions is allowed in OWL 2 DL. The rest of the argumentation follows the lines of the treatment of class expressions in step 9.
</li><li> For the treatment of RDF sequences in step 11: First, the enumeration class expressions being constructed from the RDF sequences are syntactically valid in OWL 2 DL, since all enumerated entries are IRIs by construction. Second, there is no restriction in OWL 2 DL disallowing axioms that claim equivalence of enumeration class expressions with themselves. Third, punning in OWL 2 DL allows a given non-built-in IRI of any entity type to be additionally declared as a named individual. Forth, there is no OWL 2 DL restriction forbidding to add an entity declaration for a new (i.e. not elsewhere used) IRI and to assert the denotation of this new IRI to be equivalent to some existing property or class expression. Hence, the resulting ontologies still mutually meet all syntactic restrictions on OWL 2 DL ontologies.
</li></ul>
<p><i><b>C: The Resulting Pair of RDF Graphs is Balanced.</b></i>
All the conditions of <a href="index.html#def-balanced" title=""><i>balanced</i></a> pairs of RDF graphs
are met by the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
for the following reasons:
</p>
<ul><li> Condition 1: It has already been shown in paragraph <i>B</i> that <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> mutually meet the restrictions on OWL 2 DL ontologies.
</li><li> Conditions 2.1 and 2.2 on nodes in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> are met by steps 5 and 6, respectively.
</li><li> Condition 3 on ontology headers in <i>G<sub>2</sub></i> is satisfied by step 4, always applying an anonymous ontology header.
</li><li> Conditions 4.1, 4.3 and 4.4 on annotations, deprecation statements and annotation property axioms in <i>G<sub>2</sub></i>, respectively, are all satisfied by step 3.
</li><li> Condition 4.2 on statements with ontology properties is implicitly satisfied by step 4, since the substitution of the ontology header in <i>G<sub>2</sub></i> removes all existing statements with ontology properties.
</li><li> Condition 5.1 on entity declarations in <i>G<sub>2</sub></i> being reflected in <i>G<sub>1</sub></i> is satisfied by step 7.
</li><li> Conditions 5.2, 5.3 and 5.4 on property, class and data range expressions in <i>G<sub>2</sub></i>, respectively, being reflected in <i>G<sub>1</sub></i> are met by steps 8, 9 and 10, respectively.
</li><li> Condition 5.5 on RDF sequences in <i>G<sub>2</sub></i> being reflected in <i>G<sub>1</sub></i> is satisfied by step 11.
</li></ul>
<p><i><b>D: The Resulting Ontologies are semantically equivalent with the Original Ontologies under the OWL 2 Direct Semantics.</b></i>
F(<i>G<sub>1</sub></i>) is semantically equivalent with F(<i>G<sub>1</sub><sup>*</sup></i>),
since F(<i>G<sub>1</sub></i>) differs from F(<i>G<sub>1</sub><sup>*</sup></i>) only by (potentially):
</p>
<ul><li> additional entity declarations (steps 5, 7 and 11), which have no formal meaning under the OWL 2 Direct Semantics;
</li><li> additional annotations (step 6), which have no formal meaning;
</li><li> additional tautological axioms (steps 8, 9, 10 and 11), which do not change the formal meaning;
</li></ul>
<p>F(<i>G<sub>2</sub></i>) is semantically equivalent with F(<i>G<sub>2</sub><sup>*</sup></i>),
since F(<i>G<sub>2</sub></i>) differs from F(<i>G<sub>2</sub><sup>*</sup></i>) only by (potentially):
</p>
<ul><li> differently labeled anonymous individuals (step 1), by which the formal meaning under the OWL 2 Direct Semantics keeps unchanged, since anonymous individuals are existentially interpreted;
</li><li> missing annotations, deprecation statements and annotation property axioms (step 3), which have no formal meaning;
</li><li> a modified ontology header (step 4), which has no formal meaning;
</li><li> additional entity declarations (step 5), which have no formal meaning.
</li></ul>
<p><i><b>End of Proof for the Balancing Lemma.</b></i>
</p><p>In the following,
the correspondence theorem will be proven.
</p>
<div id="topic-correspondence-proof-main"></div>
<p>Assume that the premises of the correspondence theorem are true
for a given pair
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
of RDF graphs.
This allows for applying the
<a href="index.html#thm-balancing" title=""><i>balancing lemma</i></a>,
which provides the existence of corresponding RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
that are OWL 2 DL ontologies in RDF graph form,
and which meet the
definition of <a href="index.html#def-balanced" title=""><i>balanced</i></a> graph pairs.
Let F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
be the corresponding OWL 2 DL ontologies in Functional Syntax form.
Then,
the claimed relationship 1 of the correspondence theorem
follows directly from relationship 1 of the <a href="index.html#thm-balancing" title=""><i>balancing lemma</i></a>
and from condition 1 of the definition of <a href="index.html#def-balanced" title="">balanced</a> graph pairs.
Further,
the claimed relationships 2 and 3 of the correspondence theorem
follow directly from the relationships 2 and 3 of the <a href="index.html#thm-balancing" title=""><i>balancing lemma</i></a>,
respectively.
</p><p>The rest of this proof will treat
the claimed relationship 4 of the correspondence theorem,
which states that
if F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>)
with respect to <i>F(D)</i>,
then <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>
with respect to <i>D</i>.
For this to see,
an arbitrary OWL 2 RDF-Based interpretation <i>I</i> will be selected
that OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
For <i>I</i>,
a closely corresponding OWL 2 Direct interpretation <i>J</i>
will be constructed,
and it will then be shown
that <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>1</sub></i>).
Since it was assumed that
F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>),
it will follow that <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>2</sub></i>).
Based on this result, it will then be possible to show
that <i>I</i> also OWL 2 RDF-Based satisfies <i>G<sub>2</sub></i>.
Since <i>I</i> was arbitrarily selected,
this will mean
that <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>.
</p>
<div id="topic-correspondence-proof-step1"></div>
<p><i><b>Step 1: Selection of a Pair of Corresponding Interpretations.</b></i>
</p><p>Let
F(<i>G<sub>1</sub></i>) OWL 2 Direct entail F(<i>G<sub>2</sub></i>) w.r.t. F(<i>D</i>),
and let <i>I</i> be an OWL 2 RDF-Based interpretation
of a vocabulary <i>V<sup>I</sup></i> w.r.t. <i>D</i>,
such that
<i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
</p><p>Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
there exist <i>entity declarations</i>
in F(<i>G<sub>1</sub></i>)
for each entity type
of every non-built-in IRI
occurring in <i>G<sub>1</sub></i>:
For each entity declaration
of the form
"<span class="name">Declaration</span>(<i>T</i>(<i>u</i>))"
in F(<i>G<sub>1</sub></i>),
such that <i>T</i> is the entity type for some IRI <i>u</i>,
a typing triple
of the form
"<i>u</i> <span class="name">rdf:type</span> <i>t</i>"
exists in <i>G<sub>1</sub></i>,
where <i>t</i> is the vocabulary class IRI
representing the part of the universe of <i>I</i>
that corresponds to <i>T</i>.
Since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
all these declaration typing triples are OWL 2 RDF-Based satisfied by <i>I</i>,
and thus all non-built-in IRIs in <i>G<sub>1</sub></i>
are instances of all their declared parts of the universe of <i>I</i>.
</p><p>The vocabulary
<i>V<sup>J</sup></i> := (
<i>V<sup>J</sup><sub>C</sub></i> ,
<i>V<sup>J</sup><sub>OP</sub></i> ,
<i>V<sup>J</sup><sub>DP</sub></i> ,
<i>V<sup>J</sup><sub>I</sub></i> ,
<i>V<sup>J</sup><sub>DT</sub></i> ,
<i>V<sup>J</sup><sub>LT</sub></i> ,
<i>V<sup>J</sup><sub>FA</sub></i>
)
of the OWL 2 Direct interpretation <i>J</i> w.r.t. the datatype map <i>F(D)</i> is now constructed as follows.
</p>
<ul><li> The set <i>V<sup>J</sup><sub>C</sub></i> of classes contains all IRIs in <i>V<sup>I</sup></i> that are declared as classes in F(<i>G<sub>1</sub></i>), together with all the required class names listed in <a href="../REC-owl2-direct-semantics-20091027/index.html#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
</li><li> The set <i>V<sup>J</sup><sub>OP</sub></i> of object properties contains all IRIs in <i>V<sup>I</sup></i> that are declared as object properties in F(<i>G<sub>1</sub></i>), together with all the required object property names listed in <a href="../REC-owl2-direct-semantics-20091027/index.html#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
</li><li> The set <i>V<sup>J</sup><sub>DP</sub></i> of data properties contains all IRIs in <i>V<sup>I</sup></i> that are declared as data properties in F(<i>G<sub>1</sub></i>), together with all the required data property names listed in <a href="../REC-owl2-direct-semantics-20091027/index.html#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
</li><li> The set <i>V<sup>J</sup><sub>I</sub></i> of individuals contains all IRIs in <i>V<sup>I</sup></i> that are declared as named individuals in F(<i>G<sub>1</sub></i>), and additionally all anonymous individuals occurring in F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>).
</li><li> The set <i>V<sup>J</sup><sub>DT</sub></i> of datatypes is defined according to <a href="../REC-owl2-direct-semantics-20091027/index.html#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>] w.r.t. the datatype map F(<i>D</i>), together with all other IRIs in <i>V<sup>I</sup></i> that are declared as datatypes in F(<i>G<sub>1</sub></i>).
</li><li> The set <i>V<sup>J</sup><sub>LT</sub></i> of literals is defined according to <a href="../REC-owl2-direct-semantics-20091027/index.html#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>] w.r.t. the datatype map F(<i>D</i>).
</li><li> The set <i>V<sup>J</sup><sub>FA</sub></i> of facet-literal pairs is defined according to <a href="../REC-owl2-direct-semantics-20091027/index.html#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>] w.r.t. the datatype map F(<i>D</i>).
