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
|
; ABNF (Augmented Backus-Naur Form) Library
;
; Copyright (C) 2022 Kestrel Institute (http://www.kestrel.edu)
;
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
;
; Author: Alessandro Coglio (coglio@kestrel.edu)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ABNF")
(include-book "../notation/semantics")
(local (include-book "std/lists/top" :dir :system))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defxdoc+ ambiguity
:parents (operations)
:short "Ambiguity (and unambiguity) in ABNF grammars."
:long
(xdoc::topstring-p
"This part of the ABNF formalization is work in progress.
More definitions and theorems should be added.")
:order-subtopics t
:default-parent t)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-sk rules-ambiguousp ((rules rulelistp))
:returns (yes/no booleanp)
:short "Notion of ambiguous lists of rules."
:long
(xdoc::topstring-p
"A list of rules is ambiguous iff it includes some ambiguous string.
Note that the condition that
the existentially quantified @('rulename') be defined by @('rules')
would be superfluous,
because if @('rulename') is not defined
then no parse trees can originate from it.")
(exists (string rulename)
(and (stringp string)
(rulenamep rulename)
(string-ambiguousp string rulename rules))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defrule num-val-unambiguous
:short "Numeric value notations are never ambiguous."
:long
(xdoc::topstring-p
"Any two trees that match a numeric value notation
and that have the same string at the leaves
are the same tree.")
(implies (and (tree-match-num-val-p tree1 num-val)
(tree-match-num-val-p tree2 num-val))
(equal (equal (tree->string tree1)
(tree->string tree2))
(tree-equiv tree1 tree2)))
:enable (tree-match-num-val-p tree->string tree-fix tree-leafterm->get))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defrule char-val-unambiguous
:short "Character value notations are never ambiguous."
:long
(xdoc::topstring-p
"Any two trees that match a character value notation
and that have the same string at the leaves
are the same tree.")
(implies (and (tree-match-char-val-p tree1 char-val)
(tree-match-char-val-p tree2 char-val))
(equal (equal (tree->string tree1)
(tree->string tree2))
(tree-equiv tree1 tree2)))
:enable (tree-match-char-val-p tree->string tree-fix tree-leafterm->get))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defruled prose-val-ambiguous
:short "Prose value notations are always ambiguous."
:long
(xdoc::topstring-p
"We can always construct two different trees
with the same string at the leaves
that match any prose value notation.
Recall that a prose value notation is matched by any tree.")
(implies (and (equal tree1 (make-tree-nonleaf
:rulename? nil
:branches (list (list (tree-leafterm '(1))
(tree-leafterm '(2))))))
(equal tree2 (make-tree-nonleaf
:rulename? nil
:branches (list (list (tree-leafterm '(1 2)))))))
(and (treep tree1)
(treep tree2)
(not (equal tree1 tree2))
(equal (tree->string tree1)
(tree->string tree2))
(tree-match-prose-val-p tree1 prose-val)
(tree-match-prose-val-p tree2 prose-val))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-sk element-unambiguousp ((element elementp) (rules rulelistp))
:returns (yes/no booleanp)
:short "Notion of unambiguous elements."
:long
(xdoc::topstring-p
"An element is unambiguous iff
any two trees that match the element and have the same string at the leaves
are the same tree.")
(forall (tree1 tree2)
(implies (and (treep tree1)
(treep tree2)
(tree-match-element-p tree1 element rules)
(tree-match-element-p tree2 element rules))
(equal (equal (tree->string tree1)
(tree->string tree2))
(equal tree1 tree2))))
///
(defruled element-ambiguousp-rewrite
(implies (and (element-unambiguousp element rules)
(tree-match-element-p tree1 element rules)
(tree-match-element-p tree2 element rules))
(equal (equal (tree->string tree1)
(tree->string tree2))
(tree-equiv tree1 tree2)))
:use (:instance element-unambiguousp-necc
(tree1 (tree-fix tree1))
(tree2 (tree-fix tree2)))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defrule element-num-val-unambiguous
:short "Numeric value elements are never ambiguous."
:long
(xdoc::topstring-p
"This is a simple consequnce of @(tsee num-val-unambiguous).")
