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<Chapter Label="Rewriting Systems">
<Heading>Rewriting Systems</Heading>

Rewriting systems in &GAP; are a framework for dealing with
the very general task of rewriting elements of a free (or <E>term</E>) algebra
in some normal form.  Although most rewriting systems currently in use
are <E>string rewriting systems</E> (where the algebra has only one
binary operation which is associative) the framework in &GAP;
is general enough to encompass the task of rewriting algebras of
any signature from groups to semirings.
<P/>
Rewriting systems are already implemented in  &GAP;
for finitely presented semigroups and for pc groups. The use of these
particular rewriting systems is described in the corresponding chapters.
We describe here only the general framework of rewriting systems with
a particular emphasis on material which would be
helpful for a  developer implementing a rewriting system.
<P/>
We fix some definitions and terminology for the rest of this chapter.
Let <M>T</M> be a term algebra in some signature.
A <E>term rewriting system</E> for <M>T</M> is a set of ordered pairs of
elements of <M>T</M> of the form <M>(l, r)</M>.
Viewed as a set of relations, the
rewriting system determines a presentation for a quotient algebra <M>A</M>
of <M>T</M>.
<P/>
When we take into account the fact that
the relations are expressed as <E>ordered</E> pairs, we have a way of
<E>reducing</E> the elements of <M>T</M>.
Suppose an element <M>u</M> of <M>T</M> has a
subword <M>l</M> and <M>(l, r)</M> is a rule of the rewriting system, then
we can replace the subterm <M>l</M> of <M>u</M> by the term <M>r</M>
and obtain a new word <M>v</M>.
We say that we have <E>rewritten</E> <M>u</M> as <M>v</M>.
Note that <M>u</M> and <M>v</M> represent the same element of <M>A</M>.
If <M>u</M> cannot be rewritten using any rule of the rewriting system
we sat that <M>u</M> is <E>reduced</E>.


<Section Label="Operations on rewriting systems">
<Heading>Operations on rewriting systems</Heading>

<#Include Label="IsRewritingSystem">
<#Include Label="Rules">
<#Include Label="OrderOfRewritingSystem">
<#Include Label="ReducedForm">
<#Include Label="IsConfluent">
<#Include Label="ConfluentRws">
<#Include Label="IsReduced">
<#Include Label="ReduceRules">
<#Include Label="AddRule">
<#Include Label="AddRuleReduced">
<#Include Label="MakeConfluent">
<#Include Label="GeneratorsOfRws">

</Section>


<Section Label="Operations on elements of the algebra">
<Heading>Operations on elements of the algebra</Heading>

In this section let <M>u</M> denote an element of the term algebra
<M>T</M> representing <M>[u]</M> in the quotient algebra <M>A</M>.

<ManSection>
<Oper Name="ReducedProduct" Arg='rws, u, v'/>
<Oper Name="ReducedSum" Arg='rws, left, right'/>
<Oper Name="ReducedOne" Arg='rws'/>
<Oper Name="ReducedAdditiveInverse" Arg='rws, obj'/>
<Oper Name="ReducedComm" Arg='rws, left, right'/>
<Oper Name="ReducedConjugate" Arg='rws, left, right'/>
<Oper Name="ReducedDifference" Arg='rws, left, right'/>
<Oper Name="ReducedInverse" Arg='rws, obj'/>
<Oper Name="ReducedLeftQuotient" Arg='rws, left, right'/>
<Oper Name="ReducedPower" Arg='rws, obj, pow'/>
<Oper Name="ReducedQuotient" Arg='rws, left, right'/>
<Oper Name="ReducedScalarProduct" Arg='rws, left, right'/>
<Oper Name="ReducedZero" Arg='rws'/>

<Description>
The result of <Ref Oper="ReducedProduct"/> is <M>w</M> where
<M>[w]</M> equals [<A>u</A>][<A>v</A>] in <M>A</M> and <M>w</M>
is in reduced form.
<P/>
The remaining operations are defined similarly when they
are defined (as determined by the signature of the term algebra).
</Description>
</ManSection>

</Section>


<Section Label="Properties of rewriting systems">
<Heading>Properties of rewriting systems</Heading>

<ManSection>
<Prop Name="IsBuiltFromAdditiveMagmaWithInverses" Arg='obj'/>
<Prop Name="IsBuiltFromMagma" Arg='obj'/>
<Prop Name="IsBuiltFromMagmaWithOne" Arg='obj'/>
<Prop Name="IsBuiltFromMagmaWithInverses" Arg='obj'/>
<Prop Name="IsBuiltFromSemigroup" Arg='obj'/>
<Prop Name="IsBuiltFromGroup" Arg='obj'/>

