extraBddSymm.c

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#include "extraBdd.h"
 
ABC_NAMESPACE_IMPL_START
 
 
/*---------------------------------------------------------------------------*/
/* Constant declarations                                                     */
/*---------------------------------------------------------------------------*/
 
/*---------------------------------------------------------------------------*/
/* Stucture declarations                                                     */
/*---------------------------------------------------------------------------*/
 
/*---------------------------------------------------------------------------*/
/* Type declarations                                                         */
/*---------------------------------------------------------------------------*/
 
/*---------------------------------------------------------------------------*/
/* Variable declarations                                                     */
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/*---------------------------------------------------------------------------*/
/* Macro declarations                                                        */
/*---------------------------------------------------------------------------*/
 
#define DD_GET_SYMM_VARS_TAG          0x0a /* former DD_BDD_XOR_EXIST_ABSTRACT_TAG */

 
/*---------------------------------------------------------------------------*/
/* Static function prototypes                                                */
/*---------------------------------------------------------------------------*/

 
/*---------------------------------------------------------------------------*/
/* Definition of exported functions                                          */
/*---------------------------------------------------------------------------*/

Extra_SymmInfo_t * Extra_SymmPairsCompute( 
  DdManager * dd,   /* the manager */
  DdNode * bFunc)   /* the function whose symmetries are computed */
{
    DdNode * bSupp;
    DdNode * zRes;
    Extra_SymmInfo_t * p;
 
    bSupp = Cudd_Support( dd, bFunc );                      Cudd_Ref( bSupp );
    zRes  = Extra_zddSymmPairsCompute( dd, bFunc, bSupp );  Cudd_Ref( zRes );
 
    p = Extra_SymmPairsCreateFromZdd( dd, zRes, bSupp );
 
    Cudd_RecursiveDeref( dd, bSupp );
    Cudd_RecursiveDerefZdd( dd, zRes );
 
    return p;
 
} /* end of Extra_SymmPairsCompute */
 

DdNode * Extra_zddSymmPairsCompute( 
  DdManager * dd,   /* the DD manager */
  DdNode * bF,
  DdNode * bVars) 
{
    DdNode * res;
    do {
        dd->reordered = 0;
        res = extraZddSymmPairsCompute( dd, bF, bVars );
    } while (dd->reordered == 1);
    return(res);
 
} /* end of Extra_zddSymmPairsCompute */

DdNode * Extra_zddGetSymmetricVars( 
  DdManager * dd,    /* the DD manager */
  DdNode * bF,       /* the first function  - originally, the positive cofactor */
  DdNode * bG,       /* the second fucntion - originally, the negative cofactor */
  DdNode * bVars)    /* the set of variables, on which F and G depend */
{
    DdNode * res;
    do {
        dd->reordered = 0;
        res = extraZddGetSymmetricVars( dd, bF, bG, bVars );
    } while (dd->reordered == 1);
    return(res);
 
} /* end of Extra_zddGetSymmetricVars */
 

DdNode * Extra_zddGetSingletons( 
  DdManager * dd,    /* the DD manager */
  DdNode * bVars)    /* the set of variables */
{
    DdNode * res;
    do {
        dd->reordered = 0;
        res = extraZddGetSingletons( dd, bVars );
    } while (dd->reordered == 1);
    return(res);
 
} /* end of Extra_zddGetSingletons */

DdNode * Extra_bddReduceVarSet( 
  DdManager * dd,    /* the DD manager */
  DdNode * bVars,    /* the set of variables to be reduced */
  DdNode * bF)       /* the function whose support is used for reduction */
{
    DdNode * res;
    do {
        dd->reordered = 0;
        res = extraBddReduceVarSet( dd, bVars, bF );
    } while (dd->reordered == 1);
    return(res);
 
} /* end of Extra_bddReduceVarSet */
 

Extra_SymmInfo_t * Extra_SymmPairsAllocate( int nVars )
{
    int i;
    Extra_SymmInfo_t * p;
 
    // allocate and clean the storage for symmetry info
    p = ABC_ALLOC( Extra_SymmInfo_t, 1 );
    memset( p, 0, sizeof(Extra_SymmInfo_t) );
    p->nVars     = nVars;
    p->pVars     = ABC_ALLOC( int, nVars );  
    p->pSymms    = ABC_ALLOC( char *, nVars );  
    p->pSymms[0] = ABC_ALLOC( char  , nVars * nVars );
    memset( p->pSymms[0], 0, nVars * nVars * sizeof(char) );
 
    for ( i = 1; i < nVars; i++ )
        p->pSymms[i] = p->pSymms[i-1] + nVars;
 
    return p;
} /* end of Extra_SymmPairsAllocate */

void Extra_SymmPairsDissolve( Extra_SymmInfo_t * p )
{
    ABC_FREE( p->pVars );
    ABC_FREE( p->pSymms[0] );
    ABC_FREE( p->pSymms    );
    ABC_FREE( p );
} /* end of Extra_SymmPairsDissolve */

void Extra_SymmPairsPrint( Extra_SymmInfo_t * p )
{
    int i, k;
    printf( "\n" );
    for ( i = 0; i < p->nVars; i++ )
    {
        for ( k = 0; k <= i; k++ )
            printf( " " );
        for ( k = i+1; k < p->nVars; k++ )
            if ( p->pSymms[i][k] )
                printf( "1" );
            else
                printf( "." );
        printf( "\n" );
    }
} /* end of Extra_SymmPairsPrint */
 

