Quelle cosets.xml Sprache: XML

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<Chapter Label="chap-cosets">
<Heading>The Knuth-Bendix program on cosets</Heading>

It is possible to use the Knuth-Bendix and Automatic Structure program
on cosets  of a specified subgroup <M>H</M> of <M>G</M>. Most of the
functions in the preceding chapter have analogues for cosets rather
than for elements. It is also possible sometimes to compute a complete
rewriting system or a subgroup presentation of <M>H</M>.
<P/>

<Section Label="sec-sub-pres">
<Heading>Subgroups, cosets and subgroup presentations</Heading>

The functions in this section are currently only applicable when the
rewriting system is defined from a group <M>G</M>.
<P/>

<ManSection>
   <Func Name="SubgroupOfKBMAGRewritingSystem"
     Arg = "rws H"  />
<Description>
The subgroup <M>H</M> of the group <M>G</M>
(= <C>SemigroupOfRewritingSystem(rws)</C> )
from which &rws; is defined can be specified either as
a subgroup of <M>G</M>;
or as a list of elements of <M>G</M> that generate <M>H</M>;
or as a subgroup of <M>F</M> = <C>FreeStructureOfRewritingSystem(rws)</C>
that maps onto <M>H</M>;
or as a list of elements of <M>F</M> that generate a subgroup
mapping onto <M>H</M>.
<P/>
<Ref Func="SubgroupOfKBMAGRewritingSystem"/>  returns a rewriting system <C>subrws</C> for <M>H</M>,
but <C>subrws</C> has no rules, and is only intended to be used as a parameter
in the functions that follow.
<P/>
</Description>
</ManSection>

<ManSection>
   <Func Name="ResetRewritingSystemOnCosets"
     Arg = "rws subrws"  />
<Description>
This function resets <C>subrws</C> back to its form as it
was before the application of <Ref Func="KnuthBendixOnCosets"/>  or
<Ref Func="AutomaticStructureOnCosets"/>.
The normal form and  reduction algorithms on cosets will  be
unavailable after  this call.
<P/>
Any optional control parameters set for &rws; will automatically be used when
applying the Knuth-Bendix and Automatic Structure functions on cosets,
that are now to be described.
<P/>
</Description>
</ManSection>

</Section>



<Section Label="sec-KB-cosets">
<Heading>The Knuth-Bendix program on cosets</Heading>

<Index Key="Knuth-Bendix program on cosets">
Knuth-Bendix program on cosets</Index>
<Index Key="rewriting systems on cosets" Subkey="creating"></Index>

<ManSection>
   <Oper Name="KnuthBendixOnCosets"
     Arg = "rws subrws"  />
   <Oper Name="KnuthBendixOnCosetsWithSubgroupRewritingSystem"
     Arg = "rws subrws"  />
<Description>
<C>KnuthBendixOnCosets</C> runs the external Knuth-Bendix program
on the rewriting system  &rws;
with respect to the cosets of the subgroup corresponding to <C>subrws</C>.
It returns  <C>true</C>  if it  finds   a  confluent rewriting system
on coset representatives, and otherwise <C>false</C>.
<P/>
If <Ref Func="KnuthBendixOnCosets"/>  halts without finding a confluent system,
but still manages to output the current system and update &rws;,
then it is possible to use the  resulting rewriting system to reduce coset
representatives, and count and enumerate the irreducible coset representatives;
it  cannot   be guaranteed that the
irreducible coset representatives are all in normal form, however.
<P/>
<Ref Func="KnuthBendixOnCosetsWithSubgroupRewritingSystem"/>
does the same and, in addition, tries to compute a confluent rewriting system
for the subgroup <M>H</M>.
<P/>
</Description>
</ManSection>

<ManSection>
   <Func Name="RewritingSystemOfSubgroupOfKBMAGRewritingSystem"
     Arg = "rws subrws"  />
<Description>
Use this after a successful call of
<Ref Func="KnuthBendixOnCosetsWithSubgroupRewritingSystem"/>.
It returns a confluent rewriting system for <M>H</M> on a generating set
corresponding to the generators of <M>H</M> that were used
to define <C>subrws</C>.
<P/>
</Description>
</ManSection>

