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/**************************************************************************
* *
* Regina  A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 19992016, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* As an exception, when this program is distributed through (i) the *
* App Store by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or *
* (iii) Google Play by Google Inc., then that store may impose any *
* digital rights management, device limits and/or redistribution *
* restrictions that are required by its terms of service. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public *
* License along with this program; if not, write to the Free *
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, *
* MA 021101301, USA. *
* *
**************************************************************************/
/*! \file algebra/grouppresentation.h
* \brief Deals with finite presentations of groups.
*/
#ifndef __GROUPPRESENTATION_H
#ifndef __DOXYGEN
#define __GROUPPRESENTATION_H
#endif
#include <algorithm>
#include <list>
#include <vector>
#include <set>
#include <map>
#include "reginacore.h"
#include "output.h"
#include "utilities/memutils.h"
#include "utilities/ptrutils.h"
#include "algebra/markedabeliangroup.h"
#include "algebra/abeliangroup.h"
#include <boost/noncopyable.hpp>
namespace regina {
class AbelianGroup;
class HomGroupPresentation;
class MarkedAbelianGroup;
/**
* \weakgroup algebra
* @{
*/
/**
* Represents a power of a generator in a group presentation.
*/
struct REGINA_API GroupExpressionTerm {
unsigned long generator;
/**< The number that identifies the generator in this term. */
long exponent;
/**< The exponent to which the generator is raised. */
/**
* Creates a new uninitialised term.
*/
GroupExpressionTerm();
/**
* Creates a new term initialised to the given value.
*
* @param newGen the number that identifies the generator in the new term.
* @param newExp the exponent to which this generator is raised.
*/
GroupExpressionTerm(unsigned long newGen, long newExp);
/**
* Creates a new term initialised to the given value.
*
* @param cloneMe a term whose data will be copied to the new term.
*/
GroupExpressionTerm(const GroupExpressionTerm& cloneMe);
/**
* Makes this term identical to the given term.
*
* @param cloneMe the term whose data will be copied to this term.
* @return a reference to this term.
*/
GroupExpressionTerm& operator = (const GroupExpressionTerm& cloneMe);
/**
* Determines whether this and the given term contain identical data.
*
* @param other the term with which this term will be compared.
* @return \c true if and only if this and the given term have both the
* same generator and exponent.
*/
bool operator == (const GroupExpressionTerm& other) const;
/**
* Determines whether this and the given term do not contain identical data.
*
* @param other the term with which this term will be compared.
* @return \c true if and only if this and the given term do not have
* both the same generator and exponent.
*/
bool operator != (const GroupExpressionTerm& other) const;
/**
* Imposes an ordering on terms.
* Terms are ordered lexigraphically as (generator, exponent) pairs.
*
* @param other the term to compare with this.
* @return true if and only if this term is lexicographically
* smaller than \a other.
*/
bool operator < (const GroupExpressionTerm& other) const;
/**
* Returns the inverse of this term. The inverse has the same
* generator but a negated exponent.
*
* Note that this term will remain unchanged.
*
* @return the inverse of this term.
*/
GroupExpressionTerm inverse() const;
/**
* Attempts to merge this term with the given term.
* If both terms have the same generator, the two exponents will be
* added and stored in this term. If the generators are different,
* this routine will do nothing.
*
* Note that this term might be changed but the given term will remain
* unchanged.
*
* @param other the term to merge with this term.
* @return \c true if the two terms were merged into this term, or
* \c false if the two terms have different generators.
*/
bool operator += (const GroupExpressionTerm& other);
};
/**
* Deprecated typedef for backward compatibility. This typedef will
* be removed in a future release of Regina.
*
* \deprecated The struct NGroupExpressionTerm has now been renamed to
* GroupExpressionTerm.
*/
REGINA_DEPRECATED typedef GroupExpressionTerm NGroupExpressionTerm;
/**
* Writes the given term to the given output stream.
* The term will be written in the format <tt>g3^7</tt>, where in this
* example the term represents generator number 3 raised to the 7th power.
*
* If the term has exponent 0 or 1, the output format will be
* appropriately simplified.
*
* @param out the output stream to which to write.
* @param term the term to write.
* @return a reference to the given output stream.
*/
REGINA_API std::ostream& operator << (std::ostream& out,
const GroupExpressionTerm& term);
/**
* Represents an expression involving generators from a group presentation
* or a free group. An expression is represented as word, i.e, a sequence
* of powers of generators all of which are multiplied in order. Each power
* of a generator corresponds to an individual GroupExpressionTerm.
*
* For instance, the expression <tt>g1^2 g3^1 g6</tt> contains the
* three terms <tt>g1^2</tt>, <tt>g3^1</tt> and <tt>g6^1</tt> in that
* order.
*/
class REGINA_API GroupExpression :
public ShortOutput<GroupExpression>,
public boost::noncopyable {
private:
std::list<GroupExpressionTerm> terms_;
/** The terms that make up this expression. */
public:
/**
* Creates a new expression with no terms.
*/
GroupExpression();
/**
* Creates a new expression that is a clone of the given
* expression.
*
* @param cloneMe the expression to clone.
*/
GroupExpression(const GroupExpression& cloneMe);
/**
* Attempts to interpret the given input string as a word in a group.
* Regina can recognise strings in the following four basic forms:
*
*  \c a^7b^2
*  \c aaaaaaaBB
*  \c a^7B^2
*  \c g0^7g1^2
*
* The string may contain whitespace, which will simply be ignored.
*
* The argument \a valid may be \c null, but if it is nonnull
* then the boolean it points to will be used for error reporting.
* This routine sets valid to \c true if the string was successfully
* interpreted, or \c false if the algorithm failed to interpret the
* string.
*
* Regardless of whether \a valid is \c null, if the string
* could not be interpreted then this expression will be initialised
* to the trivial word.
*
* \ifacespython The second argument \a valid is not present,
* and will be assumed to be \c null.
*
* @param input the input string that is to be interpreted.
* @param valid used for error reporting as described above, or
* \c null if no error reporting is required.
*/
GroupExpression(const std::string &input, bool* valid=NULL);
/**
* Makes this expression a clone of the given expression.
