A *logged rewrite system* is associated with a group presentation. Each *logged rewrite rule* contains, in addition to the standard rewrite rule, a record or *log component* which expresses the rule in terms of the original relators of the group. We represent such a rule by a triple `[ u, [L1,L2,..,Lk], v]`

, where `[u,v]`

is a rewrite rule and \(L_i = [n_i,w_i]\) where \(n_i\) is a group relator and \(w_i\) is a word. These three components obey the identity \(u = n_1^{w_1} \ldots n_k^{w_k} v\).

The functions in this section are the logged versions of those in the previous chapter.

`‣ InitialLoggedRulesOfPresentation` ( mon ) | ( function ) |

The \(12\) initial logged rules for \(mq8\) correspond to the \(12\) initial rules in section 2.1-4. Rules of the form \(g^{-1} \to G\) and \(gG \to id\) apply to the monoid presentation, but not to the group presentation, so are given an empty logged component. The remaining four rules, which corresppond to the relators \(r \in [a^4, b^4, abab^{-1}, a^2b^2]\) are given logged components \([r,[\,[n,id]\,], id]\) for \(n \in [9..12]\).

gap> r0 := InitialLoggedRulesOfPresentation( mq8 );; gap> PrintLnUsingLabels( r0, genfmq8, q8labs ); [ [ a^-1, [ ], A ], [ b^-1, [ ], B ], [ A^-1, [ ], a ], [ B^-1, [ ], b ], [ a*A, [ ], id ], [ b*B, [ ], id ], [ A*a, [ ], id ], [ B*b, [ ], id ], [ a^4, [ [ 1, id ] ], id ], [ a^2*b^2, [ [ 4, id ] ], id ], [ a*b*a*B, [ [ 3, id ] ], id ], [ b^4, [ [ 2, id ] ], id ] ]

`‣ LoggedOnePassKB` ( grp, loggedrules ) | ( operation ) |

Given a logged rewrite system for the group `grp`

, this function finds all the rules that would be added to complete the rewrite system of `OnePassKB`

in 2.2-3, and also the logs which relate the new rules to the originals. The result of applying this function to `loggedrules`

is to add new logged rules to the system without changing the monoid it defines.

In the example, we apply one pass of the logged Knuth-Bendix procedure to the initial set of logged rules.

gap> r1 := LoggedOnePassKB( mq8, r0 );; gap> Length( r1 ); 25 gap> PrintLnUsingLabels( r1, genfmq8, q8labs ); [ [ a^-1, [ ], A ], [ b^-1, [ ], B ], [ A^-1, [ ], a ], [ B^-1, [ ], b ], [ a*A, [ ], id ], [ b*B, [ ], id ], [ A*a, [ ], id ], [ B*b, [ ], id ], [ b^2, [ [ -4, id ], [ 2, A^2 ] ], a^2 ], [ b^2, [ [ -1, id ], [ 4, A^2 ] ], a^2 ], [ a^3, [ [ 1, id ] ], A ], [ a^3, [ [ 1, a ] ], A ], [ a^2*b, [ [ 4, id ] ], B ], [ a*b*a, [ [ 3, id ] ], b ], [ a*b^2, [ [ 4, a ] ], A ], [ b*a*B, [ [ 3, a ] ], A ], [ b^3, [ [ 2, id ] ], B ], [ b^3, [ [ 2, b ] ], B ], [ a*b^2, [ [ -1, id ], [ 4, A^3 ] ], a^3 ], [ b*a*B, [ [ -1, id ], [ 3, A^3 ] ], a^3 ], [ b^3, [ [ -4, id ], [ 2, B*A^2 ] ], a^2*b ], [ a^4, [ [ 1, id ] ], id ], [ a^2*b^2, [ [ 4, id ] ], id ], [ a*b*a*B, [ [ 3, id ] ], id ], [ b^4, [ [ 2, id ] ], id ] ]

Note that \(r1\) has length \(25\), three more than the length \(22\) of `q1`

in 2.2-3. This because the three rules \(b^2 \to a^2;~ a^3 \to A;~ b^3 \to B\) each appear twice, with alternative logged components.

If we write \(a,b,A,B\) for `M1,M2,M3,M4`

and label the four original relators as \(q=a^4,\ r=b^4,\ s=abaB,\ t=a^2b^2\) then the ninth identity (for example) says that \(b^2 = (t^{-1}r^{A^2})a^2\). To verify this, we may expand the right-hand side as follows:

\[ (B^2A^2).a^2(b^4)A^2.a^2 ~=~ B^2(A^2a^2)b^4(A^2a^2) ~=~ B^2b^4 ~=~ b^2. \]

`‣ LoggedRewriteReduce` ( grp, loggedrules ) | ( operation ) |

The function `LoggedRewriteReduce`

removes unnecessary rules from a logged rewrite system. It works on the same principle as `RewriteReduce`

in 2.2-4. Note that \(q2\) nd \(r2\) both have length \(13\).

