Fault Collapsing via Functional Dominance Vishwani D. Agrawal Rutgers University, Dept. of ECE, Piscataway, New Jersey, USA vishwani02@yahoo.com http://cm.bell-labs.com/cm/cs/who/va A. V. S. S. Prasad and M. V. Atre Agere Systems, Bangalore, India May 15, 2003 Agrawal et al.: Fault Collapsing

Test Vector Generation Flow DUT Generate fault list Collapse fault list Generate test vectors Fault Model Required fault coverage May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Background Single stuck-at fault model is the most popularly used model. Two faults f1 and f2 are equivalent if the same tests detect f1 and f2 (f1=f2) If all tests of fault f2 also detect fault f1, then f1 is said to dominate f2 (f2f1). a0 = b0 = c0 : Equivalence a1  c1 : Dominance b1  c1 : Dominance a0 a1 c0 c1 b0 b1 May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Background Both equivalence and dominance relations are transitive in nature. [ (f1  f2) and (f2  f3) => (f1  f3) ] If f1 dominates f2 and f2 dominates f1 then f1 and f2 are equivalent. [ (f1  f2) and (f2  f1) => (f1 = f2) ] Number of faults in a 2-input AND gate reduces from 6 to 4 (by equivalence) and to 3 (by dominance) collapsing. Example: c6288, #faults =12576 #equ. = 7744 (0.62), #dom. = 5824 (0.46) May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Problem Statement To devise a new method for fault collapsing with following attributes: A single procedure for equivalence and dominance Global analysis (independence from direction, and other choices, in collapsing) Use functional equivalences and dominances Hierarchical fault collapsing (collapsing in large circuits using pre-collapsed sub networks) May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Dominance Graph A fault in the circuit is represented by a node in the graph. A directed edge from f2 to f1 indicates that f1 dominates f2 (f2 f1). Edges can represent either structural or functional relations. May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Dominance Matrix Graph is represented as a connectivity matrix Each fault is assumed to be equivalent to itself Treats functional and structural relations identically (f1  f2) and (f2 f1) => f2 = f1. Appear as symmetrical components in the matrix (e.g., a0,b0,c0) #faults = 6 (dimension of dominance matrix) 2-input AND gate May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Transitive Closure Transitive closure (TC) of the dominance matrix gives all dominance relations between faults. TC is computed by the O(n3) Floyd-Warshall algorithm, where n is the dimension of the dominance matrix. May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Transitive Closure (F1  F2) and (F2  F3) => (F1  F3) F1 F2 F3   1 Transitive Closure F1 F2 F3   1 Graph May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Example A D E B C Dominance Graph E0 E1 Transitive closure edges D0 C0 D1 C1 B0 A0 B1 A1 May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Functional Dominance f1 Always 0 f0 f2 f1 dominates f2 May 15, 2003 Agrawal et al.: Fault Collapsing

Functional Equivalence Always 0 f0 f2 f1 dominates f2 and f2 dominates f1 May 15, 2003 Agrawal et al.: Fault Collapsing

Functional Equivalence Always 0 f2 f1 Always 0 f2 May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing XOR Circuit c1 h1 g1 g0 m0 i1 f1 Functional Equivalences : (c1,f1), (g1,h1,i1), (g0,m0), (d1,f0) and (e1,c0); additional dominances not shown May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing XOR Circuit Structural equivalence collapsing 16 faults May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing XOR Circuit Functional equivalence collapsing 10 faults May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing XOR Circuit Functional dominance collapsing 4 faults May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Design Hierarchy Large designs are modular and hierarchical. Advantageous to store the fault information of repeated blocks in a library. When configured as a library cell the fault list includes cell PI & PO faults for transitivity. Top module B1 B1 B0 C0 C0 C0 C0 C1 C1 May 15, 2003 Agrawal et al.: Fault Collapsing

8-bit Ripple Carry Adder May 15, 2003 Agrawal et al.: Fault Collapsing

Fault Collapsing Using Functional Dominances of xor Number of collapsed faults Flat structural only Hierarchical with functional Equ. Dom. xor cell 24 16(0.63) 13(0.54) 10(0.41) 4(0.17) Full-adder 60 38(0.63) 30(0.50) 26(0.43) 14(0.23) 8-bit adder 466 290(0.62) 226(0.49) 194(0.42) 112(0.24) c499exp 2710 1574(0.58) 1210(0.45) 950(0.35) 586(0.22) Circuit name All faults May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing References A. Lioy, “Looking for Functional Equivalence,” Proc. ITC, 1991, pp. 858-863. A. V. S. S. Prasad, V. D. Agrawal and M. V. Atre, “A New Algorithm for Global Fault Collapsing into Equivalence and Dominance Sets,” Proc. ITC, 2002, pp. 391-397. H. Al-Asaad and R. Lee, “Simulation-Based Approximate Global Fault Collapsing,” Proc. Int. Conf. VLSI, 2002, pp. 72-77. V. D. Agrawal, A. V. S. S. Prasad and M. V. Atre, “Fault Collapsing via Functional Dominance,” Proc. ITC, 2003 (accepted). May 15, 2003 Agrawal et al.: Fault Collapsing

Agrawal et al.: Fault Collapsing Conclusion A new algorithm for global fault collapsing With functional equivalence number of faults for ATPG reduces Fault set reduced below 25% with functional dominances (Caution: fault coverage not correct when redundant faults are present) Library based hierarchical fault collapsing is a useful concept Further studies are being carried out on independent fault sets May 15, 2003 Agrawal et al.: Fault Collapsing