*
* @param cloneMe the expression to clone.
* @return a reference to this expression.
*/
GroupExpression& operator = (const GroupExpression& cloneMe);
/**
* Equality operator. Checks to see whether or not these two words
* represent the same literal string.
*
* @param comp the expression to compare against this.
* @return \c true if this and the given string literal are identical.
*/
bool operator == (const GroupExpression& comp) const;
/**
* Inequality operator. Checks to see whether or not these two words
* represent different literal strings.
*
* @param comp the expression to compare against this.
* @return \c true if this and the given string literal are not
* identical.
*/
bool operator != (const GroupExpression& comp) const;
/**
* Returns the list of terms in this expression.
* These are the actual terms stored internally; any
* modifications made to this list will show up in the
* expression itself.
*
* For instance, the expression <tt>g1^2 g3^1 g6</tt> has list
* consisting of three terms <tt>g1^2</tt>, <tt>g3^1</tt> and
* <tt>g6^1</tt> in that order.
*
* \ifacespython Not present; only the const version of this
* routine is available.
*
* @return the list of terms.
*/
std::list<GroupExpressionTerm>& terms();
/**
* Returns a constant reference to the list of terms in this
* expression.
*
* For instance, the expression <tt>g1^2 g3^1 g6</tt> has list
* consisting of three terms <tt>g1^2</tt>, <tt>g3^1</tt> and
* <tt>g6^1</tt> in that order.
*
* \ifacespython This routine returns a python list of copied
* GroupExpressionTerm objects. In particular, modifying this
* list or the terms within it will not modify the group
* expression from which they came.
*
* @return the list of terms.
*/
const std::list<GroupExpressionTerm>& terms() const;
/**
* Returns the number of terms in this expression.
*
* For instance, the expression <tt>g1^2 g3^1 g6</tt> contains three
* terms. See also wordLength().
*
* @return the number of terms.
*/
size_t countTerms() const;
/**
* Returns the length of the word, i.e. the number of letters
* with exponent +1 or 1 for which this word is expressable as a
* product.
*
* For instance, the expression <tt>g1^2 g3^1 g6</tt> is a word of
* length four. See also countTerms().
*
* No attempt is made to remove redundant terms (so the word
* <tt>g g^1</tt> will count as length two).
*
* @return the length of the word.
*/
size_t wordLength() const;
/**
* Tests whether this is the trivial (unit) word.
*
* No attempt is made to remove redundant terms (so the word
* <tt>g g^1</tt> will be treated as nontrivial).
*
* @return \c true if and only if this is the trivial word.
*/
bool isTrivial() const;
/**
* Erases all terms from this this word.
* This effectively turns this word into the identity element.
*/
void erase();
/**
* Returns the term at the given index in this expression.
* Index 0 represents the first term, index 1
* represents the second term and so on.
*
* \warning This routine is <i>O(n)</i> where \a n is the number
* of terms in this expression.
*
* @param index the index of the term to return; this must be
* between 0 and countTerms()1 inclusive.
* @return the requested term.
*/
GroupExpressionTerm& term(size_t index);
/**
* Returns a constant reference to the term at the given
* index in this expression.
* Index 0 represents the first term, index 1
* represents the second term and so on.
*
* \warning This routine is <i>O(n)</i> where \a n is the number
* of terms in this expression.
*
* \ifacespython Not present; only the nonconst version of this
* routine is available.
*
* @param index the index of the term to return; this must be
* between 0 and countTerms()1 inclusive.
* @return the requested term.
*/
const GroupExpressionTerm& term(size_t index) const;
/**
* Returns the generator corresonding to the
* term at the given index in this expression.
* Index 0 represents the first term, index 1
* represents the second term and so on.
*
* \warning This routine is <i>O(n)</i> where \a n is the number
* of terms in this expression.
*
* @param index the index of the term to return; this must be
* between 0 and countTerms()1 inclusive.
* @return the number of the requested generator.
*/
unsigned long generator(size_t index) const;
/**
* Returns the exponent corresonding to the
* term at the given index in this expression.
* Index 0 represents the first term, index 1
* represents the second term and so on.
*
* \warning This routine is <i>O(n)</i> where \a n is the number
* of terms in this expression.
*
* @param index the index of the term to return; this must be
* between 0 and countTerms()1 inclusive.
* @return the requested exponent.
*/
long exponent(size_t index) const;
/**
* Adds the given term to the beginning of this expression.
*
* @param term the term to add.
*/
void addTermFirst(const GroupExpressionTerm& term);
/**
* Adds the given term to the beginning of this expression.
*
* @param generator the number of the generator corresponding to
* the new term.
* @param exponent the exponent to which the given generator is
* raised.
*/
void addTermFirst(unsigned long generator, long exponent);
/**
* Adds the given term to the end of this expression.
*
* @param term the term to add.
*/
void addTermLast(const GroupExpressionTerm& term);
/**
* Adds the given term to the end of this expression.
*
* @param generator the number of the generator corresponding to
* the new term.
* @param exponent the exponent to which the given generator is
* raised.
*/
void addTermLast(unsigned long generator, long exponent);
/**
* Multiplies this expression on the left by the given word.
* This expression will be modified directly.
*
* @param word the word to multiply with this expression.
*/
void addTermsFirst(const GroupExpression& word);
/**
* Multiplies this expression on the right by the given word.
* This expression will be modified directly.
*
* @param word the word to multiply with this expression.
*/
void addTermsLast(const GroupExpression& word);
/**
* Multiplies this expression on the left by the word
* respresented by the given string.
*
* See the stringbased constructor
* GroupExpression(const std::string&, bool*) for further
* information on how this string should be formatted.
*
* If the given string cannot be interpreted as a word in a group,
* then this expression will be left untouched.
*
* @param input a string representation of the word to multiply with
* this expression.
* @return \c true if the given string could interpreted
* (and therefore the multiplication was completed successfully), or
* \c false if the given string could not be interpreted
* (in which case this expression will be left untouched).
*/
bool addStringFirst(const std::string& input);
/**
* Multiplies this expression on the right by the word
* respresented by the given string.