gap> r2 := LoggedRewriteReduce( mq8, r1 );; gap> Length( r2 ); 13 gap> PrintLnUsingLabels( r2, genfmq8, q8labs ); [ [ a^-1, [ ], A ], [ b^-1, [ ], B ], [ A^-1, [ ], a ], [ B^-1, [ ], b ], [ a*A, [ ], id ], [ b*B, [ ], id ], [ A*a, [ ], id ], [ B*b, [ ], id ], [ b^2, [ [ -4, id ], [ 2, A^2 ] ], a^2 ], [ a^3, [ [ 1, id ] ], A ], [ a^2*b, [ [ 4, id ] ], B ], [ a*b*a, [ [ 3, id ] ], b ], [ b*a*B, [ [ 3, a ] ], A ] ]

`‣ LoggedKnuthBendix` ( grp, loggedrules ) | ( operation ) |

The function `LoggedKnuthBendix`

repeatedly applies functions `LoggedOnePassKB`

and `LoggedRewriteReduce`

until no new rules are added and no unnecessary ones are included. The output is a reduced complete logged rewrite system.

As a further example, consider the ninth rule in `r3`

which shows how \(ba\) reduces to \(aB\). For this rule `[u,L,v]`

we will verify that \(u = n_1^{w_1}n_2^{w_2}n_3^{w_3} v\), as in the introduction to this chapter. The rule is:

\[ [ ba, [ [-11,id], [12,BA] ], aB ]. \]

The relators are \(-11 \equiv s^{-1} = bABA\) and \(12 \equiv t = a^2b^2\). These are conjugated by the identity and \(BA\) respectively. So the second and third parts of the rule expand to:

\[ (bABA)(ab(aabb)BA)aB ~=~ bAB(Aa)baab(bB)(Aa)B ~=~ bA(Bb)aa(bB) ~=~ b(Aa)a ~=~ ba, \]

the first part of the rule.

gap> r3 := LoggedKnuthBendix( mq8, r0 );; gap> Length( r3 ); 20 gap> PrintLnUsingLabels( r3, genfmq8, q8labs ); [ [ a^-1, [ ], A ], [ b^-1, [ ], B ], [ A^-1, [ ], a ], [ B^-1, [ ], b ], [ a*A, [ ], id ], [ b*B, [ ], id ], [ A*a, [ ], id ], [ B*b, [ ], id ], [ b*a, [ [ -3, id ], [ 4, B*A ] ], a*B ], [ b^2, [ [ -4, id ], [ 2, A^2 ] ], a^2 ], [ b*A, [ [ -3, id ] ], a*b ], [ A*b, [ [ -1, id ], [ 4, A ] ], a*B ], [ A^2, [ [ -1, id ] ], a^2 ], [ A*B, [ [ -4, a ] ], a*b ], [ B*a, [ [ -4, id ], [ 3, A ] ], a*b ], [ B*A, [ [ -3, a*b ] ], a*B ], [ B^2, [ [ -4, id ] ], a^2 ], [ a^3, [ [ 1, id ] ], A ], [ a^2*b, [ [ 4, id ] ], B ], [ a^2*B, [ [ -4, A^2 ], [ 1, id ] ], b ] ]

`‣ LoggedRewritingSystemFpGroup` ( grp ) | ( attribute ) |

Given a group presentation, the function `LoggedRewritingSystemFpGroup`

determines a logged rewrite system based on the relators. The initial logged rewrite system associated with a group presentation consists of two types of rule. These are logged versions of the two types of rule in the monoid presentation. Corresponding to the j-th relator `rel`

of the group there is a logged rule `[rel,[[j,id]],id]`

. For each inverse relator there is a logged rule `[ gen*inv, [], id ]`

. The function then attempts a completion of the logged rewrite system. The rules in the final system are partially ordered by the function `ShorterLoggedRule`

.

gap> lrws := LoggedRewritingSystemFpGroup( q8 );; gap> PrintLnUsingLabels( lrws, genfgmon, q8labs ); [ [ a^-1, [ ], A ], [ b^-1, [ ], B ], [ A^-1, [ ], a ], [ B^-1, [ ], b ], [ a*A, [ ], id ], [ b*B, [ ], id ], [ A*a, [ ], id ], [ B*b, [ ], id ], [ b*a, [ [ -3, id ], [ 4, B*A ] ], a*B ], [ b^2, [ [ -4, id ], [ 2, A^2 ] ], a^2 ], [ b*A, [ [ -3, id ] ], a*b ], [ A*b, [ [ -1, id ], [ 4, A ] ], a*B ], [ A^2, [ [ -1, id ] ], a^2 ], [ A*B, [ [ -4, a ] ], a*b ], [ B*a, [ [ -4, id ], [ 3, A ] ], a*b ], [ B*A, [ [ -3, a*b ] ], a*B ], [ B^2, [ [ -4, id ] ], a^2 ], [ a^3, [ [ 1, id ] ], A ], [ a^2*b, [ [ 4, id ] ], B ], [ a^2*B, [ [ -4, A^2 ], [ 1, id ] ], b ] ] gap> Length( lrws ); 16