Extra_SymmInfo_t * Extra_SymmPairsCreateFromZdd( DdManager * dd, DdNode * zPairs, DdNode * bSupp )
{
    int i;
    int nSuppSize;
    Extra_SymmInfo_t * p;
    int * pMapVars2Nums;
    DdNode * bTemp;
    DdNode * zSet, * zCube, * zTemp;
    int iVar1, iVar2;
 
    nSuppSize = Extra_bddSuppSize( dd, bSupp );
 
    // allocate and clean the storage for symmetry info
    p = Extra_SymmPairsAllocate( nSuppSize );
 
    // allocate the storage for the temporary map
    pMapVars2Nums = ABC_ALLOC( int, dd->size );
    memset( pMapVars2Nums, 0, dd->size * sizeof(int) );
 
    // assign the variables
    p->nVarsMax = dd->size;
//  p->nNodes   = Cudd_DagSize( zPairs );
    p->nNodes   = 0;
    for ( i = 0, bTemp = bSupp; bTemp != b1; bTemp = cuddT(bTemp), i++ )
    {
        p->pVars[i] = bTemp->index;
        pMapVars2Nums[bTemp->index] = i;
    }
 
    // write the symmetry info into the structure
    zSet = zPairs;   Cudd_Ref( zSet );
    while ( zSet != z0 )
    {
        // get the next cube
        zCube  = Extra_zddSelectOneSubset( dd, zSet );    Cudd_Ref( zCube );
 
        // add these two variables to the data structure
        assert( cuddT( cuddT(zCube) ) == z1 );
        iVar1 = zCube->index/2;
        iVar2 = cuddT(zCube)->index/2;
        if ( pMapVars2Nums[iVar1] < pMapVars2Nums[iVar2] )
            p->pSymms[ pMapVars2Nums[iVar1] ][ pMapVars2Nums[iVar2] ] = 1;
        else
            p->pSymms[ pMapVars2Nums[iVar2] ][ pMapVars2Nums[iVar1] ] = 1;
        // count the symmetric pairs
        p->nSymms ++;
 
        // update the cuver and deref the cube
        zSet = Cudd_zddDiff( dd, zTemp = zSet, zCube );     Cudd_Ref( zSet );
        Cudd_RecursiveDerefZdd( dd, zTemp );
        Cudd_RecursiveDerefZdd( dd, zCube );
 
    } // for each cube 
    Cudd_RecursiveDerefZdd( dd, zSet );
 
    ABC_FREE( pMapVars2Nums );
    return p;
 
} /* end of Extra_SymmPairsCreateFromZdd */
 

int Extra_bddCheckVarsSymmetric( 
  DdManager * dd,   /* the DD manager */
  DdNode * bF,
  int iVar1,
  int iVar2) 
{
    DdNode * bVars;
    int Res;
 
//  return 1;
 
    assert( iVar1 != iVar2 );
    assert( iVar1 < dd->size );
    assert( iVar2 < dd->size );
 
    bVars = Cudd_bddAnd( dd, dd->vars[iVar1], dd->vars[iVar2] );   Cudd_Ref( bVars );
 
    Res = (int)( extraBddCheckVarsSymmetric( dd, bF, bVars ) == b1 );
 
    Cudd_RecursiveDeref( dd, bVars );
 
    return Res;
} /* end of Extra_bddCheckVarsSymmetric */
 

Extra_SymmInfo_t * Extra_SymmPairsComputeNaive( DdManager * dd, DdNode * bFunc )
{
    DdNode * bSupp, * bTemp;
    int nSuppSize;
    Extra_SymmInfo_t * p;
    int i, k;
 
    // compute the support
    bSupp = Cudd_Support( dd, bFunc );   Cudd_Ref( bSupp );
    nSuppSize = Extra_bddSuppSize( dd, bSupp );
//printf( "Support = %d. ", nSuppSize );
//Extra_bddPrint( dd, bSupp );
//printf( "%d ", nSuppSize );
 
    // allocate the storage for symmetry info
    p = Extra_SymmPairsAllocate( nSuppSize );
 
    // assign the variables
    p->nVarsMax = dd->size;
    for ( i = 0, bTemp = bSupp; bTemp != b1; bTemp = cuddT(bTemp), i++ )
        p->pVars[i] = bTemp->index;
 
    // go through the candidate pairs and check using Idea1
    for ( i = 0; i < nSuppSize; i++ )
    for ( k = i+1; k < nSuppSize; k++ )
    {
        p->pSymms[k][i] = p->pSymms[i][k] = Extra_bddCheckVarsSymmetricNaive( dd, bFunc, p->pVars[i], p->pVars[k] );
        if ( p->pSymms[i][k] )
             p->nSymms++;
    }
 
    Cudd_RecursiveDeref( dd, bSupp );
    return p;
 
