int main( )
{
  FPGroup G;
  cout << "Enter nilpotentcy class first then group ";
  int NilpotentcyClass;
  cin >> NilpotentcyClass;
  Chars errMsg = cin >> G;
  if (errMsg.length()>0) 
    return 1;
 
  NilpotentGroup ng(G.namesOfGenerators(),
		    NilpotentcyClass,makeVectorOf(G.getRelators()));
  ng.initialize();
  VectorOf<Word> vw;
  Word w;
  for (int i=1;true;i++){
    cout << endl << "Enter the "<<i<<" generator of subgroup"<< endl;
    cout << "Empty word to finish: ";
    w = G.readWord(cin,errMsg);
    if (w.length()==0)
      break;
    if (errMsg.length()>0) 
      return 1;
    vw.append(w);
  }
  SGOfNilpotentGroup sg(ng,vw);
  sg.initBasis();
  sg.printBasis(cout);
  cout << "The Hirsch number :" << sg.theHirschNumber() << endl;
  cout << "Index in parent group :" << sg.index() << endl;
  cout << "Is Trivial :" << sg.isTrivial() << endl;
  cout << "Is Central :" << sg.isCentral() << endl;
  cout << "Is normal :" << sg.isNormal() << endl;
  cout << "Is abelian :" << sg.isAbelian() << endl;
  cout << "Subgroup class :" << sg.subgroupClass() << endl;
  cout << "Generators of normal closure :" << endl;
  vw = sg.normalClosureGens();
  for (int i=0;i<vw.length();i++){
    G.printWord(cout,vw[i]);
    cout << endl;
  }
  PresentationForSNG sgp = sg.makePresentation();
  cout << "Presentation of subgroup :" << endl;
  sgp.print(cout);
  cout << endl << "Enter the word :";
  errMsg = "";
  w = G.readWord(cin,errMsg);
  if (errMsg.length()>0){ 
    cout << errMsg;
    return 1; 
  }
  cout << endl << "Does subgroup contain the word :" << sg.contains(w) << endl;
  PolyWord result;
  if (sg.decompose(ng.decompose(w),result))
    cout << "Decomposition in sg basis :" << sg.asDecomposition(result);
  else
    cout << "Subgroup does not contain this word";

}
Exemplo n.º 2
0
VectorOf<Chars> PresentationParser::parseGeneratorList( Chars& errMesg )
{
  VectorOf<Chars> result;

  if ( curToken == INIT ) getToken();

  while ( curToken == GENERATOR ) {

	 Chars name(tokenName);

    // Check for duplication and presence of inverses.
	 if ( result.indexOf(name) >= 0 ) {
		parseError("Duplicate generator");
		errMesg = parseErrorMessage;
		return result;
	 } else {
		invertName(tokenName);
		if ( result.indexOf(Chars(tokenName)) >= 0 ) {
		  parseError("Duplicate generator: formal inverse");
		  errMesg = parseErrorMessage;
		  return result;
		}
	 }

	 result.append(name); // Ok, it's inefficient, but time is $$

	 getToken();

	 if ( curToken == COMMA ) {
	   getToken();
	   if ( curToken == DOT ) {
	     if (!getGeneratorRange(name,result,errMesg))
	       return result;
	   } else {
	     if ( curToken != GENERATOR ) {
	       parseError("Expected a generator here");
	       errMesg = parseErrorMessage;
	       return result;
	     }
	   }
	 }
  }

  return result;
}
Exemplo n.º 3
0
//@njz:default value removed. defined in .h file
//void AbelianEquationsSolver::findSolutions( File& file , bool out = true )
void AbelianEquationsSolver::findSolutions( File& file , bool out)
//
{
  haveSol = 0;
  
  if( !A.haveCyclicDecomposition() )
    A.computeCyclicDecomposition();
  
  if( !A.havePrimaryDecomposition() )
    A.findPrimaryBasis();
  
  if( !rawA.haveCyclicDecomposition() )
    rawA.computeCyclicDecomposition();
  
  if( !rawA.havePrimaryDecomposition() )
    rawA.findPrimaryBasis();
  
  makeSystem();

  if( out )
   {
     if( system.length() > 1 )
       file << "The system of equations: " << endl << endl;
     else
       file << "The equation: " << endl << endl;
     
     printRawSystem( file );

     file << endl << "can be transformed to the one: " << endl << endl;

     printSystem( file );
  
     if( system.length() > 1 )
       file << endl << "Finding solutions of this system ..." << endl << endl;
     else
       file << endl << "Finding solutions of this equation ..." << endl << endl;
   }