</li></ul>
<p>The OWL 2 Direct interpretation
<i>J</i> := (
<i>Δ<sub>I</sub></i> ,
<i>Δ<sub>D</sub></i> ,
<i>⋅ <sup>C</sup></i> ,
<i>⋅ <sup>OP</sup></i> ,
<i>⋅ <sup>DP</sup></i> ,
<i>⋅ <sup>I</sup></i> ,
<i>⋅ <sup>DT</sup></i> ,
<i>⋅ <sup>LT</sup></i> ,
<i>⋅ <sup>FA</sup></i>
)
is now defined as follows.
The object and data domains of <i>J</i> are identified
with the universe IR and the set of data values LV of <i>I</i>,
respectively,
i.e.,
<i>Δ<sub>I</sub></i> := IR and
<i>Δ<sub>D</sub></i> := LV.
The class interpretation function <i>⋅ <sup>C</sup></i>,
the object property interpretation function <i>⋅ <sup>OP</sup></i>,
the data property interpretation function <i>⋅ <sup>DP</sup></i>,
the datatype interpretation function <i>⋅ <sup>DT</sup></i>,
the literal interpretation function <i>⋅ <sup>LT</sup></i>, and
the facet interpretation function <i>⋅ <sup>FA</sup></i>
are defined according to
<a href="../REC-owl2-direct-semantics-20091027/index.html#Interpretations" title="Direct Semantics">Section 2.2 of the OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
Specifically,
for every non-built-in IRI <i>u</i>
occurring in F(<i>G<sub>1</sub></i>):
</p>
<ul><li> If <i>u</i> is declared as a class, then set <i>u<sup>C</sup></i> := ICEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:Class</span>", i.e., <i>I</i>(<i>u</i>) ∈ IC.
</li><li> If <i>u</i> is declared as an object property, then set <i>u<sup>OP</sup></i> := IEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:ObjectProperty</span>", i.e., <i>I</i>(<i>u</i>) ∈ IP.
</li><li> If <i>u</i> is declared as a data property, then set <i>u<sup>DP</sup></i> := IEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:DatatypeProperty</span>", i.e., <i>I</i>(<i>u</i>) ∈ IODP.
</li><li> If <i>u</i> is declared as a named individual, then set <i>u<sup>I</sup></i> := <i>I</i>(<i>u</i>), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:NamedIndividual</span>", i.e., <i>I</i>(<i>u</i>) ∈ IR.
</li><li> If <i>u</i> is declared as a datatype, then set <i>u<sup>DT</sup></i> := ICEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type rdfs:Datatype</span>", i.e., <i>I</i>(<i>u</i>) ∈ IDC.
</li></ul>
<p><i>Notes:</i>
</p>
<ul><li> A <i>literal</i> occurring in <i>G<sub>1</sub></i> is mapped by the reverse RDF mapping to the same literal in F(<i>G<sub>1</sub></i>), and the formal meaning of a well-formed literal is analog for both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics.
</li><li> A <i>blank node</i> <i>b</i> occurring in <i>G<sub>1</sub></i> that represents an <i>anonymous individual</i> is written as the same blank node <i>b</i> in F(<i>G<sub>1</sub></i>). Both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics treat anonymous individuals in an analog way as <i>existential variables</i> defined locally to a given ontology, i.e. some individual <i>x</i> exists in the universe to which all occurrences of <i>b</i> in the ontology can be mapped (see <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">Section 1.5 in the RDF Semantics</a> [<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>] for the precise definition on how blank nodes are treated under the OWL 2 RDF-Based Semantics). Hence, the same mapping from <i>b</i> to <i>x</i> can be used with both <i>I</i> and <i>J</i>.
</li><li> <i>G<sub>1</sub></i> may also contain declarations for <i>annotation properties</i>. Since annotation properties have no formal meaning under the OWL 2 Direct Semantics, the OWL 2 Direct interpretation <i>J</i> does not treat them.
</li><li> With the above definition it is possible for <i>J</i> to have a <i>nonseparated vocabulary</i> according to <a href="../REC-owl2-syntax-20091027/index.html#Metamodeling" title="Syntax">Section 5.9 of the OWL 2 Structural Specification</a> [<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]. Since <i>G<sub>1</sub></i> is an OWL 2 DL ontology in RDF graph form, it is allowed that the same IRI <i>u</i> may be declared as one or more of an individual name, either a class name or a datatype name, and either an object property name or a data property name. For the OWL 2 RDF-Based interpretation <i>I</i>, the IRI <i>u</i> will always denote the same individual in the universe IR, where <i>I</i>(<i>u</i>) may additionally have a class extension or a property extension, or both. For the OWL 2 Direct interpretation <i>J</i>, however, <i>u</i> will denote as an individual name an element of <i>Δ<sub>I</sub></i>, as a class name a subset of <i>Δ<sub>I</sub></i>, as a datatype name a subset of <i>Δ<sub>D</sub></i>, as an object property name a subset of <i>Δ<sub>I</sub></i> × <i>Δ<sub>I</sub></i>, and as a data property name a subset of <i>Δ<sub>I</sub></i> × <i>Δ<sub>D</sub></i>.
</li></ul>
<div id="topic-correspondence-proof-step2"></div>
<p><b><i>Step 2: Satisfaction of F(</i>G<sub>1</sub><i>) by the OWL 2 Direct Interpretation.</i></b>
</p><p>Based on the premise that <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
it has to be shown that <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>1</sub></i>).
For this to hold,
it will be sufficient that
<i>J</i> OWL 2 Direct satisfies every axiom <i>A</i> occurring in F(<i>G<sub>1</sub></i>).
Let <i>g<sub>A</sub></i> be the sub graph of <i>G<sub>1</sub></i>
that is mapped to <i>A</i> by the reverse RDF mapping.
The basic idea can roughly be described as follows:
</p><p>Since <i>I</i> is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by <i>I</i>.
Due to the close alignment between the definitions
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the axiom <i>A</i>.
In particular,
the antecedent of one of these semantic conditions
will become true,
if the RDF-encoding of <i>A</i>,
i.e. the graph <i>g<sub>A</sub></i>,
is satisfied
(in the case of an "if-and-only-if" semantic condition
this will generally be the left-to-right direction of that condition).
Now,
all the RDF triples in <i>g<sub>A</sub></i>
are OWL 2 RDF-Based satisfied by <i>I</i>,
since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
Hence,
the antecedent of the semantic condition becomes true,
and therefore its consequent becomes true as well.
This will reveal a certain semantic relationship
that
according to <i>I</i>
holds between the denotations of the
IRIs, literals and anonymous individuals
occurring in <i>g<sub>A</sub></i>,
which,
roughly speaking,
expresses the meaning of the OWL 2 axiom <i>A</i>.
Because of the close semantic correspondence
of the OWL 2 Direct interpretation <i>J</i> to <i>I</i>,
the analog semantic relationship holds
according to <i>J</i>
between the denotations of the
IRIs, literals and anonymous individuals
occurring in <i>A</i>.
This semantic relationship
turns out to be compatible
with the formal meaning of the axiom <i>A</i>
as specified by the OWL 2 Direct Semantics,
i.e. <i>J</i> satisfies <i>A</i>.
</p><p>This basic idea is now demonstrated in more detail
for a single example axiom <i>A</i> in F(<i>G<sub>1</sub></i>),
which can be taken as a hint on
how a complete proof
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.
</p>
<div class="anexample" id="topic-correspondence-proof-example1">
<p>Let <i>A</i> be the following OWL 2 axiom in F(<i>G<sub>1</sub></i>):
</p>
<div class="indent">
<p><i>A</i> : <span class="name">SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))</span>
</p>
</div>
<p>and let <i>g<sub>A</sub></i> be the corresponding sub graph in <i>G<sub>1</sub></i>
that is being mapped to <i>A</i> via the reverse RDF mapping,
namely
</p>
<div class="indent">
<p><i>g<sub>A</sub></i> :
</p>
<div class="indent">
<p><span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
<span class="name">_:x rdf:type owl:Class .</span><br />
<span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<p>Since the pair ( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>G<sub>1</sub></i> contains the typing triples
</p>
<div class="indent">
<p><span class="name">ex:c1 rdf:type owl:Class .</span><br />
<span class="name">ex:c2 rdf:type owl:Class .</span><br />
<span class="name">ex:c3 rdf:type owl:Class .</span>
</p>
</div>
<p>that correspond to class entity declarations in F(<i>G<sub>1</sub></i>) for the IRIs
"<span class="name">ex:c1</span>",
"<span class="name">ex:c2</span>", and
"<span class="name">ex:c3</span>",
respectively.
All these declaration typing triples are OWL 2 RDF-Based satisfied by <i>I</i>,
since it has been postulated
that <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
Hence,
by applying the semantics of <span class="name">rdf:type</span>
(see
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfssemcond1" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1">Section 4.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]),
all the IRIs denote classes, precisely:
</p>
<div class="indent">
<p><i>I</i>(<span class="name">ex:c1</span>) ∈ IC ,<br />
<i>I</i>(<span class="name">ex:c2</span>) ∈ IC , and<br />
<i>I</i>(<span class="name">ex:c3</span>) ∈ IC .
</p>
</div>
<p>Since <i>I</i> is an OWL 2 RDF-Based interpretation,
it meets all the OWL 2 RDF-Based semantic conditions,
and since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
all the triples in <i>g<sub>A</sub></i> are OWL 2 RDF-Based satisfied.
This meets the left-to-right directions of the semantic conditions
for subclass axioms
("<span class="name">rdfs:subClassOf</span>",
see <a href="index.html#Semantic_Conditions_for_the_RDFS_Vocabulary" title="">Section 5.8</a>)
and union class expressions
("<span class="name">owl:unionOf</span>",
see <a href="index.html#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a>),
which results in the following semantic relationship
that holds between the extensions of the classes above
according to <i>I</i>:
</p>
<div class="indent">
<p>ICEXT(<i>I</i>(<span class="name">ex:c1</span>))
⊆
ICEXT(<i>I</i>(<span class="name">ex:c2</span>))
∪
ICEXT(<i>I</i>(<span class="name">ex:c3</span>)) .
