35 Associative Words

IsAssocWord is the category of associative words in free semigroups, IsAssocWordWithOne is the category of associative words in free monoids (which admit the operation One to compute an identity), IsAssocWordWithInverse is the category of associative words in free groups (which have an inverse). See IsWord for more general categories of words.

Different alphabets correspond to different families of associative words. There is no relation whatsoever between words in different families.

gap> f:= FreeGroup( "a", "b", "c" );
<free group on the generators [ a, b, c ]>
gap> gens:= GeneratorsOfGroup(f);
[ a, b, c ]
gap> w:= gens[1]*gens[2]/gens[3]*gens[2]*gens[1]/gens[1]*gens[3]/gens[2];
a*b*c^-1*b*c*b^-1
gap> w^-1;
b*c^-1*b^-1*c*b^-1*a^-1

Words are displayed as products of letters. The letters are usually printed like f1, f2, ¼, but it is possible to give user defined names to them, which can be arbitrary strings. These names do not necessarily identify a unique letter (generator), it is possible to have several letters --even in the same family-- that are displayed in the same way. Note also that there is no relation between the names of letters and variable names. In the example above, we might have typed

gap> a:= f.1;; b:= f.2;; c:= f.3;;

(Interactively, the function AssignGeneratorVariables (see AssignGeneratorVariables) provides a shorthand for this.) This allows us to define w more conveniently:

gap> w := a*b/c*b*a/a*c/b;
a*b*c^-1*b*c*b^-1

Using homomorphisms it is possible to express elements of a group as words in terms of generators, see Expressing group elements as words in generators.

35.2 Free Groups, Monoids and Semigroups

Usually a family of associative words will be generated by constructing the free object generated by them.

Called in the first form, FreeGroup returns a free group on rank generators. Called in the second form, FreeGroup returns a free group on rank generators, printed as name1, name2 etc. Called in the third form, FreeGroup returns a free group on as many generators as arguments, printed as name1, name2 etc. Called in the fourth form, FreeGroup returns a free group on as many generators as the length of the list names, the i-th generator being printed as names[i]. Called in the fifth form, FreeGroup returns a free group on infinitely many generators, where the first generators are printed by the names in the list init, and the other generators by name and an appended number.

If the extra argument wfilt is given, it must be either IsSyllableWordsFamily or IsLetterWordsFamily or IsWLetterWordsFamily or IsBLetterWordsFamily. The filter then specifies the representation used for the elements of the free group (see Representations for Associative Words). If no such filter is given, a letter representation is used.

Any group consisting of elements in IsAssocWordWithInverse lies in the filter IsFreeGroup; this holds in particular for any group created with FreeGroup (see FreeGroup), or any subgroup of such a group.

Also see Chapter Finitely Presented Groups.

Called in the first form, FreeMonoid returns a free monoid on rank generators. Called in the second form, FreeMonoid returns a free monoid on rank generators, printed as name1, name2 etc., that is, each name is the concatenation of the string name and an integer from 1 to range. Called in the third form, FreeMonoid returns a free monoid on as many generators as arguments, printed as name1, name2 etc. Called in the fourth form, FreeMonoid returns a free monoid on as many generators as the length of the list names, the i-th generator being printed as names[i]. Called in the fifth form, FreeMonoid returns a free monoid on infinitely many generators, where the first generators are printed by the names in the list init, and the other generators by name and an appended number.

If the extra argument wfilt is given, it must be either IsSyllableWordsFamily or IsLetterWordsFamily or IsWLetterWordsFamily or IsBLetterWordsFamily. The filter then specifies the representation used for the elements of the free group (see Representations for Associative Words). If no such filter is given, a letter representation is used.

Also see Chapter Monoids.

