May 15, 2003 Agrawal et al.: Fault Collapsing 1

Fault Collapsing via Functional Dominance

Vishwani D. AgrawalRutgers 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. AtreAgere Systems, Bangalore, India

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Test Vector Generation Flow

DUT

Generate fault list

Collapse fault list

Generate test vectors

Fault Model

Required fault coverage

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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 (f2f1).

a0 a1

b0 b1

c0 c1

a1 c1: Dominanceb1 c1: Dominance

a0 = b0 = c0 : Equivalence

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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)

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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)

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• 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.

Dominance Graph

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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

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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.

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Transitive Closure• (F1 F2) and (F2 F3) => (F1 F3)

F1 F2 F3

  F1 F2 F3

F1 1 1  

F2   1 1

F3     1

Graph

F1 F2 F3

  F1 F2 F3

F1 1 1 1

F2   1 1

F3     1

Transitive Closure

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ExampleA

B

C

D

E

A0 B0

D0

E0

C0

A1 B1

D1

E1

C1

Dominance Graph

Transitiveclosureedges

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Functional Dominance

f1

f0

f2

Always 0

f1 dominates f2

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Functional Equivalence

f1

f0

f2

Always 0

f1 dominates f2 and f2 dominates f1

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Functional Equivalence

f0

f2

Always 0

Always 0f1

f2

f1

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XOR Circuit

Functional Equivalences : (c1,f1), (g1,h1,i1), (g0,m0),

c1

f1

g1

h1

i1

g0m0

(d1,f0) and (e1,c0); additional dominances not shown

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XOR Circuit

Structural equivalence collapsing16 faults

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XOR Circuit

Functional equivalence collapsing10 faults

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XOR Circuit

Functional dominance collapsing4 faults

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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 B1B0C0 C0

C1

C0 C0

C1

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8-bit Ripple Carry Adder

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Fault Collapsing Using Functional Dominances of xor

Number of collapsed faults

Flat

structural only

Hierarchical

with functional

Equ. Dom. 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

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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).

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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