Consider now the two-generator abelian group \(T\) considered in the previous chapter (2.2-1). Using the alternative ordering on the monoid generators, `[ T_M1`

\(=a\), `T_M2`

\(=A\), `T_M3`

\(=b\), `T_M4`

\(=B\) `]`

, we obtain the following set of \(8\) logged rules. The last of these may be checked as follows:

\[ (ba(BAba)AB)ab ~=~ ba(B(A(b(aA)B)a)b) \]

and is a logged version of the rule \(ba \to ab\).

gap> lrwsT := LoggedRewritingSystemFpGroup( T );; gap> PrintLnUsingLabels( lrwsT, genfgmonT, Tlabs ); [ [ x^-1, [ ], X ], [ X^-1, [ ], x ], [ y^-1, [ ], Y ], [ Y^-1, [ ], y ], [ x*X, [ ], id ], [ X*x, [ ], id ], [ y*Y, [ ], id ], [ Y*y, [ ], id ], [ y*x, [ [ -1, X*Y ] ], x*y ], [ y*X, [ [ 1, Y ] ], X*y ], [ Y*x, [ [ 1, X ] ], x*Y ], [ Y*X, [ [ -1, id ] ], X*Y ] ]

`‣ LoggedReduceWordKB` ( word, loggedrules ) | ( operation ) |

`‣ LoggedOnePassReduceWord` ( word, loggedrules ) | ( operation ) |

`‣ ShorterLoggedRule` ( logrule1, logrule2 ) | ( operation ) |

Given a word and a logged rewrite system, the function `LoggedOnePassReduceWord`

makes one reduction pass of the word (possibly involving several reductions) (as does `OnePassReduceWord`

in 2.2-2) and records this, using the log part of the rule(s) used and the position in the original word of the replaced part.

The function `LoggedReduceWordKB`

repeatedly applies `OnePassLoggedReduceWord`

until the word can no longer be reduced. Each step of the reduction is logged, showing how the original word can be expressed in terms of the original relators and the irreducible word. When `loggedrules`

is complete the reduced word is a unique normal form for that group element. The log of the reduction depends on the order in which the rules are applied.

The function `ShorterLoggedrule`

decides whether one logged rule is better than another, using the same criteria as `ShorterRule`

in 2.2-3. In the example we perform logged reductions of \(w_0 = a^9b^{-9}\) corresponding to the ordinary reductions performed in the previous chapter (section 2.2-2).

In order to clarify the following output, note that, in the log below, \(b^9a^{-9}\) reduces to \(Bb^5aba^{-8}\) in `lw1`

, just as in section 2.2-2. This may be checked by cancelling terms in:

\[ (b^2A^2)(a^2.b^4.A^2)(a^2b^6.bABA.b^6A^2)(a^2b^2)Bb^5aba^{-8} ~=~ b^9a^9. \]

The corresponding expansion of `lw2`

is too lengthy to include here. (It's hard to believe that the logged part of this identity is the simplest possible. Further investigation is needed to determine whether or not this logged part can be simplified.)

gap> PrintLnUsingLabels( w0, genfmq8, q8labs ); b^9*a^-9 gap> lw1 := LoggedOnePassReduceWord( w0, lrws );; gap> PrintLnUsingLabels( lw1, genfmq8, q8labs ); [ [ [ -4, id ], [ 2, A^2 ], [ -3, b^-6*a^-2 ], [ 4, id ] ], B*b^5*a*b*a^-8 ] gap> lw2 := LoggedReduceWordKB( w0, lrws );; gap> PrintLnUsingLabels( lw2, genfmq8, q8labs ); [ [ [ -4, id ], [ 2, A^2 ], [ -3, b^-6*a^-2 ], [ 4, id ], [ -3, b^-3 ], [ 4, B*A*b^-3 ], [ -4, id ], [ 2, A^2 ], [ -3, B^-1*a^-1*b^-1*a^-2 ], [ -4, a^-1*b^-1*a^-2 ], [ 3, A*a^-1*b^-1*a^-2 ], [ 4, id ], [ -4, a^-2*B^-1 ], [ 2, A^2*a^-2*B^-1 ], [ -4, id ], [ 3, A ], [ 1, b^-1*a^-1 ], [ -3, a^-1 ], [ -1, b^-1*a^-2 ], [ 4, id ], [ -3, a*b ], [ -3, a*b*a^-1 ], [ -4, A^2 ], [ 1, id ], [ -3, id ] ], a*b ]

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