Called in the first form, FreeSemigroup returns a free semigroup on rank generators. Called in the second form, FreeSemigroup returns a free semigroup on rank generators, printed as name1, name2 etc., that is, each name is the concatenation of the string name and an integer from 1 to range. Called in the third form, FreeSemigroup returns a free semigroup on as many generators as arguments, printed as name1, name2 etc. Called in the fourth form, FreeSemigroup returns a free semigroup on as many generators as the length of the list names, the i-th generator being printed as names[i]. Called in the fifth form, FreeSemigroup returns a free semigroup on infinitely many generators, where the first generators are printed by the names in the list init, and the other generators by name and an appended number.

If the extra argument wfilt is given, it must be either IsSyllableWordsFamily or IsLetterWordsFamily or IsWLetterWordsFamily or IsBLetterWordsFamily. The filter then specifies the representation used for the elements of the free group (see Representations for Associative Words). If no such filter is given, a letter representation is used.

Also see Chapter Semigroups and FreeSemigroup!with examples.

Each free object defines a unique alphabet (and a unique family of words). Its generators are the letters of this alphabet, thus words of length one.

gap> FreeGroup( 5 );
<free group on the generators [ f1, f2, f3, f4, f5 ]>
gap> FreeGroup( "a", "b" );
<free group on the generators [ a, b ]>
gap> FreeGroup( infinity );
<free group with infinity generators>
gap> FreeSemigroup( "x", "y" );
<free semigroup on the generators [ x, y ]>
gap> FreeMonoid( 7 );
<free monoid on the generators [ m1, m2, m3, m4, m5, m6, m7 ]>

Remember that names are just a help for printing and do not necessarily distinguish letters. It is possible to create arbitrarily weird situations by choosing strange names for the letters.

gap> f:= FreeGroup( "x", "x" );  gens:= GeneratorsOfGroup( f );;
<free group on the generators [ x, x ]>
gap> gens[1] = gens[2];
false
gap> f:= FreeGroup( "f1*f2", "f2^-1", "Group( [ f1, f2 ] )" );
<free group on the generators [ f1*f2, f2^-1, Group( [ f1, f2 ] ) ]>
gap> gens:= GeneratorsOfGroup( f );;
gap> gens[1]*gens[2];
f1*f2*f2^-1
gap> gens[1]/gens[3];
f1*f2*Group( [ f1, f2 ] )^-1
gap> gens[3]/gens[1]/gens[2];
Group( [ f1, f2 ] )*f1*f2^-1*f2^-1^-1

If G is a group, whose generators are represented by symbols (for example a free group, a finitely presented group or a pc group) this function assigns these generators to global variables with the same names.

The aim of this function to make it easy in interactive use to work with (for example) a free group. It is a shorthand for a sequence of assignments of the form

gap> var1:=GeneratorsOfGroup(G)[1];
gap> var2:=GeneratorsOfGroup(G)[2];
...
gap> varn:=GeneratorsOfGroup(G)[n];

However, since overwriting global variables can be very dangerous, it is not permitted to use this function within a function. (If -- despite this warning -- this is done, the result is undefined.)

If the assignment overwrites existing variables a warning is given, if any of the variables if write protected, or any of the generator names would not be a proper variable name, an error is raised.

35.3 Comparison of Associative Words

Two associative words are equal if they are words over the same alphabet and if they are sequences of the same letters. This is equivalent to saying that the external representations of the words are equal, see The External Representation for Associative Words and Comparison of Words.

There is no ``universal'' empty word, every alphabet (that is, every family of words) has its own empty word.

gap> f:= FreeGroup( "a", "b", "b" );;
gap> gens:= GeneratorsOfGroup(f);
[ a, b, b ]
gap> gens[2] = gens[3];
false
gap> x:= gens[1]*gens[2];
a*b
gap> y:= gens[2]/gens[2]*gens[1]*gens[2];
a*b
gap> x = y;
true
gap> z:= gens[2]/gens[2]*gens[1]*gens[3];
a*b
gap> x = z;
false

