  
  [1X2. Double Coset Rewriting Systems[0X
  
  The  [5Xkan[0X  package provides functions for the computation of normal forms for
  double  coset  representatives  of  finitely  presented  groups.  This first
  version  of  the  package  has  been released to support the paper [BGHW06],
  which describes the algorithms used in this package.
  
  
  [1X2.1 Rewriting Systems[0X
  
  [1X2.1-1 KnuthBendixRewritingSystem[0X
  
  [2X> KnuthBendixRewritingSystem( [0X[3Xgrp, gensorder, ordering, alph[0X[2X ) ____[0Xoperation
  [2X> ReducedConfluentRewritingSystem( [0X[3Xgrp, gensorder, ordering, limit[0X[2X ) [0Xoperation
  [2X> DisplayRwsRules( [0X[3Xrws[0X[2X ) __________________________________________[0Xoperation
  
  Methods  for  [10XKnuthBendixRewritingSystem[0X and [10XReducedConfluentRewritingSystem[0X
  are  supplied  which  apply  to  a  finitely presented group. These start by
  calling  [10XIsomorphismFpMonoid[0X  and  then  work with the resulting monoid. The
  parameter  [10Xgensorder[0X will normally be [10X"shortlex"[0X or [10X"wreath"[0X, while [10Xordering[0X
  is  an  integer  list for reordering the generators, and [10Xalph[0X is an alphabet
  string  used when printing words. A [13Xpartial[0X rewriting system may be obtained
  by giving a [10Xlimit[0X to the number of rules calculated.
  
  In  the example the generators are by default ordered [A,a,B,b], so the list
  [10XL1[0X  is  used  to  specify  the  order [10X[a,A,b,B][0X to be used with the shortlex
  ordering. Specifying a limit [10X0[0X means that no limit is prescribed.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> G1 := FreeGroup( 2 );;[0X
    [4Xgap> L1 := [2,1,4,3];;[0X
    [4Xgap> order := "shortlex";;[0X
    [4Xgap> alph1 := "AaBb";;[0X
    [4Xgap> rws1 := ReducedConfluentRewritingSystem( G1, L1, order, 0, alph1 );[0X
    [4XRewriting System for Monoid( [ f1^-1, f1, f2^-1, f2 ], ... ) with rules[0X
    [4X[ [ f1^-1*f1, <identity ...> ], [ f1*f1^-1, <identity ...> ],[0X
    [4X  [ f2^-1*f2, <identity ...> ], [ f2*f2^-1, <identity ...> ] ][0X
    [4Xgap> DisplayRwsRules( rws1 );;[0X
    [4X[ [ Aa, id ], [ aA, id ], [ Bb, id ], [ bB, id ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X2.2 Example 1 -- free product of two cyclic groups[0X
  
  [1X2.2-1 DoubleCosetRewritingSystem[0X
  
  [2X> DoubleCosetRewritingSystem( [0X[3Xgrp, genH, genK, rws[0X[2X ) _______________[0Xfunction
  [2X> IsDoubleCosetRewritingSystem( [0X[3Xdcrws[0X[2X ) ____________________________[0Xproperty
  
  A  [13Xdouble  coset  rewriting  system[0X  for the double cosets H backslash G / K
  requires  as data a finitely presented group G or [10Xgrp[0X, generators [10XgenH, genK[0X
  for subgroups H, K, and a rewriting system [10Xrws[0X 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 [10Xgcd(6,4)=2[0X, we have Ha^2K=HK, so the double cosets have
  representatives  [HK,  HaK,  Ha^iba^jK, Ha^ibwba^jK] where i in [0..5], j in
  [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,
  [10X[[Aa,id],[aA,id],[Bb,id],[bB,id]][0X,  where  [10Xid[0X 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.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> genG1 := GeneratorsOfGroup( G1 );;[0X
    [4Xgap> genH1 := [ genG1[1]^6 ];;[0X
    [4Xgap> genK1 := [ genG1[1]^4 ];;[0X
    [4Xgap> dcrws1 := DoubleCosetRewritingSystem( G1, genH1, genK1, rws1 );;[0X
    [4Xgap> IsDoubleCosetRewritingSystem( dcrws1 );[0X
    [4Xtrue[0X
    [4Xgap> DisplayRwsRules( dcrws1 );;[0X
    [4XG-rules:[0X
    [4X[ [ Aa, id ], [ aA, id ], [ Bb, id ], [ bB, id ] ][0X
    [4XH-rules:[0X
    [4X[ [ Haaaa, HAA ],[0X
    [4X  [ HAAA, Haaa ] ][0X
    [4XK-rules:[0X
    [4X[ [ aaaK, AK ],[0X
    [4X  [ AAK, aaK ] ][0X
    [4XH-K-rules:[0X
    [4X[ [ HaaK, HK ],[0X
    [4X  [ HAK, HaK ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.2-2 WordAcceptorOfReducedRws[0X
  
