Quelle tietze.gd Sprache: unbekannt

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Volkmar Felsch.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declarations for finitely presented groups
## (fp groups).
##

#############################################################################
##
## Some global symbolic constants.
##
TZ_NUMGENS := 1;
TZ_NUMRELS := 2;
TZ_TOTAL := 3;
TZ_GENERATORS := 4;
TZ_INVERSES := 5;
TZ_RELATORS := 6;
TZ_LENGTHS := 7;
TZ_FLAGS := 8;
TZ_MODIFIED := 10;
TZ_NUMREDUNDS := 11;
TZ_STATUS := 15;
TZ_LENGTHTIETZE := 21;

TZ_FREEGENS := 9;
# TZ_ITERATOR := 12;
TZ_OCCUR :=21;

TR_TREELENGTH := 3;
TR_PRIMARY := 4;
TR_TREENUMS := 5;
TR_TREEPOINTERS := 6;
TR_TREELAST := 7;

#############################################################################
##
## List of option names
##
TzOptionNames := MakeImmutable([ "protected", "eliminationsLimit", "expandLimit",
 "generatorsLimit", "lengthLimit", "loopLimit", "printLevel",
 "saveLimit", "searchSimultaneous" ]);

#############################################################################
##
#A TietzeOrigin( <G> )
##
## <ManSection>
## <Attr Name="TietzeOrigin" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "TietzeOrigin", IsSubgroupFpGroup );

#############################################################################
##
#F AbstractWordTietzeWord( <word>, <fgens> )
##
## <#GAPDoc Label="AbstractWordTietzeWord">
## <ManSection>
## <Func Name="AbstractWordTietzeWord" Arg='word, fgens'/>
##
## <Description>
## assumes <A>fgens</A> to be a list of free group
## generators and <A>word</A> to be a Tietze word in these generators,
## i. e., a list of positive or negative generator numbers.
## It converts <A>word</A> to an abstract word.
## 
## This function simply calls <Ref Oper="AssocWordByLetterRep"/>.
## <Example><![CDATA[
## gap> F := FreeGroup( "a", "b", "c" ,"d");
## <free group on the generators [ a, b, c, d ]>
## gap> tzword := TietzeWordAbstractWord(
## > Comm(F.4,F.2) * (F.3^2 * F.2)^-1, GeneratorsOfGroup( F ){[2,3,4]} );
## [ -3, -1, 3, -2, -2 ]
## gap> AbstractWordTietzeWord( tzword, GeneratorsOfGroup( F ){[2,3,4]} );
## d^-1*b^-1*d*c^-2
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("AbstractWordTietzeWord");

#############################################################################
##
#F TietzeWordAbstractWord( <word>, <fgens> )
##
## <#GAPDoc Label="TietzeWordAbstractWord">
## <ManSection>
## <Oper Name="TietzeWordAbstractWord" Arg='word, fgens'/>
##
## <Description>
## assumes <A>fgens</A> to be a list of free group generators
## and <A>word</A> to be an abstract word in these generators.
## It converts <A>word</A> into a Tietze word,
## i. e., a list of positive or negative generator numbers.
## 
## This function simply calls <Ref Oper="LetterRepAssocWord"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym("TietzeWordAbstractWord",LetterRepAssocWord);

#############################################################################
##
#F TzWordAbstractWord( <word> )
#F AbstractWordTzWord(<fam>, <tzword> )
##
## <ManSection>
## <Func Name="TzWordAbstractWord" Arg='word'/>
## <Func Name="AbstractWordTzWord" Arg='fam, tzword'/>
##
## <Description>
## only supported for compatibility.
## </Description>
## </ManSection>
##
DeclareSynonym("TzWordAbstractWord",LetterRepAssocWord);
DeclareSynonym("AbstractWordTzWord",AssocWordByLetterRep);

#############################################################################
##
#F AddGenerator( )
##
## <#GAPDoc Label="AddGenerator">
## <ManSection>
## <Func Name="AddGenerator" Arg='P'/>
##
## <Description>
## extends the presentation <A>P</A> by a new generator.
## 
## Let <M>i</M> be the smallest positive integer which has not yet been used
## as a generator number in the given presentation.
## <Ref Func="AddGenerator"/> defines a new abstract generator <M>x_i</M>
## with the name <C>"_x</C><M>i</M><C>"</C> and adds it to the
## list of generators of <A>P</A>.
## 
## You may access the generator <M>x_i</M> by typing
## <A>P</A><C>!.</C><M>i</M>. However, this
## is only practicable if you are running an interactive job because you
## have to know the value of <M>i</M>. Hence the proper way to access the new
## generator is to write
## <C>Last(GeneratorsOfPresentation(P))</C>.
## <Example><![CDATA[
## gap> G := PerfectGroup(IsFpGroup, 120 );;
## gap> H := Subgroup( G, [ G.1^G.2, G.3 ] );;
## gap> P := PresentationSubgroup( G, H );
## <presentation with 4 gens and 7 rels of total length 21>
## gap> AddGenerator( P );
## #I now the presentation has 5 generators, the new generator is _x7
## gap> gens := GeneratorsOfPresentation( P );
## [ _x1, _x2, _x4, _x5, _x7 ]
## gap> gen := Last(gens);
## _x7
## gap> gen = P!.7;
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("AddGenerator");

#############################################################################
##
#F AddRelator( , <word> )
##
## <#GAPDoc Label="AddRelator">
## <ManSection>
## <Func Name="AddRelator" Arg='P, word'/>
##
## <Description>
## adds the relator <A>word</A> to the presentation <A>P</A>, probably
## changing the group defined by <A>P</A>.
## <A>word</A> must be an abstract word in the generators of <A>P</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("AddRelator");