} /* end of Extra_SymmPairsComputeNaive */

int Extra_bddCheckVarsSymmetricNaive( 
  DdManager * dd,   /* the DD manager */
  DdNode * bF,
  int iVar1,
  int iVar2) 
{
    DdNode * bCube1, * bCube2;
    DdNode * bCof01, * bCof10;
    int Res;
 
    assert( iVar1 != iVar2 );
    assert( iVar1 < dd->size );
    assert( iVar2 < dd->size );
 
    bCube1 = Cudd_bddAnd( dd, Cudd_Not( dd->vars[iVar1] ), dd->vars[iVar2] );   Cudd_Ref( bCube1 );
    bCube2 = Cudd_bddAnd( dd, Cudd_Not( dd->vars[iVar2] ), dd->vars[iVar1] );   Cudd_Ref( bCube2 );
 
    bCof01 = Cudd_Cofactor( dd, bF, bCube1 );  Cudd_Ref( bCof01 );
    bCof10 = Cudd_Cofactor( dd, bF, bCube2 );  Cudd_Ref( bCof10 );
 
    Res = (int)( bCof10 == bCof01 );
 
    Cudd_RecursiveDeref( dd, bCof01 );
    Cudd_RecursiveDeref( dd, bCof10 );
    Cudd_RecursiveDeref( dd, bCube1 );
    Cudd_RecursiveDeref( dd, bCube2 );
 
    return Res;
} /* end of Extra_bddCheckVarsSymmetricNaive */
 

DdNode* Extra_zddTuplesFromBdd( 
  DdManager * dd,   /* the DD manager */
  int K,            /* the number of variables in tuples */
  DdNode * bVarsN)   /* the set of all variables represented as a BDD */
{
    DdNode  *zRes;
    int     autoDynZ;
 
    autoDynZ = dd->autoDynZ;
    dd->autoDynZ = 0;
 
    do {
        /* transform the numeric arguments (K) into a DdNode* argument;
         * this allows us to use the standard internal CUDD cache */
        DdNode *bVarSet = bVarsN, *bVarsK = bVarsN;
        int     nVars = 0, i;
 
        /* determine the number of variables in VarSet */
        while ( bVarSet != b1 )
        {
            nVars++;
            /* make sure that the VarSet is a cube */
            if ( cuddE( bVarSet ) != b0 )
                return NULL;
            bVarSet = cuddT( bVarSet );
        }
        /* make sure that the number of variables in VarSet is less or equal 
           that the number of variables that should be present in the tuples
        */
        if ( K > nVars )
            return NULL;
 
        /* the second argument in the recursive call stannds for <n>;
         * reate the first argument, which stands for <k>
         * as when we are talking about the tuple of <k> out of <n> */
        for ( i = 0; i < nVars-K; i++ )
            bVarsK = cuddT( bVarsK );
 
        dd->reordered = 0;
        zRes = extraZddTuplesFromBdd(dd, bVarsK, bVarsN );
 
    } while (dd->reordered == 1);
    dd->autoDynZ = autoDynZ;
    return zRes;
 
} /* end of Extra_zddTuplesFromBdd */

DdNode* Extra_zddSelectOneSubset( 
  DdManager * dd,   /* the DD manager */
  DdNode * zS)      /* the ZDD */
{
    DdNode  *res;
    do {
        dd->reordered = 0;
        res = extraZddSelectOneSubset(dd, zS);
    } while (dd->reordered == 1);
    return(res);
 
} /* end of Extra_zddSelectOneSubset */
 
 
/*---------------------------------------------------------------------------*/
/* Definition of internal functions                                          */
/*---------------------------------------------------------------------------*/

DdNode * 
extraZddSymmPairsCompute( 
  DdManager * dd,   /* the manager */
  DdNode * bFunc,   /* the function whose symmetries are computed */
  DdNode * bVars )  /* the set of variables on which this function depends */
{
    DdNode * zRes;
    DdNode * bFR = Cudd_Regular(bFunc); 
 
    if ( cuddIsConstant(bFR) )
    {
        int nVars, i;
 
        // determine how many vars are in the bVars
        nVars = Extra_bddSuppSize( dd, bVars );
        if ( nVars < 2 )
            return z0;
        else
        {
            DdNode * bVarsK;
 
            // create the BDD bVarsK corresponding to K = 2;
            bVarsK = bVars;
            for ( i = 0; i < nVars-2; i++ )
                bVarsK = cuddT( bVarsK );
            return extraZddTuplesFromBdd( dd, bVarsK, bVars );
        }
    }
    assert( bVars != b1 );
 
    if ( (zRes = cuddCacheLookup2Zdd(dd, extraZddSymmPairsCompute, bFunc, bVars)) )
        return zRes;
    else
    {
        DdNode * zRes0, * zRes1;
        DdNode * zTemp, * zPlus, * zSymmVars;             
        DdNode * bF0, * bF1;             
        DdNode * bVarsNew;
        int nVarsExtra;
        int LevelF;
 
        // every variable in bF should be also in bVars, therefore LevelF cannot be above LevelV
        // if LevelF is below LevelV, scroll through the vars in bVars to the same level as F
        // count how many extra vars are there in bVars
        nVarsExtra = 0;
        LevelF = dd->perm[bFR->index];
        for ( bVarsNew = bVars; LevelF > dd->perm[bVarsNew->index]; bVarsNew = cuddT(bVarsNew) )
            nVarsExtra++; 
        // the indexes (level) of variables should be synchronized now
        assert( bFR->index == bVarsNew->index );
 