*
* See the stringbased constructor
* GroupExpression(const std::string&, bool*) for further
* information on how this string should be formatted.
*
* If the given string cannot be interpreted as a word in a group,
* then this expression will be left untouched.
*
* @param input a string representation of the word to multiply with
* this expression.
* @return \c true if the given string could interpreted
* (and therefore the multiplication was completed successfully), or
* \c false if the given string could not be interpreted
* (in which case this expression will be left untouched).
*/
bool addStringLast(const std::string& input);
/**
* Cycles this word by moving the leftmost term around to the rightmost.
* All other terms shift one step to the left.
*
* If the word is of the form
* <tt>g_i1^j1 g_i2^j2 ... g_in^jn</tt>,
* this converts it into the word
* <tt>g_i2^j2 ... g_in^jn g_i1^j1</tt>.
*/
void cycleRight();
/**
* Cycles this word by moving the rightmost term around to the leftmost.
* All other terms shift one step to the right.
*
* If the word is of the form
* <tt>g_i1^j1 g_i2^j2 ... g_in^jn</tt>,
* this converts it into the word
* <tt>g_in^jn g_i1^j1 g_i1^j1 ... g_in1^jn1</tt>.
*/
void cycleLeft();
/**
* Returns a newly created expression that is the inverse of
* this expression. The terms will be reversed and the
* exponents negated.
*
* @return the inverse of this expression.
*/
GroupExpression* inverse() const;
/**
* Inverts this expression. Does not allocate or deallocate anything.
*/
void invert();
/**
* Returns a newly created expression that is
* this expression raised to the given power.
* Note that the given exponent may be positive, zero or negative.
*
* @param exponent the power to which this expression should be raised.
* @return this expression raised to the given power.
*/
GroupExpression* power(long exponent) const;
/**
* Simplifies this expression.
* Adjacent powers of the same generator will be combined, and
* terms with an exponent of zero will be removed.
* Note that it is \e not assumed that the underlying group is
* abelian.
*
* You may declare that the expression is cyclic, in which case
* it is assumed that terms may be moved from the back to the
* front and vice versa. Thus expression <tt>g1 g2 g1 g2 g1</tt>
* simplifies to <tt>g1^2 g2 g1 g2</tt> if it is cyclic, but does not
* simplify at all if it is not cyclic.
*
* @param cyclic \c true if and only if the expression may be
* assumed to be cyclic.
* @return \c true if and only if this expression was changed.
*/
bool simplify(bool cyclic = false);
/**
* Replaces every occurrence of the given generator with the
* given substite expression. If the given generator was found,
* the expression will be simplified once the substitution is
* complete.
*
* @param generator the generator to be replaced.
* @param expansion the substitute expression that will replace
* every occurrence of the given generator.
* @param cyclic \c true if and only if the expression may be
* assumed to be cyclic; see simplify() for further details.
* @return \c true if and only if any substitutions were made.
*/
bool substitute(unsigned long generator,
const GroupExpression& expansion, bool cyclic = false);
/**
* Determines whether or not one can relabel the generators in
* this word to obtain the given other word. If so, returns a nonempty
* list of all such relabellings. If not, returns an empty list.
*
* Relabellings are partiallydefined permutations on the
* generator set, also allowing for possible inversions if
* cyclic is \c true.
*
* \apinotfinal
*
* \todo Change this to use less heavyweight types and less deep
* copying.
*
* \pre If \a cyclic is \c true, then both this word and \a other
* have been cyclically reduced.
*
* \ifacespython Not present.
*
* @param other the word to compare against this.
* @param cyclic if \c false we get a list of exact relabellings from
* this word to \a other. If \c true, it can be up to cyclic
* permutation and inversion.
* @return a list of permutations, implemented as maps from
* generator indices of this word to generator indices of \a other.
*/
std::list< std::map< unsigned long, GroupExpressionTerm > >
relabellingsThisToOther( const GroupExpression &other,
bool cyclic=false ) const;
/**
* Writes a chunk of XML containing this expression.
*
* \ifacespython Not present.
*
* @param out the output stream to which the XML should be written.
*/
void writeXMLData(std::ostream& out) const;
/**
* Returns a TeX representation of this expression.
* See writeTeX() for details on how this is formed.
*
* @return a TeX representation of this expression.
*/
std::string toTeX() const;
/**
* Writes a TeX represesentation of this expression to the given
* output stream.
*
* The text representation will be of the form
* <tt>g_2^4 g_{13}^{5} g_4</tt>.
*
* \ifacespython The parameter \a out does not exist;
* standard output will be used.
*
* @param out the output stream to which to write.
*/
void writeTeX(std::ostream& out) const;
/**
* Writes a text representation of this expression to the given
* output stream, using either numbered generators or
* alphabetic generators.
*
* The text representation will be of the form
* <tt>g2^4 g13^5 g4</tt>. If the \a shortword flag is \c true, it
* will assume your word is in an alphabet of no more than 26 letters,
* and will write the word using lowercase ASCII, i.e.,
* <tt>c^4 n^5 e</tt>. If the \a utf8 flag is \c true, all exponents
* will be written using superscript characters encoded in UTF8.
*
* \pre If \a shortword is \c true, the number of generators in
* the corresponding group must be 26 or fewer.
*
* \ifacespython The parameter \a out does not exist;
* standard output will be used.
*
* @param out the output stream to which to write.
* @param shortword indicates whether to use numbered or
* alphabetic generators, as described above.
* @param utf8 \c true if exponents should be written using
* unicode superscript characters, or \c false if they should be
* written using a caret (^) symbol.
*/
void writeText(std::ostream& out, bool shortword = false,
bool utf8 = false) const;
/**
* Writes a short text representation of this object to the
* given output stream.
*
* The text representation will be of the form
* <tt>g2^4 g13^5 g4</tt>.
*
* \ifacespython Not present.
*
* @param out the output stream to which to write.
*/
void writeTextShort(std::ostream& out) const;
};
/**
* Deprecated typedef for backward compatibility. This typedef will
* be removed in a future release of Regina.