#############################################################################
##
#F DecodeTree()
##
## <#GAPDoc Label="DecodeTree">
## <ManSection>
## <Func Name="DecodeTree" Arg='P'/>
##
## <Description>
## assumes that <A>P</A> is a subgroup presentation provided by the Reduced
## Reidemeister-Schreier or by the Modified Todd-Coxeter method (see
## <Ref Func="PresentationSubgroupRrs"
## Label="for two groups (and a string)"/>,
## <Ref Func="PresentationNormalClosureRrs"/>,
## <Ref Func="PresentationSubgroupMtc"/>).
## It eliminates the secondary generators of <A>P</A>
## (see Section <Ref Sect="Subgroup Presentations"/>) by applying the
## so called <Q>decoding tree</Q> procedure.
## 
## <Ref Func="DecodeTree"/> is called automatically by the command
## <Ref Func="PresentationSubgroupMtc"/> where it
## reduces <A>P</A> to a presentation on the given (primary) subgroup
## generators.
## <Index>secondary subgroup generators</Index>
## 
## In order to explain the effect of this command we need to insert a few
## remarks on the subgroup presentation commands described in section
## <Ref Sect="Subgroup Presentations"/>.
## All these commands have the common property that in the process of
## constructing a presentation for a given subgroup <A>H</A> of a finitely
## presented group <A>G</A> they first build up a highly
## redundant list of generators of <A>H</A> which consists of an (in general
## small) list of <Q>primary</Q> generators, followed by an (in general
## large) list of <Q>secondary</Q> generators, and then construct a
## presentation <M>P_0</M>
## <E>on a sublist of these generators</E> by rewriting
## the defining relators of <A>G</A>.
## This sublist contains all primary, but, at least in general,
## by far not all secondary generators.
## <Index>primary subgroup generators</Index>
## 
## The role of the primary generators depends on the concrete choice of the
## subgroup presentation command. If the Modified Todd-Coxeter method is
## used, they are just the given generators of <A>H</A>,
## whereas in the case of the Reduced Reidemeister-Schreier algorithm they
## are constructed by the program.
## 
## Each of the secondary generators is defined by a word of length two in
## the preceding generators and their inverses. By historical reasons, the
## list of these definitions is called the <E>subgroup generators tree</E>
## though in fact it is not a tree but rather a kind of bush.
## <Index>subgroup generators tree</Index>
## 
## Now we have to distinguish two cases. If <M>P_0</M> has been constructed
## by the Reduced Reidemeister-Schreier routines, it is a presentation of
## <A>H</A>. However, if the Modified Todd-Coxeter routines have been used
## instead, then the relators in <M>P_0</M> are valid relators of <A>H</A>,
## but they do not necessarily define <A>H</A>.
## We handle these cases in turn, starting with the latter one.
## 
## In fact, we could easily receive a presentation of <A>H</A> also in this
## case if we extended <M>P_0</M> by adding to it all the
## secondary generators which are not yet contained in it and all the
## definitions from the generators tree as additional generators and
## relators.
## Then we could recursively eliminate all secondary generators by Tietze
## transformations using the new relators.
## However, this procedure turns out to be too inefficient to
## be of interest.
## 
## Instead, we use the so called <E>decoding tree</E> procedure
## (see <Cite Key="AMW82"/>, <Cite Key="AR84"/>). It proceeds as follows.
## 
## Starting from <M>P = P_0</M>, it runs through a number of steps in each
## of which it eliminates the current <Q>last</Q> generator (with respect to
## the list of all primary and secondary generators). If the last generator
## <A>g</A> is a primary generator, then the procedure terminates.
## Otherwise it checks whether there is a relator in the current
## presentation which can be used to substitute <A>g</A> by a Tietze
## transformation. If so, this is done.
## Otherwise, and only then, the tree definition of <A>g</A> is added to
## <A>P</A> as a new relator, and the generators involved are added as new
## generators if they have not yet been contained in <A>P</A>.
## Subsequently, <A>g</A> is eliminated.
## 
## Note that the extension of <A>P</A> by one or two new generators is
## <E>not</E> a Tietze transformation.
## In general, it will change the isomorphism type
## of the group defined by <A>P</A>.
## However, it is a remarkable property of this procedure, that at the end,
## i.e., as soon as all secondary generators have been eliminated,
## it provides a presentation <M>P = P_1</M>,
## say, which defines a group isomorphic to <A>H</A>. In fact, it is this
## presentation which is returned by the command <Ref Func="DecodeTree"/>
## and hence by the command <Ref Func="PresentationSubgroupMtc"/>.
## 
## If, in the other case, the presentation <M>P_0</M> has been constructed
## by the Reduced Reidemeister-Schreier algorithm,
## then <M>P_0</M> itself is a presentation of <A>H</A>,
## and the corresponding subgroup presentation command
## (<Ref Func="PresentationSubgroupRrs"
## Label="for two groups (and a string)"/> or
## <Ref Func="PresentationNormalClosureRrs"/>) just returns <M>P_0</M>.
## 
## As mentioned in section <Ref Sect="Subgroup Presentations"/>,
## we recommend to further simplify this presentation before you use it.
## The standard way to do this is to start from <M>P_0</M> and to apply
## suitable Tietze transformations,
## e. g., by calling the commands <Ref Func="TzGo"/> or
## <Ref Func="TzGoGo"/>.
## This is probably the most efficient approach, but you will end up with a
## presentation on some unpredictable set of generators.
## As an alternative, &GAP; offers you the <Ref Func="DecodeTree"/> command
## which you can use to eliminate all secondary
## generators (provided that there are no space or time problems). For this
## purpose, the subgroup presentation commands do not only return the
## resulting presentation, but also the tree (together with some associated
## lists) as a kind of side result in a component <A>P</A><C>!.tree</C> of
## the resulting presentation <A>P</A>.
## 
## Note, however, that the decoding tree routines will not work correctly
## any more on a presentation from which generators have already been
## eliminated by Tietze transformations.
## Therefore, to prevent you from getting wrong results by calling
## <Ref Func="DecodeTree"/> in such a situation,
## &GAP; will automatically remove the subgroup generators tree
## from a presentation as soon as one of the generators is substituted by a
## Tietze transformation.
## 
## Nevertheless, a certain misuse of the command is still possible, and we
## want to explicitly warn you from this.
## The reason is that the Tietze option parameters described in
## Section <Ref Sect="Tietze Options"/> apply to
## <Ref Func="DecodeTree"/> as well.
## Hence, in case of inadequate values of these parameters, it may happen that
## <Ref Func="DecodeTree"/> stops before all the secondary generators have
## vanished. In this case &GAP;
## will display an appropriate warning. Then you should change the
## respective parameters and continue the process by calling
## <Ref Func="DecodeTree"/> again. Otherwise, if you would apply Tietze
## transformations, it might happen because of the convention described
## above that the tree is removed and that you end up with a wrong
## presentation.
## 
## After a successful run of <Ref Func="DecodeTree"/> it is convenient to
## further simplify the resulting presentation by suitable Tietze
## transformations.
## 
## As an example of an explicit call of <Ref Func="DecodeTree"/> we compute
## two presentations of a subgroup of order <M>384</M> in a group of order
## <M>6912</M>. In both cases we use the Reduced Reidemeister-Schreier
## algorithm, but in the first run we just apply the Tietze transformations
## offered by the <Ref Func="TzGoGo"/> command with its default parameters,
## whereas in the second run we call the <Ref Func="DecodeTree"/> command
## before.
## 
## <Example><![CDATA[
## gap> F2 := FreeGroup( "a", "b" );;
## gap> G := F2 / [ F2.1*F2.2^2*F2.1^-1*F2.2^-1*F2.1^3*F2.2^-1,
## > F2.2*F2.1^2*F2.2^-1*F2.1^-1*F2.2^3*F2.1^-1 ];;
## gap> a := G.1;; b := G.2;;
## gap> H := Subgroup( G, [ Comm(a^-1,b^-1), Comm(a^-1,b), Comm(a,b) ] );;
## ]]></Example>
## 
## We use the Reduced Reidemeister Schreier method and default Tietze
## transformations to get a presentation for <A>H</A>.
## 
## <Example><![CDATA[
## gap> P := PresentationSubgroupRrs( G, H );
## <presentation with 18 gens and 35 rels of total length 169>
## gap> TzGoGo( P );
## #I there are 3 generators and 20 relators of total length 488
## #I there are 3 generators and 20 relators of total length 466
## ]]></Example>
## 
## We end up with 20 relators of total length 466. Now we repeat the
## procedure, but we call the decoding tree algorithm before doing the Tietze
## transformations.
## 
## <Example><![CDATA[
## gap> P := PresentationSubgroupRrs( G, H );
## <presentation with 18 gens and 35 rels of total length 169>
## gap> DecodeTree( P );
## #I there are 9 generators and 26 relators of total length 185
## #I there are 6 generators and 23 relators of total length 213
## #I there are 3 generators and 20 relators of total length 252
## #I there are 3 generators and 20 relators of total length 244
## gap> TzGoGo( P );
## #I there are 3 generators and 19 relators of total length 168
## #I there are 3 generators and 17 relators of total length 138
## #I there are 3 generators and 15 relators of total length 114
## #I there are 3 generators and 13 relators of total length 96
## #I there are 3 generators and 12 relators of total length 84
## ]]></Example>
## 
## This time we end up with a shorter presentation.
## 
## 
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("DecodeTree");

