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<!-- rws.xml IdRel documentation Chris Wensley -->
<!-- & Anne Heyworth -->
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<!-- $Id: rws.xml,v 2.04 2008/11/03 gap Exp $ -->
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<!-- <M>Id: intro.xml,v 2.04 Exp <M> -->
<Chapter Label="chap-rws">
<Heading>Rewriting Systems</Heading>
This chapter describes functions to construct rewriting systems
for finitely presented groups which store rewriting information.
The main example used is a presentation of the quaternion group <C>q8</C>
with generators <M>a,b</M>
and relators <M>[a^4, b^4, abab^{-1}, a^2b^2]</M>.
<Section><Heading>Identity Y-sequences</Heading>
A typical input for &idrel; is an fp-group presentation.
This requires a free group <C>F</C> on a set of generators
and a set of relators <C>R</C> (words in the free group).
The module of identities among relators for this presentation
has as its elements the Peiffer equivalence classes of all products of
conjugates of relators which represent the identity in the free group.
<P/>
In this package the identities among relators are represented by Y-sequences,
which are lists <M>[[r_1, u_1],\ldots,[r_k,u_k]]</M>
where <M>r_1,\ldots,r_k</M> are the group relators or their inverses,
and <M>u_1,\ldots,u_k</M> are words in the free group <C>F</C>.
A Y-sequence is evaluated in <C>F</C> as the product
<M>(u_1^{-1}r_1u_1)\ldots(u_k^{-1}r_ku_k)</M>
and is an identity Y-sequence if it evaluates to the identity in <C>F</C>.
An identity Y-sequence represents an identity among the relators
of the group presentation.
The main function of the package is to produce a set of Y-sequences
which generate the module of identites among relators, and further,
that this set be minimal in the sense that every element in it
is needed to generate the module.
<P/>
Before starting on the main example,
we consider a simpler example illustrating the use of &idrel;.
All the functions used are described in detail in this manual.
We compute a reduced set of identities among relators
for the presentation of the symmetric group <C>s3</C>
with generators <M>a,b</M> and relators <M>[a^3 , b^2, (ab)^2]</M>.
In the listing below, <C>s3_M1</C> is the first monoid generator
for <C>s3</C>, <C>s3_R2</C> is the second relator,
while <C>s3_Y4</C> is the fourth Y-sequence for <C>s3</C>.
<P/>
<Example>
<![CDATA[
gap> F := FreeGroup( 2 );;
gap> a := F.l;; b:= F.2;;
gap> rels3 := [ a^3 , b^2, a*b*a*b];
[ fl^3 , f2^2, fl*f2*fl*f2]
gap> s3 := F/rels3;
<fp group on the generators [ fl, f2 ]>
gap> SetName( s3, "s3" );
gap> idy3 := IdentityYSequences( s3 );;
gap> Length( idy3 );
18
gap> Y4 := idy3[4];
[ [ s3_R1^-1, f1^-1 ], [ s3_R1, <identity ...> ] ]
gap> Y6 := idy3[6];
[ [ s3_R3^-1, f1^-1 ], [ s3_R1, <identity ...> ], [ s3_R3^-1, f1 ],
[ s3_R2, f1^-1*f2^-1 ], [ s3_R1, f2^-1 ], [ s3_R3^-1, f1*f2^-1 ],
[ s3_R2, <identity ...> ], [ s3_R2, f1^-1 ] ]
gap> Y7 := idy3[7];
[ [ s3_R3^-1, f1*f2^-1 ], [ s3_R2, <identity ...> ], [ s3_R3, <identity ...> ],
[ s3_R2^-1, <identity ...> ] ]
gap> Y8 := idy3[8];
[ [ s3_R2^-1, f2^-1 ], [ s3_R2, <identity ...> ] ]
]]>
</Example>
Of the <M>18</M> Y-sequences formed, <M>6</M> are empty,
and <C>Y4</C> is the <E>root identity</E>
<C>((a^3)^-1)^(a^-1).(a^3)</C>.
If we write <M>r=a^3, s=b^2, t=(ab)^2</M>
then <C>Y4</C> is <C>(r^-1)^(a^-1).r</C>.