The ordering of associative words is defined by length and lexicography (this ordering is called short-lex ordering), that is, shorter words are smaller than longer words, and words of the same length are compared w.r.t. the lexicographical ordering induced by the ordering of generators. Generators are sorted according to the order in which they were created. If the generators are invertible then each generator g is larger than its inverse g^-1, and g^-1 is larger than every generator that is smaller than g.

gap> f:= FreeGroup( 2 );;  gens:= GeneratorsOfGroup( f );;
gap> a:= gens[1];;  b:= gens[2];;
gap> One(f) < a^-1;  a^-1 < a;  a < b^-1;  b^-1 < b; b < a^2;  a^2 < a*b;
true
true
true
true
true
true

returns IsLessThanOrEqualUnder(ord, u, v) where ord is the short less ordering for the family of u and v. (This is here for compatibility with GAP 4.2.)

returns IsLessThanOrEqualUnder(ord, u, v) where ord is the basic wreath product ordering for the family of u and v. (This is here for compatibility with GAP 4.2.)

35.4 Operations for Associative Words

The product of two given associative words is defined as the freely reduced concatenation of the words; so adjacent pairs of a generator and its inverse never occur in words. Besides the multiplication *, the arithmetical operators One (if the word lies in a family with identity) and (if the generators are invertible) Inverse, /,^, Comm, and LeftQuotient are applicable to associative words (see Arithmetic Operations for Elements).

For the operation MappedWord, which is applicable to arbitrary words, see MappedWord.

There are two internal representations of associative words: By letters and by syllables (see Representations for Associative Words). Unless something else is specified, words are stored in the letter representation. Note, however, that operations to extract or act on parts of words (letter or syllables) can carry substantially different costs, depending on the representation the words are in.

For an associative word w, Length returns the number of letters in w.

gap> f := FreeGroup("a","b");; gens := GeneratorsOfGroup(f);;
gap> a := gens[1];; b := gens[2];;w := a^5*b*a^2*b^-4*a;;
gap>  w; Length( w );  Length( a^17 );  Length( w^0 );
a^5*b*a^2*b^-4*a
13
17
0

For an associative word w and a generator gen, ExponentSumWord returns the number of times gen appears in w minus the number of times its inverse appears in w. If both gen and its inverse do not occur in w then 0 is returned. gen may also be the inverse of a generator.

gap> w;  ExponentSumWord( w, a );  ExponentSumWord( w, b );
a^5*b*a^2*b^-4*a
8
-3
gap> ExponentSumWord( (a*b*a^-1)^3, a );  ExponentSumWord( w, b^-1 );
0
3

For an associative word w and two positive integers from and to, Subword returns the subword of w that begins at position from and ends at position to. Indexing is done with origin 1.

gap> w;  Subword( w, 3, 7 );
a^5*b*a^2*b^-4*a
a^3*b*a

Let w and sub be associative words, and from a positive integer. PositionWord returns the position of the first occurrence of sub as a subword of w, starting at position from. If there is no such occurrence, fail is returned. Indexing is done with origin 1.

In other words, PositionWord( w, sub, from ) is the smallest integer i larger than or equal to from such that Subword( w, i, i+Length( sub )-1 ) = sub, see Subword.

gap> w;  PositionWord( w, a/b, 1 );
a^5*b*a^2*b^-4*a
8
gap> Subword( w, 8, 9 );
a*b^-1
gap> PositionWord( w, a^2, 1 );
1
gap> PositionWord( w, a^2, 2 );
2
gap> PositionWord( w, a^2, 6 );
7
gap> PositionWord( w, a^2, 8 );
fail

Let w be an associative word.

In the first form, SubstitutedWord returns the associative word obtained by replacing the subword of w that begins at position from and ends at position to by the associative word by. from and to must be positive integers, indexing is done with origin 1. In other words, SubstitutedWord( w, from, to, by ) is the product of the three words Subword( w, 1, from-1 ), by, and Subword( w, to+1, Length( w ) ), see Subword.