#############################################################################
##
#F FpGroupPresentation( [,<nam>] )
##
## <#GAPDoc Label="FpGroupPresentation">
## <ManSection>
## <Func Name="FpGroupPresentation" Arg='P [,nam]'/>
##
## <Description>
## constructs an f. p. group as defined by the given Tietze
## presentation <A>P</A>.
## <Example><![CDATA[
## gap> h := FpGroupPresentation( p );
## <fp group on the generators [ a, b ]>
## gap> h = g;
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("FpGroupPresentation");

#############################################################################
##
#F PresentationFpGroup( <G> [,<printlevel>] ) . . . create a presentation
##
## <#GAPDoc Label="PresentationFpGroup">
## <ManSection>
## <Func Name="PresentationFpGroup" Arg='G [,printlevel]'/>
##
## <Description>
## creates a presentation, i. e., a Tietze object, for the given finitely
## presented group <A>G</A>. This presentation will be exactly as the
## presentation of <A>G</A> and <E>no</E> initial Tietze transformations
## are applied to it.
## 
## The optional <A>printlevel</A> parameter can be used to restrict or to
## extend the amount of output provided by Tietze transformation
## commands when being applied to the created presentation. The
## default value 1 is designed for interactive use and implies
## explicit messages to be displayed by most of these commands. A
## <A>printlevel</A> value of 0 will suppress these messages, whereas a
## <A>printlevel</A> value of 2 will enforce some additional output.
## <Example><![CDATA[
## gap> f := FreeGroup( "a", "b" );
## <free group on the generators [ a, b ]>
## gap> g := f / [ f.1^3, f.2^2, (f.1*f.2)^3 ];
## <fp group on the generators [ a, b ]>
## gap> p := PresentationFpGroup( g );
## <presentation with 2 gens and 3 rels of total length 11>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("PresentationFpGroup");

#############################################################################
##
#F PresentationRegularPermutationGroup(<G>)
#F PresentationRegularPermutationGroupNC(<G>)
##
## <ManSection>
## <Func Name="PresentationRegularPermutationGroup" Arg='G'/>
## <Func Name="PresentationRegularPermutationGroupNC" Arg='G'/>
##
## <Description>
## constructs a presentation from the given regular permutation group using
## the algorithm which has been described in <Cite Key="Can73"/> and <Cite Key="Neu82"/>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("PresentationRegularPermutationGroup");
DeclareGlobalFunction("PresentationRegularPermutationGroupNC");

#############################################################################
##
#F PresentationViaCosetTable( <G>[, <F>, <words>] )
##
## <#GAPDoc Label="PresentationViaCosetTable">
## <ManSection>
## <Func Name="PresentationViaCosetTable" Arg='G[, F, words]'/>
##
## <Description>
## constructs a presentation for a given concrete finite group.
## It applies the relations finding algorithm which has been described in
## <Cite Key="Can73"/> and <Cite Key="Neu82"/>.
## It automatically applies Tietze transformations to the presentation
## found.
## 
## If only a group <A>G</A> has been specified, the single stage algorithm
## is applied.
## 
## The operation <Ref Attr="IsomorphismFpGroup"/> in contrast uses a
## multiple-stage algorithm using a chief series and stabilizer chains.
## It usually should be used rather than
## <Ref Func="PresentationViaCosetTable"/>.
## (It does not apply Tietze transformations automatically.)
## 
## If the two stage algorithm is to be used,
## <Ref Func="PresentationViaCosetTable"/> expects a subgroup <A>H</A> of
## <A>G</A> to be provided in form of two additional arguments <A>F</A> and
## <A>words</A>, where <A>F</A> is a free group with the same number
## of generators as <A>G</A>, and <A>words</A> is a list of words in the
## generators of <A>F</A> which supply a list of generators of <A>H</A> if
## they are evaluated as words in the corresponding generators of <A>G</A>.
## <Example><![CDATA[
## gap> G := GeneralLinearGroup( 2, 7 );
## GL(2,7)
## gap> GeneratorsOfGroup( G );
## [ [ [ Z(7), 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ],
## [ [ Z(7)^3, Z(7)^0 ], [ Z(7)^3, 0*Z(7) ] ] ]
## gap> Size( G );
## 2016
## gap> P := PresentationViaCosetTable( G );
## <presentation with 2 gens and 5 rels of total length 46>
## gap> TzPrintRelators( P );
## #I 1. f2^3
## #I 2. f1^6
## #I 3. (f1*f2)^6
## #I 4. f1*f2*f1^-1*f2*f1*f2^-1*f1^-1*f2*f1*f2*f1^-1*f2^-1
## #I 5. f1^-3*f2*f1*f2*(f1^-1*f2^-1)^2*f1^-2*f2
## ]]></Example>
## 
## The two stage algorithm saves an essential amount of space by
## constructing two coset tables of lengths <M>|H|</M> and <M>|G|/|H|</M>
## instead of just one coset table of length <M>|G|</M>.
## The next example shows an application
## of this option in the case of a subgroup of size 7920 and index 12 in a
## permutation group of size 95040.
## 
## <Example><![CDATA[
## gap> M12 := Group( [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6),
## > (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ], () );;
## gap> F := FreeGroup( "a", "b", "c" );
## <free group on the generators [ a, b, c ]>
## gap> words := [ F.1, F.2 ];
## [ a, b ]
## gap> P := PresentationViaCosetTable( M12, F, words );
## <presentation with 3 gens and 10 rels of total length 97>
## gap> G := FpGroupPresentation( P );
## <fp group on the generators [ a, b, c ]>
## gap> RelatorsOfFpGroup( G );
## [ c^2, b^4, (a*c)^3, (a*b^-2)^3, a^11,
## a^2*b*a^-2*b^-1*(b^-1*a)^2*a*b^-1, (a*(b*a^-1)^2*b^-1)^2,
## a^2*b*a^2*b^-2*a^-1*b*(a^-1*b^-1)^2,
## a^2*b^-1*a^-1*b^-1*a*c*b*c*(a*b)^2, a^2*(a^2*b)^2*a^-2*c*a*b*a^-1*c
## ]
## ]]></Example>
## 
## Before it is returned, the resulting presentation is being simplified by
## appropriate calls of the function <Ref Func="SimplifyPresentation"/>
## (see <Ref Sect="Tietze Transformations"/>),
## but without allowing any eliminations of generators.
## This restriction guarantees that we get a bijection between the list of
## generators of <A>G</A> and the list of generators in the presentation.
## Hence, if the generators of <A>G</A> are redundant and if you don't care
## for the bijection, you may get a shorter presentation by calling the
## function <Ref Func="SimplifyPresentation"/>,
## now without this restriction, once more yourself.
## 
## <Example><![CDATA[
## gap> H := Group(
## > [ (2,5,3), (2,7,5), (1,8,4), (1,8,6), (4,8,6), (3,5,7) ], () );;
## gap> P := PresentationViaCosetTable( H );
## <presentation with 6 gens and 12 rels of total length 42>
## gap> SimplifyPresentation( P );
## #I there are 4 generators and 10 relators of total length 36
## ]]></Example>
## 
## If you apply the function <Ref Func="FpGroupPresentation"/> to the
## resulting presentation you will get a finitely presented group isomorphic
## to <A>G</A>.
## Note, however, that the function <Ref Attr="IsomorphismFpGroup"/>
## is recommended for this purpose.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("PresentationViaCosetTable");