</p>
</div>
<p>By applying the definition of <i>J</i>,
one can conclude
that the following semantic relationship
holds between the denotations of the class names occurring in <i>A</i>
according to <i>J</i>:
</p>
<div class="indent">
<p>(<span class="name">ex:c1</span>) <sup><i>C</i></sup>
⊆
(<span class="name">ex:c2</span>) <sup><i>C</i></sup>
∪
(<span class="name">ex:c3</span>) <sup><i>C</i></sup> .
</p>
</div>
<p>This semantic relationship is compatible
with the formal meaning of the axiom <i>A</i>
under the OWL 2 Direct Semantics.
Hence, <i>J</i> OWL 2 Direct satisfies <i>A</i>.
</p>
</div>
<p>Since <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>1</sub></i>),
and since it has been postulated that
F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>),
it follows that
<i>J</i> OWL 2 Direct satisfies F(<i>G<sub>2</sub></i>).
</p>
<div id="topic-correspondence-proof-step3"></div>
<p><b><i>Step 3: Satisfaction of </i>G<sub>2</sub><i> by the OWL 2 RDF-Based Interpretation.</i></b>
</p><p>The last step will be
to show that <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>2</sub></i>.
For this to hold,
<i>I</i> needs to OWL 2 RDF-Based satisfy
every triple occurring in <i>G<sub>2</sub></i>.
The basic idea can roughly be described as follows:
</p><p><i>First:</i>
According to the <i>"semantic conditions for ground graphs"</i>
in <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">Section 1.4 of the RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
all the IRIs and literals used in RDF triples in <i>G<sub>2</sub></i>
need to be in the vocabulary <i>V<sup>I</sup></i> of <i>I</i>.
This is true for the following reason:
Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
all IRIs and literals occurring in <i>G<sub>2</sub></i>
do also occur in <i>G<sub>1</sub></i>.
Since <i>I</i> satisfies <i>G<sub>1</sub></i>,
all IRIs and literals in <i>G<sub>1</sub></i>,
including those in <i>G<sub>2</sub></i>,
are contained in <i>V<sup>I</sup></i>
due to the semantic conditions for ground graphs.
</p><p><i>Second:</i>
If a set of RDF triples encodes an OWL 2 language construct
that is not interpreted by the OWL 2 Direct Semantics,
such as annotations,
then <i>G<sub>2</sub></i> should contain such a set of RDF triples
only if they are also included in <i>G<sub>1</sub></i>.
The reason is
that with such triples
there will, in general, exist OWL 2 RDF-Based interpretations
only satisfying the graph <i>G<sub>1</sub></i> but not <i>G<sub>2</sub></i>,
which will render the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
into a nonentailment
(an exception are RDF triples
that are true
under every OWL 2 RDF-Based interpretation).
Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>G<sub>2</sub></i> will not contain the RDF encoding for any
<i>annotations</i>,
statements with <i>ontology properties</i>,
<i>deprecation</i> statements or
<i>annotation property axioms</i>.
Hence,
there are no corresponding RDF triples that need to be satisfied by <i>I</i>.
</p><p><i>Third:</i>
Since <i>G<sub>2</sub></i> is an OWL 2 DL ontology in RDF graph form,
the graph is partitioned by the
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
into sub graphs corresponding to
either <i>ontology headers</i>,
<i>entity declarations</i>
or <i>axioms</i>,
where axioms may further consist of different kinds of <i>expressions</i>,
such as Boolean class expressions.
It has to be shown that all the triples in each such sub graph
are OWL 2 RDF-Based satisfied by <i>I</i>.
</p><p><i>For ontology headers:</i>
Let <i>A</i> be the ontology header of F(<i>G<sub>2</sub></i>)
and let <i>g<sub>A</sub></i> be the corresponding sub graph of <i>G<sub>2</sub></i>.
Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>g<sub>A</sub></i> is encoded as a single RDF triple of the form
"<i>x</i> <span class="name">rdf:type owl:Ontology</span>",
where <i>x</i> is either an IRI or a blank node.
Since <i>G<sub>1</sub></i> is an OWL 2 DL ontology in RDF graph form,
<i>G<sub>1</sub></i> also contains the encoding of an ontology header
including a triple <i>g<sub>1</sub></i> of the form
"<i>y</i> <span class="name">rdf:type owl:Ontology</span>",
where <i>y</i> is either an IRI or a blank node.
Since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>g<sub>1</sub></i> is satisfied by <i>I</i>.
If both <i>y</i> and <i>x</i> are IRIs,
then, due to balancing,
<i>x</i> equals <i>y</i>,
and therefore <i>g<sub>A</sub></i> equals <i>g<sub>1</sub></i>,
i.e. <i>g<sub>A</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>.
Otherwise,
balancing forces <i>x</i> to be a blank node,
i.e. <i>x</i> is treated as an existential variable
under the OWL 2 RDF-Based Semantics
according to the
<i><a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">"semantic conditions for blank nodes"</a></i>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
From this observation,
and from the premise that <i>I</i> satisfies <i>g<sub>1</sub></i>,
it follows that <i>g<sub>A</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>.
</p><p><i>For entity declarations:</i>
Let <i>A</i> be an entity declaration in F(<i>G<sub>2</sub></i>),
and let <i>g<sub>A</sub> be the corresponding sub graph of </i>G<sub>2</sub><i>.</i>
Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>A</i> occurs in F(<i>G<sub>1</sub></i>),
and hence <i>g<sub>A</sub></i> is a sub graph of <i>G<sub>1</sub></i>.
Since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>I</i> OWL 2 RDF-Based satisfies <i>g<sub>A</sub></i>.
</p><p><i>For axioms:</i>
Let <i>A</i> be an axiom in F(<i>G<sub>2</sub></i>),
and let <i>g<sub>A</sub></i> be the corresponding sub graph of <i>G<sub>2</sub></i>.
Since <i>I</i> is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by <i>I</i>.
Due to the close alignment between the definitions
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the axiom <i>A</i>.
In particular,
the consequent of one of these semantic conditions
corresponds to the RDF-encoding of <i>A</i>,
i.e. the graph <i>g<sub>A</sub></i>,
except for declaration typing triples,
for which satisfaction has already been shown
(in the case of an "if-and-only-if" semantic condition
this will generally be the right-to-left direction of that condition).
Hence,
in order to show that <i>g<sub>A</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>,
it will be sufficient to show
that the antecedent of this semantic condition is true.
In general,
several requirements have to be met to ensure this:
</p><p><i>Requirement 1:</i>
The denotations of all the non-built-in IRIs in <i>g<sub>A</sub></i>
have to be contained in the appropriate part of the universe of <i>I</i>.
This can be shown as follows.
For every non-built-in IRI <i>u</i> occurring in <i>g<sub>A</sub></i>,
<i>u</i> also occurs in <i>A</i>.
Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
there are entity declarations in F(<i>G<sub>2</sub></i>)
for all the entity types of <i>u</i>,
each being of the form
<i>D</i> := "<span class="name">Declaration</span>(<i>T</i>(<i>u</i>))"
for some entity type <i>T</i>.
From the reverse RDF mapping follows
that for each such declaration <i>D</i>
a typing triple <i>d</i> exists in <i>G<sub>2</sub></i>,
being of the form <i>d</i> := "<i>u</i> <span class="name">rdf:type</span> <i>t</i>",
where <i>t</i> is the vocabulary class IRI
representing the part of the universe of <i>I</i>
that corresponds to the entity type <i>T</i>.
It has already been shown that,
for <i>D</i> being an entity declaration in F(<i>G<sub>2</sub></i>)
and <i>d</i> being the corresponding sub graph in <i>G<sub>2</sub></i>,
<i>I</i> OWL 2 RDF-Based satisfies <i>d</i>.
Hence, <i>I</i>(<i>u</i>) is an individual
contained in the appropriate part of the universe.
</p><p><i>Requirement 2:</i>
For every expression <i>E</i> occurring in <i>A</i>,
with the RDF encoding <i>g<sub>E</sub></i> in <i>g<sub>A</sub></i>,
an individual has to exist in the universe of <i>I</i>
that appropriately represents the denotation of <i>E</i>.
Since <i>I</i> is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by <i>I</i>.
Due to the close alignment between the definitions
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the expression <i>E</i>.
In particular,
the antecedent of one of these semantic conditions
will become true,
if the RDF-encoding of <i>E</i>,
i.e. the graph <i>g<sub>E</sub></i>,
is satisfied
(in the case of an "if-and-only-if" semantic condition
this will generally be the left-to-right direction of that condition).
Now,
since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>g<sub>E</sub></i> also occurs in <i>G<sub>1</sub></i>.
So,
since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>g<sub>E</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>.
Hence,
the antecedent of the semantic condition becomes true,
and therefore its consequent becomes true as well.
This will result in the existence of an individual with the required properties,
when taking into account existential blank node semantics.
</p><p><i>Requirement 3:</i>
A semantic relationship
has to hold
between the denotations of the
IRIs, literals and anonymous individuals
occurring in <i>g<sub>A</sub></i>
with respect to <i>I</i>,
which,
roughly speaking,
expresses the meaning of the OWL 2 axiom <i>A</i>.
This is the case for the following reasons:
First,
the literals and anonymous individuals
occurring in <i>A</i> and <i>g<sub>A</sub></i>, respectively,
are interpreted in an analog way
under the OWL 2 Direct Semantics and the OWL 2 RDF-Based Semantics.
Second,
it was assumed that the OWL 2 Direct interpretation <i>J</i>
OWL 2 Direct satisfies <i>A</i>,
and therefore a semantic relationship
with the desired properties
holds with respect to <i>J</i>.
Third,
<i>J</i> has been defined in close correspondence to <i>I</i>,
so that for the semantic relationship expressed by <i>J</i>
an analog semantic relationship holds with respect to <i>I</i>.
</p><p>This basic idea is now demonstrated in more detail
for a single example axiom <i>A</i> in F(<i>G<sub>2</sub></i>),
which can be taken as a hint on
how a complete proof
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.
