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### 2 Double Coset Rewriting Systems

The Kan package provides functions for the computation of normal forms for double coset representatives of finitely presented groups. The first version of the package was released to support the paper [BGHW06], which describes the algorithms used in this package.

#### 2.1 Rewriting Systems

##### 2.1-1 KnuthBendixRewritingSystem
 ‣ KnuthBendixRewritingSystem( grp, gensorder, ordering, alph ) ( operation )
 ‣ ReducedConfluentRewritingSystem( grp, gensorder, ordering, limit ) ( operation )
 ‣ DisplayRwsRules( rws ) ( operation )

Methods for KnuthBendixRewritingSystem and ReducedConfluentRewritingSystem are supplied which apply to a finitely presented group. These start by calling IsomorphismFpMonoid and then work with the resulting monoid. The parameter gensorder will normally be "shortlex" or "wreath", while ordering is an integer list for reordering the generators, and alph is an alphabet string used when printing words. A partial rewriting system may be obtained by giving a limit to the number of rules calculated. As usual, A,B denote the inverses of a,b.

In the example the generators are by default ordered [A,a,B,b], so the list L1 is used to specify the order [a,A,b,B] to be used with the shortlex ordering. Specifying a limit 0 means that no limit is prescribed.


gap> G1 := FreeGroup( 2 );;
gap> L1 := [2,1,4,3];;
gap> order := "shortlex";;
gap> alph1 := "AaBb";;
gap> rws1 := ReducedConfluentRewritingSystem( G1, L1, order, 0, alph1 );
Rewriting System for Monoid( [ f1, f1^-1, f2, f2^-1 ], ... ) with rules
[ [ f1*f1^-1, <identity ...> ], [ f1^-1*f1, <identity ...> ],
[ f2*f2^-1, <identity ...> ], [ f2^-1*f2, <identity ...> ] ]
gap> DisplayRwsRules( rws1 );;
[ [ Aa, id ], [ aA, id ], [ Bb, id ], [ bB, id ] ]



#### 2.2 Example 1 -- free product of two cyclic groups

##### 2.2-1 DoubleCosetRewritingSystem
 ‣ DoubleCosetRewritingSystem( grp, genH, genK, rws ) ( function )
 ‣ IsDoubleCosetRewritingSystem( dcrws ) ( property )

A double coset rewriting system for the double cosets H backslash G / K requires as data a finitely presented group G =grp; generators genH, genK for subgroups H, K; and a rewriting system rws for G.

A simple example is given by taking G to be the free group on two generators a,b, a cyclic subgroup H with generator a^6, and a second cyclic subgroup K with generator a^4. (Similar examples using different powers of a are easily constructed.) Since gcd(6,4)=2, we have Ha^2K=HK, so the double cosets have representatives [HK, HaK, Ha^iba^jK, Ha^ibwba^jK] where i ∈ [0..5], j ∈ [0..3], and w is any word in a,b.

In the example the free group G is converted to a four generator monoid with relations defining the inverse of each generator, [[Aa,id],[aA,id],[Bb,id],[bB,id]], where id is the empty word. The initial rules for the double coset rewriting system comprise those of G plus those given by the generators of H,K, which are [[Ha^6,H],[a^4K,K]]. In the complete rewrite system new rules involving H or K may arise, and there may also be rules involving both H and K.

For this example,

• there are two H-rules, [[Ha^4,HA^2],[HA^3,Ha^3]],

• there are two K-rules, [[a^3K,AK],[A^2K,a^2K]],

• and there are two H-K-rules, [[Ha^2K,HK],[HAK,HaK]].

