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

is `blist`[`i`]`true`

if

is in
`set`[`i`]`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
- ListBlist
- IsBlist
- SizeBlist
- IsSubsetBlist
- UnionBlist
- IntersectionBlist
- DifferenceBlist
- UniteBlist
- IntersectBlist
- SubtractBlist
- 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

is `blist`[`i`]`true`

if

is in `list`[`i`]`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

if
`list`[`i`]

is `blist`[`i`]`true`

and does not contain the element if

is `blist`[`i`]`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

for all `blist1`[`i`] =
`blist1`[`i`] or `blist2`[`i`]`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

for all `blist1`[`i`] := `blist1`[`i`] or `blist2`[`i`]`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

for all `blist1`[`i`]:= `blist1`[`i`] and `blist2`[`i`]`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

for all `blist1`[`i`] := `blist1`[`i`] and not `blist2`[`i`]`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`

.

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 Apr 2021