  [2X> WordAcceptorOfReducedRws( [0X[3Xrws[0X[2X ) _________________________________[0Xattribute
  [2X> WordAcceptorOfDoubleCosetRws( [0X[3Xrws[0X[2X ) _____________________________[0Xattribute
  [2X> IsWordAcceptorOfDoubleCosetRws( [0X[3Xaut[0X[2X ) ____________________________[0Xproperty
  
  Using functions from the [5Xautomata[0X package, we may
  
  --    compute a [13Xword acceptor[0X for the rewriting system of G;
  
  --    compute a [13Xword acceptor[0X for the double coset rewriting system;
  
  --    test  a  list  of  words  to  see  whether  they are recognised by the
        automaton;
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> waG1 := WordAcceptorOfReducedRws( rws1 );[0X
    [4XAutomaton("det",6,"aAbB",[ [ 1, 4, 1, 4, 4, 4 ], [ 1, 3, 3, 1, 3, 3 ], [ 1, 2,\[0X
    [4X 2, 2, 1, 2 ], [ 1, 1, 5, 5, 5, 5 ] ],[ 6 ],[ 2, 3, 4, 5, 6 ]);;[0X
    [4Xgap> wadc1 := WordAcceptorOfDoubleCosetRws( dcrws1 );[0X
    [4X< deterministic automaton on 6 letters with 15 states >[0X
    [4Xgap> Print( wadc1 );[0X
    [4XAutomaton("det",15,"HKaAbB",[ [ 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],\[0X
    [4X [ 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2 ], [ 2, 2, 13, 2, 10, 5, 2, 13,\[0X
    [4X 2, 7, 11, 11, 12, 2, 2 ], [ 2, 2, 9, 2, 2, 14, 2, 9, 15, 2, 2, 2, 2, 7, 15 ],\[0X
    [4X [ 2, 2, 2, 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8 ], [ 2, 2, 3, 2, 3, 3, 3, 2, 3,\[0X
    [4X 3, 3, 3, 3, 3, 3 ] ],[ 4 ],[ 1 ]);;[0X
    [4Xgap> words1 := [ "HK","HaK","HbK","HAK","HaaK","HbbK","HabK","HbaK","HbaabK"];;[0X
    [4Xgap> valid1 := List( words1, w -> IsRecognizedByAutomaton( wadc1, w ) );[0X
    [4X[ true, true, true, false, false, true, true, true, true ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.2-3 SFAtoRatExp[0X
  
  [2X> SFAtoRatExp( [0X[3Xaut[0X[2X ) _______________________________________________[0Xfunction
  
  The  function [10XSFAtoRatExp[0X comes from the development version of the [5Xautomata[0X
  package, and has been supplied by the authors of that package. It computes a
  regular expression for the language recognised by the automaton [10Xaut[0X.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> lang1 := SFAtoRatExp( wadc1 );[0X
    [4X((H(aaaUAA)BUH(a(aBUB)UABUB))(a(a(aa*BUB)UB)UA(AA*BUB)UB)*(a(a(aa*bUb)Ub)UA(AA\[0X
    [4X*bUb))UH(aaaUAA)bUH(a(abUb)UAbUb))((a(a(aa*BUB)UB)UA(AA*BUB))(a(a(aa*BUB)UB)UA\[0X
    [4X(AA*BUB)UB)*(a(a(aa*bUb)Ub)UA(AA*bUb))Ua(a(aa*bUb)Ub)UA(AA*bUb)Ub)*((a(a(aa*BU\[0X
    [4XB)UB)UA(AA*BUB))(a(a(aa*BUB)UB)UA(AA*BUB)UB)*(a(aKUK)UAKUK)Ua(aKUK)UAKUK)U(H(a\[0X
    [4XaaUAA)BUH(a(aBUB)UABUB))(a(a(aa*BUB)UB)UA(AA*BUB)UB)*(a(aKUK)UAKUK)UH(aKUK)[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X2.3 Example 2 -- the trefoil group[0X
  