Here is how the computation may be performed.


gap> genG1 := GeneratorsOfGroup( G1 );;
gap> genH1 := [ genG1[1]^6 ];;
gap> genK1 := [ genG1[1]^4 ];;
gap> dcrws1 := DoubleCosetRewritingSystem( G1, genH1, genK1, rws1 );;
gap> IsDoubleCosetRewritingSystem( dcrws1 );
true
gap> DisplayRwsRules( dcrws1 );;
G-rules:
[ [ Aa, id ], [ aA, id ], [ Bb, id ], [ bB, id ] ]
H-rules:
[ [ Haaaa, HAA ],
[ HAAA, Haaa ] ]
K-rules:
[ [ aaaK, AK ],
[ AAK, aaK ] ]
H-K-rules:
[ [ HaaK, HK ],
[ HAK, HaK ] ]



##### 2.2-2 WordAcceptorOfReducedRws
 ‣ WordAcceptorOfReducedRws( rws ) ( attribute )
 ‣ WordAcceptorOfDoubleCosetRws( rws ) ( attribute )
 ‣ IsWordAcceptorOfDoubleCosetRws( aut ) ( property )

Using functions from the automata package, we may

• compute a word acceptor for the rewriting system of G;

• compute a word acceptor for the double coset rewriting system;

• test a list of words to see whether they are recognised by the automaton;

• obtain a rational expression for the language of the automaton.


gap> waG1 := WordAcceptorOfReducedRws( rws1 );
Automaton("det",6,"aAbB",[ [ 1, 4, 1, 4, 4, 4 ], [ 1, 3, 3, 1, 3, 3 ], [ 1, 2,\
2, 2, 1, 2 ], [ 1, 1, 5, 5, 5, 5 ] ],[ 6 ],[ 2, 3, 4, 5, 6 ]);;
gap> wadc1 := WordAcceptorOfDoubleCosetRws( dcrws1 );
< deterministic automaton on 6 letters with 15 states >
Automaton("det",15,"HKaAbB",[ [ 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],\
[ 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2 ], [ 2, 2, 13, 2, 10, 5, 2, 13,\
2, 7, 11, 11, 12, 2, 2 ], [ 2, 2, 9, 2, 2, 14, 2, 9, 15, 2, 2, 2, 2, 7, 15 ],\
[ 2, 2, 2, 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8 ], [ 2, 2, 3, 2, 3, 3, 3, 2, 3,\
3, 3, 3, 3, 3, 3 ] ],[ 4 ],[ 1 ]);;
gap> words1 := [ "HK","HaK","HbK","HAK","HaaK","HbbK","HabK","HbaK","HbaabK"];;
gap> valid1 := List( words1, w -> IsRecognizedByAutomaton( wadc1, w ) );
[ true, true, true, false, false, true, true, true, true ]
gap> lang1 := FAtoRatExp( wadc1 );
((H(aaaUAA)BUH(a(aBUB)UABUB))(a(a(aa*BUB)UB)UA(AA*BUB)UB)*(a(a(aa*bUb)Ub)UA(AA\
*bUb))UH(aaaUAA)bUH(a(abUb)UAbUb))((a(a(aa*BUB)UB)UA(AA*BUB))(a(a(aa*BUB)UB)UA\
(AA*BUB)UB)*(a(a(aa*bUb)Ub)UA(AA*bUb))Ua(a(aa*bUb)Ub)UA(AA*bUb)Ub)*((a(a(aa*BU\
B)UB)UA(AA*BUB))(a(a(aa*BUB)UB)UA(AA*BUB)UB)*(a(aKUK)UAKUK)Ua(aKUK)UAKUK)U(H(a\
aaUAA)BUH(a(aBUB)UABUB))(a(a(aa*BUB)UB)UA(AA*BUB)UB)*(a(aKUK)UAKUK)UH(aKUK)



#### 2.3 Example 2 -- the trefoil group

##### 2.3-1 PartialDoubleCosetRewritingSystem
 ‣ PartialDoubleCosetRewritingSystem( grp, Hgens, Kgens, rws, limit ) ( operation )
 ‣ WordAcceptorOfPartialDoubleCosetRws( grp, prws ) ( attribute )

It may happen that, even when G has a finite rewriting system, the double coset rewriting system is infinite. This is the case with the trefoil group T with generators [x,y] and relator [x^3 = y^2] when the wreath product ordering is used with X > x > Y > y. The group itself has a rewriting system with just 6 rules.


