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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.

`‣ 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.

`‣ 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).

`‣ 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 ]

`‣ 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

`‣ 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

`‣ 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`.

`‣ 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.

`‣ 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 ]

`‣ 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`

.

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

works like `UniteBlist(`

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

`‣ 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`

.

`‣ 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`

.

`‣ 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 ]

`‣ 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 ]

`‣ 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 ]

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.

`‣ 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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