  //@njz
  //  int **matrix = new (int*)[ system.length() ];
  int **matrix = new int*[ system.length() ];
  //
  
  int i,j,k,i1,j1;
  
  for( i = 0 ; i < system.length() ; i++ )
   {
     matrix[i] = new int[ numberOfVariables ];
     for( j = 0 ; j < numberOfVariables ; j++ )
       {
	 Integer u = (AbelianWord( numberOfVariables , system[i] ))[j];
	 matrix[i][j] = u.as_long();
       }
   }

  VectorOf< VectorOf<int> > transform;
  VectorOf<int> trans(3);

  // diagonalization of the matrix
  
  sysRank = 0;
  
  for( i = 0 ; ( i < numberOfVariables && i < system.length() ) ; i++ )
    {
      bool flag = false;
      
      for( j = i ; j < system.length() ; j++ )
	{
	  for( k = i ; k < numberOfVariables ; k++ )
	    if( matrix[j][k] )
	      {
		flag = true;
		break;
	      }
	  
	  if( flag ) break;
	}
      
      if( k == numberOfVariables && j == system.length() )
	{    
	  if( sysRank )
	    break;
	  
	  int q;
	  for( q = 0 ; q < b.length() ; q++ )
	    if( !A.isTrivial( Word( b[q] ) ) )
	      {
		haveSol = -1;
		
		if( out )
		  if( system.length() > 1 )
		    file << "while computing the canonical form of the system it was found that this system has no solutions." << endl;
		  else
		    file << "while computing the canonical form of the equation it was found that this equation has no solutions." << endl;
		
		return;
	      }
	  
	  haveSol = 0;
	  
	  if( out )
	    if( system.length() > 1 )
	      file << "while computing the canonical form of the system it was found that this system has all group as a set of solutions." << endl;
	    else
	      file << "while computing the canonical form of the equation it was found that this equation has all group as a set of solutions." << endl;
	  
	  return;
	  
	}
      
      int *tmp = matrix[i];
      matrix[i] = matrix[j];
      matrix[j] = tmp;
      
      Word r = b[i];
      b[i] = b[j];
      b[j] = r;
      
      for( j = 0 ; j < system.length() ; j++ )
	{
	  int t = matrix[j][i];
	  matrix[j][i] = matrix[j][k];
	  matrix[j][k] = t;
	}
      
      if( i != k )
      {
	trans[0] = i;
	trans[1] = k;
	trans[2] = 0;
	
	transform.append( trans );
      }
      
      while( true )
	{
	  bool check;
	  bool flag;
	  int z;
	  bool done = false;
	  int count = i + 1;
	  
	  while( !done )
	    {
	      for( j = count ; j < system.length() ; j++ )
		if( matrix[j][i] && abs(matrix[i][i]) != abs(matrix[j][i]) )
		  break;
	      
	      if( j == system.length() )
		break;
	      
	      count = j + 1;
	      
	      flag = false;
	      
	      while( !flag )
		{
		  if( abs(matrix[i][i]) > abs(matrix[j][i]) )
		    {
		      z = matrix[i][i] / matrix[j][i];
		      for( k = i ; k < numberOfVariables ; k++ )
			matrix[i][k] = matrix[i][k] - matrix[j][k] * z;
		      b[i] = b[i] * A.getFPGroup().raiseToPower(b[j],-z);
		    }
		  else
		    {
		      z = matrix[j][i] / matrix[i][i];
		      for( k = i ; k < numberOfVariables ; k++ )
			matrix[j][k] = matrix[j][k] - matrix[i][k] * z;
		      b[j] = b[j] * A.getFPGroup().raiseToPower(b[i],-z);
		    }
		  