        // cofactor the function
        if ( bFR != bFunc ) // bFunc is complemented 
        {
            bF0 = Cudd_Not( cuddE(bFR) );
            bF1 = Cudd_Not( cuddT(bFR) );
        }
        else
        {
            bF0 = cuddE(bFR);
            bF1 = cuddT(bFR);
        }
 
        // solve subproblems
        zRes0 = extraZddSymmPairsCompute( dd, bF0, cuddT(bVarsNew) );
        if ( zRes0 == NULL )
            return NULL;
        cuddRef( zRes0 );
 
        // if there is no symmetries in the negative cofactor
        // there is no need to test the positive cofactor
        if ( zRes0 == z0 )
            zRes = zRes0;  // zRes takes reference
        else
        {
            zRes1 = extraZddSymmPairsCompute( dd, bF1, cuddT(bVarsNew) );
            if ( zRes1 == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zRes0 );
                return NULL;
            }
            cuddRef( zRes1 );
 
            // only those variables are pair-wise symmetric 
            // that are pair-wise symmetric in both cofactors
            // therefore, intersect the solutions
            zRes = cuddZddIntersect( dd, zRes0, zRes1 );
            if ( zRes == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zRes0 );
                Cudd_RecursiveDerefZdd( dd, zRes1 );
                return NULL;
            }
            cuddRef( zRes );
            Cudd_RecursiveDerefZdd( dd, zRes0 );
            Cudd_RecursiveDerefZdd( dd, zRes1 );
        }
 
        // consider the current top-most variable and find all the vars
        // that are pairwise symmetric with it
        // these variables are returned as a set of ZDD singletons
        zSymmVars = extraZddGetSymmetricVars( dd, bF1, bF0, cuddT(bVarsNew) );
        if ( zSymmVars == NULL )
        {
            Cudd_RecursiveDerefZdd( dd, zRes );
            return NULL;
        }
        cuddRef( zSymmVars );
 
        // attach the topmost variable to the set, to get the variable pairs
        // use the positive polarity ZDD variable for the purpose
 
        // there is no need to do so, if zSymmVars is empty
        if ( zSymmVars == z0 )
            Cudd_RecursiveDerefZdd( dd, zSymmVars );
        else
        {
            zPlus = cuddZddGetNode( dd, 2*bFR->index, zSymmVars, z0 );
            if ( zPlus == NULL ) 
            {
                Cudd_RecursiveDerefZdd( dd, zRes );
                Cudd_RecursiveDerefZdd( dd, zSymmVars );
                return NULL;
            }
            cuddRef( zPlus );
            cuddDeref( zSymmVars );
 
            // add these variable pairs to the result
            zRes = cuddZddUnion( dd, zTemp = zRes, zPlus );
            if ( zRes == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zTemp );
                Cudd_RecursiveDerefZdd( dd, zPlus );
                return NULL;
            }
            cuddRef( zRes );
            Cudd_RecursiveDerefZdd( dd, zTemp );
            Cudd_RecursiveDerefZdd( dd, zPlus );
        }
 
        // only zRes is referenced at this point
 
        // if we skipped some variables, these variables cannot be symmetric with
        // any variables that are currently in the support of bF, but they can be 
        // symmetric with the variables that are in bVars but not in the support of bF
        if ( nVarsExtra )
        {
            // it is possible to improve this step:
            // (1) there is no need to enter here, if nVarsExtra < 2
 
            // create the set of topmost nVarsExtra in bVars
            DdNode * bVarsExtra;
            int nVars;
 
            // remove from bVars all the variable that are in the support of bFunc
            bVarsExtra = extraBddReduceVarSet( dd, bVars, bFunc );  
            if ( bVarsExtra == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zRes );
                return NULL;
            }
            cuddRef( bVarsExtra );
 
            // determine how many vars are in the bVarsExtra
            nVars = Extra_bddSuppSize( dd, bVarsExtra );
            if ( nVars < 2 )
            {
                Cudd_RecursiveDeref( dd, bVarsExtra );
            }
            else
            {
                int i;
                DdNode * bVarsK;
 
                // create the BDD bVarsK corresponding to K = 2;
                bVarsK = bVarsExtra;
                for ( i = 0; i < nVars-2; i++ )
                    bVarsK = cuddT( bVarsK );
 
                // create the 2 variable tuples
                zPlus = extraZddTuplesFromBdd( dd, bVarsK, bVarsExtra );
                if ( zPlus == NULL )
                {
                    Cudd_RecursiveDeref( dd, bVarsExtra );
                    Cudd_RecursiveDerefZdd( dd, zRes );
                    return NULL;
                }
                cuddRef( zPlus );
                Cudd_RecursiveDeref( dd, bVarsExtra );
 
                // add these to the result
                zRes = cuddZddUnion( dd, zTemp = zRes, zPlus );
                if ( zRes == NULL )
                {
                    Cudd_RecursiveDerefZdd( dd, zTemp );
                    Cudd_RecursiveDerefZdd( dd, zPlus );
                    return NULL;
                }
                cuddRef( zRes );
                Cudd_RecursiveDerefZdd( dd, zTemp );
                Cudd_RecursiveDerefZdd( dd, zPlus );
            }
        }
        cuddDeref( zRes );
 