#############################################################################
##
#F RelsViaCosetTable(<G>,<cosets>,<F>)
#F RelsViaCosetTable(<G>,<cosets>,<F>,<ggens>)
#F RelsViaCosetTable(<G>,<cosets>,<F>,<words>,<H>,<R1>)
##
## <ManSection>
## <Func Name="RelsViaCosetTable" Arg='G,cosets,F'/>
## <Func Name="RelsViaCosetTable" Arg='G,cosets,F,ggens'/>
## <Func Name="RelsViaCosetTable" Arg='G,cosets,F,words,H,R1'/>
##
## <Description>
## constructs a defining set of relators for the given
## concrete group using the algorithm
## which has been described in <Cite Key="Can73"/> and <Cite Key="Neu82"/>.
## 
## It is a subroutine of function <C>PresentationViaCosetTable</C>. Hence its
## input and output are specifically designed only for this purpose, and it
## does not check the arguments.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("RelsViaCosetTable");

#############################################################################
##
#F RemoveRelator( , <n> )
##
## <#GAPDoc Label="RemoveRelator">
## <ManSection>
## <Func Name="RemoveRelator" Arg='P, n'/>
##
## <Description>
## removes the <A>n</A>-th relator from the presentation <A>P</A>,
## probably changing the group defined by <A>P</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("RemoveRelator");

#############################################################################
##
#F SimplifiedFpGroup( <G> )
##
## <#GAPDoc Label="SimplifiedFpGroup">
## <ManSection>
## <Func Name="SimplifiedFpGroup" Arg='G'/>
##
## <Description>
## applies Tietze transformations to a copy of the presentation of the
## given finitely presented group <A>G</A> in order to reduce it
## with respect to the number of generators, the number of relators,
## and the relator lengths.
## 
## <Ref Func="SimplifiedFpGroup"/> returns a group isomorphic to
## the given one with a presentation which has been tried to simplify
## via Tietze transformations.
## 
## If the connection to the original group is important, then the operation
## <Ref Attr="IsomorphismSimplifiedFpGroup"/> should be used instead.
## 
## <Example><![CDATA[
## gap> F6 := FreeGroup( 6, "G" );;
## gap> G := F6 / [ F6.1^2, F6.2^2, F6.4*F6.6^-1, F6.5^2, F6.6^2,
## > F6.1*F6.2^-1*F6.3, F6.1*F6.5*F6.3^-1, F6.2*F6.4^-1*F6.3,
## > F6.3*F6.4*F6.5^-1, F6.1*F6.6*F6.3^-2, F6.3^4 ];;
## gap> H := SimplifiedFpGroup( G );
## <fp group on the generators [ G1, G3 ]>
## gap> RelatorsOfFpGroup( H );
## [ G1^2, (G1*G3^-1)^2, G3^4 ]
## ]]></Example>
## 
## In fact, the command
## 
## <Log><![CDATA[
## H := SimplifiedFpGroup( G );
## ]]></Log>
## 
## is an abbreviation of the command sequence
## 
## <Log><![CDATA[
## P := PresentationFpGroup( G, 0 );;
## SimplifyPresentation( P );
## H := FpGroupPresentation( P );
## ]]></Log>
## 
## which applies a rather simple-minded strategy of Tietze transformations
## to the intermediate presentation <A>P</A>.
## If, for some concrete group, the resulting presentation is unsatisfying,
## then you should try a more sophisticated, interactive use of the
## available Tietze transformation commands
## (see <Ref Sect="Tietze Transformations"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SimplifiedFpGroup");

############################################################################
##
#F TzCheckRecord
##
## <ManSection>
## <Func Name="TzCheckRecord" Arg='obj'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzCheckRecord");

#############################################################################
##
#F TzEliminate( [, <gen>] )
#F TzEliminate( [, <n>] )
##
## <#GAPDoc Label="TzEliminate">
## <ManSection>
## <Heading>TzEliminate</Heading>
## <Func Name="TzEliminate" Arg='P[, gen]'
## Label="for a presentation (and a generator)"/>
## <Func Name="TzEliminate" Arg='P[, n]'
## Label="for a presentation (and an integer)"/>
##
## <Description>
## tries to eliminate a generator from a presentation <A>P</A> via
## Tietze transformations.
## 
## Any relator which contains some generator just once can be used to
## substitute that generator by a word in the remaining generators.
## If such generators and relators exist, then
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## chooses a generator for which the product of its number of occurrences
## and the length of the substituting word is minimal,
## and then it eliminates this generator from the presentation,
## provided that the resulting total length of the relators does not exceed
## the associated Tietze option parameter <C>spaceLimit</C>
## (see <Ref Sect="Tietze Options"/>). The default value of that parameter
## is <Ref Var="infinity"/>, but you may alter it appropriately.
## 
## If a generator <A>gen</A> has been specified,
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## eliminates it if possible, i. e. if there is a relator in which
## <A>gen</A> occurs just once.
## If no second argument has been specified,
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## eliminates some appropriate generator if possible and if the resulting
## total length of the relators will not exceed the Tietze options parameter
## <C>lengthLimit</C>.
## 
## If an integer <A>n</A> has been specified,
## <Ref Func="TzEliminate" Label="for a presentation (and an integer)"/>
## tries to eliminate up to <A>n</A> generators.
## Note that the calls <C>TzEliminate(<A>P</A>)</C> and
## <C>TzEliminate(<A>P</A>,1)</C> are equivalent.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzEliminate");

#############################################################################
##
#F TzEliminateFromTree( )
##
## <ManSection>
## <Func Name="TzEliminateFromTree" Arg='P'/>
##
## <Description>
## <Ref Func="TzEliminateFromTree"/> eliminates the last Tietze generator.
## If that generator cannot be isolated in any Tietze relator,
## then its definition is taken from the tree and added as an additional
## Tietze relator, extending the set of Tietze generators appropriately,
## if necessary.
## However, the elimination will not be performed if the resulting total
## length of the relators cannot be guaranteed to not exceed the parameter
## <C>lengthLimit</C>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzEliminateFromTree");

#############################################################################
##
#F TzEliminateGen( , <n> )
##
## <ManSection>
## <Func Name="TzEliminateGen" Arg='P, n'/>
##
## <Description>
## eliminates the Tietze generator <C>GeneratorsOfPresentation(P)[n]</C>
## if possible, i. e. if that generator can be isolated in some appropriate
## Tietze relator. However, the elimination will not be performed if the
## resulting total length of the relators cannot be guaranteed to not exceed
## the parameter <C>lengthLimit</C>
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzEliminateGen");

#############################################################################
##
#F TzEliminateGen1( )
##
## <ManSection>
## <Func Name="TzEliminateGen1" Arg='P'/>
##
## <Description>
## tries to eliminate a Tietze generator: If there are
## Tietze generators which occur just once in certain Tietze relators, then
## one of them is chosen for which the product of the length of its minimal
## defining word and the number of its occurrences is minimal. However,
## the elimination will not be performed if the resulting total length of
## the relators cannot be guaranteed to not exceed the parameter
## <C>lengthLimit</C>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzEliminateGen1");

#############################################################################
##
#F TzEliminateGens( [, <decode>] )
##
## <ManSection>
## <Func Name="TzEliminateGens" Arg='P [, decode]'/>
##
## <Description>
## <Ref Func="TzEliminateGens"/> repeatedly eliminates generators from the
## presentation of the given group until at least one of the following
## conditions is violated:
## 
## <Enum>
## <Item>
## The current number of generators is not greater than the
## parameter <C>generatorsLimit</C>.
## </Item>
## <Item>
## The number of generators eliminated so far is less than
## the parameter <C>eliminationsLimit</C>.
## </Item>
## <Item>
## The total length of the relators has not yet grown to a percentage
## greater than the parameter <C>expandLimit</C>.
## </Item>
## <Item>
## The next elimination will not extend the total length to a value
## greater than the parameter <C>lengthLimit</C>.
## </Item>
## </Enum>
## 
## If a second argument has been specified, then it is assumed that we
## are in the process of decoding a tree.
## 
## If not, then the function will not eliminate any protected generators.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzEliminateGens");