  [1X2.3-1 PartialDoubleCosetRewritingSystem[0X
  
  [2X> PartialDoubleCosetRewritingSystem( [0X[3Xgrp, Hgens, Kgens, rws, limit[0X[2X ) [0Xoperation
  [2X> WordAcceptorOfPartialDoubleCosetRws( [0X[3Xgrp, prws[0X[2X ) ________________[0Xattribute
  
  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] if the wreath product
  ordering is used with X > x > Y > y. The group itself has a rewriting system
  with just 6 rules.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> FT := FreeGroup( 2 );;[0X
    [4Xgap> relsT := [ FT.1^3*FT.2^-2 ];;[0X
    [4Xgap> T := FT/relsT;;[0X
    [4Xgap> genT := GeneratorsOfGroup( T );;[0X
    [4Xgap> x := genT[1];  y := genT[2];[0X
    [4Xgap> alphT := "XxYy";;[0X
    [4Xgap> ordT := [4,3,2,1];;[0X
    [4Xgap> orderT := "wreath";;[0X
    [4Xgap> rwsT := ReducedConfluentRewritingSystem( T, ordT, orderT, 0, alphT );[0X
    [4Xgap> DisplayRwsRules( rwsT );;[0X
    [4X[ [ Yy, id ], [ yY, id ], [ X, xxYY ], [ xxx, yy ], [ yyx, xyy ], [ Yx, yxYY ]\[0X
    [4X ][0X
    [4Xgap> accT := WordAcceptorOfReducedRws( rwsT );[0X
    [4X< deterministic automaton on 4 letters with 7 states >[0X
    [4Xgap> Print( accT, "\n" );[0X
    [4XAutomaton("det",7,"yYxX",[ [ 6, 2, 2, 4, 6, 4, 6 ], [ 3, 2, 3, 2, 3, 2, 3 ], [\[0X
    [4X 7, 2, 2, 2, 2, 7, 5 ], [ 2, 2, 2, 2, 2, 2, 2 ] ],[ 1 ],[ 1, 3, 4, 5, 6, 7 ]);\[0X
    [4X;[0X
    [4Xgap> langT := SFAtoRatExp( accT );[0X
    [4X(yxUx)((xyUy)x)*((xyUy)y*UxY*UY*)Uyy*UY*[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  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   [10XPartialDoubleCosetRewritingSystem[0X  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.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> prwsT := PartialDoubleCosetRewritingSystem( T, [x], [y], rwsT, 20 );;[0X
    [4X#I WARNING: reached supplied limit 20 on number of rules[0X
    [4Xgap> DisplayRwsRules( prwsT );[0X
    [4XG-rules:[0X
    [4X[ [ X, xxYY ], [ Yx, yxYY ], [ Yy, id ], [ yY, id ], [ xxx, yy ], [ yyx, xyy ]\[0X
    [4X ][0X
    [4XH-rules:[0X
    [4X[ [ Hx, H ],[0X
    [4X  [ HY, Hy ],[0X
    [4X  [ Hyy, H ],[0X
    [4X  [ Hyxyy, Hyx ],[0X
    [4X  [ HyxY, Hyxy ],[0X
    [4X  [ Hyxyxyy, Hyxyx ],[0X
    [4X  [ Hyxxyy, Hyxx ],[0X
    [4X  [ HyxxY, Hyxxy ],[0X
    [4X  [ HyxyxY, Hyxyxy ],[0X
    [4X  [ Hyxyxyxyy, Hyxyxyx ],[0X
    [4X  [ Hyxyxxyy, Hyxyxx ],[0X
    [4X  [ Hyxxyxyy, Hyxxyx ],[0X
    [4X  [ HyxxyxYY, Hyxxyx ] ][0X
    [4XK-rules:[0X
    [4X[ [ YK, K ],[0X
    [4X  [ yK, K ] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  This  list  of  partial  rules  is  then  used  by  a modified word acceptor
  function.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> paccT := WordAcceptorOfPartialDoubleCosetRws( T, prwsT );;[0X
    [4X< deterministic automaton on 6 letters with 6 states >[0X
    [4Xgap> Print( paccT, "\n" );[0X
    [4XAutomaton("det",6,"HKyYxX",[ [ 2, 2, 2, 6, 2, 2 ], [ 2, 2, 1, 2, 2, 1 ], [ 2, \[0X
    [4X2, 5, 2, 2, 5 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 6, 2, 3, 2 ], [ 2, 2, 2, 2, 2, \[0X
    [4X2 ] ],[ 4 ],[ 1 ]);;[0X
    [4Xgap> plangT := SFAtoRatExp( paccT );[0X
    [4XH(yx(yx)*x)*(yx(yx)*KUK)[0X
    [4Xgap> wordsT := ["HK", "HxK", "HyK", "HYK", "HyxK", "HyxxK", "HyyH", "HyxYK"];;[0X
    [4Xgap> validT := List( wordsT, w -> IsRecognizedByAutomaton( paccT, w ) );[0X
    [4X[ true, false, false, false, true, true, false, false ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X2.4 Example 3 -- an infinite rewriting system[0X
  