		  if( !matrix[i][i] || !matrix[j][i] ) 
		    {
		      if( matrix[i][i] == 0 )
			{  
			  tmp = matrix[i];
			  matrix[i] = matrix[j];
			  matrix[j] = tmp;
			  
			  r = b[i];
			  b[i] = b[j];
			  b[j] = r;
			}
		      flag = true;
		    }
		}
	      
	      if( count == system.length() ) 
		done = true;
	    }
	  
	  check = true;
	  for( j = i + 1 ; j < numberOfVariables ; j++ )
	    if( matrix[i][j] % matrix[i][i] )
	      {
		check = false;
		break;
	      }
	  
	  if( check )
	    break;

	  done = false;
	  count = i + 1;
	  while( !done )
	    {
	      for( j = count ; j < numberOfVariables ; j++ )
		if( matrix[i][j] && abs(matrix[i][i]) != abs(matrix[i][j]) )
		  break;
	      
	      if( j == numberOfVariables )
		break;
	      
	      count = j + 1;
	      
	      flag = false;
	      	      
	      while( !flag )
		{
		  if( abs(matrix[i][i]) > abs(matrix[i][j]) )
		    {
		      z = matrix[i][i] / matrix[i][j];
		      for( k = i ; k < system.length() ; k++ )
			matrix[k][i] = matrix[k][i] - matrix[k][j] * z;
		     
		      trans[0] = j;
		      trans[1] = i;
		      trans[2] = -z;
		      
		      transform.append( trans );
      		    }
		  else
		    {
		      z = matrix[i][j] / matrix[i][i];
		      for( k = i ; k < system.length() ; k++ )
			matrix[k][j] = matrix[k][j] - matrix[k][i] * z;
		      
		      trans[0] = i;
		      trans[1] = j;
		      trans[2] = -z;
		      
		      transform.append( trans );
		    }
		  
		  if( !matrix[i][i] || !matrix[i][j] ) 
		    {
		      if( matrix[i][i] == 0 )
			{  
			  for( k = i ; k < system.length() ; k++ )
			    {
			      int a = matrix[k][i];
			      matrix[k][i] = matrix[k][j];
			      matrix[k][j] = a;
			    }
			  
			  if( i != j )
			    {
			      trans[0] = i;
			      trans[1] = j;
			      trans[2] = 0;
			  
			      transform.append( trans );
			    }
			}
		      flag = true;
		    }
		}
	      
	      if( count == numberOfVariables ) 
		done = true;
	    }
	  
	  check = true;
	  for( j = i + 1 ; j < system.length() ; j++ )
	    if( matrix[j][i] % matrix[i][i] )
	      {
		check = false;
		break;
	      }
	  
	  if( check )
	    break;
	}
      
      for( j = i + 1 ; j < system.length() ; j++ )
	if( matrix[j][i] )
	  {
	    int z = matrix[j][i] / matrix[i][i];
	    for( k = i ; k < numberOfVariables ; k++ )
	      matrix[j][k] = matrix[j][k] - matrix[i][k] * z;
	    
	    b[j] = b[j] * A.getFPGroup().raiseToPower(b[i],-z);
	  }
      
      for( j = i + 1 ; j < numberOfVariables ; j++ )
	if( matrix[i][j] )
	  {
	    int z = matrix[i][j] / matrix[i][i];
	    for( k = i ; k < system.length() ; k++ )
	      matrix[k][j] = matrix[k][j] - matrix[k][i] * z;
	    
	    trans[0] = i;
	    trans[1] = j;
	    trans[2] = -z;
	    
	    transform.append( trans );
	  }
      
      sysRank++;
    }
  
  for( i = sysRank ; i < system.length() ; i++ )
    {
      if( !A.isTrivial( Word( b[i] ) ) )
	{
	  haveSol = -1;
	  
	  if( out )
	    if( system.length() > 1 )
	      file << "while computing the canonical form of the system it was found that this system has no solutions." << endl;
	    else
	      file << "while computing the canonical form of the equation it was found that this equation has no solutions." << endl;
	  
