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FnMap.cpp
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FnMap.cpp
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//
/////////////////////////////////////////////////////////
#include <QChar>
#include <QList>
#include <QString>
#include "FnMap.h"
#include "WhiteheadData.h"
/////////////////////////////////////////////////////////
// CONSTRUCTORS
FnMap::FnMap(int rank) : QHash<QChar, FnWord>()
{
// constructs the identity function
image_rank = rank;
if (rank < Fn_MinRank || rank > Fn_MaxRank) { // check MIN_RANK <= rank <= MAX_RANK
insert(BASIS.at(0),Fail);
return;
}
for(int i = 0; i < 2*rank; i++) {
insert(BASIS.at(i),FnWord(BASIS.at(i)));
}
}
FnMap::FnMap(const QStringList & images, int rank) : QHash<QChar, FnWord>()
{
// rank = 0 means build an endomorphism
image_rank = (rank == 0) ? images.size() : rank;
if (image_rank < Fn_MinRank || image_rank > Fn_MaxRank) { // check MIN_RANK <= rank <= MAX_RANK
insert(BASIS.at(0),Fail);
return;
}
if (images.size() < Fn_MinRank || images.size() > Fn_MaxRank) {
insert(BASIS.at(0),Fail);
return;
}
Basis basis(image_rank);
for (int i = 0; i < images.size(); i++) {
FnWord u(images.at(i));
if (!u.checkBasis(basis)) {
insert(BASIS.at(0),Fail);
return;
}
insert(BASIS.at(2*i),u);
insert(BASIS.at(2*i+1),u.inverse());
}
}
/////////////////////////////////////////////////////////
// PUBLIC MEMBER FUNCTIONS
FnMap FnMap::isAutomorphism() const
{
/* Test to see is map defines an automorphism. Can be used to check
if a given collection words defines a basis. If the map is not
an automorphism, the Fail map is returned. */
FnMap phi(image_rank);
// check that map is an endomorphism
if (size() != 2*image_rank) {
phi.fail();
return phi;
}
int old_norm,new_norm;
FnMap tmp(image_rank);
FnMap this_min(*this);
FnMap permutation(image_rank);
FnMap whAuto(image_rank);
QChar a;
Basis basis(size()/2);
QList<WhiteheadData> whAutos = whiteheadAutos(basis);
QListIterator<WhiteheadData> move(whAutos);
bool reduced_norm = true;
while (reduced_norm) {
old_norm = this_min.norm();
new_norm = old_norm;
move.toFront();
while (old_norm <= new_norm && move.hasNext()) {
whAuto = whitehead(move.next(),basis);
tmp = whAuto*this_min;
new_norm = tmp.norm();
}
if (new_norm < old_norm) {
this_min = tmp;
phi = whAuto*phi; // build inverse
} else
reduced_norm = false;
} // end while(reduced_norm)
// now this_min = phi*this
QString basisElements = basis;
MapIterator x(this_min);
while (x.hasNext()) {
x.next();
if (x.value() == Id || x.value().length() != 1) {
phi.fail();
return phi;
}
a = x.value().at(0);
basisElements.remove(a);
permutation.insert(a,FnWord(x.key()));
}
if (!basisElements.isEmpty()) {
phi.fail();
return phi;
}
return permutation*phi;
}
FnMap FnMap::iterate(int n) const
{
//returns the nth power of the automorphism
//uses repeated squaring
int i;
FnMap phi_pow2(*this);
FnMap phi_n(image_rank);
// special cases
if (n == 1) {
return FnMap(*this);
}
if (phi_pow2.imageRank() != phi_pow2.domainRank()) {
phi_n.fail();
return phi_n;
}
if (n < 0) { // eventually want this to take you to the inverse if possible...