*
* \deprecated The struct NGroupExpression has now been renamed to
* GroupExpression.
*/
REGINA_DEPRECATED typedef GroupExpression NGroupExpression;
/**
* Represents a finite presentation of a group.
*
* A presentation consists of a number of generators and a set of
* relations between these generators that together define the group.
*
* If there are \a g generators, they will be numbered 0, 1, ..., <i>g</i>1.
*
* \todo Let's make intelligent simplify a tad more intelligent, and the GUI
* call a bit more safe. Perhaps parallelize the GUI call, and give users
* parameters to ensure it won't crash the computer. Also look at the FPGroup
* package. We should also have a simple way of creating GroupPresentation
* objects directly from text strings. We would like to have something like
* GroupPresentation( numGens, "abAAB", "bccd" ) etc., with arbitrary
* numbers of relators. Maybe std::tuple. Or "variadic templates"?
*/
class REGINA_API GroupPresentation :
public Output<GroupPresentation>,
public boost::noncopyable {
protected:
unsigned long nGenerators;
/**< The number of generators. */
std::vector<GroupExpression*> relations;
/**< The relations between the generators. */
public:
/**
* Creates a new presentation with no generators and no
* relations.
*/
GroupPresentation();
/**
* Creates a clone of the given group presentation.
*
* @param cloneMe the presentation to clone.
*/
GroupPresentation(const GroupPresentation& cloneMe);
/**
* Constructor that allows you to directly pass an arbitrary number
* of relators in string format.
*
* The first argument \a nGens is the number of generators one wants
* the group to have. The second argument \a rels is a vector
* of strings, where each string gives a single relator. See
* the GroupExpression::GroupExpression(const std::string&, bool*)
* constructor notes for information on what format these strings
* can take.
*
* If any of the given strings could not be interpreted as
* words, this routine will insert the trivial (unit) word in
* its place.
*
* If you are compiling Regina against C++11, you can use the
* C++11 initializer_list construction to construct an
* GroupPresentation directly using syntax of the form
* <tt>GroupPresentation(nGens, { "rel1", "rel2", ... })</tt>.
*
* \ifacespython Not present.
*
* @param nGens the number of generators.
* @param rels a vector of relations each given in string form,
* as outlined above.
*/
GroupPresentation(unsigned long nGens,
const std::vector<std::string> &rels);
/**
* Destroys the group presentation.
* All relations that are stored will be deallocated.
*/
~GroupPresentation();
/**
* Assignment operator.
*
* @param cloneMe the group presentation that this will become a
* copy of.
* @return a reference to this group presentation.
*/
GroupPresentation& operator=(const GroupPresentation& cloneMe);
/**
* Adds one or more generators to the group presentation.
* If the new presentation has \a g generators, the new
* generators will be numbered <i>g</i>1, <i>g</i>2 and so on.
*
* @param numToAdd the number of generators to add.
* @return the number of generators in the new presentation.
*/
unsigned long addGenerator(unsigned long numToAdd = 1);
/**
* Adds the given relation to the group presentation.
* The relation must be of the form <tt>expression = 1</tt>.
*
* This presentation will take ownership of the given
* expression, may change it and will be responsible for its
* deallocation.
*
* \warning This routine does not check whether or not your relation
* is a word only in the generators of this group. In other
* words, it does not stop you from using generators beyond the
* countGenerators() bound.
*
* \ifacespython In Python, this routine clones its argument
* instead of claiming ownership of it.
*
* @param rel the expression that the relation sets to 1; for
* instance, if the relation is <tt>g1^2 g2 = 1</tt> then this
* parameter should be the expression <tt>g1^2 g2</tt>.
*/
void addRelation(GroupExpression* rel);
/**
* Returns the number of generators in this group presentation.
*
* @return the number of generators.
*/
unsigned long countGenerators() const;
/**
* Returns the number of relations in this group presentation.
*
* @return the number of relations.
*/
size_t countRelations() const;
/**
* Returns the relation at the given index in this group
* presentation. The relation will be of the form <tt>expresson
* = 1</tt>.
*
* @param index the index of the desired relation; this must be
* between 0 and countRelations()1 inclusive.
*
* @return the expression that the requested relation sets to 1;
* for instance, if the relation is <tt>g1^2 g2 = 1</tt> then
* this will be the expression <tt>g1^2 g2</tt>.
*/
const GroupExpression& relation(size_t index) const;
/**
* Tests whether all of the relations for the group are indeed words
* in the generators. This routine returns \c false if at least
* one relator uses an outofbound generator, and \c true otherwise.
*
* This routine is intended only for sanity checking: you should
* never have an invalid group presentation in the first place.
*
* @return \c true if and only if all of the relations are words
* in the generators.
*/
bool isValid() const;
/**
* Attempts to simplify the group presentation as intelligently
* as possible without further input.
*
* See intelligentSimplifyDetail() for further details on how
* the simplification is done.
*
* @return \c true if and only if the group presentation was changed.
* You can call intelligentSimplifyDetail() to get the isomorphism.
*/
bool intelligentSimplify();
/**
* Attempts to simplify the group presentation as intelligently
* as possible without further input.
*
* The current simplification method uses a combination of small
* cancellation theory and Nielsen moves.
*
* If this routine does return a homomorphism (because the
* presentation was changed), then this homomorphsm will in fact be
* a declared isomorphism. See the HomGroupPresentation class
* notes for details on what this means.
*
* @return a newly allocated homomorphism describing the
* reduction map from the original presentation to the new
* presentation, or a null pointer if this presentation was not
* changed.
*/
std::unique_ptr<HomGroupPresentation> intelligentSimplifyDetail();
/**
* Attempts to simplify the group presentation using only small
* cancellation theory.
*
* See smallCancellationDetail() for further details on how
* the simplification is done.
*
* @return \c true if and only if the group presentation was changed.
* You can call smallCancellationDetail() to get the isomorphism.
*/
bool smallCancellation();
/**
* Attempts to simplify the group presentation using small cancellation
* theory. The simplification method is based on the Dehn algorithm
* for hyperbolic groups, i.e. small cancellation theory. This means
* we look to see if part of one relator can be used to simplify
* others. If so, make the substitution and simplify. We continue
* until no more presentationshortening substitutions are available.