</p>
<div class="anexample" id="topic-correspondence-proof-example2">
<p>Let <i>A</i> be the following OWL 2 axiom in F(<i>G<sub>2</sub></i>):
</p>
<div class="indent">
<p><i>A</i> : <span class="name">SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))</span>
</p>
</div>
<p>and let <i>g<sub>A</sub></i> be the corresponding sub graph in <i>G<sub>2</sub></i>
that is being mapped to <i>A</i> via the reverse RDF mapping,
namely
</p>
<div class="indent">
<p><i>g<sub>A</sub></i> :
</p>
<div class="indent">
<p><span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
<span class="name">_:x rdf:type owl:Class .</span><br />
<span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<p><i>First</i>,
since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>G<sub>2</sub></i> contains the typing triples
</p>
<div class="indent">
<p><span class="name">ex:c1 rdf:type owl:Class .</span><br />
<span class="name">ex:c2 rdf:type owl:Class .</span><br />
<span class="name">ex:c3 rdf:type owl:Class .</span>
</p>
</div>
<p>that correspond to class entity declarations in F(<i>G<sub>2</sub></i>) for the IRIs
"<span class="name">ex:c1</span>",
"<span class="name">ex:c2</span>", and
"<span class="name">ex:c3</span>",
respectively.
All these declaration typing triples are OWL 2 RDF-Based satisfied by <i>I</i>,
since due to balancing
the typing triples exist in <i>G<sub>1</sub></i> as well,
and since it has been postulated
that <i>I</i> OWL 2 RDF-Based satisfies all triples in <i>G<sub>1</sub></i>.
Hence,
by applying the semantics of <span class="name">rdf:type</span>
(see
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#rdfssemcond1" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1">Section 4.1 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]),
all the IRIs denote classes,
and therefore the denotations of the IRIs
are included in the appropriate part of the universe of <i>I</i>,
precisely:
</p>
<div class="indent">
<p><i>I</i>(<span class="name">ex:c1</span>) ∈ IC ,<br />
<i>I</i>(<span class="name">ex:c2</span>) ∈ IC , and<br />
<i>I</i>(<span class="name">ex:c3</span>) ∈ IC .
</p>
</div>
<p><i>Second</i>,
<i>g<sub>A</sub></i> contains the sub graph <i>g<sub>E</sub></i>,
given by
</p>
<div class="indent">
<p><i>g<sub>E</sub></i> :<br />
</p>
<div class="indent">
<p><span class="name">_:x rdf:type owl:Class .</span><br />
<span class="name">_:x owl:unionOf ( c2 c3 ) .</span>
</p>
</div>
</div>
<p>which corresponds to the union class expression <i>E</i> in <i>A</i>,
given by
</p>
<div class="indent">
<p><i>E</i> : <span class="name">ObjectUnionOf(ex:c2 ex:c3)</span>
</p>
</div>
<p>Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="index.html#def-balanced" title=""><i>balanced</i></a>,
<i>g<sub>E</sub></i> occurs as a sub graph of <i>G<sub>1</sub></i> as well.
<i>g<sub>E</sub></i> contains blank nodes
and,
since <i>I</i> satisfies <i>G<sub>1</sub></i>,
the semantic conditions for RDF graphs with blank nodes apply
(see
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">Section 1.5 of the RDF Semantics</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>]).
This provides the existence of
a mapping <i>B</i> from blank(<i>g<sub>E</sub></i>) to IR,
where blank(<i>g<sub>E</sub></i>) is
the set of all blank nodes occurring in <i>g<sub>E</sub></i>.
It follows that
the extended interpretation <i>I</i>+<i>B</i>
OWL 2 RDF-Based satisfies all the triples in <i>g<sub>E</sub></i>.
Further,
since <i>I</i> is an OWL 2 RDF-Based interpretation,
<i>I</i> meets all the OWL 2 RDF-Based semantic conditions.
Thus, the left-to-right direction
of the semantic condition for union class expressions
("<span class="name">owl:unionOf</span>",
see <a href="index.html#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a>)
applies, providing:
</p>
<div class="indent">
<p>[<i>I</i>+<i>B</i>](<span class="name">_:x</span>) ∈ IC ,<br />
ICEXT([<i>I</i>+<i>B</i>](<span class="name">_:x</span>))
=
ICEXT(<i>I</i>(<span class="name">ex:c2</span>))
∪
ICEXT(<i>I</i>(<span class="name">ex:c3</span>)) .
</p>
</div>
<p><i>Third</i>,
since the OWL 2 Direct interpretation <i>J</i> OWL 2 Direct satisfies <i>A</i>,
the following semantic relationship
holds between the denotations of the class names in <i>A</i>
according to <i>J</i>:
</p>
<div class="indent">
<p>(<span class="name">ex:c1</span>) <sup><i>C</i></sup>
⊆
(<span class="name">ex:c2</span>) <sup><i>C</i></sup>
∪
(<span class="name">ex:c3</span>) <sup><i>C</i></sup> .
</p>
</div>
<p>By applying the definition of
the OWL 2 Direct interpretation <i>J</i>,
one can conclude that the following semantic relationship
holds between the extensions of the classes above
according to <i>I</i>:
</p>
<div class="indent">
<p>ICEXT(<i>I</i>(<span class="name">ex:c1</span>))
⊆
ICEXT(<i>I</i>(<span class="name">ex:c2</span>))
∪
ICEXT(<i>I</i>(<span class="name">ex:c3</span>)) .
</p>
</div>
<p><i>Finally</i>,
combining all intermediate results gives
</p>
<div class="indent">
<p><i>I</i>(<span class="name">ex:c1</span>) ∈ IC ,<br />
[<i>I</i>+<i>B</i>](<span class="name">_:x</span>) ∈ IC ,<br />
ICEXT(<i>I</i>(<span class="name">ex:c1</span>))
⊆
ICEXT([<i>I</i>+<i>B</i>](<span class="name">_:x</span>)) .
</p>
</div>
<p>Therefore, all the premises are met
to apply the right-to-left direction of the semantic condition for subclass axioms
("<span class="name">rdfs:subClassOf</span>",
see <a href="index.html#Semantic_Conditions_for_the_RDFS_Vocabulary" title="">Section 5.8</a>),
which results in
</p>
<div class="indent">
<p>( <i>I</i>(<span class="name">ex:cl</span>) , [<i>I</i>+<i>B</i>](<span class="name">_:x</span>) )
∈
IEXT(<i>I</i>(<span class="name">rdfs:subClassOf</span>)) .
</p>
</div>
<p>So,
the remaining triple
</p>
<div class="indent">
<p><span class="name">ex:c1 rdfs:subClassOf _:x .</span>
</p>
</div>
<p>in <i>g<sub>A</sub></i>
is OWL 2 RDF-Based satisfied by <i>I</i>+<i>B</i>,
where "<span class="name">_:x</span>" is
the root blank node of the union class expression <i>g<sub>E</sub></i>.
Hence,
w.r.t. existential blank node semantics,
<i>I</i> OWL 2 RDF-Based satisfies all the triples in <i>g<sub>A</sub></i>.
</p>
</div>
<p>To conclude,
for any OWL 2 RDF-Based interpretation <i>I</i>
that OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>I</i> also OWL 2 RDF-Based satisfies <i>G<sub>2</sub></i>.
Hence,
<i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>,
and therefore relationship 4 of the correspondence theorem holds.
<i><b>Q.E.D.</b></i>
</p>
<a name="Appendix:_Comprehension_Conditions_.28Informative.29"></a><h2> <span class="mw-headline">8 Appendix: Comprehension Conditions (Informative) </span></h2>
<p>The <a href="index.html#thm-correspondence" title="">correspondence theorem</a>
in <a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>
shows
that it is possible for the OWL 2 RDF-Based Semantics
to reflect all the entailments of the
<a href="../REC-owl2-direct-semantics-20091027/index.html" title="Direct Semantics">OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>],
provided that one allows for certain "harmless" syntactic transformations
on the RDF graphs being considered.
This makes numerous potentially desirable and useful entailments available
that would otherwise be outside the scope of the OWL 2 RDF-Based Semantics,
for the technical reasons discussed in
<a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>.
It seems natural to ask for similar entailments
even
when an entailment query
does not consist of OWL 2 DL ontologies in RDF graph form.
However,
the correspondence theorem does not apply to such cases,
and thus the OWL 2 Direct Semantics cannot be taken
as a reference frame
for "desirable" and "useful" entailments,
or for when a graph transformation
can be considered "harmless" or not.
</p><p>As discussed in
<a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>,
a core obstacle for the correspondence theorem to hold
was the RDF encoding of OWL 2 expressions,
such as union class expressions,
when they appear on the right hand side of an entailment query.
Under the OWL 2 RDF-Based Semantics
it is not generally ensured that an individual exists,
which represents the denotation of such an expression.
The <i>"comprehension conditions"</i> defined in this section
are additional semantic conditions
that provide the necessary individuals
for <i>every</i> sequence, class and property expression.
By this,
the combination
of the normative semantic conditions of the OWL 2 RDF-Based Semantics
(<a href="index.html#Semantic_Conditions" title="">Section 5</a>)
and the comprehension conditions
can be regarded to "simulate" the semantic expressivity
of the OWL 2 Direct Semantics
on entailment queries consisting of <i>arbitrary</i> RDF graphs.
</p><p>The combined semantics is,
however,
not primarily intended for use in actual implementations.
The comprehension conditions add significantly
to the complexity and expressivity
of the basic semantics
and,
in fact,
have proven to
<a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/119" title="http://www.w3.org/2007/OWL/tracker/issues/119">lead to formal inconsistency</a>.
But
the combined semantics
can still be seen as a generalized reference frame
for "desirable" and "useful" entailments,
and this can be used,
for example,
to evaluate methods that syntactically transform <i>unrestricted</i> entailment queries
in order to receive additional entailments under the OWL 2 RDF-Based Semantics.
Such a concrete method is, however,
outside the scope of this specification.
</p><p><i>Note:</i>
The <a href="index.html#topic-semcond-conventions" title="">conventions</a>
in the introduction of
<a href="index.html#Semantic_Conditions" title="">Section 5 ("Semantic Conditions")</a>
apply to the current section as well.