 
        /* insert the result into cache */
        cuddCacheInsert2(dd, extraZddSymmPairsCompute, bFunc, bVars, zRes);
        return zRes;
    }
} /* end of extraZddSymmPairsCompute */

DdNode * extraZddGetSymmetricVars( 
  DdManager * dd,    /* the DD manager */
  DdNode * bF,       /* the first function  - originally, the positive cofactor */
  DdNode * bG,       /* the second function - originally, the negative cofactor */
  DdNode * bVars)    /* the set of variables, on which F and G depend */
{
    DdNode * zRes;
    DdNode * bFR = Cudd_Regular(bF); 
    DdNode * bGR = Cudd_Regular(bG); 
 
    if ( cuddIsConstant(bFR) && cuddIsConstant(bGR) )
    {
        if ( bF == bG )
            return extraZddGetSingletons( dd, bVars );
        else 
            return z0;
    }
    assert( bVars != b1 );
 
    if ( (zRes = cuddCacheLookupZdd(dd, DD_GET_SYMM_VARS_TAG, bF, bG, bVars)) )
        return zRes;
    else
    {
        DdNode * zRes0, * zRes1;             
        DdNode * zPlus, * zTemp;
        DdNode * bF0, * bF1;             
        DdNode * bG0, * bG1;             
        DdNode * bVarsNew;
    
        int LevelF = cuddI(dd,bFR->index);
        int LevelG = cuddI(dd,bGR->index);
        int LevelFG;
 
        if ( LevelF < LevelG )
            LevelFG = LevelF;
        else
            LevelFG = LevelG;
 
        // at least one of the arguments is not a constant
        assert( LevelFG < dd->size );
 
        // every variable in bF and bG should be also in bVars, therefore LevelFG cannot be above LevelV
        // if LevelFG is below LevelV, scroll through the vars in bVars to the same level as LevelFG
        for ( bVarsNew = bVars; LevelFG > dd->perm[bVarsNew->index]; bVarsNew = cuddT(bVarsNew) );
        assert( LevelFG == dd->perm[bVarsNew->index] );
 
        // cofactor the functions
        if ( LevelF == LevelFG )
        {
            if ( bFR != bF ) // bF is complemented 
            {
                bF0 = Cudd_Not( cuddE(bFR) );
                bF1 = Cudd_Not( cuddT(bFR) );
            }
            else
            {
                bF0 = cuddE(bFR);
                bF1 = cuddT(bFR);
            }
        }
        else 
            bF0 = bF1 = bF;
 
        if ( LevelG == LevelFG )
        {
            if ( bGR != bG ) // bG is complemented 
            {
                bG0 = Cudd_Not( cuddE(bGR) );
                bG1 = Cudd_Not( cuddT(bGR) );
            }
            else
            {
                bG0 = cuddE(bGR);
                bG1 = cuddT(bGR);
            }
        }
        else 
            bG0 = bG1 = bG;
 
        // solve subproblems
        zRes0 = extraZddGetSymmetricVars( dd, bF0, bG0, cuddT(bVarsNew) );
        if ( zRes0 == NULL ) 
            return NULL;
        cuddRef( zRes0 );
 
        // if there is not symmetries in the negative cofactor
        // there is no need to test the positive cofactor
        if ( zRes0 == z0 )
            zRes = zRes0;  // zRes takes reference
        else
        {
            zRes1 = extraZddGetSymmetricVars( dd, bF1, bG1, cuddT(bVarsNew) );
            if ( zRes1 == NULL ) 
            {
                Cudd_RecursiveDerefZdd( dd, zRes0 );
                return NULL;
            }
            cuddRef( zRes1 );
 
            // only those variables should belong to the resulting set 
            // for which the property is true for both cofactors
            zRes = cuddZddIntersect( dd, zRes0, zRes1 );
            if ( zRes == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zRes0 );
                Cudd_RecursiveDerefZdd( dd, zRes1 );
                return NULL;
            }
            cuddRef( zRes );
            Cudd_RecursiveDerefZdd( dd, zRes0 );
            Cudd_RecursiveDerefZdd( dd, zRes1 );
        }
 
        // add one more singleton if the property is true for this variable
        if ( bF0 == bG1 )
        {
            zPlus = cuddZddGetNode( dd, 2*bVarsNew->index, z1, z0 );
            if ( zPlus == NULL ) 
            {
                Cudd_RecursiveDerefZdd( dd, zRes );
                return NULL;
            }
            cuddRef( zPlus );
 
            // add these variable pairs to the result
            zRes = cuddZddUnion( dd, zTemp = zRes, zPlus );
            if ( zRes == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zTemp );
                Cudd_RecursiveDerefZdd( dd, zPlus );
                return NULL;
            }
            cuddRef( zRes );
            Cudd_RecursiveDerefZdd( dd, zTemp );
            Cudd_RecursiveDerefZdd( dd, zPlus );
        }
 
        if ( bF == bG && bVars != bVarsNew )
        { 
            // if the functions are equal, so are their cofactors
            // add those variables from V that are above F and G 
 
            DdNode * bVarsExtra;
 
            assert( LevelFG > dd->perm[bVars->index] );
 