gap> FT := FreeGroup( 2 );;
gap> relsT := [ FT.1^3*FT.2^-2 ];;  T := FT/relsT;;
gap> genT := GeneratorsOfGroup( T );;  x := genT[1];  y := genT[2];
gap> alphT := "XxYy";;  ordT := [4,3,2,1];;  orderT := "wreath";;
gap> rwsT := ReducedConfluentRewritingSystem( T, ordT, orderT, 0, alphT );;
gap> DisplayRwsRules( rwsT );;
[ [ Yy, id ], [ yY, id ], [ xxx, yy ], [ yyx, xyy ], [ X, xxYY ], [ Yx, yxYY ]\
]
gap> accT := WordAcceptorOfReducedRws( rwsT );
< deterministic automaton on 4 letters with 7 states >
gap> Print( "accT = ", accT );
accT = Automaton("det",7,"yYxX",[ [ 1, 2, 1, 5, 2, 5, 5 ], [ 1, 1, 3, 3, 1, 3,\
3 ], [ 1, 1, 1, 7, 7, 1, 6 ], [ 1, 1, 1, 1, 1, 1, 1 ] ],[ 4 ],[ 2, 3, 4, 5, 6\
, 7 ]);;
gap> langT := FAtoRatExp( accT );
(yxUx)((xyUy)x)*((xyUy)(yy*U@)Ux(YY*U@)UYY*U@)Uy(yy*U@)UYY*U@
gap> IsRecognizedByAutomaton( accT, "X" );
false
gap> IsRecognizedByAutomaton( accT, "yxyxyxYY" );
true



In versions of Kan, from about 1.01 up to 1.21, the complementary automaton and language were returned in the example above. This error has now been rectified.

In earlier versions of Kan (in 0.95 for example) a shorter rational expression for the language was obtained from Automata. In what follows, we check that the two expressions define the same language.


gap> alph := AlphabetOfRatExpAsList( langT );;
gap> a1 := RatExpOnnLetters( alph, [ ], [1] );;   ## y
gap> a2 := RatExpOnnLetters( alph, [ ], [2] );;   ## Y
gap> a3 := RatExpOnnLetters( alph, [ ], [3] );;   ## x
gap> a4 := RatExpOnnLetters( alph, [ ], [4] );;   ## X
gap> s1 := RatExpOnnLetters( alph, "star", a1 );; ## y*
gap> s2 := RatExpOnnLetters( alph, "star", a2 );; ## Y*
gap> a1a3 := RatExpOnnLetters( alph, "product", [ a1, a3 ] );;  ## yx
gap> u1 := RatExpOnnLetters( alph, "union", [ a1a3, a3 ] );;    ## yxUx
gap> a3a1 := RatExpOnnLetters( alph, "product", [ a3, a1 ] );;  ## xy
gap> u2 := RatExpOnnLetters( alph, "union", [ a3a1, a1 ] );;    ## xyUy
gap> u2a3 := RatExpOnnLetters( alph, "product", [ u2, a3 ] );;  ## (xyUy)x
gap> su2a3 := RatExpOnnLetters( alph, "star", u2a3 );;          ## ((xyUy)x)*
gap> u2s1 := RatExpOnnLetters( alph, "product", [ u2, s1 ] );;  ## (xyUy)y*
gap> a3s2 := RatExpOnnLetters( alph, "product", [ a3, s2 ] );;  ## xY*
gap> u3 := RatExpOnnLetters( alph, "union", [u2s1,a3s2,s2] );;
gap> prod := RatExpOnnLetters( alph, "product", [u1,su2a3,u3] );;
gap> a1s1 := RatExpOnnLetters( alph, "product", [ a1, s1 ] );;  ## yy*
gap> r := RatExpOnnLetters( alph, "union", [ prod, a1s1, s2] );
(yxUx)((xyUy)x)*((xyUy)y*UxY*UY*)Uyy*UY*
gap> AreEqualLang( langT, r );
true