phi_n.fail();
return phi_n;
}
QString n_base2 = QString::number(n,2); // convert n to base 2
for (i = n_base2.length() - 1; i > 0; i--) {
// if the ith digit in the base 2 representation of n is 1,
// multiply phi_n by this power
if(n_base2.at(i).digitValue() == 1) phi_n *= phi_pow2;
// now square
phi_pow2 *= phi_pow2;
}
// the first digit is always nonzero
phi_n *= phi_pow2;
return phi_n;
}
/////////////////////////////////////////////////////////
// OPERATORS
FnWord FnMap::operator () (const FnWord &u, int niter) const
{
// returns the image of u upon applying the map niter times
// special cases
if (niter == 0) return u;
if (niter < 0) return Fail;
if (u == Id) return Id;
if (niter > 1 && image_rank != domainRank()) return Fail;
int i;
FnMap phi_n;
FnWord image;
phi_n = iterate(niter);
for (i = 0; i < u.length(); i++) {
image += phi_n.value(u.at(i));
}
image.tighten();
return image;
}
FnMap & FnMap::operator *= (const FnMap &phi) {
// composes phi * this
FnMap this_map(*this); // make a copy of this morphism
FnMap phi_copy(phi); // make a copy of phi incase phi = this
// check that morphisms can be composed
if (image_rank != phi_copy.domainRank()) {
fail();
return *this;
}
// set new image rank
image_rank = phi_copy.imageRank();
int i;
FnWord this_image,phi_image;
MapIterator x(this_map);
while (x.hasNext()) {
x.next();
this_image = x.value();
if (this_image == Id) {
phi_image = Id;
} else {
phi_image = FnWord(QString("")); // reset phi_image
for (i = 0; i < this_image.length(); i++) {
phi_image += phi_copy.value(this_image.at(i));
}
phi_image.tighten();
}
insert(x.key(),phi_image);
}
return *this;
}
//FRIEND OPERATORS
FnMap operator * (const FnMap &phi, const FnMap &psi)
{
// composes the maps phi and psi
FnMap phi_psi(psi);
phi_psi *= phi;
return phi_psi;
}
/////////////////////////////////////////////////////////
// ADDITIONAL FUNCTIONS
FnMap minimizeLength(const FnWord &u, const Basis &basis)
{
/* Returns an automorphism phi such that phi(u) has minimal
length. */
int r = basis.getRank();
FnMap phi(r);
if (u == Id) return phi;
if (u.length() == 1) return phi;
int old_norm,new_norm;
FnWord u_min(u),tmp;
FnMap whAuto(r);
QList<WhiteheadData> whAutos = whiteheadAutos(basis);
QListIterator<WhiteheadData> move(whAutos);
bool reduced_norm = true;
while (reduced_norm) {
old_norm = u_min.length();
new_norm = old_norm;
move.toFront();
while (old_norm <= new_norm && move.hasNext()) {
whAuto = whitehead(move.next(),basis);
tmp = whAuto(u_min);
new_norm = tmp.length();
}
if (new_norm < old_norm) {
u_min = tmp;
phi = whAuto*phi;
} else
reduced_norm = false;
} // end while (reduced_norm)
// now u_min = phi(u)
return phi;
}
FnMap isPrimitiveElement(const FnWord &u, const Basis &basis)
{
/* Determines if u is a primitive element. If it is, an automorphism
phi such that phi(u) has length 1 is returned. If not, the map
returned is the Fail map. */
FnMap phi;
if (u == Id) {
phi.fail();
return phi;
}
phi = minimizeLength(u,basis);
FnWord u_min = phi(u);
if (u_min.length() != 1) { // u is not a basis element
phi.fail();
}
//else u is a primitve element
return phi;
}
FnMap whiteheadProblem(const FnWord &u, const FnWord &v,
const Basis &basis)
{
/* Determines if there is an automorphism phi such that u = phi(v).
If yes, the map is returned. Else, the Fail map is returned. */
int i,r = basis.getRank();
int min_len;
FnWord tmp,v_tmp;
FnWord u_min,v_min;
FnMap phi(r);
FnMap whAuto(r);
FnMap rho(r),sigma(r),tau(r);
FnMap phi_u(r),phi_v(r),phi_u_inv(r);
// get minimal length representatives
phi_u = minimizeLength(u,basis);
phi_v = minimizeLength(v,basis);
u_min = phi_u(u);
v_min = phi_v(v);
// easy case, minimum lengths are different
if (u_min.length() != v_min.length()) {
phi.fail();
return phi;
}
min_len = u_min.length();
phi_u_inv = phi_u.isAutomorphism();
// other easy case, u_min = v_min
if (u_min == v_min) { // u = phi_u^-1 (phi_v(v))
phi = phi_u_inv*phi_v;
return phi;
}
// now the hard case... need to find all words of length u_min related
// by Whitehead moves and permutations...
