# 29 Boolean Lists

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).

## 29.1 BlistList

`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`.

## 29.2 ListBlist

`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`.

## 29.3 IsBlist

`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 ```

## 29.4 SizeBlist

`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 ```

## 29.5 IsSubsetBlist

`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 ```

## 29.6 UnionBlist

`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).

## 29.7 IntersectionBlist

`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).

## 29.8 DifferenceBlist

`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).

## 29.9 UniteBlist

`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`.

## 29.10 IntersectBlist

`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`.

## 29.11 SubtractBlist

`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
24 Jun 2022