Version abc50831

5 files changed, 1525 insertions, 3 deletions

diff --git a/src/misc/extra/extra.h b/src/misc/extra/extra.h
index d298a204..14a3d950 100644
--- a/src/misc/extra/extra.h
+++ b/src/misc/extra/extra.h
@@ -117,6 +117,57 @@ extern DdNode *     Extra_bddFindOneCube( DdManager * dd, DdNode * bF );
 extern DdNode *     Extra_bddGetOneCube( DdManager * dd, DdNode * bFunc );
 extern DdNode *     Extra_bddComputeRangeCube( DdManager * dd, int iStart, int iStop );
 
+/*=== extraBddSymm.c =================================================================*/
+
+typedef struct Extra_SymmInfo_t_  Extra_SymmInfo_t;
+struct Extra_SymmInfo_t_ {
+    int nVars;      // the number of variables in the support
+    int nVarsMax;   // the number of variables in the DD manager
+    int nSymms;     // the number of pair-wise symmetries
+    int nNodes;     // the number of nodes in a ZDD (if applicable)
+    int * pVars;    // the list of all variables present in the support
+    char ** pSymms; // the symmetry information
+};
+
+/* computes the classical symmetry information for the function - recursive */
+extern Extra_SymmInfo_t *  Extra_SymmPairsCompute( DdManager * dd, DdNode * bFunc );
+/* computes the classical symmetry information for the function - using naive approach */
+extern Extra_SymmInfo_t *  Extra_SymmPairsComputeNaive( DdManager * dd, DdNode * bFunc );
+extern int         Extra_bddCheckVarsSymmetricNaive( DdManager * dd, DdNode * bF, int iVar1, int iVar2 );
+
+/* allocates the data structure */
+extern Extra_SymmInfo_t *  Extra_SymmPairsAllocate( int nVars );
+/* deallocates the data structure */
+extern void        Extra_SymmPairsDissolve( Extra_SymmInfo_t * );
+/* print the contents the data structure */
+extern void        Extra_SymmPairsPrint( Extra_SymmInfo_t * );
+/* converts the ZDD into the Extra_SymmInfo_t structure */
+extern Extra_SymmInfo_t *  Extra_SymmPairsCreateFromZdd( DdManager * dd, DdNode * zPairs, DdNode * bVars );
+
+/* computes the classical symmetry information as a ZDD */
+extern DdNode *    Extra_zddSymmPairsCompute( DdManager * dd, DdNode * bF, DdNode * bVars );
+extern DdNode *     extraZddSymmPairsCompute( DdManager * dd, DdNode * bF, DdNode * bVars );
+/* returns a singleton-set ZDD containing all variables that are symmetric with the given one */
+extern DdNode *    Extra_zddGetSymmetricVars( DdManager * dd, DdNode * bF, DdNode * bG, DdNode * bVars );
+extern DdNode *     extraZddGetSymmetricVars( DdManager * dd, DdNode * bF, DdNode * bG, DdNode * bVars );
+/* converts a set of variables into a set of singleton subsets */
+extern DdNode *    Extra_zddGetSingletons( DdManager * dd, DdNode * bVars );
+extern DdNode *     extraZddGetSingletons( DdManager * dd, DdNode * bVars );
+/* filters the set of variables using the support of the function */
+extern DdNode *    Extra_bddReduceVarSet( DdManager * dd, DdNode * bVars, DdNode * bF );
+extern DdNode *     extraBddReduceVarSet( DdManager * dd, DdNode * bVars, DdNode * bF );
+
+/* checks the possibility that the two vars are symmetric */
+extern int         Extra_bddCheckVarsSymmetric( DdManager * dd, DdNode * bF, int iVar1, int iVar2 );
+extern DdNode *     extraBddCheckVarsSymmetric( DdManager * dd, DdNode * bF, DdNode * bVars );
+
+/* build the set of all tuples of K variables out of N from the BDD cube */
+extern DdNode *    Extra_zddTuplesFromBdd( DdManager * dd, int K, DdNode * bVarsN );
+extern DdNode *     extraZddTuplesFromBdd( DdManager * dd, DdNode * bVarsK, DdNode * bVarsN );
+/* selects one subset from a ZDD representing the set of subsets */
+extern DdNode *    Extra_zddSelectOneSubset( DdManager * dd, DdNode * zS );
+extern DdNode *     extraZddSelectOneSubset( DdManager * dd, DdNode * zS );
+
 /*=== extraUtilBitMatrix.c ================================================================*/
 
 typedef struct Extra_BitMat_t_ Extra_BitMat_t;
diff --git a/src/misc/extra/extraUtilBdd.c b/src/misc/extra/extraBddMisc.c
index 38e3345c..373ce5c5 100644
--- a/src/misc/extra/extraUtilBdd.c
+++ b/src/misc/extra/extraBddMisc.c
@@ -1,6 +1,6 @@
 /**CFile****************************************************************
 
-  FileName    [extraUtilBdd.c]
+  FileName    [extraBddMisc.c]
 
   SystemName  [ABC: Logic synthesis and verification system.]
 
