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# 22 Boolean Lists

### Sections

1. Boolean Lists Representing Subsets
2. Set Operations via Boolean Lists
3. Function that Modify Boolean Lists
4. More about 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 Booleans). In function names we call boolean lists blist for brevity.

- `IsBlist(` obj`) C`

  A boolean list (``blist'') is a list that has no holes and contains only `true` and `false`. If a list is known to be a boolean list by a test with `IsBlist` it is stored in a compact form. See More about Boolean Lists.

      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.

  ## 22.1 Boolean Lists Representing Subsets

- `BlistList(` list`,` sub`) F`

  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 Sorted Lists and 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 ]

  See also UniteBlistList.

- `ListBlist(` list`,` blist`) O`

  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`) F`

  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`) F`

  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 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

  ## 22.2 Set Operations via Boolean Lists

- `UnionBlist(` blist1`,` blist2`[, ...] ) F`
- `UnionBlist(` list`) F`

  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 ...`.

  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`[, ...] ) F`
- `IntersectionBlist(` list`) F`

  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 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.

- `DifferenceBlist(` blist1`,` blist2`) F`

  returns the asymmetric set difference (exclusive or) 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> UnionBlist( blist1, blist2 );
      [ true, true, true, false ]
      gap> IntersectionBlist( blist1, blist2 );
      [ true, false, false, false ]
      gap> DifferenceBlist( blist1, blist2 );
      [ false, true, false, false ]

  ## 22.3 Function that Modify Boolean Lists

- `UniteBlist(` blist1`,` blist2`) F`

  `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 ]

- `UniteBlistList(` list`,` blist`,` sub`) F`

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

  The function `UnionBlist` (see UnionBlist) is the nondestructive counterpart to the procedure `UniteBlist`.

- `IntersectBlist(` blist1`,` blist2`) F`

  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`) F`

  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`.

  ## 22.4 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 (at least) 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.

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  GAP 4 manual
  October 2007