22 Boolean Lists

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.

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

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.

22.1 Boolean Lists Representing Subsets

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.

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 ]

See also UniteBlistList.

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.

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 ]

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.

gap> SizeBlist( [ false, true, false, true, false ] );
2

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.

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

22.2 Set Operations via Boolean Lists

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.

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.

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].

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 ]

22.3 Function that Modify Boolean Lists

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.

gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> UniteBlist( blist1, blist2 );
gap> blist1;
[ true, true, true, false ]

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.

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.

gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> IntersectBlist( blist1, blist2 );
gap> blist1;
[ true, false, false, false ]

The function IntersectionBlist (see IntersectionBlist) is the nondestructive counterpart to the procedure IntersectBlist.

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.

gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> SubtractBlist( blist1, blist2 );
gap> blist1;
[ false, true, false, false ]

The function DifferenceBlist (see DifferenceBlist) is the nondestructive counterpart to the procedure SubtractBlist.

22.4 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.

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.

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GAP 4 manual
March 2006