// build Whitehead data
QList<WhiteheadData> whAutos = whiteheadAutos(basis);
QListIterator<WhiteheadData> move(whAutos);
// define the permutation automorphisms
for (i = 0; i < r-1; i++) { // i -> i+1
rho.insert(basis.at(2*i),FnWord(basis.at(2*i + 2)));
rho.insert(basis.at(2*i + 1),FnWord(basis.at(2*i + 3)));
}
rho.insert(basis.at(2*r - 2),FnWord(basis.at(0)));
rho.insert(basis.at(2*r - 1),FnWord(basis.at(1)));
sigma.insert(basis.at(0),FnWord(basis.at(1))); // a -> A
sigma.insert(basis.at(1),FnWord(basis.at(0))); // A -> a
tau.insert(basis.at(0),FnWord(basis.at(2))); // a -> b
tau.insert(basis.at(1),FnWord(basis.at(3))); // A -> B
tau.insert(basis.at(2),FnWord(basis.at(0))); // b -> a
tau.insert(basis.at(3),FnWord(basis.at(1))); // B -> A
int n = 0;
QList<FnWord> v_rel;
QHash<FnWord, FnMap> wh_v; // wh_v[x](v_min) = x
v_rel.append(v_min);
wh_v.insert(v_min,FnMap(r));
// looped over words with the same length as v
while (n < v_rel.size()) {
v_tmp = v_rel.at(n++);
move.toFront();
// loop over Whitehead moves
while (move.hasNext()) {
whAuto = whitehead(move.next(),basis);
tmp = whAuto(v_tmp);
if (tmp.length() == min_len) {
if (!v_rel.contains(tmp)) {
v_rel.append(tmp);
wh_v.insert(tmp,whAuto*wh_v.value(v_tmp));
}
if (tmp == u_min) { // found it!
phi_v = wh_v.value(tmp)*phi_v; // phi_u(u) = tmp = phi_v(v)
phi = phi_u_inv*phi_v; // u = phi_u^-1 phi_v(v)
return phi;
}
}
} // end while (move.hasNext())
// loop over permutation moves, these don't change length
tmp = rho(v_tmp);
if (!v_rel.contains(tmp)) {
v_rel.append(tmp);
wh_v.insert(tmp,rho*wh_v.value(v_tmp));
}
if (tmp == u_min) { // found it!
phi_v = wh_v.value(tmp)*phi_v; // phi_u(u) = tmp = phi_v(v)
phi = phi_u_inv*phi_v; // u = phi_u^-1 phi_v(v)
return phi;
}
tmp = sigma(v_tmp);
if (!v_rel.contains(tmp)) {
v_rel.append(tmp);
wh_v.insert(tmp,sigma*wh_v.value(v_tmp));
}
if (tmp == u_min) { // found it!
phi_v = wh_v.value(tmp)*phi_v; // phi_u(u) = tmp = phi_v(v)
phi = phi_u_inv*phi_v; // u = phi_u^-1 phi_v(v)
return phi;
}
tmp = tau(v_tmp);
if (!v_rel.contains(tmp)) {
v_rel.append(tmp);
wh_v.insert(tmp,tau*wh_v.value(v_tmp));
}
if (tmp == u_min) { // found it!
phi_v = wh_v.value(tmp)*phi_v; // phi_u(u) = tmp = phi_v(v)
phi = phi_u_inv*phi_v; // u = phi_u^-1 phi_v(v)
return phi;
}
} // end while (x.hasNext())
// didn't find it
phi.fail();
return phi;
}
/////////////////////////////////////////////////////////