<ManSection>
   <Oper Name="IsConfluentOnCosets"
         Arg = "rws" />
<Description>
This operation returns <C>true</C> if &rws; is a rewriting system
on cosets that is known to be confluent.
<P/>
</Description>
</ManSection>

</Section>



<Section Label="sec-aut-cosets">
<Heading>The automatic cosets program</Heading>
<Index Key="automatic cosets program">
automatic cosets program</Index>

<ManSection>
   <Func Name="AutomaticStructureOnCosets"
      Arg = "rws subrws [large filestore diff1]"  />
   <Func Name="AutomaticStructureOnCosetsWithSubgroupPresentation"
      Arg = "rws subrws [large filestore diff1]"  />
<Description>
<C>AutomaticStructureOnCosets</C> runs the external automatic cosets program
on the rewriting system &rws; with respect to the cosets
of the subgroup <M>H</M> from which <C>subrws</C> was defined.
It returns <C>true</C> if successful and <C>false</C> otherwise.
<P/>
The optional parameters to <Ref Func="AutomaticStructureOnCosets"/>
are the same as for <Ref Func="AutomaticStructure"/>.
<P/>
The ordering of &rws; must be &shortlex;.
<P/>
<Ref Func="AutomaticStructureOnCosetsWithSubgroupPresentation"/>
does the same and, in addition,
tries to compute a presentation of the subgroup <M>H</M>.
<P/>
</Description>
</ManSection>

<ManSection>
   <Func Name="PresentationOfSubgroupOfKBMAGRewritingSystem"
     Arg = "rws subrws"  />
<Description>
Use this after a successful call of
<Ref Func="AutomaticStructureOnCosetsWithSubgroupPresentation"/>.
It returns a presentation for <M>H</M>, but this is not on the generators
used to define <M>H</M>.
<P/>
</Description>
</ManSection>

</Section>



<Section Label="sec-coset-red">
<Heading>Word reduction on cosets</Heading>

<ManSection>
   <Oper Name="IsReducedCosetRepresentative"
     Arg = "rws subrws w"  />
<Description>
This operation tests whether the word <M>w</M> in  the generators
of  the freestructure <C>FreeStructure(rws)</C> of the rewriting system system
&rws; is reduced or not as a coset representative of the subgroup <M>H</M>
of <M>G</M>. It returns <C>true</C> or <C>false</C>.
<P/>
<Ref Func="IsReducedCosetRepresentative"/>  can only be used after
<Ref Func="KnuthBendixOnCosets"/>  or <Ref Func="AutomaticStructureOnCosets"/>
has been run successfully on &rws; and <C>subrws</C>.
In the former case, if <C>KnuthBendixOnCosets</C>
halted without a  confluent set of rules, then irreducible words are not
necessarily in  normal form
(but  reducible  words are  definitely  not in  normal form).
If  <C>KnuthBendixOnCosets</C>  completes
with    a  confluent rewriting system  or
<C>AutomaticStructureOnCosets</C> completes  successfully,
then  it is guaranteed that  all irreducible words are in normal form.
<P/>
</Description>
</ManSection>

<ManSection>
   <Oper Name="ReducedCosetRepresentative"
     Arg = "rws subrws w"  />
   <Oper Name="ReducedFormOfCosetRepresentative"
     Arg = "rws subrws w"  />
<Description>
<C>ReducedCosetRepresentative</C> reduces the word  <M>w</M>
in the generators of the free structure
<C>FreeStructure(rws)</C>  of the rewriting system  &rws;
as a coset representative of the subgroup <M>H</M> from which <C>subrws</C>
was defined, and returns the result.
<P/>
<Ref Func="ReducedFormOfCosetRepresentative"/>  can only be used after
<Ref Func="KnuthBendixOnCosets"/>  or <Ref Func="AutomaticStructureOnCosets"/>  has been run successfully on &rws; and <C>subrws</C>.
In the former case, if  <C>KnuthBendixOnCosets</C>  halted
without  a confluent  set of  rules, then the   irreducible word returned is
not necessarily in  normal   form.
If <C>KnuthBendixOnCosets</C>   completes
with  a confluent rewriting  system or  <C>AutomaticStructureOnCosets</C>
completes  successfully, then it  is guaranteed that all irreducible words are
in normal form.
<P/>
</Description>
</ManSection>