#############################################################################
##
#F TzFindCyclicJoins( )
##
## <#GAPDoc Label="TzFindCyclicJoins">
## <ManSection>
## <Func Name="TzFindCyclicJoins" Arg='P'/>
##
## <Description>
## searches for power and commutator relators in order
## to find pairs of generators which generate a common cyclic subgroup.
## It uses these pairs to introduce new relators, but it does not introduce
## any new generators as is done by <Ref Func="TzSubstituteCyclicJoins"/>.
## 
## More precisely:
## <Ref Func="TzFindCyclicJoins"/> searches for pairs of generators <M>a</M>
## and <M>b</M> such that (possibly after inverting or conjugating some
## relators) the set of relators contains the commutator <M>[a,b]</M>,
## a power <M>a^n</M>, and a product of the form <M>a^s b^t</M>
## with <M>s</M> prime to <M>n</M>.
## For each such pair, <Ref Func="TzFindCyclicJoins"/> uses the
## Euclidean algorithm to express <M>a</M> as a power of <M>b</M>,
## and then it eliminates <M>a</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzFindCyclicJoins");

############################################################################
##
#F TzGeneratorExponents()
##
## <ManSection>
## <Func Name="TzGeneratorExponents" Arg='P'/>
##
## <Description>
## <Ref Func="TzGeneratorExponents"/> tries to find exponents for the
## Tietze generators and returns them in a list parallel to the list of the
## generators.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzGeneratorExponents");

#############################################################################
##
#F TzGo( [, <silent>] )
##
## <#GAPDoc Label="TzGo">
## <ManSection>
## <Func Name="TzGo" Arg='P[, silent]'/>
##
## <Description>
## automatically performs suitable Tietze transformations of the given
## presentation <A>P</A>. It is perhaps the most convenient one among the
## interactive Tietze transformation commands. It offers a kind of default
## strategy which, in general, saves you from explicitly calling the
## lower-level commands it involves.
## 
## If <A>silent</A> is specified as <K>true</K>,
## the printing of the status line by <Ref Func="TzGo"/> is suppressed
## if the Tietze option <C>printLevel</C>
## (see <Ref Sect="Tietze Options"/>) has a value less than <M>2</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzGo");

############################################################################
##
#F SimplifyPresentation()
##
## <#GAPDoc Label="SimplifyPresentation">
## <ManSection>
## <Func Name="SimplifyPresentation" Arg='P'/>
##
## <Description>
## <Ref Func="SimplifyPresentation"/> is a synonym for <Ref Func="TzGo"/>.
## <Example><![CDATA[
## gap> F2 := FreeGroup( "a", "b" );;
## gap> G := F2 / [ F2.1^9, F2.2^2, (F2.1*F2.2)^4, (F2.1^2*F2.2)^3 ];;
## gap> a := G.1;; b := G.2;;
## gap> H := Subgroup( G, [ (a*b)^2, (a^-1*b)^2 ] );;
## gap> Index( G, H );
## 408
## gap> P := PresentationSubgroup( G, H );
## <presentation with 8 gens and 36 rels of total length 111>
## gap> PrimaryGeneratorWords( P );
## [ b, a*b*a ]
## gap> TzOptions( P ).protected := 2;
## 2
## gap> TzOptions( P ).printLevel := 2;
## 2
## gap> SimplifyPresentation( P );
## #I eliminating _x7 = _x5^-1
## #I eliminating _x5 = _x4
## #I eliminating _x18 = _x3
## #I eliminating _x8 = _x3
## #I there are 4 generators and 8 relators of total length 21
## #I there are 4 generators and 7 relators of total length 18
## #I eliminating _x4 = _x3^-1*_x2^-1
## #I eliminating _x3 = _x2*_x1^-1
## #I there are 2 generators and 4 relators of total length 14
## #I there are 2 generators and 4 relators of total length 13
## #I there are 2 generators and 3 relators of total length 9
## gap> TzPrintRelators( P );
## #I 1. _x1^2
## #I 2. _x2^3
## #I 3. (_x2*_x1)^2
## ]]></Example>
## 
## Roughly speaking, <Ref Func="TzGo"/> consists of a loop over a
## procedure which involves two phases: In the <E>search phase</E> it calls
## <Ref Func="TzSearch"/> and <Ref Func="TzSearchEqual"/> described below
## which try to reduce the relator lengths by substituting common subwords
## of relators, in the <E>elimination phase</E> it calls the command
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## described below (or, more precisely, a subroutine of
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## in order to save some administrative overhead) which tries to eliminate
## generators that can be expressed as words in the remaining generators.
## 
## If <Ref Func="TzGo"/> succeeds in reducing the number of generators,
## the number of relators, or the total length of all relators, it
## displays the new status before returning (provided that you did not set
## the print level to zero). However, it does not provide any output if all
## these three values have remained unchanged, even if the command
## <Ref Func="TzSearchEqual"/> involved has changed the presentation
## such that another call of <Ref Func="TzGo"/> might provide further
## progress.
## Hence, in such a case it makes sense to repeat the call of the command
## for several times (or to call the command <Ref Func="TzGoGo"/> instead).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
SimplifyPresentation := TzGo;

############################################################################
##
#F TzGoGo()
##
## <#GAPDoc Label="TzGoGo">
## <ManSection>
## <Func Name="TzGoGo" Arg='P'/>
##
## <Description>
## calls the command <Ref Func="TzGo"/> again and again until it does not
## reduce the presentation any more.
## 
## The result of the Tietze transformations can be affected substantially by
## the options parameters (see <Ref Sect="Tietze Options"/>).
## To demonstrate the effect of the <C>eliminationsLimit</C> parameter,
## we will give an example in which we handle a subgroup of index 240 in a
## group of order 40320 given by a presentation due to B. H. Neumann.
## First we construct a presentation of the subgroup, and then we apply to
## it the command <Ref Func="TzGoGo"/> for different
## values of the parameter <C>eliminationsLimit</C>
## (including the default value 100). In fact, we also alter the
## <C>printLevel</C> parameter, but this is only done in order to suppress
## most of the output. In all cases the resulting presentations cannot be
## improved any more by applying the command <Ref Func="TzGoGo"/> again,
## i.e., they are the best results which we can get without substituting new
## generators.
## 
## <Example><![CDATA[
## gap> F3 := FreeGroup( "a", "b", "c" );;
## gap> G := F3 / [ F3.1^3, F3.2^3, F3.3^3, (F3.1*F3.2)^5,
## > (F3.1^-1*F3.2)^5, (F3.1*F3.3)^4, (F3.1*F3.3^-1)^4,
## > F3.1*F3.2^-1*F3.1*F3.2*F3.3^-1*F3.1*F3.3*F3.1*F3.3^-1,
## > (F3.2*F3.3)^3, (F3.2^-1*F3.3)^4 ];;
## gap> a := G.1;; b := G.2;; c := G.3;;
## gap> H := Subgroup( G, [ a, c ] );;
## gap> for i in [ 61, 62, 63, 90, 97 ] do
## > Pi := PresentationSubgroup( G, H );
## > TzOptions( Pi ).eliminationsLimit := i;
## > Print("#I eliminationsLimit set to ",i,"\n");
## > TzOptions( Pi ).printLevel := 0;
## > TzGoGo( Pi );
## > TzPrintStatus( Pi );
## > od;
## #I eliminationsLimit set to 61
## #I there are 2 generators and 104 relators of total length 7012
## #I eliminationsLimit set to 62
## #I there are 2 generators and 7 relators of total length 56
## #I eliminationsLimit set to 63
## #I there are 3 generators and 97 relators of total length 5998
## #I eliminationsLimit set to 90
## #I there are 3 generators and 11 relators of total length 68
## #I eliminationsLimit set to 97
## #I there are 4 generators and 109 relators of total length 3813
## ]]></Example>
## 
## Similarly, we demonstrate the influence of the <C>saveLimit</C> parameter
## by just continuing the preceding example for some different values of the
## <C>saveLimit</C> parameter (including its default value 10), but without
## changing the <C>eliminationsLimit</C> parameter which keeps its default
## value 100.
## 
## <Example><![CDATA[
## gap> for i in [ 7 .. 11 ] do
## > Pi := PresentationSubgroup( G, H );
## > TzOptions( Pi ).saveLimit := i;
## > Print( "#I saveLimit set to ", i, "\n" );
## > TzOptions( Pi ).printLevel := 0;
## > TzGoGo( Pi );
## > TzPrintStatus( Pi );
## > od;
## #I saveLimit set to 7
## #I there are 3 generators and 99 relators of total length 2713
## #I saveLimit set to 8
## #I there are 2 generators and 103 relators of total length 11982
## #I saveLimit set to 9
## #I there are 2 generators and 6 relators of total length 41
## #I saveLimit set to 10
## #I there are 3 generators and 118 relators of total length 13713
## #I saveLimit set to 11
## #I there are 3 generators and 11 relators of total length 58
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzGoGo");