In the second form, SubstitutedWord returns the associative word obtained by replacing the first occurrence of the associative word sub of w, starting at position from, by the associative word by; if there is no such occurrence, fail is returned.

gap> w;  SubstitutedWord( w, 3, 7, a^19 );
a^5*b*a^2*b^-4*a
a^22*b^-4*a
gap> SubstitutedWord( w, a, 6, b^7 );
a^5*b^8*a*b^-4*a
gap> SubstitutedWord( w, a*b, 6, b^7 );
fail

For an associative word w, a generator gen, and an associative word by, EliminatedWord returns the associative word obtained by replacing each occurrence of gen in w by by.

gap> w;  EliminatedWord( w, a, a^2 );  EliminatedWord( w, a, b^-1 );
a^5*b*a^2*b^-4*a
a^10*b*a^4*b^-4*a^2
b^-11

35.5 Operations for Associative Words by their Syllables

For an associative word w = x₁^n₁ x₂^n₂ ¼x_k^n_k over an alphabet containing x₁, x₂, ¼, x_k, such that x_i ¹ x_i+1^±1 for 1 £ i £ k-1, the subwords x_i^e_i are uniquely determined; these powers of generators are called the syllables of w.

NumberSyllables returns the number of syllables of the associative word w.

ExponentSyllable returns the exponent of the i-th syllable of the associative word w.

GeneratorSyllable returns the number of the generator that is involved in the i-th syllable of the associative word w.

SubSyllables returns the subword of the associative word w that consists of the syllables from positions from to to, where from and to must be positive integers, and indexing is done with origin 1.

gap> w;  NumberSyllables( w );
a^5*b*a^2*b^-4*a
5
gap> ExponentSyllable( w, 3 );
2
gap> GeneratorSyllable( w, 3 );
1
gap> SubSyllables( w, 2, 3 );
b*a^2

There are two internal representations of associative words: By letters and by syllables (see Representations for Associative Words). Unless something else is specified, words are stored in the letter representation. Note, however, that operations to extract or act on parts of words (letter or syllables) can carry substantially different costs, depending on the representation the words are in.

35.6 Representations for Associative Words

GAP provides two different internal kinds of representations of associative words. The first one are ``syllable representations'' in which words are stored in syllable (i.e. generator,exponent) form. (Older versions of GAP only used this representation.) The second kind are ``letter representations'' in which each letter in a word is represented by its index number. Negative numbers are used for inverses. Unless the syllable representation is specified explicitly when creating the free group/monoid or semigroup, a letter representation is used by default.

Depending on the task in mind, either of these two representations will perform better in time or in memeory use and algorithms that are syllable or letter absed (for example GeneratorSyllable and Subword) perform substantially better in the corresponding representation. For example when creating pc groups (see Pc groups), it is advantageous to use a syllable representation while calculations in free groups usually benefit from using a letter representation.

A word in letter representation stores a list of generator/inverses numbers (as given by LetterRepAssocWord). Letter access is fast, syllable access is slow for such words.

A letter word family stores words by default in letter form.

Internally, there are letter representations that use integers (4 Byte) to represent a generator and letter representations that use single bytes to represent a character. The latter are more memory efficient, but can only be used if there are less than 128 generators (in which case they are used by default).

these two subrepresentations of IsLetterAssocWordRep indicate whether the word is stored as a list of bytes (in a string) or as a list of integers)

These two subcategories of IsLetterWordsFamily specify the type of letter representation to be used.

A word in syllable representation stores generator/exponents pairs (as given by ExtRepOfObj. Syllable access is fast, letter access is slow for such words.

A syllable word family stores words by default in syllable form.

There are also different versions of syllable representations, which compress a generator exponent pair in 8,16 or 32 bits or use a pair of integers. Internal mechanisms try to make this as memory efficient as possible.