            // create the BDD of the extra variables
            bVarsExtra = cuddBddExistAbstractRecur( dd, bVars, bVarsNew );
            if ( bVarsExtra == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zRes );
                return NULL;
            }
            cuddRef( bVarsExtra );
 
            zPlus = extraZddGetSingletons( dd, bVarsExtra );
            if ( zPlus == NULL )
            {
                Cudd_RecursiveDeref( dd, bVarsExtra );
                Cudd_RecursiveDerefZdd( dd, zRes );
                return NULL;
            }
            cuddRef( zPlus );
            Cudd_RecursiveDeref( dd, bVarsExtra );
 
            // add these to the result
            zRes = cuddZddUnion( dd, zTemp = zRes, zPlus );
            if ( zRes == NULL )
            {
                Cudd_RecursiveDerefZdd( dd, zTemp );
                Cudd_RecursiveDerefZdd( dd, zPlus );
                return NULL;
            }
            cuddRef( zRes );
            Cudd_RecursiveDerefZdd( dd, zTemp );
            Cudd_RecursiveDerefZdd( dd, zPlus );
        }
        cuddDeref( zRes );
 
        cuddCacheInsert( dd, DD_GET_SYMM_VARS_TAG, bF, bG, bVars, zRes );
        return zRes;
    }
}   /* end of extraZddGetSymmetricVars */
 

DdNode * extraZddGetSingletons( 
  DdManager * dd,    /* the DD manager */
  DdNode * bVars)    /* the set of variables */
{
    DdNode * zRes;
 
    if ( bVars == b1 )
//    if ( bVars == b0 )  // bug fixed by Jin Zhang, Jan 23, 2004
        return z1;
 
    if ( (zRes = cuddCacheLookup1Zdd(dd, extraZddGetSingletons, bVars)) )
        return zRes;
    else
    {
        DdNode * zTemp, * zPlus;          
 
        // solve subproblem
        zRes = extraZddGetSingletons( dd, cuddT(bVars) );
        if ( zRes == NULL ) 
            return NULL;
        cuddRef( zRes );
        
        zPlus = cuddZddGetNode( dd, 2*bVars->index, z1, z0 );
        if ( zPlus == NULL ) 
        {
            Cudd_RecursiveDerefZdd( dd, zRes );
            return NULL;
        }
        cuddRef( zPlus );
 
        // add these to the result
        zRes = cuddZddUnion( dd, zTemp = zRes, zPlus );
        if ( zRes == NULL )
        {
            Cudd_RecursiveDerefZdd( dd, zTemp );
            Cudd_RecursiveDerefZdd( dd, zPlus );
            return NULL;
        }
        cuddRef( zRes );
        Cudd_RecursiveDerefZdd( dd, zTemp );
        Cudd_RecursiveDerefZdd( dd, zPlus );
        cuddDeref( zRes );
 
        cuddCacheInsert1( dd, extraZddGetSingletons, bVars, zRes );
        return zRes;
    }
}   /* end of extraZddGetSingletons */
 

DdNode * extraBddReduceVarSet( 
  DdManager * dd,    /* the DD manager */
  DdNode * bVars,    /* the set of variables to be reduced */
  DdNode * bF)       /* the function whose support is used for reduction */
{
    DdNode * bRes;
    DdNode * bFR = Cudd_Regular(bF);
 
    if ( cuddIsConstant(bFR) || bVars == b1 )
        return bVars;
 
    if ( (bRes = cuddCacheLookup2(dd, extraBddReduceVarSet, bVars, bF)) )
        return bRes;
    else
    {
        DdNode * bF0, * bF1;             
        DdNode * bVarsThis, * bVarsLower, * bTemp;
        int LevelF;
 
        // if LevelF is below LevelV, scroll through the vars in bVars 
        LevelF = dd->perm[bFR->index];
        for ( bVarsThis = bVars; LevelF > cuddI(dd,bVarsThis->index); bVarsThis = cuddT(bVarsThis) );
        // scroll also through the current var, because it should be not be added
        if ( LevelF == cuddI(dd,bVarsThis->index) )
            bVarsLower = cuddT(bVarsThis);
        else
            bVarsLower = bVarsThis;
 
        // cofactor the function
        if ( bFR != bF ) // bFunc is complemented 
        {
            bF0 = Cudd_Not( cuddE(bFR) );
            bF1 = Cudd_Not( cuddT(bFR) );
        }
        else
        {
            bF0 = cuddE(bFR);
            bF1 = cuddT(bFR);
        }
 
        // solve subproblems
        bRes = extraBddReduceVarSet( dd, bVarsLower, bF0 );
        if ( bRes == NULL ) 
            return NULL;
        cuddRef( bRes );
 
        bRes = extraBddReduceVarSet( dd, bTemp = bRes, bF1 );
        if ( bRes == NULL ) 
        {
            Cudd_RecursiveDeref( dd, bTemp );
            return NULL;
        }
        cuddRef( bRes );
        Cudd_RecursiveDeref( dd, bTemp );
 
        // the current var should not be added
        // add the skipped vars
        if ( bVarsThis != bVars )
        {
            DdNode * bVarsExtra;
 
