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
|
% This file was created automatically from blist.msk.
% DO NOT EDIT!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%A blist.msk GAP documentation Martin Schoenert
%%
%A @(#)$Id: blist.msk,v 1.9 2002/04/15 10:02:27 sal Exp $
%%
%Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
%Y Copyright (C) 2002 The GAP Group
%%
\Chapter{Boolean Lists}
This chapter describes boolean lists. A *boolean list* is a list that
has no holes and contains only the boolean values `true' and `false'
(see Chapter~"Booleans").
In function names we call boolean lists *blist* for brevity.
\>IsBlist( <obj> ) C
A boolean list (``blist'') is a list that has no holes and contains only
`true' and `false'. If a list is known to be a boolean list by a test
with `IsBlist' it is stored in a compact form. See "More about Boolean
Lists".
\beginexample
gap> IsBlist( [ true, true, false, false ] );
true
gap> IsBlist( [] );
true
gap> IsBlist( [false,,true] ); # has holes
false
gap> IsBlist( [1,1,0,0] ); # contains not only boolean values
false
gap> IsBlist( 17 ); # is not even a list
false
\endexample
Boolean lists are lists and all operations for lists are therefore
applicable to boolean lists.
Boolean lists can be used in various ways, but maybe the most important
application is their use for the description of *subsets* of finite sets.
Suppose <set> is a finite set, represented as a list. Then a subset
<sub> of <set> is represented by a boolean list <blist> of the same
length as <set> such that `<blist>[<i>]' is `true' if `<set>[<i>]' is in
<sub> and `false' otherwise.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Boolean Lists Representing Subsets}
\>BlistList( <list>, <sub> ) F
returns a new boolean list that describes the list <sub> as a sublist of
the dense list <list>.
That is `BlistList' returns a boolean list <blist> of the same length as
<list> such that `<blist>[<i>]' is `true' if `<list>[<i>]' is in
<sub> and `false' otherwise.
<list> need not be a proper set (see~"Sorted Lists and Sets"),
even though in this case `BlistList' is most efficient.
In particular <list> may contain duplicates.
<sub> need not be a proper sublist of <list>,
i.e., <sub> may contain elements that are not in <list>.
Those elements of course have no influence on the result of `BlistList'.
\beginexample
gap> BlistList( [1..10], [2,3,5,7] );
[ false, true, true, false, true, false, true, false, false, false ]
gap> BlistList( [1,2,3,4,5,2,8,6,4,10], [4,8,9,16] );
[ false, false, false, true, false, false, true, false, true, false ]
\endexample
See also~"UniteBlistList".
\>ListBlist( <list>, <blist> ) O
returns the sublist <sub> of the list <list>, which must have no holes,
represented by the boolean list <blist>, which must have the same
length as <list>. <sub> contains the element `<list>[<i>]' if
`<blist>[<i>]' is `true' and does not contain the element
if `<blist>[<i>]' is `false'. The order of the elements in <sub> is
the same as the order of the corresponding elements in <list>.
\beginexample
gap> ListBlist([1..8],[false,true,true,true,true,false,true,true]);
[ 2, 3, 4, 5, 7, 8 ]
gap> ListBlist( [1,2,3,4,5,2,8,6,4,10],
> [false,false,false,true,false,false,true,false,true,false] );
[ 4, 8, 4 ]
\endexample
\>SizeBlist( <blist> ) F
returns the number of entries of the boolean list <blist> that are
`true'. This is the size of the subset represented by the boolean
list <blist>.
\beginexample
gap> SizeBlist( [ false, true, false, true, false ] );
2
\endexample
\>IsSubsetBlist( <blist1>, <blist2> ) F
returns `true' if the boolean list <blist2> is a subset of the boolean
list <list1>, which must have equal length, and `false' otherwise.
<blist2> is a subset of <blist1> if `<blist1>[<i>] =
<blist1>[<i>] or <blist2>[<i>]' for all <i>.
\beginexample
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> IsSubsetBlist( blist1, blist2 );
false
gap> blist2 := [ true, false, false, false ];;
gap> IsSubsetBlist( blist1, blist2 );
true
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Set Operations via Boolean Lists}
\>UnionBlist( <blist1>, <blist2>[, ...] ) F
\>UnionBlist( <list> ) F
In the first form `UnionBlist' returns the union of the boolean
lists <blist1>, <blist2>, etc., which must have equal length. The
*union* is a new boolean list such that `<union>[<i>] = <blist1>[<i>] or
<blist2>[<i>] or ...'.
The second form takes the union of all blists (which
as for the first form must have equal length) in the list <list>.
\>IntersectionBlist( <blist1>, <blist2>[, ...] ) F
\>IntersectionBlist( <list> ) F
In the first form `IntersectionBlist' returns the intersection of
the boolean lists <blist1>, <blist2>, etc., which must have equal
length. The *intersection* is a new blist such that
`<inter>[<i>] = <blist1>[<i>] and <blist2>[<i>] and ...'.