Similarly, <C>Y8</C> is the second root identity <C>(s^-1)^(b^-1).s</C>,
while <C>Y7</C> is the third root identity <C>(t^-1)^(ab^-1).t</C>.
The identity <C>Y6</C>, which is not a root identity,
is obtained by walking around the Schreier diagram of the presentation,
which is a somewhat truncated triangular prism,
taking the appropriate conjugate of each face in turn:
<C>
Y6=(t^-1)^(a^-1).r.(t^-1)^a.s^(a^-1b^-1).r^(b^-1).(t^-1)^(ab^-1).s.s^(a^-1)
</C>.
These four identities generate the module of identities for <C>s3</C>.
<P/>
<Example>
<![CDATA[
gap> idrels3 := IdentitiesAmongRelators( s3 );;
gap> Display( idrels3[1] );
[ ( s3_Y4*( s3_M2*s3_M1), s3_R1*( s3_M1 - <identity ...>) ),
( s3_Y8*( s3_M2*s3_M1), s3_R2*( s3_M2 - <identity ...>) ),
( s3_Y7*( s3_M2*s3_M1), s3_R3*( s3_M2 - s3_M1) ),
( s3_Y6*( -s3_M2*s3_M1), s3_R1*( -s3_M2*s3_M1 - s3_M1) + s3_R2*( -s3_M1*s3_M\
2 - s3_M1 - <identity ...>) + s3_R3*( s3_M3 + s3_M2 + <identity ...>) )
]
]]>
</Example>
</Section>
<Section><Heading>Monoid Presentations of FpGroups</Heading>
<ManSection>
<Attr Name="FreeRelatorGroup"
Arg="grp" />
<Attr Name="FreeRelatorHomomorphism"
Arg="grp" />
<Description>
The function <C>FreeRelatorGroup</C> returns a free group
on the set of relators of the given fp-group <C>G</C>.
If <C>HasName(G)</C> is <C>true</C> then a name is automatically assigned
to the free group.
<P/>
The function <C>FreeRelatorHomomorphism</C> returns the group homomorphism
from the free group on the relators
to the free group on the generators of <C>G</C>,
mapping each generator to the corresponding word.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> F := FreeGroup( 2 );;
gap> a := F.1;; b:= F.2;;
gap> rels := [ a^4, b^4, a*b*a*b^-1, a^2*b^2];;
gap> q8 := F/rels;;
gap> SetName( q8, "q8" );
gap> frq8 := FreeRelatorGroup( q8 );
q8_R
gap> GeneratorsOfGroup( frq8 );
[ q8_R1, q8_R2, q8_R3, q8_R4]
gap> frhomq8 := FreeRelatorHomomorphism( q8 );
[ q8_R1, q8_R2, q8_R3, q8_R4] -> [ f1^4, f2^4, f1*f2*f1*f2^-1, f1^2*f2^2]
]]>
</Example>
<ManSection>
<Attr Name="MonoidPresentationFpGroup"
Arg="grp" />
<Attr Name="FreeGroupOfPresentation"
Arg="mon" />
<Attr Name="GroupRelatorsOfPresentation"
Arg="mon" />
<Attr Name="InverseRelatorsOfPresentation"
Arg="mon" />
<Attr Name="HomomorphismOfPresentation"
Arg="mon" />
<Description>
A monoid presentation for a finitely presented group <C>G</C>
has two monoid generators <M>g^+,g^-</M> for each group generator <M>g</M>.
The relators of the monoid presentation comprise the group relators,
and relators <M>g^+g^-</M> specifying the inverses.
The function <C>MonoidPresentationFpGroup</C> returns the monoid presentation
derived in this way from an fp-presentation.
(Note: this function should always be followed by a double semicolon --
<C>MonoidPresentationFpGroup(G);;</C> --
because an error occurs in attempting to display
the results on the screen: the <C>ElementsFamily</C> needs to be fixed.)
<P/>
The function <C>FreeGroupOfPresentation</C>
returns the free group on the monoid generators.
<P/>
The function <C>GroupRelatorsOfPresentation</C> returns those relators
of the monoid which correspond to the relators of the group.