	  return;
	}
    }

  // finding solutions and output in file

  if( out )
    {
      file << "The canonical form: ";
      file << endl << endl;
    }
  
  for( int p = 0 ; p < sysRank ; p++ )
    {
      if( out )
	{      
	  file << "x" << p + 1;
	  
	  if( matrix[p][p] != 1 )
	    file << "^" << matrix[p][p];
	  
	  file << " = ";
	}
      
      AbelianWord w = A.oldInAbelianForm( b[p] );
      w = A.oldToNewGens( w );
      w = A.newToOldGens( w );    
      b[p] = w.getWord().freelyReduce();
      
      if( out )
	{
	  A.getFPGroup().printWord( file , b[p] );
	  file << endl;
	}
    }
  
  VectorOf<int> xNums( numberOfVariables );
  
  for( i = 0 ; i < sysRank ; i++ )
    if( root( b[i] , matrix[i][i] ) )
      {   
	x[i] = b[i];
	xNums[i] = i;
	
	if( !A.isFree() )
	  {
	    VectorOf<int> tmp(2);
	    tmp[0] = ( matrix[i][i] > 0 ) ? matrix[i][i] : -matrix[i][i];
	    tmp[1] = 1;
	    torsion[i].append( tmp );
	  }
      }
    else
      {
	haveSol = -1;
	
	if( out )
	{
	  if( system.length() > 1 )
	    file << endl << "The system is unsolvable because there are no solutions for: " << endl << endl;
	  else
	    file << endl << "The equation is unsolvable because there are no solutions for: " << endl << endl;
	      
	  file << "x" << i + 1;
	
	  if( matrix[i][i] != 1 )
	    file << "^" << matrix[i][i];
	
	  file << " = ";
	  A.getFPGroup().printWord( file , b[i].freelyReduce() );
	}
	
	return;
      }
  
  for( j = sysRank ; j < numberOfVariables ; j++ )
    {
      x[j] = Word();
      xNums[j] = j;

      VectorOf<int> tmp(2);
      tmp[0] = j - sysRank + 1;
      tmp[1] = 1;
      params[j].append( tmp );
    }
  
  for( i = transform.length() - 1 ; i >= 0 ; i-- )
    {
      trans = transform[i];
      
      if( !trans[2] )
	{
	  int ind1 = xNums.indexOf( trans[0] );
	  int ind2 = xNums.indexOf( trans[1] );
	  
	  xNums[ind1] = trans[1];
	  xNums[ind2] = trans[0];
	}
      else
	{
	  int ind1 = xNums.indexOf( trans[0] );
	  int ind2 = xNums.indexOf( trans[1] );
	  
	  x[ind1] *= A.getFPGroup().raiseToPower( x[ind2] , trans[2] ); 
	  
	  int len = torsion[ind1].length();
	  for( j = 0 ; j < torsion[ind2].length() ; j++ )
	    {
	      bool f = false;
	      
	      for( k = 0 ; k < len ; k++ )
		if( torsion[ind1][k][0] == torsion[ind2][j][0] )
		  {
		    f = true;
		    break;
		  }
		  
	      if( f )
		torsion[ind1][k][1] += torsion[ind2][j][1] * trans[2];
	      else
		{
		  VectorOf<int> tmp = torsion[ind2][j];
		  tmp[1] *= trans[2];
		  torsion[ind1].append( tmp );
		}
	    }
	     
	  len = params[ind1].length();
	  for( j = 0 ; j < params[ind2].length() ; j++ )
	    {
	      bool f = false;
	      
	      for( k = 0 ; k < len ; k++ )
		if( params[ind1][k][0] == params[ind2][j][0] )
		  {
		    f = true;
		    break;
		  }
		  
	      if( f )
		params[ind1][k][1] += params[ind2][j][1] * trans[2];
	      else
		{
		  VectorOf<int> tmp = params[ind2][j];
		  tmp[1] *= trans[2];
		  params[ind1].append( tmp );
		}
	    }
	}
    }

  for( i = 0 ; i < x.length() ; i++ )
    {
      AbelianWord w = A.oldInAbelianForm( x[i] );
      w = A.oldToNewGens( w );
      w = A.newToOldGens( w );
      x[i] = w.getWord();
    }
  
