GAP (ref) - Chapter 22: Boolean Lists

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22 Boolean Lists
22.1 IsBlist (Filter)
22.1-1 IsBlist
22.2 Boolean Lists Representing Subsets
22.2-1 BlistList
22.2-2 ListBlist
22.2-3 SizeBlist
22.2-4 IsSubsetBlist
22.3 Set Operations via Boolean Lists
22.3-1 UnionBlist
22.3-2 IntersectionBlist
22.3-3 DifferenceBlist
22.4 Function that Modify Boolean Lists
22.4-1 UniteBlist
22.4-2 UniteBlistList
22.4-3 IntersectBlist
22.4-4 SubtractBlist
22.4-5 MeetBlist
22.4-6 FlipBlist
22.4-7 SetAllBlist
22.4-8 ClearAllBlist
22.5 More about Boolean Lists
22.5-1 IsBlistRep

### 22 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 20). In function names we call boolean lists blists for brevity.

#### 22.1 IsBlist (Filter)

##### 22.1-1 IsBlist

`‣ IsBlist`( obj ) | ( category )

A boolean list (blist) is a list that has no holes and contains only `true` and `false`. Boolean lists can be represented in an efficient compact form, see 22.5 for details.

    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.

#### 22.2 Boolean Lists Representing Subsets

##### 22.2-1 BlistList

`‣ BlistList`( list, sub ) | ( function )

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 21.19), 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` (22.4-2).

##### 22.2-2 ListBlist

`‣ ListBlist`( list, blist ) | ( function )

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 ]

##### 22.2-3 SizeBlist

`‣ SizeBlist`( blist ) | ( function )

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

##### 22.2-4 IsSubsetBlist

`‣ IsSubsetBlist`( blist1, blist2 ) | ( function )

returns `true` if the boolean list blist2 is a subset of the boolean list blist1, 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

#### 22.3 Set Operations via Boolean Lists

##### 22.3-1 UnionBlist

`‣ UnionBlist`( blist1, blist2[, ...] ) | ( function )

`‣ UnionBlist`( list ) | ( function )

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 that contains at position \(i\) the value blist1\([i]\) `or` blist2\([i]\) `or` \(\ldots\).

The second form takes the union of all blists (which as for the first form must have equal length) in the list list.

##### 22.3-2 IntersectionBlist

`‣ IntersectionBlist`( blist1, blist2[, ...] ) | ( function )

`‣ IntersectionBlist`( list ) | ( function )

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 that contains at position \(i\) the value blist1\([i]\) `and` blist2\([i]\) `and` \(\ldots\).

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.

##### 22.3-3 DifferenceBlist

`‣ DifferenceBlist`( blist1, blist2 ) | ( function )

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 that contains at position \(i\) the value 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 ]

#### 22.4 Function that Modify Boolean Lists

##### 22.4-1 UniteBlist

`‣ UniteBlist`( blist1, blist2 ) | ( function )

`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` (22.3-1) is the nondestructive counterpart to `UniteBlist`.

##### 22.4-2 UniteBlistList

`‣ UniteBlistList`( list, blist, sub ) | ( function )

works like `UniteBlist(blist,BlistList(list,sub))`. As no intermediate blist is created, the performance is better than the separate function calls.

##### 22.4-3 IntersectBlist

`‣ IntersectBlist`( blist1, blist2 ) | ( function )

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` (22.3-2) is the nondestructive counterpart to `IntersectBlist`.

##### 22.4-4 SubtractBlist

`‣ SubtractBlist`( blist1, blist2 ) | ( function )

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` (22.3-3) is the nondestructive counterpart to `SubtractBlist`.

##### 22.4-5 MeetBlist

`‣ MeetBlist`( blist1, blist2 ) | ( function )

Returns `true` if there is a position at which both blist1 and blist2 contain `true`, and `false` otherwise. It is equivalent to, but faster than `SizeBlist(IntersectionBlist(blist1, blist2)) <> 0`. An error is thrown if the lists do not have the same length.

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

##### 22.4-6 FlipBlist

`‣ FlipBlist`( blist ) | ( function )

Changes every entry in blist that equals `true` to `false` and vice versa. If `blist1` and `blist2` are boolean lists with equal length and every value in `blist2` is `true`, then `FlipBlist( blist1 )` is equivalent to `SubtractBlist( blist2, blist1 ); blist1 := blist2;` but `FlipBlist` is faster, and simpler to type.

`FlipBlist` returns nothing, it is only called to change blist in-place.

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

##### 22.4-7 SetAllBlist

`‣ SetAllBlist`( blist ) | ( function )

Changes every entry in blist to `true`. If `blist1` and `blist2` are boolean lists with equal length and every value in `blist2` is `true`, then `SetAllBlist( blist1 )` is equivalent to `UniteBlist( blist1, blist2 );` but is faster, and simpler to type.

`SetAllBlist` returns nothing, it is only called to change blist in-place.

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

##### 22.4-8 ClearAllBlist

`‣ ClearAllBlist`( blist ) | ( function )

Changes every entry in blist to `false`. If `blist1` and `blist2` are boolean lists with equal length and every value in `blist2` is `false`, then `ClearAllBlist( blist1 )` is equivalent to `IntersectBlist( blist1, blist2 );` but is faster, and simpler to type.

`ClearAllBlist` returns nothing, it is only called to change blist in-place.

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

#### 22.5 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 32 or 64 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.

If a dense list containing only `true` and `false` is read, it is stored in the compact representation. Furthermore, the results of `BlistList` (22.2-1), `UnionBlist` (22.3-1), `IntersectionBlist` (22.3-2) and `DifferenceBlist` (22.3-3) are known to be boolean lists by construction, and thus are represented in the compact representation upon creation.

If an argument of `IsSubsetBlist` (22.2-4), `ListBlist` (22.2-2), `UnionBlist` (22.3-1), `IntersectionBlist` (22.3-2), `DifferenceBlist` (22.3-3), `UniteBlist` (22.4-1), `IntersectBlist` (22.4-3) and `SubtractBlist` (22.4-4) 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, an error is signalled. 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 21.4) or `Add` (21.4-2) 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.

##### 22.5-1 IsBlistRep

`‣ IsBlistRep`( obj ) | ( representation )

`‣ ConvertToBlistRep`( blist ) | ( function )

Returns: `true` or `false`

The first function is a filter that returns `true` if the object obj is a boolean list in compact representation and `false` otherwise, see 22.5.

The second function converts the object blist to a boolean list in compact representation and returns `true` if this is possible. Otherwise blist is unchanged and `false` is returned.

    gap> l := [true, false, true];
    [ true, false, true ]
    gap> IsBlistRep(l);
    true
    gap> l := [true, false, 1];
    [ true, false, 1 ]
    gap> l[3] := false;
    false
    gap> IsBlistRep(l);
    false
    gap> ConvertToBlistRep(l);
    true

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