Regardless of the internal representation used, it is possible to convert a word in a list of numbers in letter or syllable representation and vice versa:

The letter representation of an associated word is as a list of integers, each entry corresponding to a group generator. Inverses of the generators are represented by negative numbers. The generator numbers are as associated to the family.

This operation returns the letter representation of the associative word w.

In the second variant, the generator numbers correspond to the generator order given in the list gens.

(For words stored in syllable form the letter representation has to be comnputed.)

takes a letter representation lrep (see LetterRepAssocWord, section LetterRepAssocWord) and returns an associative word in family fam. corresponding to this letter representation.

If gens is given, the numbers in the letter rerpresentation correspond to gens.

gap> w:=AssocWordByLetterRep( FamilyObj(a), [-1,2,1,-2,-2,-2,1,1,1,1]);
a^-1*b*a*b^-3*a^4
gap> LetterRepAssocWord( w^2 );
[ -1, 2, 1, -2, -2, -2, 1, 1, 1, 2, 1, -2, -2, -2, 1, 1, 1, 1 ]

The external representation (see section The External Representation for Associative Words) can be used if a syllable representation is needed.

35.7 The External Representation for Associative Words

The external representation of the associative word w is defined as follows. If w = g_i1^e₁ * g_i2^e₂ * ¼* g_ik^e_k is a word over the alphabet g₁, g₂, ¼, i.e., g_i denotes the i-th generator of the family of w, then w has external representation [ i₁, e₁, i₂, e₂, ¼, i_k, e_k ]. The empty list describes the identity element (if exists) of the family. Exponents may be negative if the family allows inverses. The external representation of an associative word is guaranteed to be freely reduced; for example, g₁ * g₂ * g₂^-1 * g₁ has the external representation [ 1, 2 ].

Regardless of the family preference for letter or syllable representations (see Representations for Associative Words), ExtRepOfObj and ObjByExtRep can be used and interface to this ``syllable''-like representation.

gap> w:= ObjByExtRep( FamilyObj(a), [1,5,2,-7,1,3,2,4,1,-2] );
a^5*b^-7*a^3*b^4*a^-2
gap> ExtRepOfObj( w^2 );
[ 1, 5, 2, -7, 1, 3, 2, 4, 1, 3, 2, -7, 1, 3, 2, 4, 1, -2 ]

35.8 Straight Line Programs

Straight line programs describe an efficient way for evaluating an abstract word at concrete generators, in a more efficient way than with MappedWord (see MappedWord). For example, the associative word ababbab of length 7 can be computed from the generators a, b with only four multiplications, by first computing c = ab, then d = cb, and then cdc; Alternatively, one can compute c = ab, e = bc, and aee. In each step of these computations, one forms words in terms of the words computed in the previous steps.

A straight line program in GAP is represented by an object in the category IsStraightLineProgram (see IsStraightLineProgram) that stores a list of ``lines'' each of which has one of the following three forms.

1. a nonempty dense list l of integers,
2. a pair [ l, i ] where l is a list of form 1. and i is a positive integer,
3. a list [ l₁, l₂, ¼, l_k ] where each l_i is a list of form 1.; this may occur only for the last line of the program.

The lists of integers that occur are interpreted as external representations of associative words (see The External Representation for Associative Words); for example, the list [ 1, 3, 2, -1 ] represents the word g₁³ g₂^-1, with g₁ and g₂ the first and second abstract generator, respectively.

Straight line programs can be constructed using StraightLineProgram (see StraightLineProgram).

Defining attributes for straight line programs are NrInputsOfStraightLineProgram (see NrInputsOfStraightLineProgram) and LinesOfStraightLineProgram (see LinesOfStraightLineProgram). Another operation for straight line programs is ResultOfStraightLineProgram (see ResultOfStraightLineProgram).

Special methods applicable to straight line programs are installed for the operations Display, IsInternallyConsistent, PrintObj, and ViewObj.