(implies (element-case element :num-val)
(element-unambiguousp element rules))
:enable (element-unambiguousp tree-match-element-p))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defrule element-char-val-unambiguous
:short "Character value elements are never ambiguous."
:long
(xdoc::topstring
"This is a simple consequnce of @(tsee char-val-unambiguous).")
(implies (element-case element :char-val)
(element-unambiguousp element rules))
:enable (element-unambiguousp tree-match-element-p))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defrule element-prose-val-ambiguous
:short "Prose value elements are always ambiguous."
:long
(xdoc::topstring-p
"This is a simple consequence of @(tsee prose-val-ambiguous).")
(implies (element-case element :prose-val)
(not (element-unambiguousp element rules)))
:enable tree-match-element-p
:use (:instance element-unambiguousp-necc
(tree1 (make-tree-nonleaf
:rulename? nil
:branches (list (list (tree-leafterm '(1))
(tree-leafterm '(2))))))
(tree2 (make-tree-nonleaf
:rulename? nil
:branches (list (list (tree-leafterm '(1 2))))))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-sk repetition-unambiguousp ((repetition repetitionp) (rules rulelistp))
:returns (yes/no booleanp)
:short "Notion of unambiguous repetitions."
:long
(xdoc::topstring
(xdoc::p
"A repetition is unambiguous iff
any two lists of trees that match the repetition
and have the same string at the leaves
are the same list of trees.")
(xdoc::p
"A repetition of 0 elements is always unambiguous."))
(forall (trees1 trees2)
(implies (and (tree-listp trees1)
(tree-listp trees2)
(tree-list-match-repetition-p trees1 repetition rules)
(tree-list-match-repetition-p trees2 repetition rules))
(equal (equal (tree-list->string trees1)
(tree-list->string trees2))
(equal trees1 trees2))))
///
(defruled repetition-unambiguousp-rewrite
(implies (and (repetition-unambiguousp repetition rules)
(tree-list-match-repetition-p trees1 repetition rules)
(tree-list-match-repetition-p trees2 repetition rules))
(equal (equal (tree-list->string trees1)
(tree-list->string trees2))
(tree-list-equiv trees1 trees2)))
:use (:instance repetition-unambiguousp-necc
(trees1 (tree-list-fix trees1))
(trees2 (tree-list-fix trees2))))
(defrule empty-repetition-umabiguous
(implies (equal (repetition->range repetition)
(repeat-range 0 (nati-finite 0)))
(repetition-unambiguousp repetition rules))
:enable tree-list-match-repetition-p))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-sk concatenation-unambiguousp ((concatenation concatenationp)
(rules rulelistp))
:returns (yes/no booleanp)
:short "Notion of unambiguous concatenations."
:long
(xdoc::topstring
(xdoc::p
"A concatenation is unambiguous iff
any two lists of lists of trees that match the concatenation
and have the same string at the leaves
are the same list of lists of trees.")
(xdoc::p
"An empty concatenation is always unambiguous."))
(forall (treess1 treess2)
(implies (and (tree-list-listp treess1)
(tree-list-listp treess2)
(tree-list-list-match-concatenation-p
treess1 concatenation rules)
(tree-list-list-match-concatenation-p
treess2 concatenation rules))
(equal (equal (tree-list-list->string treess1)
(tree-list-list->string treess2))
(equal treess1 treess2))))
///
(defruled concatenation-unambiguousp-rewrite
(implies (and (concatenation-unambiguousp concatenation rules)
(tree-list-list-match-concatenation-p treess1
concatenation
rules)
(tree-list-list-match-concatenation-p treess2
concatenation
rules))
(equal (equal (tree-list-list->string treess1)
(tree-list-list->string treess2))
(tree-list-list-equiv treess1 treess2)))
:use (:instance concatenation-unambiguousp-necc
(treess1 (tree-list-list-fix treess1))
(treess2 (tree-list-list-fix treess2))))
(defrule empty-concatenation-unambiguous
(concatenation-unambiguousp nil rules)
:enable tree-list-list-match-concatenation-p))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-sk alternation-unambiguousp ((alternation alternationp)
(rules rulelistp))
:returns (yes/no booleanp)
:short "Notion of unambiguous alternations."
