Solution 3: (a) True. Proof.

Transcription

1 Solution 3: (a) True Proof. f = (x + f x ) (x + f x ) + f x f x + f x f x (x + x) f x + x f x + x f x f x (Boolean absorption) This is equal to Shannon s expansion of f. (b) True Proof. f = x f x x f x f = x f x (x f x ) + x f x (x f x ) f = x f x (x + f x ) + x f x (x + f x ) (DeMorgan) f x + x f x + x f x f x (Boolean absorption) This is equal to Shannon s expansion of f. (c) True Proof. (Shannon s expansion) f = (x + f x ) (x + f x ) (DeMorgan) + f x f x + f x f x (x + x) f x + x f x + x f x f x (Boolean absorption)

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3 Problem 6 (Pseudo Code) ECE/CS 6745 Fall 2014 Solution HW1 September 25, 2014 Problem 6 Problem description: Given a ROBDD f with variables (x 1,x 2,...,x n )whichisordered by x 1 >x 2 > >x n. Our objective is to transform f to ROBDD f xi eliminating variable x i (or f x 0 i when x 0 i is negative cofactor) Algorithm 1 Arbitrary variable elimination algorithm on ROBDD 1: function ROBDDVarElim(f,i) 2: if v = top(f) then. v is the variable of top node 3: return f v or f v 0. Directly return f v or f v 0 when requiring negative cofactor 4: else 5: while BFS Traverse(f) do 6: if idx(v) =i then. Reach nodes of variable x i 7: Edge.Delete(v, low(v)) 8: Edge.Delete(v, high(v)). Delete its edges to children 9: for all parent(v) do. For all of its parent nodes 10: if x i is positive cofactor then 11: Redirect edge hparent(v), vi to hparent(v), high(v)i 12: else 13: Redirect edge hparent(v),vi to hparent(v),low(v)i 14: end if 15: Clean-up if a node has no reference. Please refer to Example 1 16: end for 17: Node.Delete(v) 18: end if 19: end while 20: f xi Reduce(top(f)). Please refer to Example 2 21: return f xi 22: end if 23: end function Note: low(), high(), idx() means the child on FALSE edge, child on TRUE edge, and index of variables of this node. Their definitions and function Reduce() can be found in Graph-Based Algorithms for Boolean Function Manipulation by R. E. Bryant, which is linked on class webpage. 1

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5 int main() { FILE* fp; // Initializing the Manager // Cudd_Init(numVars, numvarsz, numslots, cachesize, maxmemory) DdManager* manager=cudd_init(0, 0, CUDD_UNIQUE_SLOTS, CUDD_CACHE_SLOTS, 0); if(manager == NULL) { printf("error while initializing CUDD\n"); return 1; } DdNode *one = Cudd_ReadOne(manager); DdNode *zero = Cudd_Not(one); char const * outputnames[1]; outputnames[0] = "BDD"; char const * inputnames[3]; inputnames[0] = "a"; inputnames[1] = "b"; inputnames[2] = "c"; DdNode *a, *b, *c; a = Cudd_bddNewVar(manager); Cudd_Ref(a); // a variable will have index 0 b = Cudd_bddNewVar(manager); Cudd_Ref(b); // 1 c = Cudd_bddNewVar(manager); Cudd_Ref(c); // 2 //Part 1 f1 = ab + ac + bc DdNode *ft11 = Cudd_bddOr(manager,b,c); Cudd_Ref(ft11); DdNode *ft12 = Cudd_bddAnd(manager,a,ft11); Cudd_Ref(ft12); DdNode *ft13 = Cudd_bddAnd(manager,b,c); Cudd_Ref(ft13); DdNode *f1 = Cudd_bddOr(manager, ft13, ft12); Cudd_Ref(f1); Cudd_RecursiveDeref(manager,ft11); Cudd_RecursiveDeref(manager,ft12); Cudd_RecursiveDeref(manager,ft13); // Some details about the function f1 Cudd_PrintDebug( manager, f1, 3, 3); // Printing out f1 fp = fopen("dot_files/f_maj.dot","w"); Cudd_DumpDot(manager, 1, &f1, inputnames, outputnames, fp); fclose(fp); // Positive Cofactor of f1 w.r.t. b DdNode *f1_b = Cudd_Cofactor(manager, f1, b); Cudd_Ref(f1_b); // Negative Cofactor of f1 w.r.t. b DdNode *f1_bbar = Cudd_Cofactor(manager, f1, Cudd_Not(b) ); Cudd_Ref(f1_bbar); DdNode *f1_check = Cudd_bddOr(manager,f1_b,Cudd_Not(f1_bbar)); Cudd_Ref(f1_check); // Checking for containment f1_bbar \subset f1_b if(f1_check == one) printf("f1_b contains f1_bbar\n"); else printf("f1_b does not contains f1_bbar\n");

6 //Dereferencing all the variables that are no longer needed Cudd_RecursiveDeref(manager,f1_check); Cudd_RecursiveDeref(manager,f1_bbar); Cudd_RecursiveDeref(manager,f1_b); Cudd_RecursiveDeref(manager,f1); //Part 2 f2 = a xor b xor c DdNode *ft21 = Cudd_bddXor(manager,a,b); Cudd_Ref(ft21); DdNode *f2 = Cudd_bddXor(manager, c, ft21); Cudd_Ref(f2); Cudd_RecursiveDeref(manager,ft21); // Some details about the function f2 Cudd_PrintDebug( manager, f2, 3, 3); // Printing out f2 fp = fopen("dot_files/f_xor.dot","w"); Cudd_DumpDot(manager, 1, &f2, inputnames, outputnames, fp); fclose(fp); // Positive Cofactor of f2 w.r.t. b DdNode *f2_b = Cudd_Cofactor(manager, f2, b); Cudd_Ref(f2_b); // Negative Cofactor of f2 w.r.t. b DdNode *f2_bbar = Cudd_Cofactor(manager, f2, Cudd_Not(b) ); Cudd_Ref(f2_bbar); DdNode *f2_check = Cudd_bddOr(manager,f2_b,Cudd_Not(f2_bbar)); Cudd_Ref(f2_check); // Checking for containment f2_bbar \subset f2_b if(f2_check == one) printf("f2_b contains f2_bbar\n"); else printf("f2_b does not contains f2_bbar\n"); //Dereferencing all the variables that are no longer needed Cudd_RecursiveDeref(manager,f2_check); Cudd_RecursiveDeref(manager,f2_bbar); Cudd_RecursiveDeref(manager,f2_b); Cudd_RecursiveDeref(manager,f2); //Dereferencing all the inputs Cudd_RecursiveDeref(manager, a); Cudd_RecursiveDeref(manager, b); Cudd_RecursiveDeref(manager, c); // Checking if ref-deref was done right int nonzref = Cudd_CheckZeroRef(manager); printf("no. of non-zero refs = %d\n", nonzref ); // Exiting the manager Cudd_Quit(manager); return 0; }

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