</p>
<a name="Comprehension_Conditions_for_Sequences"></a><h3> <span class="mw-headline">8.1 Comprehension Conditions for Sequences </span></h3>
<p><a href="index.html#table-comprehension-lists" title="">Table 8.1</a>
lists the comprehension conditions for sequences,
i.e. RDF lists.
These comprehension conditions provide the existence
of sequences
built from any finite combination of individuals
contained in the universe.
</p>
<div class="left" id="table-comprehension-lists">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.1: Comprehension Conditions for Sequences</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-lists"></span>if
</th><th style="text-align: center"> then exists z<sub>1</sub> , … , z<sub>n</sub> ∈ IR
</th></tr>
<tr>
<td> <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR
</td><td> ( <i>z<sub>1</sub></i> , <i>a<sub>1</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:first</span>)) , ( <i>z<sub>1</sub></i> , <i>z<sub>2</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:rest</span>)) , … ,<br />( <i>z<sub>n</sub></i> , <i>a<sub>n</sub></i> ) ∈ IEXT(<i>I</i>(<span class="name">rdf:first</span>)) , ( <i>z<sub>n</sub></i> , <i>I</i>(<span class="name">rdf:nil</span>) ) ∈ IEXT(<i>I</i>(<span class="name">rdf:rest</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Boolean_Connectives"></a><h3> <span class="mw-headline">8.2 Comprehension Conditions for Boolean Connectives </span></h3>
<p><a href="index.html#table-comprehension-booleans" title="">Table 8.2</a>
lists the comprehension conditions for
Boolean connectives
(see <a href="index.html#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a>
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of complements for any class and datatype,
and of intersections and unions
built from any finite set of classes
contained in the universe.
</p>
<div class="left" id="table-comprehension-booleans">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.2: Comprehension Conditions for Boolean Connectives</span>
</caption>
<tr>
<th style="text-align: center"> if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <span id="item-comprehension-booleans-intersectionof"></span><i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC
</td><td> ( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:intersectionOf</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-booleans-unionof"></span><i>s</i> sequence of <i>c<sub>1</sub></i> , … , <i>c<sub>n</sub></i> ∈ IC
</td><td> ( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:unionOf</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-booleans-complementof"></span><i>c</i> ∈ IC
</td><td> ( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:complementOf</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-booleans-datatypecomplement"></span><i>d</i> ∈ IDC
</td><td> ( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:datatypeComplementOf</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Enumerations"></a><h3> <span class="mw-headline">8.3 Comprehension Conditions for Enumerations </span></h3>
<p><a href="index.html#table-comprehension-enums" title="">Table 8.3</a>
lists the comprehension conditions for
enumerations
(see <a href="index.html#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of enumeration classes
built from any finite set of individuals
contained in the universe.
</p>
<div class="left" id="table-comprehension-enums">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.3: Comprehension Conditions for Enumerations</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-enums"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , … , <i>a<sub>n</sub></i> ∈ IR
</td><td> ( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:oneOf</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Property_Restrictions"></a><h3> <span class="mw-headline">8.4 Comprehension Conditions for Property Restrictions </span></h3>
<p><a href="index.html#table-comprehension-restrictions" title="">Table 8.4</a>
lists the comprehension conditions for
property restrictions
(see <a href="index.html#Semantic_Conditions_for_Property_Restrictions" title="">Section 5.6</a>
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of cardinality restrictions
on any property and for any nonnegative integer,
as well as value restrictions
on any property and on any class
contained in the universe.
</p>
<div id="topic-comprehension-restrictions-self"></div>
<p>Note that the comprehension conditions for self restrictions
constrains the right hand side of
the produced <span class="name">owl:hasSelf</span> assertions
to be the Boolean value
<span class="name">"true"^^xsd:boolean</span>.
This is in accordance with
<a href="../REC-owl2-mapping-to-rdf-20091027/index.html#Parsing_of_Expressions" title="Mapping to RDF Graphs">Table 13 in Section 3.2.4 of the OWL 2 RDF Mapping</a>
[<cite><a href="index.html#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>].
</p>
<div id="topic-comprehension-restrictions-narydatatype"></div>
<p>Implementations are <i>not</i> required
to support the comprehension conditions for
<span class="name">owl:onProperties</span>,
but
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
support them
in order to realize
<i>n-ary dataranges</i> with arity ≥ 2
(see
Sections
<a href="../REC-owl2-syntax-20091027/index.html#Data_Ranges" title="Syntax">7</a>
and
<a href="../REC-owl2-syntax-20091027/index.html#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for further information).
</p>
<div class="left" id="table-comprehension-restrictions">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.4: Comprehension Conditions for Property Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <span id="item-comprehension-restrictions-somevaluesfrom"></span><i>c</i> ∈ IC ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-somevaluesfrom-nary"></span><i>c</i> ∈ IC ,<br /><i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP , <i>n</i> ≥ 1
</td><td> ( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-allvaluesfrom"></span><i>c</i> ∈ IC ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-allvaluesfrom-nary"></span><i>c</i> ∈ IC ,<br /><i>s</i> sequence of <i>p<sub>1</sub></i> , … , <i>p<sub>n</sub></i> ∈ IP , <i>n</i> ≥ 1
</td><td> ( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-hasvalue"></span><i>a</i> ∈ IR ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>a</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:hasValue</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-hasself"></span><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>I</i>(<span class="name">"true"^^xsd:boolean</span>) ) ∈ IEXT(<i>I</i>(<span class="name">owl:hasSelf</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-mincardinality"></span><i>n</i> ∈ INNI ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:minCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-maxcardinality"></span><i>n</i> ∈ INNI ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:maxCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-cardinality"></span><i>n</i> ∈ INNI ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:cardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-minqualifiedcardinality"></span><i>n</i> ∈ INNI ,<br /><i>c</i> ∈ IC ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onClass</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-minqualifiedcardinality-data"></span><i>n</i> ∈ INNI ,<br /><i>d</i> ∈ IDC ,<br /><i>p</i> ∈ IODP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-maxqualifiedcardinality"></span><i>n</i> ∈ INNI ,<br /><i>c</i> ∈ IC ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onClass</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-maxqualifiedcardinality-data"></span><i>n</i> ∈ INNI ,<br /><i>d</i> ∈ IDC ,<br /><i>p</i> ∈ IODP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-qualifiedcardinality"></span><i>n</i> ∈ INNI ,<br /><i>c</i> ∈ IC ,<br /><i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>c</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onClass</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-qualifiedcardinality-data"></span><i>n</i> ∈ INNI ,<br /><i>d</i> ∈ IDC ,<br /><i>p</i> ∈ IODP
</td><td> ( <i>z</i> , <i>n</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>)) ,<br />( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Datatype_Restrictions"></a><h3> <span class="mw-headline">8.5 Comprehension Conditions for Datatype Restrictions </span></h3>
<p><a href="index.html#table-comprehension-facets" title="">Table 8.5</a>
lists the comprehension conditions for
datatype restrictions
(see <a href="index.html#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of datatypes
built from restricting any datatype
contained in the universe
by any finite set of facet-value pairs
contained in the facet space
(see <a href="index.html#Datatype_Maps" title="">Section 4.1</a>)
of the original datatype.
</p><p>The set IFS is defined in
<a href="index.html#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>.
</p>
<div class="left" id="table-comprehension-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.5: Comprehension Conditions for Datatype Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-facets"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR , <i>s</i> sequence of <i>z<sub>1</sub></i> , … , <i>z<sub>n</sub></i> ∈ IR
</th></tr>
<tr>
<td> <i>d</i> ∈ IDC ,<br /><i>f<sub>1</sub></i> , … , <i>f<sub>n</sub></i> ∈ IODP ,<br /><i>v<sub>1</sub></i> , … , <i>v<sub>n</sub></i> ∈ LV ,<br />( <i>f<sub>1</sub></i> , <i>v<sub>1</sub></i> ) , … , ( <i>f<sub>n</sub></i> , <i>v<sub>n</sub></i> ) ∈ IFS(<i>d</i>)
</td><td> ( <i>z</i> , <i>d</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:onDatatype</span>)) ,<br />( <i>z</i> , <i>s</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:withRestrictions</span>)) ,<br />( <i>z<sub>1</sub></i> , <i>v<sub>1</sub></i> ) ∈ IEXT(<i>f<sub>1</sub></i>) , … , ( <i>z<sub>n</sub></i> , <i>v<sub>n</sub></i> ) ∈ IEXT(<i>f<sub>n</sub></i>)
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Inverse_Properties"></a><h3> <span class="mw-headline">8.6 Comprehension Conditions for Inverse Properties </span></h3>
<p><a href="index.html#table-comprehension-inverses" title="">Table 8.6</a>
lists the comprehension conditions for
inverse property expressions.
These comprehension conditions provide the existence
of an inverse property for any property
contained in the universe.
</p><p>Inverse property expressions can be used
to build axioms with anonymous inverse properties,
such as in the graph
</p>
<div class="indent">
<p><span class="name">_:x owl:inverseOf ex:p .</span><br />
<span class="name">_:x rdfs:subPropertyOf owl:topObjectProperty .</span>
</p>
</div>
<p>Note that,
to some extent,
the OWL 2 RDF-Based Semantics already covers the use of inverse property expressions
by means of the semantic conditions of inverse property axioms
(see <a href="index.html#Semantic_Conditions_for_Inverse_Properties" title="">Section 5.12</a>),
since these semantic conditions also apply to an existential variable
on the left hand side of an inverse property axiom.
Nevertheless,
not all relevant cases are covered by this semantic condition.
For example,
one might expect the above example graph
to be generally true.
However,
the normative semantic conditions
do not permit this conclusion,
since it is not ensured that
for every property <i>p</i>
there is an individual in the universe
with a property extension being inverse to that of <i>p</i>.