  [1X2.4-1 KBMagRewritingSystem[0X
  
  [2X> KBMagRewritingSystem( [0X[3Xfpgrp[0X[2X ) ___________________________________[0Xattribute
  [2X> KBMagWordAcceptor( [0X[3Xfpgrp[0X[2X ) ______________________________________[0Xattribute
  [2X> KBMagFSAtoAutomataDFA( [0X[3Xfsa, alph[0X[2X ) ______________________________[0Xoperation
  [2X> WordAcceptorByKBMag( [0X[3Xgrp, alph[0X[2X ) ________________________________[0Xoperation
  [2X> WordAcceptorByKBMagOfDoubleCosetRws( [0X[3Xgrp, dcrws[0X[2X ) _______________[0Xoperation
  
  When  the  group  G  has  an  infinite  rewriting  system,  the double coset
  rewriting  system  will  also  be infinite. In this case we try the function
  [10XKBMagWordAcceptor[0X  which  calls the package [5XKBMAG[0X to compute a word acceptor
  for  G,  and  [10XKBMagFSAtoAutomataDFA[0X  to  convert  this  to  a  deterministic
  automaton  as  used by the [5Xautomata[0X package. The resulting [10Xdfa[0X forms part of
  the  double  coset  automaton, together with sufficient H-rules, K-rules and
  H-K-rules  found  by  the  function [10XPartialDoubleCosetRewritingSystem[0X. (Note
  that these five attributes and operations will not be available if the [5Xkbmag[0X
  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 [5XKBMAG[0X has a word acceptor with 17 states.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> F3 := FreeGroup("a","b");;[0X
    [4Xgap> rels3 := [ F3.1^3, F3.2^3, (F3.1*F3.2)^3 ];;[0X
    [4Xgap> G3 := F3/rels3;;[0X
    [4Xgap> alph3 := "AaBb";;[0X
    [4Xgap> waG3 := WordAcceptorByKBMag( G3, alph3 );;[0X
    [4Xgap> Print( waG3, "\n");[0X
    [4XAutomaton("det",18,"aAbB",[ [ 2, 18, 18, 8, 10, 12, 13, 18, 18, 18, 18, 18, 18\[0X
    [4X, 8, 17, 12, 18, 18 ], [ 3, 18, 18, 9, 11, 9, 12, 18, 18, 18, 18, 18, 18, 11, \[0X
    [4X18, 11, 18, 18 ], [ 4, 6, 6, 18, 18, 18, 18, 18, 6, 12, 16, 18, 12, 18, 18, 18\[0X
    [4X, 12, 18 ], [ 5, 5, 7, 18, 18, 18, 18, 14, 15, 5, 18, 18, 7, 18, 18, 18, 15, 1\[0X
    [4X8 ] ],[ 1 ],[ 1 .. 17 ]);;[0X
    [4Xgap> langG3 := SFAtoRatExp( waG3 );[0X
    [4X((abUAb)AUbA)(bA)*(b(aU@)UB(aB)*(a(bU@)U@)U@)U(abUAb)(aU@)U((aBUB)(aB)*AUba(Ba\[0X
    [4X)*BA)(bA)*(b(aU@)U@)U(aBUB)(aB)*(a(bU@)U@)Uba(Ba)*(BU@)UbUaUA(B(aB)*(a(bU@)UAU\[0X
    [4X@)U@)U@[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.4-2 DCrules[0X
  