Taking subgroups H, K to be generated by x and y respectively, the double coset rewriting system has an infinite number of H-rules. It turns out that only a finite number of these are needed to produce the required automaton. The function PartialDoubleCosetRewritingSystem allows a limit to be specified on the number of rules to be computed. In the listing below a limit of 20 is used, but in fact 10 is sufficient.


gap> prwsT := PartialDoubleCosetRewritingSystem( T, [x], [y], rwsT, 20 );;
#I WARNING: reached supplied limit 20 on number of rules
gap> DisplayRwsRules( prwsT );
G-rules:
[ [ X, xxYY ], [ Yx, yxYY ], [ Yy, id ], [ yY, id ], [ xxx, yy ], [ yyx, xyy ]\
]
H-rules:
[ [ Hx, H ],
[ HY, Hy ],
[ Hyy, H ],
[ Hyxyy, Hyx ],
[ HyxY, Hyxy ],
[ Hyxyxyy, Hyxyx ],
[ Hyxxyy, Hyxx ],
[ HyxxY, Hyxxy ],
[ HyxyxY, Hyxyxy ],
[ Hyxyxyxyy, Hyxyxyx ],
[ Hyxyxxyy, Hyxyxx ],
[ Hyxxyxyy, Hyxxyx ],
[ HyxxyxYY, Hyxxyx ] ]
K-rules:
[ [ YK, K ],
[ yK, K ] ]



This list of partial rules is then used by a modified word acceptor function.


gap> paccT := WordAcceptorOfPartialDoubleCosetRws( T, prwsT );;
< deterministic automaton on 6 letters with 6 states >
gap> Print( paccT, "\n" );
Automaton("det",6,"HKyYxX",[ [ 2, 2, 2, 6, 2, 2 ], [ 2, 2, 1, 2, 2, 1 ], [ 2, \
2, 5, 2, 2, 5 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 6, 2, 3, 2 ], [ 2, 2, 2, 2, 2, \
2 ] ],[ 4 ],[ 1 ]);;
gap> plangT := FAtoRatExp( paccT );
H(yx(yx)*x)*(yx(yx)*KUK)
gap> wordsT := ["HK", "HxK", "HyK", "HYK", "HyxK", "HyxxK", "HyyH", "HyxYK"];;
gap> validT := List( wordsT, w -> IsRecognizedByAutomaton( paccT, w ) );
[ true, false, false, false, true, true, false, false ]



#### 2.4 Example 3 -- an infinite rewriting system

##### 2.4-1 KBMagRewritingSystem
 ‣ KBMagRewritingSystem( fpgrp ) ( attribute )
 ‣ KBMagWordAcceptor( fpgrp ) ( attribute )
 ‣ KBMagFSAtoAutomataDFA( fsa, alph ) ( operation )
 ‣ WordAcceptorByKBMag( grp, alph ) ( operation )
 ‣ WordAcceptorByKBMagOfDoubleCosetRws( grp, dcrws ) ( operation )

When the group G has an infinite rewriting system, the double coset rewriting system will also be infinite. In this case we may use the function KBMagWordAcceptor which calls KBMAG to compute a word acceptor for G, and KBMagFSAtoAutomataDFA to convert this to a deterministic automaton as used by the automata package. The resulting dfa forms part of the double coset automaton, together with sufficient H-rules, K-rules and H-K-rules found by the function PartialDoubleCosetRewritingSystem. (Note that these five attributes and operations will not be available if the kbmag package has not been loaded.)