</p>
<div class="left" id="table-comprehension-inverses">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.6: Comprehension Conditions for Inverse Properties</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-inverses"></span>if
</th><th style="text-align: center"> then exists <i>z</i> ∈ IR
</th></tr>
<tr>
<td> <i>p</i> ∈ IP
</td><td> ( <i>z</i> , <i>p</i> ) ∈ IEXT(<i>I</i>(<span class="name">owl:inverseOf</span>))
</td></tr>
</table>
</div>
<a name="Appendix:_Changes_from_OWL_1_.28Informative.29"></a><h2> <span class="mw-headline">9 Appendix: Changes from OWL 1 (Informative) </span></h2>
<p>This section lists relevant differences
between the OWL 2 RDF-Based Semantics and the original specification of the
<a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html"><i>OWL 1 RDF-Compatible Semantics</i></a>
[<cite><a href="index.html#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>].
Significant effort has been spent
in keeping the design of the OWL 2 RDF-Based Semantics
as close as possible
to that of the OWL 1 RDF-Compatible Semantics.
While this aim was achieved to a large degree,
the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects.
In most cases this is because of serious technical problems
that would have arisen
from a conservative
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DefSemanticExtension" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension">semantic extension</a>.
Not listed are
the new language constructs and the new datatypes of OWL 2.
</p>
<div id="topic-languagechanges-markers"></div>
<p>The following markers are used:
</p>
<ul><li> <b>[DEV]</b>: a deviation from OWL 1 that breaks backward compatibility
</li><li> <b>[EXT]</b>: a backward compatible extension to OWL 1
</li><li> <b>[NOM]</b>: a change of the nomenclature originally used in OWL 1
</li><li> <b>[DPR]</b>: a feature of OWL 1 that has been deprecated as of OWL 2
</li></ul>
<div id="topic-languagechanges-graphsyntax"></div>
<p><b>Generalized Graph Syntax [EXT].</b>
The OWL 2 RDF-Based Semantics
allows RDF graphs to contain
<a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt"><i>IRIs</i></a>
[<cite><a href="index.html#ref-rfc-3987" title="">RFC 3987</a></cite>]
(see <a href="index.html#Syntax" title="">Section 2.1</a>),
whereas the OWL 1 RDF-Compatible Semantics was restricted to RDF graphs with
<a class="external text" href="http://www.ietf.org/rfc/rfc2396.txt" title="http://www.ietf.org/rfc/rfc2396.txt"><i>URIs</i></a>
[<cite><a href="index.html#ref-rfc-2396" title="">RFC 2396</a></cite>].
This change is in accordance with the rest of the OWL 2 specification
(see
<a href="../REC-owl2-syntax-20091027/index.html#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
In addition,
the OWL 2 RDF-Based Semantics
is now explicitly allowed to
be applied to RDF graphs containing
<i>"generalized" RDF triples</i>,
i.e. triples that can consist of
IRIs, literals or blank nodes
in all three positions
(<a href="index.html#Syntax" title="">Section 2.1</a>),
although implementations are not required to support this.
In contrast,
the OWL 1 RDF-Compatible Semantics was restricted to RDF graphs
conforming to the
<a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html#section-Graph-syntax" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax">RDF Concepts specification</a>
[<cite><a href="index.html#ref-rdf-concepts" title="">RDF Concepts</a></cite>].
These limitations of the OWL 1 RDF-Compatible Semantics
were actually inherited from the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#graphsyntax" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#graphsyntax">RDF Semantics specification</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
The relaxations are intended to warrant interoperability
with existing and future technologies and tools.
Both changes are compatible with OWL 1,
since all RDF graphs that were legal under the OWL 1 RDF-Compatible Semantics
are still legal under the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-languagechanges-facets"></div>
<p><b>Facets for Datatypes [EXT].</b>
The basic definitions of a <i>datatype</i> and a <i>D-interpretation</i>,
as defined by the RDF Semantics specification
and as applied by the OWL 1 RDF-Compatible Semantics,
have been extended
to take into account <i>constraining facets</i>
(see <a href="index.html#Interpretations" title="">Section 4</a>),
in order to allow for <i>datatype restrictions</i>
as specified in <a href="index.html#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>.
This change is compatible with OWL 1,
since <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">Section 5.1</a>
of the RDF Semantics specification
explicitly allows for extending the minimal datatype definition provided there.
</p>
<div id="topic-languagechanges-correspondence"></div>
<p><b>Correspondence Theorem and Comprehension Conditions [DEV].</b>
The semantic conditions of the OWL 1 RDF-Compatible Semantics included
a set of so called
<a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html#comprehension_principles" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#comprehension_principles"><i>"comprehension conditions"</i></a>,
which allowed to prove the original
<a class="external text" href="../../2004/REC-owl-semantics-20040210/rdfs.html#theorem-2" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#theorem-2"><i>"correspondence theorem"</i></a>
stating that every entailment of OWL 1 DL was also an entailment of OWL 1 Full.
The document at hand adds comprehension conditions
for the new language constructs of OWL 2
(see <a href="index.html#Appendix:_Comprehension_Conditions_.28Informative.29" title="">Section 8</a>).
However,
the comprehension conditions
are <i>not</i> a normative aspect of the OWL 2 RDF-Based Semantics
anymore.
It has turned out
that combining the comprehension conditions
with the normative set of semantic conditions in
<a href="index.html#Semantic_Conditions" title="">Section 5</a>
would lead to formal inconsistency of the resulting semantics
(<a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/119" title="http://www.w3.org/2007/OWL/tracker/issues/119">Issue 119</a>).
In addition,
it became clear that
a correspondence theorem along the lines of the original theorem
would not work for the relationship between the
OWL 2 RDF-Based Semantics and the
<a href="../REC-owl2-direct-semantics-20091027/index.html" title="Direct Semantics">OWL 2 Direct Semantics</a>
[<cite><a href="index.html#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>],
since it is not possible to "balance" the differences between
the two semantics
solely by means of additional comprehension conditions
(see <a href="index.html#Example_on_Semantic_Differences" title="">Section 7.1</a>).
Consequently,
the correspondence theorem
of the OWL 2 RDF-Based Semantics
(<a href="index.html#Correspondence_Theorem" title="">Section 7.2</a>)
follows an alternative approach
that replaces the use of the comprehension conditions
and can be seen as a technical refinement
of an idea
originally discussed by the WebOnt Working Group
(<a class="external text" href="http://lists.w3.org/Archives/Public/www-webont-wg/2002Mar/0179.html" title="http://lists.w3.org/Archives/Public/www-webont-wg/2002Mar/0179.html">email</a>).
This change is an <i>incompatible deviation</i> from OWL 1,
since certain aspects of the originally normative definition of the semantics
have been removed.
</p>
<div id="topic-languagechanges-arglists"></div>
<p><b>Flawed Semantics of Language Constructs with Argument Lists [DEV].</b>
In the OWL 1 RDF-Compatible Semantics,
the semantic conditions for
unions, intersections and enumerations of classes
were defined in a flawed form,
which lead to formal inconsistency of the semantics
(<a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/120" title="http://www.w3.org/2007/OWL/tracker/issues/120">Issue 120</a>;
see also this
<a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=BrokenOwl1FullFeaturesWithArgumentLists&oldid=26002" title="http://www.w3.org/2007/OWL/wiki/index.php?title=BrokenOwl1FullFeaturesWithArgumentLists&oldid=26002">unofficial problem description</a>).
The affected semantic conditions have been revised;
see
<a href="index.html#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a>
and
<a href="index.html#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>.
This change is an <i>incompatible deviation</i> from OWL 1,
since the semantics has formally been weakened
in order to eliminate a source of inconsistency.
</p>
<div id="topic-languagechanges-ndis"></div>
<p><b>Incomplete Semantics of <span class="name">owl:AllDifferent</span> [EXT].</b>
The OWL 1 RDF-Compatible Semantics missed a certain semantic condition
for axioms based on the vocabulary term "<span class="name">owl:AllDifferent</span>"
(see also this
<a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=MissingOwl1FullAllDifferentSemanticCondition&oldid=26003" title="http://www.w3.org/2007/OWL/wiki/index.php?title=MissingOwl1FullAllDifferentSemanticCondition&oldid=26003">unofficial problem description</a>).
The missing semantic condition
has been added to the OWL 2 RDF-Based Semantics
(see <a href="index.html#Semantic_Conditions_for_N-ary_Disjointness" title="">Section 5.10</a>).
This change is compatible with OWL 1,
since the semantics has been conservatively extended.
</p>
<div id="topic-languagechanges-datarange"></div>
<p><b>Aligned Semantics of <span class="name">owl:DataRange</span> and <span class="name">rdfs:Datatype</span> [EXT].</b>
The class
<span class="name">owl:DataRange</span>
has been made an <i>equivalent class</i>
to <span class="name">rdfs:Datatype</span>
(see <a href="index.html#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>).
The main purpose for this change was
to allow for the deprecation of the term
<span class="name">owl:DataRange</span>
in favor of <span class="name">rdfs:Datatype</span>.
This change is compatible with OWL 1
according to an analysis
of the relationship between the two classes
in the OWL 1 RDF-Compatible Semantics
(<a class="external text" href="http://lists.w3.org/Archives/Public/public-owl-wg/2008Jan/0229.html" title="http://lists.w3.org/Archives/Public/public-owl-wg/2008Jan/0229.html">email</a>).
</p>
<div id="topic-languagechanges-ontoprop"></div>
<p><b>Ontology Properties as Annotation Properties [EXT].</b>
Several properties
that have been ontology properties in OWL 1,
such as <span class="name">owl:priorVersion</span>,
have now been specified
as both ontology properties and annotation properties,
in order to be in line
with the rest of the OWL 2 specification
(see
<a href="../REC-owl2-syntax-20091027/index.html#Annotation_Properties" title="Syntax">Section 5.5 of the OWL 2 Structural Specification</a>
[<cite><a href="index.html#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
This change is compatible with OWL 1,
since the semantics has been conservatively extended:
all the ontology properties of OWL 1 are still ontology properties in OWL 2.
</p>
<div id="topic-languagechanges-dataenums"></div>
<p><b>Nonempty Data Value Enumerations [DEV].</b>
The semantic condition for enumerations of data values
in <a href="index.html#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>
is now restricted to <i>nonempty</i> sets of data values.