* We follow that by killing any available generators using words
* where generators appear a single time.
*
* If this routine does return a homomorphism (because the
* presentation was changed), then this homomorphsm will in fact be
* a declared isomorphism. See the HomGroupPresentation class
* notes for details on what this means.
*
* \todo \optlong This routine could use some small tweaks 
* recognition of utility of some score==0 moves, such as
* commutators, for example.
*
* @return a newly allocated homomorphism describing the
* reduction map from the original presentation to the new
* presentation, or a null pointer if this presentation was not
* changed.
*/
std::unique_ptr<HomGroupPresentation> smallCancellationDetail();
/**
* Uses small cancellation theory to reduce the input word,
* using the current presentation of the group. The input word
* will be modified directly.
*
* \warning This routine is only as good as the relator table for the
* group. You might want to consider running intelligentSimplify(),
* possibly in concert with proliferateRelators(), before using this
* routine for any significant tasks.
*
* @param input is the word you would like to simplify.
* This must be a word in the generators of this group.
* @return \c true if and only if the input word was modified.
*/
bool simplifyWord(GroupExpression &input) const;
/**
* A routine to help escape local wells when simplifying
* presentations, which may be useful when small cancellation theory
* can't find the simplest relators.
*
* Given a presentation <g_i  r_i>, this routine appends
* consequences of the relators {r_i} to the presentation that
* are of the form ab, where both a and b are cyclic permutations
* of relators from the collection {r_i}.
*
* Passing depth=1 means it will only form products of two
* relators. Depth=2 means products of three, etc. Depth=4 is
* typically the last depth before the exponential growth of
* the operation grows out of hand. It also conveniently trivializes
* all the complicated trivial group presentations that we've come
* across so far.
*
* \warning Do not call this routine with depth n before having called
* it at depth n1 first. Depth=0 is invalid, and depth=1 should be
* your first call to this routine. This routine gobbles up an
* exponential amount of memory (exponential in your presentation
* size times n). So do be careful when using it.
*
* @param depth controls the depth of the proliferation, as
* described above; this must be strictly positive.
*/
void proliferateRelators(unsigned long depth=1);
/**
* Attempts to recognise the group corresponding to this
* presentation. This routine is much more likely to be
* successful if you have already called intelligentSimplify().
*
* Currently, the groups this routine recognises include:
* the trivial group, abelian groups, free groups,
* extensions over the integers, and free products of any group
* the algorithm can recognise (inductively).
*
* The string returned from this routine may use some unicode
* characters, which will be encoding using UTF8. If \a moreUtf8
* is passed as \c false then unicode will be used sparingly;
* if \a moreUtf8 is \c true then unicode will be use more liberally,
* resulting in strings that look nicer but require more complex
* fonts to be available on the user's machine.
*
* Examples of the format of the returned string are:
*
*  <tt>0</tt> for the trivial group;
*  <tt>Z_n</tt> for cyclic groups with \a n > 1;
*  <tt>Free(n)</tt> for free groups with \a n > 1 generators  see
* AbelianGroup::str() for how abelian groups are presented;
*  <tt>FreeProduct(G1, G2, ... , Gk)</tt> for free products, where
* one replaces \a G1 through \a Gk by text strings representing the
* free summands;
*  <tt>Z~G w/ monodromy H</tt> for extensions over Z,
* where \a G is a description of the kernel of the homomorphism
* to the integers, and \a H is a text string representing the
* monodromy  see HomMarkedAbelianGroup.str() for details on
* how these are presented.
*
* \todo \featurelong Make this recognition more effective.
*
* @return a simple string representation of the group if it is
* recognised, or an empty string if the group is not
* recognised.
*/
std::string recogniseGroup(bool moreUtf8 = false) const;
/**
* Writes a chunk of XML containing this group presentation.
*
* \ifacespython Not present.
*
* @param out the output stream to which the XML should be written.
*/
void writeXMLData(std::ostream& out) const;
/**
* The sum of the word lengths of the relators.
* Word lengths are computing using GroupExpression::wordLength().
* Used as a coarse measure of the complexity of the presentation.
*
* @return the sum of word lengths.
*/
size_t relatorLength() const;
/**
* Computes the abelianisation of this group.
*
* @return a newly allocated abelianisation of this group.
*/
std::unique_ptr<AbelianGroup> abelianisation() const;
/**
* Computes the abelianisation of this group.
* The coordinates in the chain complex correspond
* to the generators and relators for this group.
*
* @return a newly allocated abelianisation of this group.
*/
std::unique_ptr<MarkedAbelianGroup> markedAbelianisation() const;
/**
* Attempts to determine if the group is abelian.
*
* A return value of \c true indicates that this routine
* successfully certified that the group is abelian.
* A return value of \c false indicates an inconclusive result:
* either the group is nonabelian, or the group
* is abelian but this routine could not prove so.
*
* If the group is abelian, then markedAbelianization() is the easiest
* way to see precisely
* which abelian group it is, and how the generators sit in that group.
*
* You will have better results from this algorithm if the
* presentation has been simplified, since this algorithm uses small
* cancellation theory in an attempt to reduce the commutators of all
* pairs of generators.
*
* \warning If you have not adequately simplified this presentation
* this routine will most likely return \c false. Consider running
* intelligentSimplify, possibly in concert with proliferateRelators(),
* in order to discover adequately many commutators.
*
* @return \c true if the group is shown to be abelian, or
* \c false if the result is inconclusive.
*/
bool identifyAbelian() const;
/**
* Switches the generators in the presentation indexed by \a i
* and \a j respectively, and recomputes the appropriate presentation.
* It is one of the standard Nielsen moves, which is the first of
* three generator types of the automorphism group of a free group.
*
* \pre Both \a i and \a j are strictly less than
* countGenerators().
*
* @param i indicates the first of the two generators to switch.
* @param j indicates the second of the two generators to switch.
* @return \c true if and only if the Nielsen automorphism had an
* effect on at least one relation.