############################################################################
##
#F TzGoElim(,<len>)
##
## <#GAPDoc Label="TzGoElim">
## <ManSection>
## <Func Name="TzGoElim" Arg='P,len'/>
##
## <Description>
## A variant for the TzGoXXX functions for the MTC. Tries to reduce down to
## <C>len</C> generators and does not try so hard to reduce.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzGoElim");

#############################################################################
##
#F TzHandleLength1Or2Relators( )
##
## <ManSection>
## <Func Name="TzHandleLength1Or2Relators" Arg='P'/>
##
## <Description>
## <Ref Func="TzHandleLength1Or2Relators"/> searches for relators of length 1 or 2 and
## performs suitable Tietze transformations for each of them:
## 
## Generators occurring in relators of length 1 are eliminated.
## 
## Generators occurring in square relators of length 2 are marked to be
## involutions.
## 
## If a relator of length 2 involves two different generators, then the
## generator with the larger number is substituted by the other one in all
## relators and finally eliminated from the set of generators.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzHandleLength1Or2Relators");

#############################################################################
##
#F GeneratorsOfPresentation()
##
## <#GAPDoc Label="GeneratorsOfPresentation">
## <ManSection>
## <Func Name="GeneratorsOfPresentation" Arg='P'/>
##
## <Description>
## returns a list of free generators that is a shallow copy
## (see <Ref Oper="ShallowCopy"/>) of the current
## generators of the presentation <A>P</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("GeneratorsOfPresentation");

#############################################################################
##
#F TzInitGeneratorImages( )
##
## <#GAPDoc Label="TzInitGeneratorImages">
## <ManSection>
## <Func Name="TzInitGeneratorImages" Arg='P'/>
##
## <Description>
## expects <A>P</A> to be a presentation. It defines the current generators
## to be the <Q>old generators</Q> of <A>P</A> and initializes the
## (pre)image tracing.
## See <Ref Func="TzImagesOldGens"/> and <Ref Func="TzPreImagesNewGens"/>
## for details.
## 
## You can reinitialize the tracing of the generator images at any later
## state by just calling the function <Ref Func="TzInitGeneratorImages"/>
## again.
## 
## Note:
## A subsequent call of the function <Ref Func="DecodeTree"/> will imply
## that the images and preimages are deleted and reinitialized
## after decoding the tree.
## 
## Moreover, if you introduce a new generator by calling the function
## <Ref Func="AddGenerator"/> described
## in Section <Ref Sect="Changing Presentations"/>, this
## new generator cannot be traced in the old generators.
## Therefore <Ref Func="AddGenerator"/> will terminate the tracing of the
## generator images and preimages and delete the respective lists
## whenever it is called.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzInitGeneratorImages");

#############################################################################
##
#F OldGeneratorsOfPresentation()
##
## <#GAPDoc Label="OldGeneratorsOfPresentation">
## <ManSection>
## <Func Name="OldGeneratorsOfPresentation" Arg='P'/>
##
## <Description>
## assumes that <A>P</A> is a presentation for which the generator images
## and preimages are being traced under Tietze transformations. It
## returns the list of old generators of <A>P</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("OldGeneratorsOfPresentation");

#############################################################################
##
#F TzImagesOldGens()
##
## <#GAPDoc Label="TzImagesOldGens">
## <ManSection>
## <Func Name="TzImagesOldGens" Arg='P'/>
##
## <Description>
## assumes that <A>P</A> is a presentation for which the generator images
## and preimages are being traced under Tietze transformations. It
## returns a list <M>l</M> of words in the (current)
## <Ref Func="GeneratorsOfPresentation"/> value of <A>P</A>
## such that the <M>i</M>-th word
## <M>l[i]</M> represents the <M>i</M>-th old generator of <A>P</A>, i. e.,
## the <M>i</M>-th entry of the <Ref Func="OldGeneratorsOfPresentation"/>
## value of <A>P</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzImagesOldGens");

#############################################################################
##
#F TzPreImagesNewGens()
##
## <#GAPDoc Label="TzPreImagesNewGens">
## <ManSection>
## <Func Name="TzPreImagesNewGens" Arg='P'/>
##
## <Description>
## assumes that <A>P</A> is a presentation for which the generator images
## and preimages are being traced under Tietze transformations.
## It returns a list <M>l</M> of words in the old generators of <A>P</A>
## (the <Ref Func="OldGeneratorsOfPresentation"/> value of <A>P</A>)
## such that the <M>i</M>-th entry of <M>l</M>
## represents the <M>i</M>-th (current) generator of <A>P</A>
## (the <Ref Func="GeneratorsOfPresentation"/> value of <A>P</A>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPreImagesNewGens");

#############################################################################
##
#F TzMostFrequentPairs( , <n> )
##
## <ManSection>
## <Func Name="TzMostFrequentPairs" Arg='P, n'/>
##
## <Description>
## <Ref Func="TzMostFrequentPairs"/> returns a list describing the <A>n</A>
## most frequently occurring relator subwords of the form <M>g_1 g_2</M>,
## where <M>g_1</M> and <M>g_2</M> are different generators or their
## inverses.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzMostFrequentPairs");

############################################################################
##
#F TzNewGenerator()
##
## <#GAPDoc Label="TzNewGenerator">
## <ManSection>
## <Func Name="TzNewGenerator" Arg='P'/>
##
## <Description>
## is an internal function which defines a new abstract generator and
## adds it to the presentation <A>P</A>.
## It is called by <Ref Func="AddGenerator"/> and
## by several Tietze transformation commands. As it does not know which
## global lists have to be kept consistent, you should not call it.
## Instead, you should call the function <Ref Func="AddGenerator"/>,
## if needed.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzNewGenerator");