This prevents the class <span class="name">owl:Nothing</span>
from unintentionally becoming an instance
of the class <span class="name">rdfs:Datatype</span>,
as analyzed in
(<a class="external text" href="http://lists.w3.org/Archives/Public/public-webont-comments/2008May/0001.html" title="http://lists.w3.org/Archives/Public/public-webont-comments/2008May/0001.html">email</a>).
This restriction of the semantics
is an <i>incompatible deviation</i> from OWL 1.
Note, however,
that it is still possible
to define a datatype as an empty enumeration of data values,
as explained in <a href="index.html#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>.
</p>
<div id="topic-languagechanges-nameing"></div>
<p><b>Terminological Clarifications [NOM].</b>
This document uses the term <i>"OWL 2 RDF-Based Semantics"</i>
to refer to the specified semantics only.
According to <a href="index.html#Syntax" title="">Section 2.1</a>,
the term <i>"OWL 2 Full"</i>
refers to the language
that is determined
by the set of all RDF graphs
(also called <i>"OWL 2 Full ontologies"</i>)
being interpreted using the OWL 2 RDF-Based Semantics.
OWL 1 has not been particularly clear on this distinction.
Where the OWL 1 RDF-Compatible Semantics specification talked about
<i>"OWL Full interpretations"</i>,
<i>"OWL Full satisfaction"</i>,
<i>"OWL Full consistency"</i>
and
<i>"OWL Full entailment"</i>,
the OWL 2 RDF-Based Semantics Specification talks
in <a href="index.html#Interpretations" title="">Section 4</a>
about
<i>"OWL 2 RDF-Based interpretations"</i>,
<i>"OWL 2 RDF-Based satisfaction"</i>,
<i>"OWL 2 RDF-Based consistency"</i>
and
<i>"OWL 2 RDF-Based entailment"</i>,
respectively,
since these terms are primarily meant to be related to
the semantics
rather than the whole language.
</p>
<div id="topic-languagechanges-abbreviations"></div>
<p><b>Modified Abbreviations [NOM].</b>
The names
"R<sub>I</sub>", "P<sub>I</sub>", "C<sub>I</sub>",
"EXT<sub>I</sub>", "CEXT<sub>I</sub>",
"S<sub>I</sub>", "L<sub>I</sub>" and "LV<sub>I</sub>",
which have been used in the
OWL 1 RDF-Compatible Semantics specification,
have been replaced by the corresponding names
defined in the
<a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp">RDF Semantics document</a>
[<cite><a href="index.html#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
namely "IR", "IP", "IC", "IEXT", "ICEXT", "IS", "IL" and "LV", respectively.
Furthermore,
all uses of the IRI mapping "IS"
have been replaced by the more general interpretation mapping "<i>I</i>",
following the conventions in the RDF Semantics document.
These changes are intended to support
the use of the OWL 2 RDF-Based Semantics document
as an incremental extension
of the RDF Semantics document.
Names for the <a href="index.html#Parts_of_the_Universe" title="">"parts of the universe"</a>
that were exclusively used in the OWL 1 RDF-Compatible Semantics document,
such as "IX" or "IODP",
have not been changed.
Other abbreviations,
such as "IAD" for the class extension of <span class="name">owl:AllDifferent</span>,
have in general not been reused in the document at hand,
but the explicit nonabbreviated form,
such as
"IEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>))",
is used instead.
</p><p><b>Modified Tuple Notation Style [NOM].</b>
Tuples are written in the form
"( … )"
instead of "< … >",
as in the other OWL 2 documents.
</p>
<div id="topic-languagechanges-deprecated"></div>
<p><b>Deprecated Vocabulary Terms [DPR].</b>
The following vocabulary terms have been deprecated as of OWL 2
by the Working Group,
and <em class="RFC2119" title="SHOULD NOT in RFC 2119 context">SHOULD NOT</em> be used
in new ontologies anymore:
</p>
<ul><li> <span class="name">owl:DataRange</span> (per <a class="external text" href="http://www.w3.org/2007/OWL/wiki/Teleconference.2008.01.23/Minutes" title="http://www.w3.org/2007/OWL/wiki/Teleconference.2008.01.23/Minutes">resolution</a> of <a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/29" title="http://www.w3.org/2007/OWL/tracker/issues/29">Issue 29</a>)
</li></ul>
<div id="changelog">
<a name="Appendix:_Change_Log_.28Informative.29"></a><h2> <span class="mw-headline">10 Appendix: Change Log (Informative) </span></h2>
<a name="Changes_Since_Proposed_Recommendation"></a><h3> <span class="mw-headline">10.1 Changes Since Proposed Recommendation </span></h3>
<p>This section summarizes the changes to this document since the <a class="external text" href="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/" title="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/">Proposed Recommendation of 22 September, 2009</a>.
</p>
<ul><li> [editorial] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=25950&oldid=25949" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=25950&oldid=25949">Correction of grammar</a> (punctuation, word order, etc.), mainly in the <a href="index.html#Introduction_.28Informative.29" title="">Introduction section</a>.
</li><li> [editorial] Updated and corrected several hyperlinks.
</li></ul>
<a name="Changes_Since_Candidate_Recommendation"></a><h3> <span class="mw-headline">10.2 Changes Since Candidate Recommendation </span></h3>
<p>This section summarizes the changes to this document since the <a class="external text" href="http://www.w3.org/TR/2009/CR-owl2-rdf-based-semantics-20090611/" title="http://www.w3.org/TR/2009/CR-owl2-rdf-based-semantics-20090611/">Candidate Recommendation of 11 June, 2009</a>.
</p>
<ul><li> [resolution] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24829&oldid=24826" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24829&oldid=24826">Re-definition of several ontology properties</a> to be both ontology properties and annotation properties, in order to align the RDF-Based Semantics with the rest of the <a href="../REC-owl2-syntax-20091027/index.html#Annotation_Properties" title="Syntax">OWL 2 specification</a>, and in particular to avoid an equivocal definition of the <a href="../REC-owl2-profiles-20091027/index.html#Reasoning_in_OWL_2_RL_and_RDF_Graphs_using_Rules" title="Profiles">OWL 2 RL/RDF rules</a> (per <a class="external text" href="http://www.w3.org/2007/OWL/meeting/2009-07-15#resolution_2" title="http://www.w3.org/2007/OWL/meeting/2009-07-15#resolution_2">WG resolution</a>).
</li><li> [correction] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24865&oldid=24864" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24865&oldid=24864">Correction of the type of facets</a>: Facets are intended to be data properties and have been used as such <a href="index.html#table-semcond-facets" title="">elsewhere in the document</a>, but they were wrongly specified as unrestricted properties so far.
</li><li> [correction] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24758&oldid=24757" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24758&oldid=24757">Correction of a mismatch</a> between the <a href="index.html#topic-int-rdfinterpretation" title="">definition of D-interpretations</a> in the document at hand and the RDF Semantics specification: according to the <a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html#interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp">definition of simple interpretations</a>, LV contains all plain literals in the vocabulary <i>V</i>. The missing reference to "<i>V</i>" has been added.
</li><li> [nonnormative] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24755&oldid=24670" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24755&oldid=24670">Correction of an error</a> in the formulation of the <a href="index.html#thm-correspondence" title="">correspondence theorem</a>.
</li><li> [nonnormative] The <a href="index.html#Appendix:_Axiomatic_Triples_.28Informative.29" title="">section on Axiomatic Triples</a> has been extended by an explicit <a href="index.html#A_Set_of_Axiomatic_Triples" title="">set of axiomatic triples</a>, based on the discussion in the rest of the section.
</li><li> [nonnormative] The <a href="index.html#Appendix:_Axiomatic_Triples_.28Informative.29" title="">section on Axiomatic Triples</a> now explicitly mentions axiomatic triples for <a href="index.html#topic-axiomatic-classes-datatypes" title="">datatypes</a> and <a href="index.html#topic-axiomatic-properties-facets" title="">facets</a> corresponding to the semantic conditions for <a href="index.html#topic-semcond-classes-datatypes" title="">datatypes</a> and <a href="index.html#topic-semcond-properties-facets" title="">facets</a>, respectively.
</li><li> [nonnormative] Refinement of the <a href="index.html#Proof_for_the_Correspondence_Theorem" title="">proof for the correspondence theorem</a> and correction of several errors. Motivated by these changes, the <a href="index.html#Example_on_Semantic_Differences" title="">example in Section 7.1</a> has been slightly revised as well.
</li><li> [editorial] Added a description and ALT-attribute text to <a href="index.html#fig-partshierarchy" title="">Figure 1 on the parts hierarchy</a>.
</li><li> [editorial] Distinction between <a href="index.html#References" title="">normative and nonnormative references</a>, as in other OWL 2 documents.
</li><li> [editorial] Added some clarification to the <a href="index.html#Introduction_.28Informative.29" title="">introduction section</a>.
</li><li> [editorial] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24766&oldid=24765" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&diff=24766&oldid=24765">Removed a redundant conclusion</a> from the table presenting the <a href="index.html#table-semcond-facets" title="">semantic conditions for datatype restrictions</a>, since this conclusion already follows from the <a href="index.html#item-semcond-properties-ondatatype" title="">semantic conditions for the vocabulary properties</a>, and having the conclusion repeated would not match the <a href="index.html#topic-semcond-conditionform-deducedrhs" title="">general approach</a> that is applied when presenting "if-then" semantic conditions in this document.
</li><li> [editorial] Reworded the description of the <a href="index.html#topic-languagechanges-markers" title="">markers in the section on changes from OWL 1</a>, and added a marker "DPR" for the <a href="index.html#topic-languagechanges-deprecated" title="">deprecated features</a>.
</li><li> [editorial] Changed the presentation style of references and citations to a form used in all OWL 2 documents.
</li><li> [editorial] Changed the presentation style for tuples from "⟨ … ⟩" to "( … )", to follow the conventions used in the other OWL 2 documents.
</li><li> [editorial] Numerous minor corrections and stylistic improvements.