@@ -14,7 +14,7 @@
 
   Date        [Ver. 1.0. Started - June 20, 2005.]
 
-  Revision    [$Id: extraUtilBdd.c,v 1.0 2003/09/01 00:00:00 alanmi Exp $]
+  Revision    [$Id: extraBddMisc.c,v 1.0 2003/09/01 00:00:00 alanmi Exp $]
 
 ***********************************************************************/
 
diff --git a/src/misc/extra/extraBddSymm.c b/src/misc/extra/extraBddSymm.c
new file mode 100644
index 00000000..474f663f
--- /dev/null
+++ b/src/misc/extra/extraBddSymm.c
@@ -0,0 +1,1469 @@
+/**CFile****************************************************************
+
+  FileName    [extraBddSymm.c]
+
+  PackageName [extra]
+
+  Synopsis    [Efficient methods to compute the information about
+  symmetric variables using the algorithm presented in the paper:
+  A. Mishchenko. Fast Computation of Symmetries in Boolean Functions. 
+  Transactions on CAD, Nov. 2003.]
+
+  Author      [Alan Mishchenko]
+  
+  Affiliation [UC Berkeley]
+
+  Date        [Ver. 2.0. Started - September 1, 2003.]
+
+  Revision    [$Id: extraBddSymm.c,v 1.0 2003/09/01 00:00:00 alanmi Exp $]
+
+***********************************************************************/
+
+#include "extra.h"
+
+/*---------------------------------------------------------------------------*/
+/* Constant declarations                                                     */
+/*---------------------------------------------------------------------------*/
+
+/*---------------------------------------------------------------------------*/
+/* Stucture declarations                                                     */
+/*---------------------------------------------------------------------------*/
+
+/*---------------------------------------------------------------------------*/
+/* Type declarations                                                         */
+/*---------------------------------------------------------------------------*/
+
+/*---------------------------------------------------------------------------*/
+/* Variable declarations                                                     */
+/*---------------------------------------------------------------------------*/
+
+/*---------------------------------------------------------------------------*/
+/* Macro declarations                                                        */
+/*---------------------------------------------------------------------------*/
+
+#define DD_GET_SYMM_VARS_TAG          0x0a /* former DD_BDD_XOR_EXIST_ABSTRACT_TAG */
+
+/**AutomaticStart*************************************************************/
+
+/*---------------------------------------------------------------------------*/
+/* Static function prototypes                                                */
+/*---------------------------------------------------------------------------*/
+
+/**AutomaticEnd***************************************************************/
+
+/*---------------------------------------------------------------------------*/
+/* Definition of exported functions                                          */
+/*---------------------------------------------------------------------------*/
+
+/**Function********************************************************************
+
+  Synopsis    [Computes the classical symmetry information for the function.]
+
+  Description [Returns the symmetry information in the form of Extra_SymmInfo_t structure.]
+
+  SideEffects [If the ZDD variables are not derived from BDD variables with
+  multiplicity 2, this function may derive them in a wrong way.]
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Computes the classical symmetry information as a ZDD.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Returns a singleton-set ZDD containing all variables that are symmetric with the given one.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Converts a set of variables into a set of singleton subsets.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Filters the set of variables using the support of the function.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Allocates symmetry information structure.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+Extra_SymmInfo_t * Extra_SymmPairsAllocate( int nVars )
+{
+    int i;
+    Extra_SymmInfo_t * p;
+
+    // allocate and clean the storage for symmetry info
+    p = ALLOC( Extra_SymmInfo_t, 1 );
+    memset( p, 0, sizeof(Extra_SymmInfo_t) );
+    p->nVars     = nVars;
+    p->pVars     = ALLOC( int, nVars );  
+    p->pSymms    = ALLOC( char *, nVars );  
+    p->pSymms[0] = 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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Delocates symmetry information structure.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+void Extra_SymmPairsDissolve( Extra_SymmInfo_t * p )
+{
+    free( p->pVars );
+    free( p->pSymms[0] );
+    free( p->pSymms    );
+    free( p );
+} /* end of Extra_SymmPairsDissolve */
+
+/**Function********************************************************************
+
+  Synopsis    [Allocates symmetry information structure.]
+
+  Description []
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Creates the symmetry information structure from ZDD.]
+
+  Description [ZDD representation of symmetries is the set of cubes, each
+  of which has two variables in the positive polarity. These variables correspond
+  to the symmetric variable pair.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 = 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 );