For a straight line program prog, the default Display method prints the interpretation of prog as a sequence of assignments of associative words; a record with components gensnames (with value a list of strings) and listname (a string) may be entered as second argument of Display, in this case these names are used, the default for gensnames is [ g1, g2, ¼], the default for listname is r.

Each straight line program in GAP lies in the category IsStraightLineProgram.

In the first form, lines must be a list of lists that defines a unique straight line program (see IsStraightLineProgram); in this case StraightLineProgram returns this program, otherwise an error is signalled. The optional argument nrgens specifies the number of input generators of the program; if a line of form 1. (that is, a list of integers) occurs in lines except in the last position, this number is not determined by lines and therefore must be specified by the argument nrgens; if not then StraightLineProgram returns fail.

In the second form, string must be a string describing an arithmetic expression in terms of the strings in the list gens, where multiplication is denoted by concatenation, powering is denoted by ^, and round brackets (, ) may be used. Each entry in gens must consist only of (uppercase or lowercase) letters (i.e., letters in IsAlphaChar, see IsAlphaChar) such that no entry is an initial part of another one. Called with this input, StraightLineProgramNC returns a straight line program that evaluates to the word corresponding to string when called with generators corresponding to gens.

StraightLineProgramNC does the same as StraightLineProgram, except that the internal consistency of the program is not checked.

For a straight line program prog, LinesOfStraightLineProgram returns the list of program lines. There is no default method to compute these lines if they are not stored.

For a straight line program prog, NrInputsOfStraightLineProgram returns the number of generators that are needed as input.

If a line of form 1. (that is, a list of integers) occurs in the lines of prog except the last line then the number of generators is not determined by the lines, and must be set in the construction of the straight line program (see StraightLineProgram). So if prog contains a line of form 1. other than the last line and does not store the number of generators then NrInputsOfStraightLineProgram signals an error.

ResultOfStraightLineProgram evaluates the straight line program (see IsStraightLineProgram) prog at the group elements in the list gens.

The result of a straight line program with lines p₁, p₂, ¼, p_k when applied to gens is defined as follows.

1. First a list r of intermediate results is initialized with a shallow copy of gens .
2. For i < k, before the i-th step, let r be of length n. If p_i is the external representation of an associative word in the first n generators then the image of this word under the homomorphism that is given by mapping r to these first n generators is added to r; if p_i is a pair [ l, j ], for a list l, then the same element is computed, but instead of being added to r, it replaces the j-th entry of r.
3. For i = k, if p_k is the external representation of an associative word then the element described in (b) is the result of the program, if p_k is a pair [ l, j ], for a list l, then the result is the element described by l, and if p_k is a list [ l₁, l₂, ¼, l_k ] of lists then the result is a list of group elements, where each l_i is treated as in (b).

Here are some examples.

gap> f:= FreeGroup( "x", "y" );;  gens:= GeneratorsOfGroup( f );;
gap> x:= gens[1];;  y:= gens[2];;
gap> prg:= StraightLineProgram( [ [] ] );
<straight line program>
gap> ResultOfStraightLineProgram( prg, [] );
[  ]