*/
bool nielsenTransposition(unsigned long i, unsigned long j);
/**
* Replaces a generator in a presentation by its inverse, and
* recomputes the appropriate presentation. This is the second
* generator type of the automorphism group of a free group.
*
* \pre \a i is strictly less than countGenerators().
*
* @param i indicates the generator to invert.
* @return \c true if and only if the Nielsen automorphism had an
* effect on at least one relation.
*/
bool nielsenInvert(unsigned long i);
/**
* Replaces a generator \c gi by either
* <tt>(gi)(gj)^k</tt> or <tt>(gj)^k(gi)</tt> in the presentation. It
* it is the third type of Nielsen move one can apply to a presentation.
*
* This means that, if the new generator \c Gi is the old
* <tt>(gi)(gj)^k</tt> or <tt>(gj)^k(gi)</tt>, then we can construct
* the new presentation from the old by replacing occurrences of \c Gi
* by <tt>(Gi)(gj)^(k)</tt> or <tt>(gj)^(k)(Gi)</tt> respectively.
*
* \pre Both \a i and \a j are strictly less than countGenerators().
*
* @param i indicates the generator to replace.
* @param j indicates the generator to combine with \c gi.
* @param k indicates the power to which we raise \c gj when
* performing the replacement; this may be positive or negative
* (or zero, but this will have no effect).
* @param rightMult \c true if we should replace \c gi by
* <tt>(gi)(gj)^k</tt>, or \c false if we should replace \c gi by
* <tt>(gj)^k(gi)</tt>.
* @return \c true if and only if the nielsen automorphism had an
* effect on at least one relation.
*/
bool nielsenCombine(unsigned long i, unsigned long j,
long k, bool rightMult=true);
/**
* Looks for Nielsen moves that will simplify the presentation.
* Performs one of the mosteffective moves, if it can find any.
*
* @return \c true if and only if it performed a Nielsen move.
* You can call intelligentNielsen() to get the isomorphism.
*/
bool intelligentNielsen();
/**
* Looks for Nielsen moves that will simplify the presentation.
* Performs one of the mosteffective moves, if it can find any.
*
* If this routine does return a homomorphism (because some
* move was performed), then this homomorphsm will in fact be
* a declared isomorphism. See the HomGroupPresentation class
* notes for details on what this means.
*
* @return a newly allocated homomorphism describing the
* map from the original presentation to the new presentation,
* or a null pointer if no move was performed.
*/
std::unique_ptr<HomGroupPresentation> intelligentNielsenDetail();
/**
* Rewrites the presentation so that generators
* of the group map to generators of the abelianisation, with any
* leftover generators mapping to zero (if possible). Consider this a
* \e homologicalalignment of the presentation.
*
* See homologicalAlignmentDetail() for further details on what
* this routine does.
*
* @return \c true if presentation was changed, or \c false if
* the presentation was already homologically aligned.
* See homologicalAlignmentDetail() if you wish to get the isomorphism.
*/
bool homologicalAlignment();
/**
* Rewrites the presentation so that generators
* of the group map to generators of the abelianisation, with any
* leftover generators mapping to zero (if possible). Consider this a
* \e homologicalalignment of the presentation.
*
* If the abelianisation of this group has rank \a N and \a M invariant
* factors <tt>d0  d2  ...  d(M1)</tt>,
* this routine applies Nielsen moves
* to the presentation to ensure that under the markedAbelianisation()
* routine, generators 0 through \a M1 are mapped to generators of the
* relevant \c Z_di group. Similarly, generators \a M through
* <i>M</i>+<i>N</i>1 are mapped to +/1 in the appropriate factor.
* All further generators will be mapped to zero.
*
* If this routine does return a homomorphism (because the
* presentation was changed), then this homomorphsm will in fact be
* a declared isomorphism. See the HomGroupPresentation class
* notes for details on what this means.
*
* @return a newly allocated homomorphism giving the reduction map
* from the old presentation to the new, or a null pointer if
* this presentation was not changed.
*/
std::unique_ptr<HomGroupPresentation> homologicalAlignmentDetail();
/**
* An entirely cosmetic rewriting of the presentation, which is
* fast and superficial.
*
* See prettyRewritingDetail() for further details on what
* this routine does.
*
* @return \c true if and only if the choice of generators for the
* group has changed. You can call prettyRewritingDetail() to get the
* the isomorphism.
*/
bool prettyRewriting();
/**
* An entirely cosmetic rewriting of the presentation, which is
* fast and superficial.
*
* 1. If there are any length 1 relators, those generators are
* deleted, and the remaining relators simplified.
* 2. It sorts the relators by number of generator indices that
* appear, followed by relator numbers (lexico) followed by
* relator length.
* 3. inverts relators if net sign of the generators is negative.
* 4. Given each generator, it looks for the smallest word where that
* generator appears with nonzero weight. If negative weight,
* it inverts that generator.
* 5. It cyclically permutes relators to start with smallest gen.
*
* If this routine does return a homomorphism (because the choice of
* generators was changed), then this homomorphsm will in fact be
* a declared isomorphism. See the HomGroupPresentation class
* notes for details on what this means.
*
* \todo As a final step, make elementary simplifications to aid in
* seeing standard relators like commutators.
*
* @return a newly allocated homomorphism describing the
* map from the original presentation to the new presentation,
* or a null pointer if the choice of generators did not change.
*/
std::unique_ptr<HomGroupPresentation> prettyRewritingDetail();
/**
* Attempts to prove that this and the given group presentation are
* <i>simply isomorphic</i>.
*
* A <i>simple isomorphism</i> is an isomorphism where each generator
* <i>g<sub>i</sub></i> of this presentation is sent to
* some generator <i>g<sub>j</sub></i><sup>+/1</sup> of the
* other presentation. Moreover, at present this routine only
* looks for maps where both presentations have the same number
* of generators, and where distinct generators <i>g<sub>i</sub></i>
* of this presentation correspond to distinct generators
* <i>g<sub>j</sub></i> of the other presentation (possibly with
* inversion, as noted above).
*
* If this routine returns \c true, it means that the two
* presentations are indeed simply isomorphic.