  [2X> DCrules( [0X[3Xdcrws[0X[2X ) ________________________________________________[0Xoperation
  [2X> Hrules( [0X[3Xdcrws[0X[2X ) _________________________________________________[0Xattribute
  [2X> Krules( [0X[3Xdcrws[0X[2X ) _________________________________________________[0Xattribute
  [2X> HKrules( [0X[3Xdcrws[0X[2X ) ________________________________________________[0Xattribute
  
  We 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.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> lim := 100;;[0X
    [4Xgap> genG3 := GeneratorsOfGroup( G3 );;[0X
    [4Xgap> a := genG3[1];;  b := genG3[2];; [0X
    [4Xgap> gH3 := [ a*b ];;  gK3 := [ b*a ];;[0X
    [4Xgap> rwsG3 := KnuthBendixRewritingSystem( G3, "shortlex", [2,1,4,3], alph3 );;[0X
    [4Xgap> dcrws3 := PartialDoubleCosetRewritingSystem( G3, gH3, gK3, rwsG3, lim );;[0X
    [4X#I using PartialDoubleCosetRewritingSystem with limit 100[0X
    [4X#I  51 rules, and 1039 pairs[0X
    [4X#I WARNING: reached supplied limit 100 on number of rules[0X
    [4Xgap> Print( Length( Rules( dcrws3 ) ), " rules found.\n" );[0X
    [4X101 rules found.[0X
    [4Xgap> dcaut3 := WordAcceptorByKBMagOfDoubleCosetRws( G3, dcrws3 );;[0X
    [4Xgap> Print( "Double Coset Minimalized automaton:\n", dcaut3 );[0X
    [4XDouble Coset Minimalized automaton:[0X
    [4XAutomaton("det",40,"HKaAbB",[ [ 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2\[0X
    [4X, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ \[0X
    [4X2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, \[0X
    [4X1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1 ], [ 2, 2, 2, 2, 3, 22, 2, 2, 2, 2, 2\[0X
    [4X, 2, 2, 2, 2, 2, 2, 2, 39, 2, 39, 2, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2\[0X
    [4X, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 40, 3, 27, 2, 8, 2, 10, 2, 12, 2, 14, 2, 16, 2, \[0X
    [4X18, 2, 20, 2, 2, 2, 2, 24, 2, 27, 2, 29, 2, 31, 2, 33, 2, 35, 2, 37, 2, 2 ], [\[0X
    [4X 2, 2, 2, 2, 19, 2, 2, 26, 2, 9, 2, 11, 2, 13, 2, 15, 2, 17, 2, 38, 2, 3, 2, 2\[0X
    [4X6, 3, 2, 7, 2, 28, 2, 30, 2, 32, 2, 34, 2, 36, 2, 2, 26 ], [ 2, 2, 2, 2, 2, 2,\[0X
    [4X 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 23, 23, 2, 2, 2, 2, 2, 2, \[0X
    [4X2, 2, 2, 2, 2, 2, 2, 21, 6 ] ],[ 4 ],[ 1 ]);;[0X
    [4Xgap> dclang3 := SFAtoRatExp( dcaut3 );;[0X
    [4Xgap> Print( "Double Coset language of acceptor:\n", dclang3, "\n" );[0X
    [4XDouble Coset language of acceptor:[0X
    [4X(HbAbAbAbAbAbAbUHAb)(Ab)*(A(Ba(Ba)*bKUK)UK)UHbAbA(bA(bA(bA(bAKUK)UK)UK)UK)UH(A\[0X
    [4X(B(aB)*(abUA)KUK)UaKUb(a(Ba)*BA(bA(bA(bA(bA(bA(bA(bA)*(bKUK)UK)UK)UK)UK)UK)UK)\[0X
    [4XUAK)UK)[0X
    [4Xgap> ok := DCrules(dcrws3);;[0X
    [4Xgap> alph3e := dcrws3!.alphabet;;[0X
    [4Xgap> Print("H-rules:\n");  DisplayAsString( Hrules(dcrws3), alph3e, true );[0X
    [4XH-rules:[0X
    [4X[ HB, Ha ][0X
    [4X[ HaB, Hb ][0X
    [4X[ Hab, H ][0X
    [4X[ HbAB, HAba ][0X
    [4X[ HbAbAB, HAbAba ][0X
    [4X[ HbAbAbAB, HAbAbAba ][0X
    [4X[ HbAbAbAbAB, HAbAbAbAba ][0X
    [4X[ HbAbAbAbAbAB, HAbAbAbAbAba ][0X
    [4X[ HbAbAbAbAbAbAB, HAbAbAbAbAbAba ][0X
    [4Xgap> Print("K-rules:\n");  DisplayAsString( Krules(dcrws3), alph3e, true );[0X
    [4XK-rules:[0X
    [4X[ BK, aK ][0X
    [4X[ BaK, bK ][0X
    [4X[ baK, K ][0X
    [4X[ BAbK, abAK ][0X
    [4X[ BAbAbK, abAbAK ][0X
    [4X[ BAbAbAbK, abAbAbAK ][0X
    [4X[ BAbAbAbAbK, abAbAbAbAK ][0X
    [4X[ BAbAbAbAbAbK, abAbAbAbAbAK ][0X
    [4X[ BAbAbAbAbAbAbK, abAbAbAbAbAbAK ][0X
    [4Xgap> Print("HK-rules:\n");  DisplayAsString( HKrules(dcrws3), alph3e, true );[0X
    [4XHK-rules:[0X
    [4X[ HbK, HAK ][0X
    [4X[ HbAbK, HAbAK ][0X
    [4X[ HbAbAbK, HAbAbAK ][0X
    [4X[ HbAbAbAbK, HAbAbAbAK ][0X
    [4X[ HbAbAbAbAbK, HAbAbAbAbAK ][0X
    [4X[ HbAbAbAbAbAbK, HAbAbAbAbAbAK ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X2.4-3 NextWord[0X
  