</Section>



<Section Label="sec-enum-cosets">
<Heading>Counting and enumerating irreducible words for cosets</Heading>

<ManSection>
   <Meth Name="Index"
     Arg = "rws subrws" />
<Description>
Returns the number of irreducible words for coset representatives of the
subgroup <M>H</M> of <M>G</M> corresponding to <C>subrws</C>.
<P/>
<Ref Func="Index"/> can only be  used  after <Ref Func="KnuthBendixOnCosets"/>
or <Ref Func="AutomaticStructureOnCosets"/>  has been run
successfully  on &rws; and <C>subrws</C>.
In  the former case, if  <C>KnuthBendixOnCosets</C>  halted without a
confluent  set of rules, then the  number of  irreducible words may be
greater than the number of words in normal form (which is equal to the
index  of <M>H</M> in <M>G</M>).
If <C>KnuthBendixOnCosets</C>   completes with a
confluent rewriting system  or <C>AutomaticStructureOnCosets</C>
completes successfully, then it is  guaranteed that <Ref Func="Index"/>
will  return the correct index of <M>H</M> in <M>G</M>.
<P/>
</Description>
</ManSection>

<ManSection>
   <Oper Name="EnumerateReducedCosetRepresentatives"
     Arg = "rws subrws min max"  />
<Description>
Enumerate  all  irreducible words for coset representatives of <M>H</M>
in <M>G</M>, that have   lengths between <C>min</C> and <C>max</C>
(inclusive), which should be non-negative integers.
The result is returned  as  a list of words.
The enumeration is by depth-first search  of a finite state automaton,
and so  the words in the  list returned  are ordered lexicographically
(not by &shortlex;).

<Ref Func="EnumerateReducedCosetRepresentatives"/>  can only be used after
<Ref Func="KnuthBendixOnCosets"/>  or <Ref Func="AutomaticStructureOnCosets"/>  has been run successfully on &rws; and <C>subrws</C>.
In  the former case,  if <C>KnuthBendixOnCosets</C>  halted  without a
confluent set of  rules, then not all irreducible   words in the  list
returned will necessarily be in normal form.
If <C>KnuthBendixOnCosets</C>  completes with a confluent rewriting  system
or <C>AutomaticStructureOnCosets</C> completes successfully, then
it is guaranteed that all words in the list will be in normal form.
<P/>
</Description>
</ManSection>

<ManSection>
   <Func Name="GrowthFunctionOfCosetRepresentatives"
     Arg = "rws subrws"  />
<Description>
Returns the growth function of the set of irreducible words for coset
representatives of <M>H</M> in <M>G</M>, where <C>subrws</C> and &rws; are
the rewriting systems for <M>H</M> and <M>G</M>.
This is a rational function, of which
the coefficient of <M>x^n</M> in its Taylor expansion is equal to the number of
coset representatives words of length <M>n</M>.
<P/>
If the coefficients in this rational function are larger than about <M>16000</M>
then strange error messages will appear and <C>fail</C> will be returned.
<P/>
<Ref Func="GrowthFunctionOfCosetRepresentatives"/>  can only be used after
<Ref Func="KnuthBendixOnCosets"/>  or <Ref Func="AutomaticStructureOnCosets"/>  has been run successfully on &rws; and <C>subrws</C>.
In  the former case,  if <C>KnuthBendixOnCosets</C>  halted  without a
confluent set of  rules, then not all irreducible   words in the  list
returned will necessarily be in normal form.
If <C>KnuthBendixOnCosets</C>  completes with a confluent rewriting system or
<C>AutomaticStructureOnCosets</C>  completes successfully, then
it is guaranteed that all words in the list will be in normal form.
<P/>
</Description>
</ManSection>

</Section>



<Section Label="sec-ex-cosets">
<Heading>Examples of the use of Rewriting System on Cosets</Heading>

<Index Key="rewriting systems on cosets" Subkey="examples"></Index>

Here are two examples to illustrate the operations described above.

