This chapter describes boolean lists. A boolean list is a list that
has no holes and contains only boolean values, i.e., true and false.
In function names we call boolean lists blist for brevity.
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.
This package contains functions to switch between the representations of
subsets of a finite set either as sets or as boolean lists (see
BlistList, ListBlist), to test if a list is a boolean list (see
IsBlist), and to count the number of true entries in a boolean list
(see SizeBlist).
Next there are functions for the standard set operations for the subsets
represented by boolean lists (see IsSubsetBlist, UnionBlist,
IntersectionBlist, and DifferenceBlist). There are also the
corresponding destructive procedures that change their first argument
(see UniteBlist, IntersectBlist, and SubtractBlist). Note that
there is no function to add or delete a single element to a subset
represented by a boolean list, because this can be achieved by assigning
true or false to the corresponding position in the boolean list (see
List Assignment).
Since boolean lists are just a special case of lists, all the operations
and functions for lists, can be used for boolean lists just as well (see
Lists). For example Position (see Position) can be used to find
the true entries in a boolean list, allowing you to loop over the
elements of the subset represented by the boolean list.
There is also a section about internal details (see More about Boolean Lists).
BlistList( list, sub )
BlistList returns a new boolean list that describes the list sub as a
sublist of the list list, which must have no holes. 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 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 ]
ListBlist (see ListBlist) is the inverse function to BlistList.
ListBlist( list, blist )
ListBlist 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 ]
BlistList (see BlistList) is the inverse function to ListBlist.
IsBlist( obj )
IsBlist returns true if obj, which may be an object of arbitrary
type, is a boolean list and false otherwise. A boolean list is a list
that has no holes and contains only true and false.
gap> IsBlist( [ true, true, false, false ] );
true
gap> IsBlist( [] );
true
gap> IsBlist( [false,,true] );
false # has holes
gap> IsBlist( [1,1,0,0] );
false # contains not only boolean values
gap> IsBlist( 17 );
false # is not even a list
SizeBlist( blist )
SizeBlist 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( [ true, true, false, false ] );
2
IsSubsetBlist( blist1, blist2 )
IsSubsetBlist 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 if 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
UnionBlist( blist1, blist2.. )
UnionBlist( list )
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 ...
In the second form list must be a list of boolean lists blist1,
blist2, etc., which must have equal length, and Union returns the
union of those boolean list.
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> UnionBlist( blist1, blist2 );
[ true, true, true, false ]
Note that UnionBlist is implemented in terms of the procedure
UniteBlist (see UniteBlist).
IntersectionBlist( blist1, blist2.. )
IntersectionBlist( 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 boolean list 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.
gap> blist1 := [ true, true, false, false ];;
gap> blist2 := [ true, false, true, false ];;
gap> IntersectionBlist( blist1, blist2 );
[ true, false, false, false ]
Note that IntersectionBlist is implemented in terms of the procedure
IntersectBlist (see IntersectBlist).
DifferenceBlist( blist1, blist2 )
DifferenceBlist returns the asymmetric set difference 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> DifferenceBlist( blist1, blist2 );
[ false, true, false, false ]
Note that DifferenceBlist is implemented in terms of the procedure
SubtractBlist (see SubtractBlist).
UniteBlist( blist1, blist2 )
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 ]
The function UnionBlist (see UnionBlist) is the nondestructive
counterpart to the procedure UniteBlist.
IntersectBlist( blist1, blist2 )
IntersectBlist 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.
SubtractBlist( blist1, blist2 )
SubtractBlist 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.
29.12 More about Boolean Lists
In the previous section (see 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 every 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 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.
gap3-jm