  // output in file
  if( out )
    {
      file << endl << "The set of solutions can be presented as follows: " << endl << endl;
      FPGroup G = rawA.getFPGroup();
      FPGroup G1 = A.getFPGroup();
      
      for( i = 0 ; i < numberOfVariables ; i++ )
	{
	  int n = xNums.indexOf( i );
	  bool flag = false;
	  
	  G.printWord( file , Generator( i + 1 ) );
	  file << " -> ";
	  
	  if( x[n].length() )
	    {
	      G1.printWord( file , x[n].freelyReduce() );
	      flag = true;
	    }
	  
	  for( j = 0 ; j < torsion[n].length() ; j++ )
	    {
	      VectorOf<int> tmp = torsion[n][j];
	      int z = tmp[1] % tmp[0];
	      
	      if( z )
		if( tmp[1] > 0 )
		  {
		    if( flag )
		      file << " + ";
		    
		    if( tmp[1] != 1 )
		      file << tmp[1] << " p( " << tmp[0] << " )";
		    else
		      file << "p( " << tmp[0] << " )";
		    
		    flag = true;
		  }
		else
		  {
		    file << " - ";
		    
		    if( tmp[1] != -1 )
		      file << -tmp[1] << " p( " << tmp[0] << " )";
		    else
		      file << "p( " << tmp[0] << " )";
		    
		    flag = true;
		  }
	    }
	  
	  for( j = 0 ; j < params[n].length() ; j++ )
	    {
	      VectorOf<int> tmp = params[n][j];
	      
	      if( tmp[1] )
		if( tmp[1] > 0 )
		  {
		    if( flag )
		      file << " + ";
		    
		    if( tmp[1] != 1 )
		      file << tmp[1] << " t" << tmp[0];
		    else
		      file << "t" << tmp[0];
		    
		    flag = true;
		  }
		else
		  {
		    file << " - ";
		    
		    if( tmp[1] != -1 )
		      file << -tmp[1] << " t" << tmp[0];
		    else
		      file << "t" << tmp[0];
		    
		    flag = true;
		  }
	    }
	  
	  if( !flag )
	    file << "1 ";
	      
	  if( i != numberOfVariables )
	    file << ",";
	  
	  file << endl;
	}
  
      file << endl << "where  p( n ) is any element of n-heigth and t_i - any element of the group." << endl;
    }
  
  VectorOf< VectorOf< VectorOf<int> > > t( numberOfVariables );
  VectorOf< VectorOf< VectorOf<int> > > p( numberOfVariables );
  VectorOf<Word> x1( numberOfVariables );
  
  for( i = 0 ; i < numberOfVariables ; i++ )
    {
      int n = xNums.indexOf( i );
      x1[i] = x[n];
      t[i] = torsion[n];
      p[i] = params[n];
    }
  
  torsion = t;
  params = p;
  x = x1;
  
  for( i = 0 ; i < system.length() ; i++ )
    delete [] matrix[i];
  delete [] matrix;
}
Exemplo n.º 4
0
main()
{
  FPGroup G;
  VectorOf<int> order;
  cin >> G;
  char ch;
  cin >> ch;
  if (ch != '[') { cerr << "Unexpected input, aborted"<<endl; exit(1);}
  do {
    int i;
    cin >> i;
    order.append(i);
    cin>> ch;
    if (ch==']') break;
    else if (ch!=','){ cerr << "Unexpected input, aborted"<<endl; exit(1);}
  } while (ch==',');
  WordOrder word_order("ShortLex",order);
  KBmagPackage kbmag(G.namesOfGenerators(),G.getRelators(),word_order,20);
  if ( !kbmag.sanityCheck() ) {
    error("kbmag failed sanity check.\n");
  }
  if (kbmag.autgroup()==yes){
    cout << "Group "<< G << " is proved shortlex automatic"<<endl;
    GroupDFSA WA = kbmag.wordAcceptor();
    WA.setName("Word_acceptor");
    WA.printOn();
    GenMult GM = kbmag.generalMultiplier();
    GM.setName("General_multiplier");
    GM.printOn();
    DiffMachine D1 = kbmag.differenceMachine(1);
    D1.setName("1stDiffMachine");
    D1.printOn();
    DiffMachine D2 = kbmag.differenceMachine(2);
    D2.setName("2ndDiffMachine");
    D2.printOn();
  }
  else 
    cout << "Group "<< G << " is not proved shortlex automatic"<<endl;
  cout << endl;
  