</li></ul>
<a name="Changes_Since_Last_Call"></a><h3> <span class="mw-headline">10.3 Changes Since Last Call </span></h3>
<p>This section summarizes the changes to this document since the <a class="external text" href="http://www.w3.org/TR/2009/WD-owl2-rdf-based-semantics-20090421/" title="http://www.w3.org/TR/2009/WD-owl2-rdf-based-semantics-20090421/">Last Call Working Draft of 21 April, 2009</a>.
</p>
<ul><li> [resolution] Renamed the annotation vocabulary terms "<span class="name">owl:subject</span>", "<span class="name">owl:predicate</span>" and "<span class="name">owl:object</span>" to "<span class="name">owl:annotatedSource</span>", "<span class="name">owl:annotatedProperty</span>" and "<span class="name">owl:annotatedTarget</span>", respectively (per <a class="external text" href="http://www.w3.org/2007/OWL/meeting/2009-05-20#resolution_2" title="http://www.w3.org/2007/OWL/meeting/2009-05-20#resolution_2">WG resolution</a>).
</li><li> [resolution] Replaced the datatype "<span class="name">rdf:text</span>" by "<span class="name">rdf:PlainLiteral</span>" (per <a class="external text" href="http://www.w3.org/2007/OWL/meeting/2009-05-27#resolution_2" title="http://www.w3.org/2007/OWL/meeting/2009-05-27#resolution_2">WG resolution</a>).
</li><li> [resolution] Replaced the facet "<span class="name">rdf:langPattern</span>" by "<span class="name">rdf:langRange</span>", following the same replacement in the original <a class="external text" href="http://www.w3.org/TR/2009/WD-rdf-text-20090421/" title="http://www.w3.org/TR/2009/WD-rdf-text-20090421/">rdf:PlainLiteral specification</a>.
</li><li> [correction] Changed the range of the property "<span class="name">owl:annotatedProperty</span>" from IP to IR in order to avoid undesired semantic side effects from annotations. This was an oversight when the original semantic conditions for annotations of axioms and annotations were removed from the document.
</li><li> [nonnormative] The semantic conditions and comprehension conditions for the n-ary property restrictions have been changed to only cover property sequences of length greater than 0, since the meaning of an expression with an empty property set is not clear.
</li><li> [editorial] Explained the optional status of the semantic conditions concerned with the IRI "<span class="name">owl:onProperties</span>", in accordance with the rest of the OWL 2 specification.
</li><li> [editorial] Shortened and clarified some section titles, moved the section on <a href="index.html#Semantic_Conditions_for_Sub_Property_Chains" title="">semantic conditions for sub property chains</a> within <a href="index.html#Semantic_Conditions" title="">Section 5</a>, and aligned the entry order of all tables in <a href="index.html#Appendix:_Comprehension_Conditions_.28Informative.29" title="">Section 8</a> with those in <a href="index.html#Semantic_Conditions" title="">Section 5</a>.
</li><li> [editorial] Several clarifications, minor corrections and cosmetic changes.
</li></ul>
</div>
<a name="Acknowledgments"></a><h2> <span class="mw-headline">11 Acknowledgments </span></h2>
<p>The starting point for the development of OWL 2 was the <a class="external text" href="http://www.w3.org/Submission/2006/10/" title="http://www.w3.org/Submission/2006/10/">OWL1.1 member submission</a>, itself a result of user and developer feedback, and in particular of information gathered during the <a class="external text" href="http://www.webont.org/owled/" title="http://www.webont.org/owled/">OWL Experiences and Directions (OWLED) Workshop series</a>. The working group also considered <a class="external text" href="http://www.w3.org/2001/sw/WebOnt/webont-issues.html" title="http://www.w3.org/2001/sw/WebOnt/webont-issues.html">postponed issues</a> from the <a class="external text" href="http://www.w3.org/2004/OWL/" title="http://www.w3.org/2004/OWL/">WebOnt Working Group</a>.
</p><p>This document has been produced by the OWL Working Group (see below), and its contents reflect extensive discussions within the Working Group as a whole.
The editors extend special thanks to
Jie Bao (RPI),
Ivan Herman (W3C/ERCIM),
Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent) and
Zhe Wu (Oracle Corporation)
for their thorough reviews.
</p><p>The regular attendees at meetings of the OWL Working Group at the time of publication of this document were:
Jie Bao (RPI),
Diego Calvanese (Free University of Bozen-Bolzano),
Bernardo Cuenca Grau (Oxford University Computing Laboratory),
Martin Dzbor (Open University),
Achille Fokoue (IBM Corporation),
Christine Golbreich (Université de Versailles St-Quentin and LIRMM),
Sandro Hawke (W3C/MIT),
Ivan Herman (W3C/ERCIM),
Rinke Hoekstra (University of Amsterdam),
Ian Horrocks (Oxford University Computing Laboratory),
Elisa Kendall (Sandpiper Software),
Markus Krötzsch (FZI),
Carsten Lutz (Universität Bremen),
Deborah L. McGuinness (RPI),
Boris Motik (Oxford University Computing Laboratory),
Jeff Pan (University of Aberdeen),
Bijan Parsia (University of Manchester),
Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent),
Sebastian Rudolph (FZI),
Alan Ruttenberg (Science Commons),
Uli Sattler (University of Manchester),
Michael Schneider (FZI),
Mike Smith (Clark & Parsia),
Evan Wallace (NIST),
Zhe Wu (Oracle Corporation), and
Antoine Zimmermann (DERI Galway).
We would also like to thank past members of the working group:
Jeremy Carroll,
Jim Hendler,
Vipul Kashyap.
</p>
<a name="References"></a><h2> <span class="mw-headline">12 References </span></h2>
<a name="Normative_References"></a><h3> <span class="mw-headline">12.1 Normative References </span></h3>
<dl><dt> <span id="ref-owl-2-specification">[OWL 2 Specification]</span>
</dt><dd><span><cite><a href="../REC-owl2-syntax-20091027/index.html">OWL 2 Web Ontology Language: <span>Structural Specification and Functional-Style Syntax</span></a></cite> Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Recommendation, 27 October 2009, <a href="../REC-owl2-syntax-20091027/index.html">http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/</a>. Latest version available at <a href="http://www.w3.org/TR/owl2-syntax/">http://www.w3.org/TR/owl2-syntax/</a>.</span></dd><dt> <span id="ref-rdf-concepts">[RDF Concepts]</span>
</dt><dd> <cite><a class="external text" href="../../2004/REC-rdf-concepts-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/">Resource Description Framework (RDF): Concepts and Abstract Syntax</a></cite>. Graham Klyne and Jeremy J. Carroll, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/. Latest version available as http://www.w3.org/TR/rdf-concepts/.
</dd><dt> <span id="ref-rdf-semantics">[RDF Semantics]</span>
</dt><dd> <cite><a class="external text" href="../../2004/REC-rdf-mt-20040210/index.html" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics</a></cite>. Patrick Hayes, ed., W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-mt-20040210/. Latest version available as http://www.w3.org/TR/rdf-mt/.
</dd><dt> <span id="ref-rfc-2119">[RFC 2119]</span>
</dt><dd> <cite><a class="external text" href="http://www.ietf.org/rfc/rfc2119.txt" title="http://www.ietf.org/rfc/rfc2119.txt">RFC 2119: Key words for use in RFCs to Indicate Requirement Levels</a></cite>. Network Working Group, S. Bradner. IETF, March 1997, http://www.ietf.org/rfc/rfc2119.txt
</dd><dt> <span id="ref-rfc-3987">[RFC 3987]</span>
</dt><dd> <cite><a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt">RFC 3987: Internationalized Resource Identifiers (IRIs)</a></cite>. M. Duerst and M. Suignard. IETF, January 2005, http://www.ietf.org/rfc/rfc3987.txt
</dd></dl>
<a name="Nonnormative_References"></a><h3> <span class="mw-headline">12.2 Nonnormative References </span></h3>
<dl><dt> <span id="ref-owl-2-direct-semantics">[OWL 2 Direct Semantics]</span>
</dt><dd><span><cite><a href="../REC-owl2-direct-semantics-20091027/index.html">OWL 2 Web Ontology Language: <span>Direct Semantics</span></a></cite> Boris Motik, Peter F. Patel-Schneider, Bernardo Cuenca Grau, eds. W3C Recommendation, 27 October 2009, <a href="../REC-owl2-direct-semantics-20091027/index.html">http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/</a>. Latest version available at <a href="http://www.w3.org/TR/owl2-direct-semantics/">http://www.w3.org/TR/owl2-direct-semantics/</a>.</span></dd><dt> <span id="ref-owl-2-rdf-mapping">[OWL 2 RDF Mapping]</span>
</dt><dd><span><cite><a href="../REC-owl2-mapping-to-rdf-20091027/index.html">OWL 2 Web Ontology Language: <span>Mapping to RDF Graphs</span></a></cite> Peter F. Patel-Schneider, Boris Motik, eds. W3C Recommendation, 27 October 2009, <a href="../REC-owl2-mapping-to-rdf-20091027/index.html">http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/</a>. Latest version available at <a href="http://www.w3.org/TR/owl2-mapping-to-rdf/">http://www.w3.org/TR/owl2-mapping-to-rdf/</a>.</span></dd><dt> <span id="ref-owl-1-rdf-semantics">[OWL 1 RDF-Compatible Semantics]</span>
</dt><dd> <cite><a class="external text" href="http://www.w3.org/TR/owl-semantics/rdfs.html" title="http://www.w3.org/TR/owl-semantics/rdfs.html">OWL Web Ontology Language: Semantics and Abstract Syntax, Section 5. RDF-Compatible Model-Theoretic Semantics</a></cite>. Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, eds., W3C Recommendation, 10 February 2004.
</dd><dt> <span id="ref-rfc-2396">[RFC 2396]</span>
</dt><dd> <cite><a class="external text" href="http://www.ietf.org/rfc/rfc2396.txt" title="http://www.ietf.org/rfc/rfc2396.txt">RFC 2396 - Uniform Resource Identifiers (URI): Generic Syntax</a></cite>. T. Berners-Lee, R. Fielding, U.C. Irvine and L. Masinter. IETF, August 1998.
</dd></dl>
</body>
</html>
|
