% This file was created automatically from blist.msk. % DO NOT EDIT! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A blist.msk GAP documentation Martin Schoenert %% %A @(#)$Id: blist.msk,v 1.9 2002/04/15 10:02:27 sal Exp $ %% %Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland %Y Copyright (C) 2002 The GAP Group %% \Chapter{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". \beginexample 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 \endexample 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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'. \beginexample 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 ] \endexample 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>. \beginexample 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 ] \endexample \>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>. \beginexample gap> SizeBlist( [ false, true, false, true, false ] ); 2 \endexample \>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>. \beginexample 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 \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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>]'. \beginexample 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 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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>. \beginexample gap> blist1 := [ true, true, false, false ];; gap> blist2 := [ true, false, true, false ];; gap> UniteBlist( blist1, blist2 ); gap> blist1; [ true, true, true, false ] \endexample \>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>. \beginexample gap> blist1 := [ true, true, false, false ];; gap> blist2 := [ true, false, true, false ];; gap> IntersectBlist( blist1, blist2 ); gap> blist1; [ true, false, false, false ] \endexample 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>. \beginexample gap> blist1 := [ true, true, false, false ];; gap> blist2 := [ true, false, true, false ];; gap> SubtractBlist( blist1, blist2 ); gap> blist1; [ false, true, false, false ] \endexample The function `DifferenceBlist' (see "DifferenceBlist") is the nondestructive counterpart to the procedure `SubtractBlist'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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. %T Add a note about internal representation of plain lists (preferably in %T the chapter ``Lists''), %T in order to allow a user to estimate the space needed for %T computations with lists; %T then add cross-references from and to the other available list %T representations! 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.