In the following example we take a two generator group G3 with relators [a^3,b^3,(a*b)^3], the normal forms of whose elements are some of the strings with a or a^-1 alternating with b or b^-1. The automatic structure computed by KBMAG has a word acceptor with 17 states.


gap> F3 := FreeGroup("a","b");;
gap> rels3 := [ F3.1^3, F3.2^3, (F3.1*F3.2)^3 ];;
gap> G3 := F3/rels3;;
gap> alph3 := "AaBb";;
gap> waG3 := WordAcceptorByKBMag( G3, alph3 );;
gap> Print( waG3, "\n");
Automaton("det",18,"aAbB",[ [ 2, 18, 18, 8, 10, 12, 13, 18, 18, 18, 18, 18, 18\
, 8, 17, 12, 18, 18 ], [ 3, 18, 18, 9, 11, 9, 12, 18, 18, 18, 18, 18, 18, 11, \
18, 11, 18, 18 ], [ 4, 6, 6, 18, 18, 18, 18, 18, 6, 12, 16, 18, 12, 18, 18, 18\
, 12, 18 ], [ 5, 5, 7, 18, 18, 18, 18, 14, 15, 5, 18, 18, 7, 18, 18, 18, 15, 1\
8 ] ],[ 1 ],[ 1 .. 17 ]);;
gap> langG3 := FAtoRatExp( waG3 );
((abUAb)AUbA)(bA)*(b(aU@)UB(aB)*(a(bU@)U@)U@)U(abUAb)(aU@)U((aBUB)(aB)*AUba(Ba\
)*BA)(bA)*(b(aU@)U@)U(aBUB)(aB)*(a(bU@)U@)Uba(Ba)*(BU@)UbUaUA(B(aB)*(a(bU@)UAU\
@)U@)U@



##### 2.4-2 DCrules
 ‣ DCrules( dcrws ) ( operation )
 ‣ Hrules( dcrws ) ( attribute )
 ‣ Krules( dcrws ) ( attribute )
 ‣ HKrules( dcrws ) ( attribute )

We now take H to be generated by ab and K to be generated by ba. If we specify a limits of 50, 75, 100, 200 for the number of rules in a partial double coset rewrite system, we obtain lists of H-rules, K-rules and H-K-rules of increasing length. The numbers of states in the resulting automata also increase. We may deduce by hand (but not computationally -- see [BGHW06]) three infinite sets of rules and a limit for the automata.