  [2X> NextWord( [0X[3Xdcrws, word[0X[2X ) _________________________________________[0Xoperation
  [2X> WordToString( [0X[3Xword, alph[0X[2X ) ______________________________________[0Xoperation
  [2X> DisplayAsString( [0X[3Xword, alph[0X[2X ) ___________________________________[0Xoperation
  [2X> IdentityDoubleCoset( [0X[3Xdcrws[0X[2X ) ____________________________________[0Xoperation
  
  These  functions  may  be used to find normal forms of increasing length for
  double coset representatives.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> len := 30;;[0X
    [4Xgap> L3 := 0*[1..len];;[0X
    [4Xgap> L3[1] := IdentityDoubleCoset( dcrws3 );;[0X
    [4Xgap> for i in [2..len] do[0X
    [4Xgap>     L3[i] := NextWord( dcrws3, L3[i-1] );[0X
    [4Xgap> od;[0X
    [4Xgap> ## List of 30 normal forms for double cosets:[0X
    [4Xgap> DisplayAsString( L3, alph3e, true );[0X
    [4X[ HK, HAK, HaK, HAbK, HbAK, HABAK, HAbAK, HABabK, HAbAbK, HbAbAK, HbaBAK, HABa\[0X
    [4XBAK, HAbAbAK, HABaBabK, HAbABabK, HAbAbAbK, HbAbAbAK, HbaBAbAK, HbaBaBAK, HABa\[0X
    [4XBaBAK, HAbAbAbAK, HABaBaBabK, HAbABaBabK, HAbAbABabK, HAbAbAbAbK, HbAbAbAbAK, \[0X
    [4XHbaBAbAbAK, HbaBaBAbAK, HbaBaBaBAK, HABaBaBaBAK ][0X
    [4Xgap> w := NextWord( dcrws3, L3[30] );[0X
    [4Xm1*m3*m6*m3*m6*m3*m6*m3*m6*m3*m2[0X
    [4Xgap> s := WordToString( w, alph3e );[0X
    [4X"HAbAbAbAbAK"[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