All negative powers in the group relators are converted to positive powers
of the <M>g^-</M>.
<P/>
The function <C>InverseRelatorsOfPresentation</C> returns relators
which specify the inverse pairs of the monoid generators.
<P/>
The function <C>HomomorphismOfPresentation</C> returns the homomorphism
from the free group of the monoid presentation to the free group
of the group presentation.
<P/>
In the example below, the four monoid generators <M>a^+, b^+, a^-, b^-</M>
are named <C>q8\_M1, q8\_M2, q8\_M3, q8\_M4</C>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> mon := MonoidPresentationFpgroup( q8 );;
gap> fgmon := FreeGroupOfPresentation( mon);
<free group on the generators [ q8_Ml, q8_M2, q8_M3, q8_M4]>
gap> genfgmon := GeneratorsOfGroup( fgmon);
[ q8_Ml, q8_M2, q8_M3, q8_M4]
gap> gprels := GroupRelatorsOfPresentation( mon );
[ q8_Ml^4, q8_M2^4, q8_Ml*q8_M2*q8_Ml*q8_M4, q8_Ml^2*q8_M2^2]
gap> invrels := InverseRelatorsOfPresentation( mon);
[ q8_Ml*q8_M3, q8_M2*q8_M4, q8_M3*q8_Ml, q8_M4*q8_M2]
gap> hompres := HomomorphismOfPresentation( mon );
[ q8_Ml, q8_M2, q8_M3, q8_M4] -> [ fl, f2, fl^-l, f2^-1 ]
]]>
</Example>
</Section>
<Section><Heading>Rewriting systems for FpGroups</Heading>
These functions duplicate the standard Knuth Bendix functions
which are available in the &GAP; library.
There are two reasons for this:
(1) these functions were first written before the standard functions
were available;
(2) we require logged versions of the functions, and these are most
conveniently extended versions of the non-logged code.
<ManSection>
<Attr Name="RewritingSystemFpGroup"
Arg="grp" />
<Description>
This function attempts to return a complete rewrite system
for the group <C>G</C> obtained from the monoid presentation <C>mon</C>,
with a length-lexicographical ordering on the words in <C>fgmon</C>,
by applying Knuth-Bendix completion.
Such a rewrite system can be obtained for all finite groups.
The rewrite rules are (partially) ordered, starting with the inverse relators,
followed by the rules which reduce the word length the most.
<P/>
In our <C>q8</C> example there are 16 rewrite rules.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> rws := RewritingSystemFpGroup( q8 );
[ [q8_Ml*q8_M3, <identity ...>], [ q8_M2*q8_M4, <identity ...> ],
[q8_M3*q8_Ml, <identity ...>], [ q8_M4*q8_M2, <identity ...> ],
[q8_M1^2*q8_M4, q8_M2], [q8_Ml^2*q8_M2, q8_M4], [ q8_Ml^3, q8_M3 ],
[ q8_M4^2, q8_Ml^2], [ q8_M4*q8_M3, q8_Ml*q8_M4],
[ q8_M4*q8_Ml, q8_Ml*q8_M2], [q8_M3*q8_M4, q8_Ml*q8_M2],
[ q8_M3^2, q8_Ml^2], [q8_M3*q8_M2, q8_Ml*q8_M4],
[ q8_M2*q8_M3, q8_Ml*q8_M2], [q8_M2^2, q8_Ml^2],
[ q8_M2*q8_Ml, q8_Ml*q8_M4] ]
]]>
</Example>
The functions called by <C>RewritingSystemFpGroup</C> are as follows.
<ManSection>
<Oper Name="OnePassReduceWord"
Arg="word, rules" />
<Oper Name="ReduceWordKB"
Arg="word, rules" />
<Description>
Assuming that <C>word</C> is an element of a free monoid
and <C>rules</C> is a list of ordered pairs of such words,
the function <C>OnePassReduceWord</C> searches the list of rules
until it finds that the left-hand side of a <C>rule</C>
is a <C>subword</C> of <C>word</C>,
whereupon it replaces that <C>subword</C> with the right-hand side
of the matching rule.