            // extract the skipped variables
            bVarsExtra = cuddBddExistAbstractRecur( dd, bVars, bVarsThis );
            if ( bVarsExtra == NULL )
            {
                Cudd_RecursiveDeref( dd, bRes );
                return NULL;
            }
            cuddRef( bVarsExtra );
 
            // add these variables
            bRes = cuddBddAndRecur( dd, bTemp = bRes, bVarsExtra );
            if ( bRes == NULL ) 
            {
                Cudd_RecursiveDeref( dd, bTemp );
                Cudd_RecursiveDeref( dd, bVarsExtra );
                return NULL;
            }
            cuddRef( bRes );
            Cudd_RecursiveDeref( dd, bTemp );
            Cudd_RecursiveDeref( dd, bVarsExtra );
        }
        cuddDeref( bRes );
 
        cuddCacheInsert2( dd, extraBddReduceVarSet, bVars, bF, bRes );
        return bRes;
    }
}   /* end of extraBddReduceVarSet */
 

DdNode * extraBddCheckVarsSymmetric( 
  DdManager * dd,    /* the DD manager */
  DdNode * bF,
  DdNode * bVars) 
{
    DdNode * bRes;
 
    if ( bF == b0 )
        return b1;
 
    assert( bVars != b1 );
    
    if ( (bRes = cuddCacheLookup2(dd, extraBddCheckVarsSymmetric, bF, bVars)) )
        return bRes;
    else
    {
        DdNode * bRes0, * bRes1;
        DdNode * bF0, * bF1;             
        DdNode * bFR = Cudd_Regular(bF);
        int LevelF = cuddI(dd,bFR->index);
 
        DdNode * bVarsR = Cudd_Regular(bVars);
        int fVar1Pres;
        int iLev1;
        int iLev2;
 
        if ( bVarsR != bVars ) // cube's pointer is complemented
        {
            assert( cuddT(bVarsR) == b1 );
            fVar1Pres = 1;                    // the first var is present on the path
            iLev1 = -1;                       // we are already below the first var level
            iLev2 = dd->perm[bVarsR->index];  // the level of the second var
        }
        else  // cube's pointer is NOT complemented
        {
            fVar1Pres = 0;                    // the first var is absent on the path
            if ( cuddT(bVars) == b1 )
            {
                iLev1 = -1;                             // we are already below the first var level 
                iLev2 = dd->perm[bVars->index];         // the level of the second var
            }
            else
            {
                assert( cuddT(cuddT(bVars)) == b1 );
                iLev1 = dd->perm[bVars->index];         // the level of the first var 
                iLev2 = dd->perm[cuddT(bVars)->index];  // the level of the second var
            }
        }
 
        // cofactor the function
        // the cofactors are needed only if we are above the second level
        if ( LevelF < iLev2 )
        {
            if ( bFR != bF ) // bFunc is complemented 
            {
                bF0 = Cudd_Not( cuddE(bFR) );
                bF1 = Cudd_Not( cuddT(bFR) );
            }
            else
            {
                bF0 = cuddE(bFR);
                bF1 = cuddT(bFR);
            }
        }
        else
            bF0 = bF1 = NULL;
 
        // consider five cases:
        // (1) F is above iLev1
        // (2) F is on the level iLev1
        // (3) F is between iLev1 and iLev2
        // (4) F is on the level iLev2
        // (5) F is below iLev2
 
        // (1) F is above iLev1
        if ( LevelF < iLev1 )
        {
            // the returned result cannot have the hash attribute
            // because we still did not reach the level of Var1;
            // the attribute never travels above the level of Var1
            bRes0 = extraBddCheckVarsSymmetric( dd, bF0, bVars );
//          assert( !Hash_IsComplement( bRes0 ) );
            assert( bRes0 != z0 );
            if ( bRes0 == b0 )
                bRes = b0;
            else
                bRes = extraBddCheckVarsSymmetric( dd, bF1, bVars );
//          assert( !Hash_IsComplement( bRes ) );
            assert( bRes != z0 );
        }
        // (2) F is on the level iLev1
        else if ( LevelF == iLev1 )
        {
            bRes0 = extraBddCheckVarsSymmetric( dd, bF0, Cudd_Not( cuddT(bVars) ) );
            if ( bRes0 == b0 )
                bRes = b0;
            else
            {
                bRes1 = extraBddCheckVarsSymmetric( dd, bF1, Cudd_Not( cuddT(bVars) ) );
                if ( bRes1 == b0 )
                    bRes = b0;
                else
                {
//                  if ( Hash_IsComplement( bRes0 ) || Hash_IsComplement( bRes1 ) )
                    if ( bRes0 == z0 || bRes1 == z0 )
                        bRes = b1;
                    else
                        bRes = b0;
                }
            }
        }
        // (3) F is between iLev1 and iLev2
        else if ( LevelF < iLev2 )
        {
            bRes0 = extraBddCheckVarsSymmetric( dd, bF0, bVars );
            if ( bRes0 == b0 )
                bRes = b0;
            else
            {
                bRes1 = extraBddCheckVarsSymmetric( dd, bF1, bVars );
                if ( bRes1 == b0 )
                    bRes = b0;
                else
                {
//                  if ( Hash_IsComplement( bRes0 ) || Hash_IsComplement( bRes1 ) )
//                      bRes = Hash_Not( b1 );
                    if ( bRes0 == z0 || bRes1 == z0 )
                        bRes = z0;
                    else
                        bRes = b1;
                }
            }
        }
        // (4) F is on the level iLev2
        else if ( LevelF == iLev2 )
        {
            // this is the only place where the hash attribute (Hash_Not) can be added 
            // to the result; it can be added only if the path came through the node
            // lebeled with Var1; therefore, the hash attribute cannot be returned
            // to the caller function
            if ( fVar1Pres )
//              bRes = Hash_Not( b1 );
                bRes = z0;
            else 
                bRes = b0;
        }
        // (5) F is below iLev2
        else // if ( LevelF > iLev2 )
        {
            // it is possible that the path goes through the node labeled by Var1
            // and still everything is okay; we do not label with Hash_Not here
            // because the path does not go through node labeled by Var2
            bRes = b1;
        }
 