+
+    FREE( pMapVars2Nums );
+    return p;
+
+} /* end of Extra_SymmPairsCreateFromZdd */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Checks the possibility of two variables being symmetric.]
+
+  Description [Returns 0 if vars are not symmetric. Return 1 if vars can be symmetric.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Computes the classical symmetry information for the function.]
+
+  Description [Uses the naive way of comparing cofactors.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Checks if the two variables are symmetric.]
+
+  Description [Returns 0 if vars are not symmetric. Return 1 if vars are symmetric.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Builds ZDD representing the set of fixed-size variable tuples.]
+
+  Description [Creates ZDD of all combinations of variables in Support that
+  is represented by a BDD.]
+
+  SideEffects [New ZDD variables are created if indices of the variables
+               present in the combination are larger than the currently
+               allocated number of ZDD variables.]
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Selects one subset from the set of subsets represented by a ZDD.]
+
+  Description []
+
+  SideEffects [None]
+
+  SeeAlso     []
+
+******************************************************************************/
+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                                          */
+/*---------------------------------------------------------------------------*/
+
+/**Function********************************************************************
+
+  Synopsis    [Performs a recursive step of Extra_SymmPairsCompute.]
+
+  Description [Returns the set of symmetric variable pairs represented as a set 
+  of two-literal ZDD cubes. Both variables always appear in the positive polarity
+  in the cubes. This function works without building new BDD nodes. Some relatively 
+  small number of ZDD nodes may be built to ensure proper bookkeeping of the 
+  symmetry information.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Performs a recursive step of Extra_zddGetSymmetricVars.]
+
+  Description [Returns the set of ZDD singletons, containing those positive
+  ZDD variables that correspond to BDD variables x, for which it is true 
+  that bF(x=0) == bG(x=1).]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Performs a recursive step of Extra_zddGetSingletons.]
+
+  Description [Returns the set of ZDD singletons, containing those positive 
+  polarity ZDD variables that correspond to the BDD variables in bVars.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Performs a recursive step of Extra_bddReduceVarSet.]
+
+  Description [Returns the set of all variables in the given set that are not in the 
+  support of the given function.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Performs the recursive step of Extra_bddCheckVarsSymmetric().]
+
+  Description [Returns b0 if the variables are not symmetric. Returns b1 if the
+  variables can be symmetric. The variables are represented in the form of a 
+  two-variable cube. In case the cube contains one variable (below Var1 level),
+  the cube's pointer is complemented if the variable Var1 occurred on the 
+  current path; otherwise, the cube's pointer is regular. Uses additional 
+  complemented bit (Hash_Not) to mark the result if in the BDD rooted that this
+  node there is a branch passing though the node labeled with Var2.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+/**Function********************************************************************
+
+  Synopsis    [Performs the reordering-sensitive step of Extra_zddTupleFromBdd().]
+
+  Description [Generates in a bottom-up fashion ZDD for all combinations
+               composed of k variables out of variables belonging to Support.]
+
+  SideEffects []
+
+  SeeAlso     []
+
+******************************************************************************/
+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 */
+
+
+/**Function********************************************************************
+
+  Synopsis    [Performs the recursive step of Extra_zddSelectOneSubset.]
+
+  Description []
+
+  SideEffects [None]
+
+  SeeAlso     []
+
+******************************************************************************/
+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                                            */
+/*---------------------------------------------------------------------------*/
diff --git a/src/misc/extra/extraUtilReader.c b/src/misc/extra/extraUtilReader.c
index 2dc597bf..042dfaca 100644
--- a/src/misc/extra/extraUtilReader.c
+++ b/src/misc/extra/extraUtilReader.c
@@ -321,6 +321,7 @@ void * Extra_FileReaderGetTokens_int( Extra_FileReader_t * p )
     // through EXTRA_OFFSET_SIZE chars till the end of the buffer
     if ( p->pBufferStop == p->pBufferEnd ) // end of file
     {
+        *pChar = 0;
         p->fStop = 1;
         return p->vTokens;
     }
diff --git a/src/misc/extra/module.make b/src/misc/extra/module.make
index 3098a23c..6cbf5d2c 100644
--- a/src/misc/extra/module.make
+++ b/src/misc/extra/module.make
@@ -1,4 +1,5 @@
-SRC +=    src/misc/extra/extraUtilBdd.c \
+SRC +=    src/misc/extra/extraBddMisc.c \
+    src/misc/extra/extraBddSymm.c \
     src/misc/extra/extraUtilBitMatrix.c \
     src/misc/extra/extraUtilCanon.c \
     src/misc/extra/extraUtilFile.c \