The search is continued from the next <C>rule</C> in <C>rules</C>,
but using the new <C>word</C>.
When the end of <C>rules</C> is reached,
one pass is considered to have been made
and the reduced <C>word</C> is returned.
If no matches are found then the original <C>word</C> is returned.
<P/>
The function <C>ReduceWordKB</C> repeatedly applies the function
<C>OnePassReduceWord</C>
until the <C>word</C> remaining contains no left-hand side of a <C>rule</C>
as a <C>subword</C>.
If <C>rules</C> is a complete rewrite system,
then the irreducible <C>word</C> that is returned is unique,
otherwise the order of the rules in <C>rules</C> will determine
which irreducible word is returned.
In the example we see that <M>b^9a^9</M> reduces to <M>ba</M>,
and we shall see later that this is not a normal form.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> monrels := Concatenation( gprels, invrels );
[ q8_Ml^4, q8_M2^4, q8_Ml*q8_M2*q8_Ml*q8_M4, q8_Ml^2*q8_M2^2, q8_Ml*q8_M3,
q8_M2*q8_M4, q8_M3*q8_Ml, q8_M4*q8_M2]
gap> id := One( monrels[l] );;
gap> r0 := List( monrels, r -> [ r, id ] );
[ [ q8_Ml^4, <identity ...> ], [ q8_M2^4, <identity. ..> ],
[ q8_Ml*q8_M2*q8_Ml*q8_M4, <identity ...> ],
[ q8_Ml^2*q8_M2^2, <identity. ..>], [ q8_Ml*q8_M3, <identity ...> ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity. ..>],
[ q8_M4*q8_M2, <identity ...> ] ]
gap> ap := genfgmon[l];; bp := genfgmon[2];; ## p for plus
gap> am := genfgmon[3];; bm := genfgmon[4];; ## m for minus
gap> w0 := bp^9 * ap^9;
q8_M2^9*q8_M1^9
gap> w1 := OnePassReduceWord( w0, r0 );
q8_M2^5*q8_M1^5
gap> w2 := ReduceWordKB( w0, r0 );
q8_M2*q8_M1
]]>
</Example>
<ManSection>
<Oper Name="OnePassKB"
Arg="rules" />
<Oper Name="RewriteReduce"
Arg="rules" />
<Oper Name="KnuthBendix"
Arg="rules" />
<Oper Name="ShorterRule"
Arg="rule1, rule2" />
<Description>
The function <C>OnePassKB</C> implements the main loop of the Knuth-Bendix
completion algorithm.
Rules are compared with each other; all critical pairs are calculated;
and the irreducible critical pairs are orientated with respect
to the length-lexicographical ordering and added to the rewrite system.
<P/>
The function <C>RewriteReduce</C> will remove unnecessary rules from
a rewrite system.
A rule is deemed to be unnecessary if it is implied by the other rules,
i.e. if both sides can be reduced to the same thing by the remaining rules.
<P/>
The function <C>KnuthBendix</C> implements the Knuth-Bendix algorithm,
attempting to complete a rewrite system with respect to
a length-lexicographic ordering.
It calls first <C>OnePassKB</C>, which adds rules, and then (for efficiency)
<C>RewriteReduce</C> which removes any unnecessary ones.
This procedure is repeated until <C>OnePassKB</C> adds no more rules.
It will not always terminate, but for many examples (all finite groups)
it will be successful.
The rewrite system returned is complete, that is:
it will rewrite any given word in the free monoid to a unique irreducible;
there is one irreducible for each element of the quotient monoid;
and any two elements of the free monoid which are in the same class
will rewrite to the same irreducible.
<P/>
The function <C>ShorterRule</C> gives an ordering on rules.
Rules <M>(g_lg_2,id)</M> that identify two generators
(or one generator with the inverse of another) come first in the ordering.
Otherwise one precedes another if it reduces the length of a word
by a greater amount.
<P/>
One pass of this procedure for our <C>q8</C> example
adds 13 relators to the original 8, and these 21 are then reduced to 9.