        cuddCacheInsert2(dd, extraBddCheckVarsSymmetric, bF, bVars, bRes);
        return bRes;
    }
} /* end of extraBddCheckVarsSymmetric */

DdNode* extraZddTuplesFromBdd( 
  DdManager * dd,   /* the DD manager */
  DdNode * bVarsK,   /* the number of variables in tuples */
  DdNode * bVarsN)   /* the set of all variables */
{
    DdNode *zRes, *zRes0, *zRes1;
    statLine(dd); 
 
    /* terminal cases */
/*  if ( k < 0 || k > n )
 *      return dd->zero;
 *  if ( n == 0 )
 *      return dd->one; 
 */
    if ( cuddI( dd, bVarsK->index ) < cuddI( dd, bVarsN->index ) )
        return z0;
    if ( bVarsN == b1 )
        return z1;
 
    /* check cache */
    zRes = cuddCacheLookup2Zdd(dd, extraZddTuplesFromBdd, bVarsK, bVarsN);
    if (zRes)
        return(zRes);
 
    /* ZDD in which this variable is 0 */
/*  zRes0 = extraZddTuplesFromBdd( dd, k,     n-1 ); */
    zRes0 = extraZddTuplesFromBdd( dd, bVarsK, cuddT(bVarsN) );
    if ( zRes0 == NULL ) 
        return NULL;
    cuddRef( zRes0 );
 
    /* ZDD in which this variable is 1 */
/*  zRes1 = extraZddTuplesFromBdd( dd, k-1,          n-1 ); */
    if ( bVarsK == b1 )
    {
        zRes1 = z0;
        cuddRef( zRes1 );
    }
    else
    {
        zRes1 = extraZddTuplesFromBdd( dd, cuddT(bVarsK), cuddT(bVarsN) );
        if ( zRes1 == NULL ) 
        {
            Cudd_RecursiveDerefZdd( dd, zRes0 );
            return NULL;
        }
        cuddRef( zRes1 );
    }
 
    /* compose Res0 and Res1 with the given ZDD variable */
    zRes = cuddZddGetNode( dd, 2*bVarsN->index, zRes1, zRes0 );
    if ( zRes == NULL ) 
    {
        Cudd_RecursiveDerefZdd( dd, zRes0 );
        Cudd_RecursiveDerefZdd( dd, zRes1 );
        return NULL;
    }
    cuddDeref( zRes0 );
    cuddDeref( zRes1 );
 
    /* insert the result into cache */
    cuddCacheInsert2(dd, extraZddTuplesFromBdd, bVarsK, bVarsN, zRes);
    return zRes;
 
} /* end of extraZddTuplesFromBdd */
 

DdNode * extraZddSelectOneSubset( 
  DdManager * dd, 
  DdNode * zS )
// selects one subset from the ZDD zS
// returns z0 if and only if zS is an empty set of cubes
{
    DdNode * zRes;
 
    if ( zS == z0 )    return z0;
    if ( zS == z1 )    return z1;
    
    // check cache
    if ( (zRes = cuddCacheLookup1Zdd( dd, extraZddSelectOneSubset, zS )) )
        return zRes;
    else
    {
        DdNode * zS0, * zS1, * zTemp; 
 
        zS0 = cuddE(zS);
        zS1 = cuddT(zS);
 
        if ( zS0 != z0 )
        {
            zRes = extraZddSelectOneSubset( dd, zS0 );
            if ( zRes == NULL )
                return NULL;
        }
        else // if ( zS0 == z0 )
        {
            assert( zS1 != z0 );
            zRes = extraZddSelectOneSubset( dd, zS1 );
            if ( zRes == NULL )
                return NULL;
            cuddRef( zRes );
 
            zRes = cuddZddGetNode( dd, zS->index, zTemp = zRes, z0 );
            if ( zRes == NULL ) 
            {
                Cudd_RecursiveDerefZdd( dd, zTemp );
                return NULL;
            }
            cuddDeref( zTemp );
        }
 
        // insert the result into cache
        cuddCacheInsert1( dd, extraZddSelectOneSubset, zS, zRes );
        return zRes;
    }       
} /* end of extraZddSelectOneSubset */
 
 
/*---------------------------------------------------------------------------*/
/* Definition of static Functions                                            */
/*---------------------------------------------------------------------------*/
ABC_NAMESPACE_IMPL_END