gap> StraightLineProgram( [ [1,2,2,3], [3,-1] ] );
fail
gap> prg:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 );
<straight line program>
gap> LinesOfStraightLineProgram( prg );
[ [ 1, 2, 2, 3 ], [ 3, -1 ] ]
gap> prg:= StraightLineProgram( "(a^2b^3)^-1", [ "a", "b" ] );
<straight line program>
gap> LinesOfStraightLineProgram( prg );
[ [ [ 1, 2, 2, 3 ], 3 ], [ [ 3, -1 ], 4 ] ]
gap> res:= ResultOfStraightLineProgram( prg, gens );
y^-3*x^-2
gap> res = (x^2 * y^3)^-1;
true
gap> NrInputsOfStraightLineProgram( prg );
2
gap> Print( prg, "\n" );
StraightLineProgram( [ [ [ 1, 2, 2, 3 ], 3 ], [ [ 3, -1 ], 4 ] ], 2 )
gap> Display( prg );
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[1]^2*r[2]^3;
r[4]:= r[3]^-1;
# return value:
r[4]
gap> IsInternallyConsistent( StraightLineProgramNC( [ [1,2] ] ) );
true
gap> IsInternallyConsistent( StraightLineProgramNC( [ [1,2,3] ] ) );
false
gap> prg1:= StraightLineProgram( [ [1,1,2,2], [3,3,1,1] ], 2 );;
gap> prg2:= StraightLineProgram( [ [ [1,1,2,2], 2 ], [2,3,1,1] ] );;
gap> res1:= ResultOfStraightLineProgram( prg1, gens );
x*y^2*x*y^2*x*y^2*x
gap> res1 = (x*y^2)^3*x;
true
gap> res2:= ResultOfStraightLineProgram( prg2, gens );
x*y^2*x*y^2*x*y^2*x
gap> res2 = (x*y^2)^3*x;
true
gap> prg:= StraightLineProgram( [ [2,3], [ [3,1,1,4], [1,2,3,1] ] ], 2 );;
gap> res:= ResultOfStraightLineProgram( prg, gens );
[ y^3*x^4, x^2*y^3 ]

StringOfResultOfStraightLineProgram returns a string that describes the result of the straight line program (see IsStraightLineProgram) prog as word(s) in terms of the strings in the list gensnames. If the result of prog is a single element then the return value of StringOfResultOfStraightLineProgram is a string consisting of the entries of gensnames, opening and closing brackets ( and ), and powering by integers via ^. If the result of prog is a list of elements then the return value of StringOfResultOfStraightLineProgram is a comma separated concatenation of the strings of the single elements, enclosed in square brackets [, ].

gap> prg:= StraightLineProgram( [ [ 1, 2, 2, 3 ], [ 3, -1 ] ], 2 );;
gap> StringOfResultOfStraightLineProgram( prg, [ "a", "b" ] );
"(a^2b^3)^-1"
gap> StringOfResultOfStraightLineProgram( prg, [ "a", "b" ], "LaTeX" );
"(a^{2}b^{3})^{-1}"

For two straight line programs prog1 and prog2, CompositionOfStraightLinePrograms returns a straight line program prog with the properties that prog1 and prog have the same number of inputs, and the result of prog when applied to given generators gens equals the result of prog2 when this is applied to the output of prog1 applied to gens.

(Of course the number of outputs of prog1 must be the same as the number of inputs of prog2.)

gap> prg1:= StraightLineProgram( "a^2b", [ "a","b" ] );;
gap> prg2:= StraightLineProgram( "c^5", [ "c" ] );;
gap> comp:= CompositionOfStraightLinePrograms( prg2, prg1 );
<straight line program>
gap> StringOfResultOfStraightLineProgram( comp, [ "a", "b" ] );
"(a^2b)^5"
gap> prg:= StraightLineProgram( [ [2,3], [ [3,1,1,4], [1,2,3,1] ] ], 2 );;
gap> StringOfResultOfStraightLineProgram( prg, [ "a", "b" ] );
"[ b^3a^4, a^2b^3 ]"
gap> comp:= CompositionOfStraightLinePrograms( prg, prg );
<straight line program>
gap> StringOfResultOfStraightLineProgram( comp, [ "a", "b" ] );
"[ (a^2b^3)^3(b^3a^4)^4, (b^3a^4)^2(a^2b^3)^3 ]"

For a nonempty dense list listofprogs of straight line programs that have the same number n, say, of inputs (see NrInputsOfStraightLineProgram) and for which the results (see ResultOfStraightLineProgram) are single elements (i.e., not lists of elements), IntegratedStraightLineProgram returns a straight line program prog with n inputs such that for each n-tuple gens of generators, ResultOfStraightLineProgram( prog, gens ) is equal to the list List( listofprogs, p -> ResultOfStraightLineProgram( p, gens ).