#############################################################################
##
#F TzPrint( [, <list>] )
##
## <#GAPDoc Label="TzPrint">
## <ManSection>
## <Func Name="TzPrint" Arg='P[, list]'/>
##
## <Description>
## prints the current generators of the given presentation <A>P</A>,
## and prints the relators of <A>P</A> as Tietze words (without converting
## them back to abstract words as the functions
## <Ref Func="TzPrintRelators"/> and <Ref Func="TzPrintPresentation"/> do).
## The optional second argument can be used to specify the numbers of the
## relators to be printed.
## Default: all relators are printed.
## <Example><![CDATA[
## gap> TzPrint( P );
## #I generators: [ f1, f2, f3 ]
## #I relators:
## #I 1. 2 [ 3, 3 ]
## #I 2. 4 [ 2, 2, 2, 2 ]
## #I 3. 4 [ 2, 3, 2, 3 ]
## #I 4. 5 [ 1, 1, 1, 1, 1 ]
## #I 5. 5 [ 1, 1, 2, 1, -2 ]
## #I 6. 8 [ 1, -2, -2, 3, 1, 3, -1, 3 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrint");

#############################################################################
##
#F TzPrintGeneratorImages()
##
## <#GAPDoc Label="TzPrintGeneratorImages">
## <ManSection>
## <Func Name="TzPrintGeneratorImages" Arg='P'/>
##
## <Description>
## assumes that <A>P</A> is a presentation for which the generator images
## and preimages are being traced under Tietze transformations. It
## displays the preimages of the current generators as Tietze words in
## the old generators, and the images of the old generators as Tietze
## words in the current generators.
## <Example><![CDATA[
## gap> G := PerfectGroup( IsSubgroupFpGroup, 960, 1 );
## A5 2^4
## gap> P := PresentationFpGroup( G );
## <presentation with 6 gens and 21 rels of total length 84>
## gap> TzInitGeneratorImages( P );
## gap> TzGo( P );
## #I there are 3 generators and 11 relators of total length 96
## #I there are 3 generators and 10 relators of total length 81
## gap> TzPrintGeneratorImages( P );
## #I preimages of current generators as Tietze words in the old ones:
## #I 1. [ 1 ]
## #I 2. [ 2 ]
## #I 3. [ 4 ]
## #I images of old generators as Tietze words in the current ones:
## #I 1. [ 1 ]
## #I 2. [ 2 ]
## #I 3. [ 1, -2, 1, 3, 1, 2, 1 ]
## #I 4. [ 3 ]
## #I 5. [ -2, 1, 3, 1, 2 ]
## #I 6. [ 1, 3, 1 ]
## gap> gens := GeneratorsOfPresentation( P );
## [ a, b, t ]
## gap> oldgens := OldGeneratorsOfPresentation( P );
## [ a, b, s, t, u, v ]
## gap> TzImagesOldGens( P );
## [ a, b, a*b^-1*a*t*a*b*a, t, b^-1*a*t*a*b, a*t*a ]
## gap> for i in [ 1 .. Length( oldgens ) ] do
## > Print( oldgens[i], " = ", TzImagesOldGens( P )[i], "\n" );
## > od;
## a = a
## b = b
## s = a*b^-1*a*t*a*b*a
## t = t
## u = b^-1*a*t*a*b
## v = a*t*a
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintGeneratorImages");

#############################################################################
##
#F TzPrintGenerators( [, <list>] )
##
## <#GAPDoc Label="TzPrintGenerators">
## <ManSection>
## <Func Name="TzPrintGenerators" Arg='P[, list]'/>
##
## <Description>
## prints the generators of the given Tietze presentation <A>P</A> together
## with the number of their occurrences in the relators. The optional second
## argument can be used to specify the numbers of the generators to be
## printed. Default: all generators are printed.
## <Example><![CDATA[
## gap> G := Group( [ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ], () );
## Group([ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ])
## gap> P := PresentationViaCosetTable( G );
## <presentation with 3 gens and 6 rels of total length 28>
## gap> TzPrintGenerators( P );
## #I 1. f1 11 occurrences
## #I 2. f2 10 occurrences
## #I 3. f3 7 occurrences involution
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintGenerators");

#############################################################################
##
#F TzPrintLengths( )
##
## <#GAPDoc Label="TzPrintLengths">
## <ManSection>
## <Func Name="TzPrintLengths" Arg='P'/>
##
## <Description>
## prints just a list of all relator lengths of the given presentation
## <A>P</A>.
## <Example><![CDATA[
## gap> TzPrintLengths( P );
## [ 2, 4, 4, 5, 5, 8 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintLengths");

#############################################################################
##
#A TzOptions()
##
## <#GAPDoc Label="TzOptions">
## <ManSection>
## <Attr Name="TzOptions" Arg='P'/>
##
## <Description>
## is a record whose components direct the heuristics applied by the Tietze
## transformation functions.
## 
## You may alter the value of any of these Tietze options by just assigning
## a new value to the respective record component.
## 
## The following Tietze options are recognized by &GAP;:
## 
## <List>
## <C>protected</C>:
## <Item>
## The first <C>protected</C> generators in a presentation <A>P</A> are
## protected from being eliminated by the Tietze transformations
## functions. There are only two exceptions: The option
## <C>protected</C> is ignored by the functions
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## and <Ref Func="TzSubstitute" Label="for a presentation and a word"/>
## because they explicitly specify the generator to be eliminated.
## The default value of <C>protected</C> is 0.
## </Item>
## <C>eliminationsLimit</C>:
## <Item>
## Whenever the elimination phase of the <Ref Func="TzGo"/> command is
## entered for a presentation <A>P</A>, then it will eliminate at most
## <C>eliminationsLimit</C> generators (except for further ones which
## have turned out to be trivial). Hence you may use the
## <C>eliminationsLimit</C> parameter as a break criterion for the
## <Ref Func="TzGo"/> command. Note, however, that it is ignored by the
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## command. The default value of <C>eliminationsLimit</C> is 100.
## </Item>
## <C>expandLimit</C>:
## <Item>
## Whenever the routine for eliminating more than 1 generator is
## called for a presentation <A>P</A> by the
## <Ref Func="TzEliminate" Label="for a presentation (and a generator)"/>
## command or the elimination phase of the <Ref Func="TzGo"/> command,
## then it saves the given total length of the relators,
## and subsequently it checks the current total length against its value
## before each elimination.
## If the total length has increased to more than <C>expandLimit</C>
## per cent of its original value, then the routine returns instead
## of eliminating another generator.
## Hence you may use the <C>expandLimit</C> parameter as a break criterion
## for the <Ref Func="TzGo"/> command.
## The default value of <C>expandLimit</C> is 150.
## </Item>
## <C>generatorsLimit</C>:
## <Item>
## Whenever the elimination phase of the <Ref Func="TzGo"/> command is
## entered for a presentation <A>P</A> with <M>n</M> generators,
## then it will eliminate at most <M>n - </M><C>generatorsLimit</C>
## generators (except for generators which turn out to be trivial).
## Hence you may use the <C>generatorsLimit</C> parameter as a break
## criterion for the <Ref Func="TzGo"/> command.
## The default value of <C>generatorsLimit</C> is 0.
## </Item>
## <C>lengthLimit</C>:
## <Item>
## The Tietze transformation commands will never eliminate a
## generator of a presentation <A>P</A>, if they cannot exclude the
## possibility that the resulting total length of the relators
## exceeds the maximal &GAP; list length of <M>2^{31}-1</M> or the value
## of the option <C>lengthLimit</C>.
## The default value of <C>lengthLimit</C> is <M>2^{31}-1</M>.
## </Item>
## <C>loopLimit</C>:
## <Item>
## Whenever the <Ref Func="TzGo"/> command is called for a presentation
## <A>P</A>, then it will loop over at most <C>loopLimit</C> of its basic
## steps. Hence you may use the <C>loopLimit</C> parameter as a break
## criterion for the <Ref Func="TzGo"/> command. The default value of
## <C>loopLimit</C> is <Ref Var="infinity"/>.
## </Item>
## <C>printLevel</C>:
## <Item>
## Whenever Tietze transformation commands are called for a
## presentation <A>P</A> with <C>printLevel</C> <M>= 0</M>, they will not
## provide any output except for error messages. If <C>printLevel</C>
## <M>= 1</M>, they will display some reasonable amount of output which
## allows you to watch the progress of the computation and to decide
## about your next commands. In the case <C>printLevel</C> <M>= 2</M>, you
## will get a much more generous amount of output. Finally, if
## <C>printLevel</C> <M>= 3</M>, various messages on internal details will
## be added. The default value of <C>printLevel</C> is 1.
## </Item>
## <C>saveLimit</C>:
## <Item>
## Whenever the <Ref Func="TzSearch"/> command has finished its main loop
## over all relators of a presentation <A>P</A>, then it checks whether
## during this loop the total length of the relators has been reduced by
## at least <C>saveLimit</C> per cent. If this is the case, then
## <Ref Func="TzSearch"/> repeats its procedure instead of returning.
## Hence you may use the <C>saveLimit</C> parameter as a break criterion
## for the <Ref Func="TzSearch"/> command and, in particular,
## for the search phase of the <Ref Func="TzGo"/> command.
## The default value of <C>saveLimit</C> is 10.
## </Item>
## <C>searchSimultaneous</C>:
## <Item>
## Whenever the <Ref Func="TzSearch"/> or the <Ref Func="TzSearchEqual"/>
## command is called for a presentation <A>P</A>, then it is allowed to
## handle up to <C>searchSimultaneous</C> short relators simultaneously
## (see the description of the <Ref Func="TzSearch"/> command for more
## details).
## The choice of this parameter may heavily influence the performance as
## well as the result of the <Ref Func="TzSearch"/> and the
## <Ref Func="TzSearchEqual"/> commands and hence also of the search phase
## of the <Ref Func="TzGo"/> command.
## The default value of <C>searchSimultaneous</C> is 20.
## </Item>
## </List>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("TzOptions",IsPresentation,"mutable");