  FPGroup G2;
  cin >> G2;
  KBmagPackage kbmag2(G2.namesOfGenerators(),G2.getRelators());
  if ( !kbmag2.sanityCheck() ) {
    error("kbmag2 failed sanity check.\n");
  }
  if (kbmag2.kbprog()==yes && kbmag2.gpmakefsa()==yes && kbmag2.gpaxioms()==yes){
    cout << "Group "<< G2 << " is proved shortlex automatic"<<endl;
    GroupDFSA WA = kbmag2.wordAcceptor();
    WA.setName("Word_acceptor");
    WA.printOn();
    GenMult GM = kbmag2.generalMultiplier();
    GM.setName("General_multiplier");
    GM.printOn();
    DiffMachine D1 = kbmag2.differenceMachine(1);
    D1.setName("1stDiffMachine");
    D1.printOn();
    DiffMachine D2 = kbmag2.differenceMachine(2);
    D2.setName("2ndDiffMachine");
    D2.printOn();
  }
  else 
    cout << "Group "<< G2 << " is not proved shortlex automatic"<<endl;
  cout << endl;
  FPGroup G3;
  cin >> G3;
  KBmagPackage kbmag3(G3.namesOfGenerators(),G3.getRelators());
  if ( !kbmag3.sanityCheck() ) {
    error("kbmag3 failed sanity check.\n");
  }
  if (kbmag3.kbprog()>0){
    Bool abort=NO;
    int loop=0;
    do {
      loop++;
      cout << "Trying to built automata."<< endl;
      if (loop>1) cout << "Pass no. " << loop <<" though loop."<< endl; 
      if (kbmag3.gpwa()!=yes || kbmag3.gpgenmult()!=yes) abort=YES; 
      if (abort){ cout << "Failed, giving up!"<< endl; break;}
      GroupDFSA WA = kbmag3.wordAcceptor();
      WA.setName("Word_acceptor");
      WA.printOn();
      GenMult GM = kbmag3.generalMultiplier();
      GM.setName("GeneralMultiplier");
      GM.printOn();
      DiffMachine D1 = kbmag3.differenceMachine(1);
      D1.setName("1stDiffMachine");
      D1.printOn();
      DiffMachineRep D2 = kbmag3.differenceMachineRep(2);
      D2.setName("2ndDiffMachine");
      D2.printOn();
    } while (kbmag3.gpcheckmult()==no);
    if (abort==NO && kbmag3.gpaxioms()==yes)
      cout << "Group "<< G3 << " is proved shortlex automatic"<<endl;
    else 
      cout << "Group "<< G3 << " is not proved shortlex automatic"<<endl;
   }
   GroupDFSARep WA2 = kbmag3.wordAcceptorRep();
   DiffMachineRep D3 = kbmag3.differenceMachineRep(2);
   KBmagPackage kbmag4(G3.namesOfGenerators(),G3.getRelators());
   kbmag4.setWordAcceptor(WA2);
   kbmag4.setDifferenceMachine(D3,2);
   kbmag4.gpgenmult();
   DFSA F;
   F.readFrom();
   kbmag4.minimize(F);
   F.printOn();
   GroupDFSARep F2;
   F2.readFrom();
   kbmag4.minimize(F2);
   F2.printOn();
   GroupDFSA M1;
   M1.readFrom();
   GroupDFSA M2;
   M2.readFrom();
   GroupDFSA M3;
   kbmag4.gpcomp(M1,M2,M3);
   M3.printOn();
   GroupDFSARep N1;
   N1.readFrom();
   GroupDFSARep N2;
   N2.readFrom();
   GroupDFSARep N3;
   kbmag4.gpcomp(N1,N2,N3);
   N3.printOn();
}