gap> f:= FreeGroup( "x", "y" );;  gens:= GeneratorsOfGroup( f );;
gap> prg1:= StraightLineProgram( [ [ [ 1, 2 ], 1 ], [ 1, 2, 2, -1 ] ], 2 );;
gap> prg2:= StraightLineProgram( [ [ [ 2, 2 ], 3 ], [ 1, 3, 3, 2 ] ], 2 );;
gap> prg3:= StraightLineProgram( [ [ 2, 2 ], [ 1, 3, 3, 2 ] ], 2 );;
gap> prg:= IntegratedStraightLineProgram( [ prg1, prg2, prg3 ] );;
gap> ResultOfStraightLineProgram( prg, gens );
[ x^4*y^-1, x^3*y^4, x^3*y^4 ]
gap> prg:= IntegratedStraightLineProgram( [ prg2, prg3, prg1 ] );;
gap> ResultOfStraightLineProgram( prg, gens );
[ x^3*y^4, x^3*y^4, x^4*y^-1 ]
gap> prg:= IntegratedStraightLineProgram( [ prg3, prg1, prg2 ] );;
gap> ResultOfStraightLineProgram( prg, gens );
[ x^3*y^4, x^4*y^-1, x^3*y^4 ]

35.9 Straight Line Program Elements

When computing with very large (in terms of memory) elements, for example permutations of degree a few hundred thousands, it can be helpful (in terms of memory usage) to represent them via straight line programs in terms of an original generator set. (So every element takes only small extra storage for the straight line program.)

A straight line program element has a seed (a list of group elements) and a straight line program on the same number of generators as the length of this seed, its value is the value of the evaluated straight line program.

At the moment, the entries of the straight line program have to be simple lists (i.e. of the first form).

Straight line program elements are in the same categories and families as the elements of the seed, so they should work together with existing algorithms.

Note however, that due to the different way of storage some normally very cheap operations (such as testing for element equality) can become more expensive when dealing with straight line program elements. This is essentially the tradeoff for using less memory.

A straight line program element is a group element given (for memory reasons) as a straight line program. Straight line program elements are positional objects, the first component is a record with a component seeds, the second component the straight line program. we need to rank higher than default methods

Creates a straight line program element for seed seed and program prog.

returns a set of straight line program elements corresponding to the generators in gens. If gens is a set of permutations then base can be given which must be a base for the group generated by gens. (Such a base will be used to speed up equality tests.)

evaluates a straight line program element slpel from its seeds.

If elm is a straight line program element whose straight line representation is very long, this operation changes the representation of elm to a straight line program element, equal to elm, whose seed contains the evaluation of elm and whose straight line program has length 1.

For other objects nothing happens.

This operation permits to designate ``important'' elements within an algorithm (elements that wil be referred to often), which will be represented by guaranteed short straight line program elements.

gap> gens:=StraightLineProgGens([(1,2,3,4),(1,2)]);
[ <[ [ 2, 1 ] ]|(1,2,3,4)>, <[ [ 1, 1 ] ]|(1,2)> ]
gap> g:=Group(gens);;
gap> (gens[1]^3)^gens[2];
<[ [ 1, -1, 2, 3, 1, 1 ] ]|(1,2,4,3)>
gap> Size(g);
24
gap> Random(g);
<[ [ 1, -1, 2, -1, 1, 1, 2, -1, 1, -1, 2, 1, 1, 1, 2, 1, 1, -1, 2, 2, 1, 1 ], 
   [ 3, -2, 2, -2, 1, -1, 2, -2, 1, 1, 2, -1, 3, -2, 1, -1, 2, -2, 1, 1 ] ]
  | (1,4,3,2)>

See also Section Working with large degree permutation groups.

[Top] [Up] [Previous] [Next] [Index]

GAP 4 manual
May 2002