In the second form <list> must be a list of boolean lists
<blist1>, <blist2>, etc., which must have equal length, and
`IntersectionBlist' returns the intersection of those boolean lists.
\>DifferenceBlist( <blist1>, <blist2> ) F
returns the asymmetric set difference (exclusive or) of the two
boolean lists <blist1> and <blist2>, which must have equal length.
The *asymmetric set difference* is a new boolean list such that
`<union>[<i>] = <blist1>[<i>] and not <blist2>[<i>]'.
\beginexample
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> UnionBlist( blist1, blist2 );
[ true, true, true, false ]
gap> IntersectionBlist( blist1, blist2 );
[ true, false, false, false ]
gap> DifferenceBlist( blist1, blist2 );
[ false, true, false, false ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Function that Modify Boolean Lists}
\>UniteBlist( <blist1>, <blist2> ) F
`UniteBlist' unites the boolean list <blist1> with the boolean
list <blist2>, which must have the same length. This is
equivalent to assigning `<blist1>[<i>] := <blist1>[<i>] or
<blist2>[<i>]' for all <i>. `UniteBlist' returns nothing, it is only
called to change <blist1>.
\beginexample
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> UniteBlist( blist1, blist2 );
gap> blist1;
[ true, true, true, false ]
\endexample
\>UniteBlistList( <list>, <blist>, <sub> ) F
works like `UniteBlist(<blist>,BlistList(<list>,<sub>))'. As no
intermediate blist is created, the performance is better than the
separate function calls.
The function `UnionBlist' (see "UnionBlist") is the nondestructive
counterpart to the procedure `UniteBlist'.
\>IntersectBlist( <blist1>, <blist2> ) F
intersects the boolean list <blist1> with the boolean list <blist2>,
which must have the same length. This is equivalent to assigning
`<blist1>[<i>]:= <blist1>[<i>] and <blist2>[<i>]' for all <i>.
`IntersectBlist' returns nothing, it is only called to change <blist1>.
\beginexample
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> IntersectBlist( blist1, blist2 );
gap> blist1;
[ true, false, false, false ]
\endexample
The function `IntersectionBlist' (see "IntersectionBlist") is the
nondestructive counterpart to the procedure `IntersectBlist'.
\>SubtractBlist( <blist1>, <blist2> ) F
subtracts the boolean list <blist2> from the boolean list <blist1>,
which must have equal length. This is equivalent to assigning
`<blist1>[<i>] := <blist1>[<i>] and not <blist2>[<i>]' for all
<i>. `SubtractBlist' returns nothing, it is only called to change
<blist1>.
\beginexample
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> SubtractBlist( blist1, blist2 );
gap> blist1;
[ false, true, false, false ]
\endexample
The function `DifferenceBlist' (see "DifferenceBlist") is the
nondestructive counterpart to the procedure `SubtractBlist'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{More about Boolean Lists}
We defined a boolean list as a list that has no holes and contains only
`true' and `false'.
There is a special internal representation for boolean lists that needs
only 1 bit for each entry.
This bit is set if the entry is `true' and reset if the entry is `false'.
This representation is of course much more compact than the ordinary
representation of lists, which needs (at least) 32 bits per entry.
%T Add a note about internal representation of plain lists (preferably in
%T the chapter ``Lists''),
%T in order to allow a user to estimate the space needed for
%T computations with lists;
%T then add cross-references from and to the other available list
%T representations!
Not every boolean list is represented in this compact representation. It
would be too much work to test every time a list is changed, whether this
list has become a boolean list. This section tells you under which
circumstances a boolean list is represented in the compact
representation, so you can write your functions in such a way that you make
best use of the compact representation.
The results of `BlistList', `UnionBlist', `IntersectionBlist' and
`DifferenceBlist' are known to be boolean lists by construction, and thus
are represented in the compact representation upon creation.
If an argument of `IsBlist', `IsSubsetBlist', `ListBlist', `UnionBlist',
`IntersectionBlist', `DifferenceBlist', `UniteBlist', `IntersectBlist' and
`SubtractBlist' is a list represented in the ordinary representation, it is
tested to see if it is in fact a boolean list. If it is not, `IsBlist'
returns `false' and the other functions signal an error. If it is, the
representation of the list is changed to the compact representation.
If you change a boolean list that is represented in the compact
representation by assignment (see "List Assignment") or `Add' (see "Add")
in such a way that the list remains a boolean list it will remain
represented in the compact representation. Note that changing a list
that is not represented in the compact representation, whether it is a
boolean list or not, in such a way that the resulting list becomes a
boolean list, will never change the representation of the list.
|