#############################################################################
##
#F TzPrintOptions( )
##
## <#GAPDoc Label="TzPrintOptions">
## <ManSection>
## <Func Name="TzPrintOptions" Arg='P'/>
##
## <Description>
## prints the current values of the Tietze options of the presentation
## <A>P</A>.
## <Example><![CDATA[
## gap> TzPrintOptions( P );
## #I protected = 0
## #I eliminationsLimit = 100
## #I expandLimit = 150
## #I generatorsLimit = 0
## #I lengthLimit = 2147483647
## #I loopLimit = infinity
## #I printLevel = 1
## #I saveLimit = 10
## #I searchSimultaneous = 20
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintOptions");

#############################################################################
##
#F TzPrintPairs( [,<n>] )
##
## <#GAPDoc Label="TzPrintPairs">
## <ManSection>
## <Func Name="TzPrintPairs" Arg='P [,n]'/>
##
## <Description>
## prints the <A>n</A> most often occurring relator subwords of the form
## <M>a b</M>,
## where <M>a</M> and <M>b</M> are different generators or inverses of
## generators, together with the number of their occurrences. The default
## value of <A>n</A> is 10.
## A value <A>n</A> = 0 is interpreted as <Ref Var="infinity"/>.
## 
## The function <Ref Func="TzPrintPairs"/> is useful in the context of
## Tietze transformations which introduce new generators by substituting
## words in the current generators
## (see <Ref Sect="Tietze Transformations that introduce new Generators"/>).
## It gives some evidence for an appropriate choice of
## a word of length 2 to be substituted.
## <Example><![CDATA[
## gap> TzPrintPairs( P, 3 );
## #I 1. 3 occurrences of f2^-1 * f3
## #I 2. 2 occurrences of f2 * f3
## #I 3. 2 occurrences of f1^-1 * f3
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintPairs");

############################################################################
##
#F TzPrintPresentation()
##
## <#GAPDoc Label="TzPrintPresentation">
## <ManSection>
## <Func Name="TzPrintPresentation" Arg='P'/>
##
## <Description>
## prints the generators and the relators of a Tietze presentation.
## In fact, it is an abbreviation for the successive call of the three
## commands <Ref Func="TzPrintGenerators"/>,
## <Ref Func="TzPrintRelators"/>, and <Ref Func="TzPrintStatus"/>,
## each with the presentation <A>P</A> as only argument.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintPresentation");

############################################################################
##
#F TzPrintRelators([, <list>])
##
## <#GAPDoc Label="TzPrintRelators">
## <ManSection>
## <Func Name="TzPrintRelators" Arg='P[, list]'/>
##
## <Description>
## prints the relators of the given Tietze presentation <A>P</A>.
## The optional second argument <A>list</A> can be used to specify the
## numbers of the relators to be printed.
## Default: all relators are printed.
## <Example><![CDATA[
## gap> TzPrintRelators( P );
## #I 1. f3^2
## #I 2. f2^4
## #I 3. (f2*f3)^2
## #I 4. f1^5
## #I 5. f1^2*f2*f1*f2^-1
## #I 6. f1*f2^-2*f3*f1*f3*f1^-1*f3
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintRelators");

#############################################################################
##
#F TzPrintStatus( [, <norepeat>] )
##
## <#GAPDoc Label="TzPrintStatus">
## <ManSection>
## <Func Name="TzPrintStatus" Arg='P[, norepeat]'/>
##
## <Description>
## is an internal function which is used by the Tietze transformation
## routines to print the number of generators, the number of relators,
## and the total length of all relators in the given Tietze presentation
## <A>P</A>.
## If <A>norepeat</A> is specified as <K>true</K>, the printing is
## suppressed if none of the three values has changed since the last call.
## <Example><![CDATA[
## gap> TzPrintStatus( P );
## #I there are 3 generators and 6 relators of total length 28
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TzPrintStatus");

############################################################################
##
#f TzRecoverFromFile
##
#T DeclareGlobalFunction("TzRecoverFromFile");
#T up to now no function is installed

############################################################################
##
#F TzRelator( , <word> )
##
## <ManSection>
## <Func Name="TzRelator" Arg='P, word'/>
##
## <Description>
## <Ref Func="TzRelator"/> assumes <A>word</A> to be an abstract word in the
## group generators associated to the given presentation, and converts it to
## a Tietze relator, i.e. a free and cyclically reduced Tietze word.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzRelator");

############################################################################
##
#F TzRemoveGenerators()
##
## <ManSection>
## <Func Name="TzRemoveGenerators" Arg='P'/>
##
## <Description>
## <C>TzRemoveGenerators</C> deletes the redundant Tietze generators and
## renumbers the non-redundant ones accordingly. The redundant generators
## are assumed to be marked in the inverses list by an entry
## <C>invs[numgens+1-i] <> i</C>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TzRemoveGenerators");

############################################################################
##
#F TzSearch()
##
## <#GAPDoc Label="TzSearch">
## <ManSection>
## <Func Name="TzSearch" Arg='P'/>
##
## <Description>
## searches for relator subwords which, in some relator, have a complement
## of shorter length and which occur in other relators, too, and uses them
## to reduce these other relators.
## 
## The idea is to find pairs of relators <M>r_1</M> and <M>r_2</M> of length
## <M>l_1</M> and <M>l_2</M>, respectively,
## such that <M>l_1 \leq l_2</M> and <M>r_1</M> and <M>r_2</M>
## coincide (possibly after inverting or conjugating one of them) in some
## maximal subword <M>w</M> of length greater than <M>l_1/2</M>,
--> --------------------

--> maximum size reached

--> --------------------

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