*
* If this routine returns \c false, it could mean one of many
* things:
*
*  The groups are not isomorphic;
*  The groups are isomorphic, but not simply isomorphic;
*  The groups are simply isomorphic but this routine could not
* prove it, due to difficulties with the word problem.
*
* @param other the group presentation to compare with this.
* @return \c true if this routine could certify that the two group
* presentations are simply isomorphic, or \c false if it could not.
*/
bool identifySimplyIsomorphicTo(const GroupPresentation& other) const;
/**
* Returns a TeX representation of this group presentation.
* See writeTeX() for details on how this is formed.
*
* @return a TeX representation of this group presentation.
*/
std::string toTeX() const;
/**
* Writes a TeX represesentation of this group presentation
* to the given output stream.
*
* The output will be of the form < generators  relators >.
* There will be no final newline.
*
* \ifacespython The parameter \a out does not exist;
* standard output will be used.
*
* @param out the output stream to which to write.
*/
void writeTeX(std::ostream& out) const;
/**
* Returns a compact oneline representation of this group presentation,
* including details of all generators and relations.
* See writeTextCompact() for details on how this is formed.
*
* @return a compact representation of this group presentation.
*/
std::string compact() const;
/**
* Writes a compact represesentation of this group to the given
* output stream.
*
* The output will be of the form < generators  relators >.
* The full relations will be included, and the entire output
* will be written on a single line. There will be no final newline.
*
* \ifacespython The parameter \a out does not exist;
* standard output will be used.
*
* @param out the output stream to which to write.
*/
void writeTextCompact(std::ostream& out) const;
/**
* Writes a short text representation of this object to the
* given output stream.
*
* \ifacespython Not present.
*
* @param out the output stream to which to write.
*/
void writeTextShort(std::ostream& out) const;
/**
* Writes a detailed text representation of this object to the
* given output stream.
*
* \ifacespython Not present.
*
* @param out the output stream to which to write.
*/
void writeTextLong(std::ostream& out) const;
private:
/**
* Attempts to rewrite the presentation as a group extension.
* In particular, this routine attempts to rewrite this group
* as a semidirect product of the integers and another
* finitelypresented group, i.e., an extension of the form:
*
* < a, r1,...,rn  R1,...,RM, ar1a^1 = w1, ... arna^1 = wn >
*
* This is an algorithmic
* implementation of the ReidemeisterSchrier algorithm, which isn't
* actually an algorithm. So sometimes this procedure works, and
* sometimes it does not. The return value is an allocated unique_ptr
* if and only if the algorithm is successful. Even if the algorithm
* is unsuccessful, its application will likely result in a
* modification of the presentation.
*
* \apinotfinal This routine may very well either be eliminated
* in future versions of this software, perhaps incorporated into a
* biggerandbetter future algorithm.
*
* @return a newly allocated homomorphism if and only
* if the algorithm is successful. When this pointer is allocated
* it will be an automorphism of a presentation of the kernel of the
* map this to the integers.
*/
std::unique_ptr< HomGroupPresentation > identifyExtensionOverZ();
/**
* Attempts to determine if this group is clearly a free
* product of other groups. This is an unsophisticated algorithm
* and will likely only have success if one has preprocessed the
* presentation with simplification routines beforehand.
*
* If this routine succeeds then the group is definitely a free
* product. If this routine fails (by returning an empty list)
* then the result is inconclusive: the group might not be a
* free product, or it might be a free product but this routine
* could not prove so.
*
* \apinotfinal Reconsider how the enduser should see this routine.
*
* @return a list of newly allocated group presentations giving
* the factors of this free product, or an empty list if this
* routine fails (i.e., the result is inconclusive).
*/
std::list< GroupPresentation* > identifyFreeProduct() const;
/**
* A structure internal to the small cancellation simplification
* algorithm.
*
* Given two words, A and B, one wants to know how one can make
* substitutions into A using variants of the word B. This
* structure holds that data. For example, if:
*
* A == a^5b^2abababa^4b^1 and B == bababa^1
* == aaaaabbabababaaaab
* start_sub_at == 6, start_from == 0, sub_length == 5 makes sense,
* this singles out the subword aaaaab[babab]abaaaab. Since it would
* reduce the length by four, the score is 4.
*
* Similarly, if A == baba^4b^1a^5b^2aba == babaaaabaaaaabbaba
* and B == baba^1ba start_sub_at == 14, start_from == 5,
* sub_length == 5 makes sense, and is a cyclic variation
* on the above substitution, so the score is also 4.
*/
struct NWordSubstitutionData {
unsigned long start_sub_at;
/**< Where in A do we start? */
unsigned long start_from;
/**< Where in B do we start? */
unsigned long sub_length;
/**< The number of letters from B to use. */
bool invertB;
/**< Invert B before making the substitution? */
long int score;
/**< The score, i.e., the decrease in the word letter count
provided this substitution is made. */
bool operator<( const NWordSubstitutionData &other ) const {
if (score < other.score) return false;
if (score > other.score) return true;
if (sub_length < other.sub_length) return false;
if (sub_length > other.sub_length) return true;
if ( (invertB == true) && (other.invertB == false) )
return false;
if ( (invertB == false) && (other.invertB == true) )
return true;
if (start_from < other.start_from) return false;
if (start_from > other.start_from) return true;
if (start_sub_at < other.start_sub_at) return false;
if (start_sub_at > other.start_sub_at) return true;
return false;
}
void writeTextShort(std::ostream& out) const
{
out<<"Target position "<<start_sub_at<<
" length of substitution "<<sub_length<<(invertB ?
" inverse reducer position " : " reducer position ")
<<start_from<<" score "<<score;
}
};
/**
* A routine internal to the small cancellation simplification
* algorithm.
*
* This is the core of the Dehn algorithm for hyperbolic groups.
* Given two words, this_word and that_word, this routine searches for
* subwords of that_word (in the cyclic sense), and builds a table
* of substitutions one can make from that_word into this_word. The
* table is refined so that one knows the "value" of each
* substitution  the extent to which the substitution would shorten
* this_word. This is to allow for intelligent choices of
* substitutions by whichever algorithms call this one.