A second pass and reduction gives a list of 16 rules
which forms a complete rewrite system for the group.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> r1 := OnePassKB( r0 );
[ [ q8_Ml^4, <identity ...> ], [ q8_M2^4, <identity ...> ],
[ q8_Ml*q8_M2*q8_Ml*q8_M4, <identity ...> ],
[ q8_Ml^2*q8_M2^2, <identity. ..> ], [ q8_Ml*q8_M3, <identity ...> ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity ...> ],
[ q8_M4*q8_M2, <identity ...> ], [ q8_M2*q8_Ml*q8_M4, q8_Ml^3],
[ q8_Ml*q8_M2^2, q8_Ml^3 ], [ q8_M2^2, q8_Ml^2 ], [q8_Ml^3, q8_M3 ],
[ q8_M2^3, q8_M4 ], [ q8_Ml*q8_M2*q8_Ml, q8_M2],
[ q8_M2^3, q8_Ml^2*q8_M2], [ q8_M2^2, q8_Ml^2 ], [q8_Ml^2*q8_M2, q8_M4 ],
[ q8_Ml^3, q8_M3 ], [ q8_M2*q8_Ml*q8_M4, q8_M3 ], [q8_Ml*q8_M2^2, q8_M3 ],
[ q8_M2^3, q8_M4 ] ]
gap> r1 := RewriteReduce( r1 );
[ [ q8_Ml*q8_M3, <identity ...> ], [ q8_M2^2, q8_Ml^2 ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity ...> ],
[ q8_M4*q8_M2, <identity ...> ], [ q8_Ml^3, q8_M3 ],
[ q8_Ml^2*q8_M2, q8_M4 ], [ q8_Ml*q8_M2*q8_Ml, q8_M2 ],
[ q8_M2*q8_Ml*q8_M4, q8_M3 ] ]
gap> r2 := KnuthBendix( r1 );
[ [ q8_Ml*q8_M3, <identity ...> ], [ q8_M2*q8_Ml, q8_Ml*q8_M4 ],
[ q8_M2^2, q8_Ml^2], [ q8_M2*q8_M3, q8_Ml*q8_M2 ],
[ q8_M2*q8_M4, <identity ...> ], [ q8_M3*q8_Ml, <identity ...> ],
[ q8_M3*q8_M2, q8_Ml*q8_M4 ], [ q8_M3^2, q8_Ml^2 ],
[ q8_M3*q8_M4, q8_Ml*q8_M2 ], [ q8_M4*q8_Ml, q8_Ml*q8_M2 ],
[ q8_M4*q8_M2, <identity ...> ], [ q8_M4*q8_M3, q8_Ml*q8_M4 ],
[ q8_M4^2, q8_Ml^2], [ q8_Ml^3, q8_M3 ], [q8_Ml^2*q8_M2, q8_M4 ],
[ q8_Ml^2*q8_M4, q8_M2 ] ]
gap> w2 := ReduceWordKB( w0, r2 );
q8_M1*q8_M4
]]>
</Example>
</Section>
<Section><Heading>Enumerating elements</Heading>
<ManSection>
<Attr Name="ElementsOfMonoidPresentation"
Arg="mon" />
<Description>
The function <C>ElementsOfMonoidPresentation</C> returns a list
of normal forms for the elements of the group given by
the monoid presentation <C>mon</C>.
The normal forms are the least elements in each equivalence class
(with respect to length-lex order).
<!-- The function <C>EnumerateKB</C> builds a catalogue of irreducible -->
<!-- words in the generators of a monoid with respect to a set of rules. -->
When <C>rules</C> is a complete rewrite system for <C>G</C>
the list returned is a set of normal forms for the group elements.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> elq8 := Elements( q8 );
[ <identity .. .>, fl, f2, fl^2, fl*f2, fl^3*f2, fl^3, fl^2*f2 ]
gap> elmonq8 := ElementsOfMonoidPresentation( monq8 );
[ <identity. ..>, q8_Ml, q8_M2, q8_M3, q8_M4, q8_Ml^2, q8_Ml*q8_M2,
q8_Ml*q8_M4 ]
]]>
</Example>
</Section>
</Chapter>