:long
(xdoc::topstring
(xdoc::p
"An alternation is unambiguous iff
any two lists of lists of trees that match the alternation
and have the same string at the leaves
are the same list of lists of trees.")
(xdoc::p
"An empty alternation is always unambiguous."))
(forall (treess1 treess2)
(implies (and (tree-list-listp treess1)
(tree-list-listp treess2)
(tree-list-list-match-alternation-p
treess1 alternation rules)
(tree-list-list-match-alternation-p
treess2 alternation rules))
(equal (equal (tree-list-list->string treess1)
(tree-list-list->string treess2))
(equal treess1 treess2))))
///
(defruled alternation-unambiguousp-rewrite
(implies (and (alternation-unambiguousp alternation rules)
(tree-list-list-match-alternation-p treess1
alternation
rules)
(tree-list-list-match-alternation-p treess2
alternation
rules))
(equal (equal (tree-list-list->string treess1)
(tree-list-list->string treess2))
(tree-list-list-equiv treess1 treess2)))
:use (:instance alternation-unambiguousp-necc
(treess1 (tree-list-list-fix treess1))
(treess2 (tree-list-list-fix treess2))))
(defrule alternation-unambiguousp-of-nil
(alternation-unambiguousp nil rules)
:enable tree-list-list-match-alternation-p))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-sk concatenation-alternation-disjointp ((concatenation concatenationp)
(alternation alternationp)
(rules rulelistp))
:returns (yes/no booleanp)
:short "Notion of disjoint concatenation-alternation pairs."
:long
(xdoc::topstring
(xdoc::p
"A concatenation is disjoint from an alternation iff
the concatenation and the alternation are matched
by disjoint sets of lists of lists of trees.
That is, there is no list of lists of trees that matches
both the concatenation and the alternation.")
(xdoc::p
"The empty alternation is disjoint from the empty concatenation."))
(forall (treess1 treess2)
(implies (and (tree-list-listp treess1)
(tree-list-listp treess2)
(tree-list-list-match-concatenation-p
treess1 concatenation rules)
(tree-list-list-match-alternation-p
treess2 alternation rules))
(not (equal (tree-list-list->string treess1)
(tree-list-list->string treess2)))))
:rewrite (implies (and (concatenation-alternation-disjointp
concatenation alternation rules)
(tree-list-listp treess1)
(tree-list-listp treess2)
(tree-list-list-match-concatenation-p
treess1 concatenation rules)
(tree-list-list-match-alternation-p
treess2 alternation rules))
(and (not (equal (tree-list-list->string treess1)
(tree-list-list->string treess2)))
(not (equal (tree-list-list->string treess2)
(tree-list-list->string treess1)))))
///
(defrule concatenation-alternation-disjointp-of-nil
(concatenation-alternation-disjointp concatenation nil rules)
:enable tree-list-list-match-alternation-p))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defrule alternation-unambiguousp-of-cons-when-disjointp
:short "If an umambiguous concatenation
is disjoint from an unambiguous alternation,
then adding the concatenation maintains the alternation unambiguous."
:long
(xdoc::topstring
(xdoc::p
"This theorem can be used to show that an alternation is unambiguous,
one constituting concatenation at a time,
starting with "
(xdoc::seetopic "concatenation-alternation-disjointp"
"@('concatenation-alternation-disjointp-of-nil')")
". In other words, it must be showed that the alternatives of the alternation
are all disjoint, i.e. they have no lists of lists of trees in common."))
(implies (and (concatenation-unambiguousp concatenation rules)
(alternation-unambiguousp alternation rules)
(concatenation-alternation-disjointp
concatenation alternation rules))
(alternation-unambiguousp (cons concatenation alternation) rules))
:expand ((alternation-unambiguousp (cons concatenation alternation) rules))
:enable (tree-list-list-match-alternation-p
concatenation-unambiguousp-necc
alternation-unambiguousp-necc
concatenation-alternation-disjointp-necc))
|