gap> lim := 100;;
gap> genG3 := GeneratorsOfGroup( G3 );;
gap> a := genG3[1];;  b := genG3[2];;
gap> gH3 := [ a*b ];;  gK3 := [ b*a ];;
gap> rwsG3 := KnuthBendixRewritingSystem( G3, "shortlex", [2,1,4,3], alph3 );;
gap> dcrws3 := PartialDoubleCosetRewritingSystem( G3, gH3, gK3, rwsG3, lim );;
#I using PartialDoubleCosetRewritingSystem with limit 100
#I  51 rules, and 1039 pairs
#I WARNING: reached supplied limit 100 on number of rules
gap> Print( Length( Rules( dcrws3 ) ), " rules found.\n" );
101 rules found.
gap> dcaut3 := WordAcceptorByKBMagOfDoubleCosetRws( G3, dcrws3 );;
gap> Print( "Double Coset Minimalized automaton:\n", dcaut3 );
Double Coset Minimalized automaton:
Automaton("det",44,"HKaAbB",[ [ 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2\
, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2\
, 2, 2 ], [ 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, \
2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1 ], [ 2, 2, 2,\
2, 3, 24, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 43, 2, 43, 2, 27, 2, 2, 2\
, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 44, 3, 29, 2\
, 8, 2, 10, 2, 12, 2, 14, 2, 16, 2, 18, 2, 20, 2, 22, 2, 2, 2, 2, 26, 2, 29, 2\
, 31, 2, 33, 2, 35, 2, 37, 2, 39, 2, 41, 2, 2 ], [ 2, 2, 2, 2, 21, 2, 2, 28, 2\
, 9, 2, 11, 2, 13, 2, 15, 2, 17, 2, 19, 2, 42, 2, 3, 2, 28, 3, 2, 7, 2, 30, 2,\
32, 2, 34, 2, 36, 2, 38, 2, 40, 2, 2, 28 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2\
, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 25, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2,\
2, 2, 2, 2, 2, 2, 23, 6 ] ],[ 4 ],[ 1 ]);;
gap> dclang3 := FAtoRatExp( dcaut3 );;
gap> Print( "Double Coset language of acceptor:\n", dclang3, "\n" );
Double Coset language of acceptor:
(HbAbAbAbAbAbAbAbUHAb)(Ab)*(A(Ba(Ba)*bKUK)UK)UHbAbA(bA(bA(bA(bA(bAKUK)UK)UK)UK\
)UK)UH(A(B(aB)*(abUA)KUK)UaKUb(a(Ba)*BA(bA(bA(bA(bA(bA(bA(bA(bA)*(bKUK)UK)UK)U\
K)UK)UK)UK)UK)UAK)UK)
gap> ok := DCrules(dcrws3);;
gap> alph3e := dcrws3!.alphabet;;
gap> Print("H-rules:\n");  DisplayAsString( Hrules(dcrws3), alph3e, true );
H-rules:
[ HB, Ha ]
[ HaB, Hb ]
[ Hab, H ]
[ HbAB, HAba ]
[ HbAbAB, HAbAba ]
[ HbAbAbAB, HAbAbAba ]
[ HbAbAbAbAB, HAbAbAbAba ]
[ HbAbAbAbAbAB, HAbAbAbAbAba ]
[ HbAbAbAbAbAbAB, HAbAbAbAbAbAba ]
[ HbAbAbAbAbAbAbAB, HAbAbAbAbAbAbAba ]
gap> Print("K-rules:\n");  DisplayAsString( Krules(dcrws3), alph3e, true );;
K-rules:
[ BK, aK ]
[ BaK, bK ]
[ baK, K ]
[ BAbK, abAK ]
[ BAbAbK, abAbAK ]
[ BAbAbAbK, abAbAbAK ]
[ BAbAbAbAbK, abAbAbAbAK ]
[ BAbAbAbAbAbK, abAbAbAbAbAK ]
[ BAbAbAbAbAbAbK, abAbAbAbAbAbAK ]
[ BAbAbAbAbAbAbAbK, abAbAbAbAbAbAbAK ]
gap> Print("HK-rules:\n");  DisplayAsString( HKrules(dcrws3), alph3e, true );;
HK-rules:
[ HbK, HAK ]
[ HbAbK, HAbAK ]
[ HbAbAbK, HAbAbAK ]
[ HbAbAbAbK, HAbAbAbAK ]
[ HbAbAbAbAbK, HAbAbAbAbAK ]
[ HbAbAbAbAbAbK, HAbAbAbAbAbAK ]
[ HbAbAbAbAbAbAbK, HAbAbAbAbAbAbAK ]



##### 2.4-3 NextWord
 ‣ NextWord( dcrws, word ) ( operation )
 ‣ WordToString( word, alph ) ( operation )
 ‣ DisplayAsString( word, alph ) ( operation )
 ‣ IdentityDoubleCoset( dcrws ) ( operation )

These functions may be used to find normal forms of increasing length for double coset representatives.


gap> len := 30;;
gap> L3 := 0*[1..len];;
gap> L3[1] := IdentityDoubleCoset( dcrws3 );;
gap> for i in [2..len] do
gap>     L3[i] := NextWord( dcrws3, L3[i-1] );
gap> od;
gap> ## List of 30 normal forms for double cosets:
gap> DisplayAsString( L3, alph3e, true );
[ HK, HAK, HaK, HAbK, HbAK, HABAK, HAbAK, HABabK, HAbAbK, HbAbAK, HbaBAK, HABa\
BAK, HAbAbAK, HABaBabK, HAbABabK, HAbAbAbK, HbAbAbAK, HbaBAbAK, HbaBaBAK, HABa\
BaBAK, HAbAbAbAK, HABaBaBabK, HAbABaBabK, HAbAbABabK, HAbAbAbAbK, HbAbAbAbAK, \
HbaBAbAbAK, HbaBaBAbAK, HbaBaBaBAK, HABaBaBaBAK ]
gap> w := NextWord( dcrws3, L3[30] );
m1*(m3*m6)^4*m3*m2
gap> s := WordToString( w, alph3e );
"HAbAbAbAbAK"


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