<Subsection Label="subsec-example1-cosets">
<Heading>Example 1</Heading>

<Example>
<![CDATA[
gap> F := FreeGroup( "a", "b", "c" );;
gap> a := F.1;;  b := F.2;;  c := F.3;;
gap> G := F/[b^3,c^3,(b*c)^4,(b*c^-1)^5,a^-1*b^-1*c*b*c*b^-1*c*b*c^-1];
<fp group on the generators [ a, b, c ]>
gap> R := KBMAGRewritingSystem( G );
rec(
       isRWS := true,
      silent := true,
  generatorOrder := [_g1,_g2,_g3,_g4,_g5,_g6],
    inverses := [_g2,_g1,_g4,_g3,_g6,_g5],
    ordering := "shortlex",
       equations := [
     [_g3^2,_g4],
     [_g5^2,_g6],
     [_g3*_g5*_g3*_g5,_g6*_g4*_g6*_g4],
     [_g3*_g6*_g3*_g6*_g3,_g5*_g4*_g5*_g4*_g5],
     [_g2*_g4*_g5*_g3*_g5,_g5*_g4*_g6*_g3]
       ]
)
gap> S := SubgroupOfKBMAGRewritingSystem( R, [ a^3, c*a^2 ] );
rec(
       isRWS := true,
      silent := true,
  generatorOrder := [_x1,_X1,_x2,_X2],
    inverses := [_X1,_x1,_X2,_x2],
    ordering := "shortlex",
       equations := [
       ]
)
gap> KnuthBendixOnCosetsWithSubgroupRewritingSystem( R, S );
true
gap> Index( R, S );
18
gap> IsReducedCosetRepresentative( R, S, b*a*b*a );
false
gap> ReducedFormOfCosetRepresentative( R, S, b*a*b*a );
b^-1*a^-1
gap> EnumerateReducedCosetRepresentatives( R, S, 0, 4 );
[ <identity ...>, a, a*b, a*b*c, a*b^-1, a^-1, a^-1*b, a^-1*b*c, a^-1*b^-1,
  b, b*c, b*c*a, b*c*a^-1, b*c^-1, b^-1, b^-1*a, b^-1*a^-1, b^-1*a^-1*b ]
gap> SS := RewritingSystemOfSubgroupOfKBMAGRewritingSystem( R, S );;
gap> Size( SS );
60
]]>
</Example>
</Subsection>

<Subsection Label="subsec-example2-cosets">
<Heading>Example 2</Heading>

We find a free subgroup of the Fibonacci group <M>F(2,8)</M>.
This example may take about <M>20</M> minutes to run on a typical WorkStation.
<P/>
<Example>
<![CDATA[
gap> F := FreeGroup( 8 );;
gap> a := F.1;  b := F.2;  c := F.3;  d := F.4;
gap> e := F.5;  f := F.6;  g := F.7;  h := F.8;
gap> G := F/[a*b*c^-1, b*c*d^-1, c*d*e^-1, d*e*f^-1,
>            e*f*g^-1, f*g*h^-1, g*h*a^-1, h*a*b^-1];
gap> R := KBMAGRewritingSystem( G );;
gap> S := SubgroupOfKBMAGRewritingSystem( R, [a,e] );;
gap> AutomaticStructureOnCosetsWithSubgroupPresentation( R, S );
gap> P := PresentationOfSubgroupOfKBMAGRewritingSystem( R, S );
<fp group on the generators [ f1, f3 ]>
gap> RelatorsOfFpGroup( P );
[  ]
gap> Index( R, S );
infinity
]]>
</Example>

</Subsection>

</Section>

</Chapter>

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