<Description>
These properties may be used to identify the type of term algebra
over which the rewriting system is defined.
</Description>
</ManSection>

</Section>


<Section Label="Rewriting in Groups and Monoids">
<Heading>Rewriting in Groups and Monoids</Heading>

One application of rewriting is to reduce words in finitely presented groups
and monoids. The rewriting system still has to be built for a finitely
presented monoid (using <C>IsomorphismFpMonoid</C> for conversion). Rewriting
then can take place for words in the underlying free monoid. (These can be
obtained from monoid elements with the command <C>UnderlyingElement</C>.)
<P/>
<Example><![CDATA[
gap> f:=FreeGroup(3);;
gap> rels:=[f.1*f.2^2/f.3,f.2*f.3^2/f.1,f.3*f.1^2/f.2];;
gap> g:=f/rels;
<fp group on the generators [ f1, f2, f3 ]>
gap> mhom:=IsomorphismFpMonoid(g);
MappingByFunction( <fp group on the generators
[ f1, f2, f3 ]>, <fp monoid on the generators
[ f1, f1^-1, f2, f2^-1, f3, f3^-1
]>, function( x ) ... end, function( x ) ... end )
gap> mon:=Image(mhom);
<fp monoid on the generators [ f1, f1^-1, f2, f2^-1, f3, f3^-1 ]>
gap> k:=KnuthBendixRewritingSystem(mon);
Knuth Bendix Rewriting System for Monoid(
[ f1, f1^-1, f2, f2^-1, f3, f3^-1 ] ) with rules
[ [ f1*f1^-1, <identity ...> ], [ f1^-1*f1, <identity ...> ],
  [ f2*f2^-1, <identity ...> ], [ f2^-1*f2, <identity ...> ],
  [ f3*f3^-1, <identity ...> ], [ f3^-1*f3, <identity ...> ],
  [ f1*f2^2*f3^-1, <identity ...> ], [ f2*f3^2*f1^-1, <identity ...> ]
    , [ f3*f1^2*f2^-1, <identity ...> ] ]
gap> MakeConfluent(k);
gap> a:=Product(GeneratorsOfMonoid(mon));
f1*f1^-1*f2*f2^-1*f3*f3^-1
gap> ReducedForm(k,UnderlyingElement(a));
<identity ...>
]]></Example>
<P/>
To rewrite a word in the finitely presented group, one has to convert it to
a word in the monoid first, rewrite in the underlying free monoid and
convert back (by forming first again an element of the fp monoid) to the
finitely presented group.
<P/>


<Log><![CDATA[
gap> r:=PseudoRandom(g);;
gap> Length(r);
3704
gap> melm:=Image(mhom,r);;
gap> red:=ReducedForm(k,UnderlyingElement(melm));
f1^-1^3*f2^-1*f1^2
gap> melm:=ElementOfFpMonoid(FamilyObj(One(mon)),red);
f1^-1^3*f2^-1*f1^2
gap> gpelm:=PreImagesRepresentative(mhom,melm);
f1^-3*f2^-1*f1^2
gap> r=gpelm;
true
gap> CategoriesOfObject(red);
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement",
  "IsMultiplicativeElementWithOne", "IsAssociativeElement", "IsWord" ]
gap> CategoriesOfObject(melm);
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement",
  "IsMultiplicativeElementWithOne", "IsAssociativeElement",
  "IsElementOfFpMonoid" ]
gap> CategoriesOfObject(gpelm);
[ "IsExtLElement", "IsExtRElement", "IsMultiplicativeElement",
  "IsMultiplicativeElementWithOne", "IsMultiplicativeElementWithInverse",
  "IsAssociativeElement", "IsElementOfFpGroup" ]
]]></Log>
<P/>
Note, that the elements <C>red</C> (free monoid) <C>melm</C> (fp monoid) and <C>gpelm</C>
(group) differ, though they are displayed identically.
<P/>
Under Unix, it is possible to use the <Package>kbmag</Package> package
to replace the built-in rewriting by this packages efficient C implementation.
You can do this (after loading the <Package>kbmag</Package> package)
by assigning the variable <Ref Var="KB_REW"/> to <C>KBMAG_REW</C>.
Assignment to <C>GAPKB_REW</C> reverts to the built-in implementation.
<Log><![CDATA[
gap> LoadPackage("kbmag");
true
gap> KB_REW:=KBMAG_REW;;
]]></Log>

</Section>


<Section Label="Developing rewriting systems">
<Heading>Developing rewriting systems</Heading>

<#Include Label="[2]{rws}">

</Section>
</Chapter>

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