*
* This algorithm assumes that this_word and that_word are cyclically
* reduced words. If you feed it noncyclically reduced words it
* will give you suggestions although they will not be as strong
* as if the words were cyclically reduced. It also only adds
* to sub_list, so in normal usage one would pass it an empty sublist.
*
* The default argument step==1 assumes you are looking for
* substitutions that shorten the length of a word, and that
* you only want to make an immediate substitution. Setting
* step==2 assumes after you make your first substitution you
* will want to attempt a further substitution, etc. step>1
* is used primarily when building relator tables for group
* recognition.
*/
static void dehnAlgorithmSubMetric(
const GroupExpression &this_word,
const GroupExpression &that_word,
std::set< NWordSubstitutionData > &sub_list,
unsigned long step=1 );
/**
* A routine internal to the small cancellation simplification
* algorithm.
*
* Given a word this_word and that_word, apply the substitution
* specified by sub_data to *this. See dehnAlgorithm() and struct
* NWordSubstitutionData. In particular sub_data needs to be a
* valid substitution, usually it will be generated by
* dehnAlgorithmSubMetric.
*/
static void applySubstitution(
GroupExpression& this_word,
const GroupExpression &that_word,
const NWordSubstitutionData &sub_data );
};
/**
* Deprecated typedef for backward compatibility. This typedef will
* be removed in a future release of Regina.
*
* \deprecated The struct NGroupPresentation has now been renamed to
* GroupPresentation.
*/
REGINA_DEPRECATED typedef GroupPresentation NGroupPresentation;
/*@}*/
// Inline functions for GroupExpressionTerm
inline GroupExpressionTerm::GroupExpressionTerm() {
}
inline GroupExpressionTerm::GroupExpressionTerm(unsigned long newGen,
long newExp) : generator(newGen), exponent(newExp) {
}
inline GroupExpressionTerm::GroupExpressionTerm(
const GroupExpressionTerm& cloneMe) :
generator(cloneMe.generator), exponent(cloneMe.exponent) {
}
inline GroupExpressionTerm& GroupExpressionTerm::operator = (
const GroupExpressionTerm& cloneMe) {
generator = cloneMe.generator;
exponent = cloneMe.exponent;
return *this;
}
inline bool GroupExpressionTerm::operator == (
const GroupExpressionTerm& other) const {
return (generator == other.generator) && (exponent == other.exponent);
}
inline bool GroupExpressionTerm::operator != (
const GroupExpressionTerm& other) const {
return (generator != other.generator)  (exponent != other.exponent);
}
inline GroupExpressionTerm GroupExpressionTerm::inverse() const {
return GroupExpressionTerm(generator, exponent);
}
inline bool GroupExpressionTerm::operator += (
const GroupExpressionTerm& other) {
if (generator == other.generator) {
exponent += other.exponent;
return true;
} else
return false;
}
inline bool GroupExpressionTerm::operator < (
const GroupExpressionTerm& other) const {
return ( (generator < other.generator) 
( (generator == other.generator) &&
( exponent < other.exponent ) ) );
}
// Inline functions for GroupExpression
inline GroupExpression::GroupExpression() {
}
inline GroupExpression::GroupExpression(const GroupExpression& cloneMe) :
terms_(cloneMe.terms_) {
}
inline bool GroupExpression::operator ==(const GroupExpression& comp) const {
return terms_ == comp.terms_;
}
inline bool GroupExpression::operator !=(const GroupExpression& comp) const {
return terms_ != comp.terms_;
}
inline GroupExpression& GroupExpression::operator=(
const GroupExpression& cloneMe) {
terms_ = cloneMe.terms_;
return *this;
}
inline std::list<GroupExpressionTerm>& GroupExpression::terms() {
return terms_;
}
inline const std::list<GroupExpressionTerm>& GroupExpression::terms()
const {
return terms_;
}
inline size_t GroupExpression::countTerms() const {
return terms_.size();
}
inline bool GroupExpression::isTrivial() const {
return terms_.empty();
}
inline size_t GroupExpression::wordLength() const {
size_t retval(0);
std::list<GroupExpressionTerm>::const_iterator it;
for (it = terms_.begin(); it!=terms_.end(); it++)
retval += labs((*it).exponent);
return retval;
}
inline unsigned long GroupExpression::generator(size_t index) const {
return term(index).generator;
}
inline long GroupExpression::exponent(size_t index) const {
return term(index).exponent;
}
inline void GroupExpression::addTermFirst(const GroupExpressionTerm& term) {
terms_.push_front(term);
}
inline void GroupExpression::addTermFirst(unsigned long generator,
long exponent) {
terms_.push_front(GroupExpressionTerm(generator, exponent));
}
inline void GroupExpression::addTermLast(const GroupExpressionTerm& term) {
terms_.push_back(term);
}
inline void GroupExpression::addTermLast(unsigned long generator,
long exponent) {
terms_.push_back(GroupExpressionTerm(generator, exponent));
}
inline void GroupExpression::erase() {
terms_.clear();
}
// Inline functions for GroupPresentation
inline GroupPresentation::GroupPresentation() : nGenerators(0) {
}
inline GroupPresentation::~GroupPresentation() {
for_each(relations.begin(), relations.end(),
FuncDelete<GroupExpression>());
}
inline unsigned long GroupPresentation::addGenerator(unsigned long num) {
return (nGenerators += num);
}
inline void GroupPresentation::addRelation(GroupExpression* rel) {
relations.push_back(rel);
}
inline unsigned long GroupPresentation::countGenerators() const {
return nGenerators;
}
inline size_t GroupPresentation::countRelations() const {
return relations.size();
}
inline const GroupExpression& GroupPresentation::relation(
size_t index) const {
return *relations[index];
}
inline void GroupPresentation::writeTextShort(std::ostream& out) const {
out << "Group presentation: " << nGenerators << " generators, "
<< relations.size() << " relations";
}
inline size_t GroupPresentation::relatorLength() const {
size_t retval(0);
for (size_t i=0; i<relations.size(); i++)
retval += relations[i]>wordLength();
return retval;
}
} // namespace regina
#endif
