1. Diagnostic and Detection Fault Collapsing for Multiple Output Circuits Raja K. K. R. Sandireddy and Vishwani D. Agrawal Dept. Of Electrical and Computer Engineering, Auburn University, Auburn, AL-36849 USA.

2. 2 Outline  Introduction  Fault Equivalence and Fault Dominance  Functional collapsing  Fault Equivalence and Dominance definitions  Results of functional collapsing  Hierarchical fault collapsing  Conclusions and Future work.

3. 3 Equivalence R Structural equivalence 1 : Two faults f 1 and f 2 are said to be R structurally equivalent if they produce the same reduced circuit graph [netlist] when faulty values are implied and constant edges [signals] are removed. Functional equivalence 1 : Two faults f 1 and f 2 are said to be functionally equivalent if they modify the Boolean function of the circuit in the same way, i.e., they yield the same output functions. 1 E. J. McCluskey and F. W. Clegg, “Fault Equivalence in Combinational Logic Networks,” IEEE Trans. Computers, vol. C-20, no. 11, Nov. 1971, pp. 1286-1293.

4. 4 Structural Dominance A fault f i is said to dominate fault f j if the faults are equivalent with respect to test set of fault f j. a 0 a 1 b 0 b 1 c 0 c 1 Equivalence collapsed set = {a 0, b 0, c 0, c 1 } Dominance collapsed set = {a 0, b 0, c 1 } Example: Full adder circuit. Total faults: 60 Structural equivalence collapsed set 2, 3 = 38 (0.63) Structural dominance collapsed set 3 = 30 (0.5) 2 Using Hitec, 3 Using Fastest

5. 5 Functional Dominance 4 F1F1 F0F0 F2F2 Z If the fault introduced in block F 1 dominates the fault in block F 2, then Z is always 0. 4 V. D. Agrawal, A. V. S. S. Prasad, and M. V. Atre, “Fault Collapsing via Functional Dominance,” Proc. International Test Conf., 2003, pp. 274-280. 1 1 0 For the full adder, functional dominance collapsed set = 12 (0.20) {Structural equiv. = 38, Structural dom. = 30, Functional equiv.= 23}

6. 6 Equivalence Definitions For multiple output circuits,  Diagnostic Equivalence - Two faults of a Boolean circuit are called diagnostically equivalent if and only if the functions of the two faulty circuits are identical at each output.  Detection Equivalence - Two faults are called detection equivalent if and only if all tests that detect one fault also detect the other fault, not necessarily at the same output. Y Z A B c s-a-0 The faults c 0 and Y 0 are detection equivalent faults, but not diagnostic equivalent. For the full adder, diagnostic equivalence collapsed set = 26 (0.43), detection equivalence collapsed set = 23 (0.38) {Structural equiv. = 38, Structural dom. = 30, Functional equiv.= 26, Functional dom.= 12}

7. 7 Dominance Definitions  Fault Dominance 5 - A fault f i is said to dominate fault f j if (a) the set of all vectors that detects fault f j is a subset of all vectors that detects fault f i and (b) each vector that detects f j implies identical values at the corresponding outputs of faulty versions of the circuit. Conventionally dominance is defined as:  A fault f i is said to dominate fault f j if the faults are equivalent with respect to test set of fault f j.  If all tests of fault f j detect another fault f i, then f i is said to dominate f j. 5 J. F. Poage, “Derivation of Optimum Tests to Detect Faults in Combinational Circuits", Proc. Symposium on Mathematical Theory of Automata, 1962, pp. 483-528.

8. 8 Dominance Definitions contd. For multiple output circuits, the two possible interpretations of dominance:  Diagnostic dominance - If all tests of a fault f 1 detect another fault f 2 on the exact same outputs where f 1 was detected, then f 2 is said to diagnostically dominate f 1.  Detection dominance - If all tests of a fault f 1 detect another fault f 2, irrespective of the output where f 1 was detected, then f 2 is said to detection dominate f 1. Diagnostic dominance implies detection dominance. For the full adder, diagnostic dominance collapsed set = 12 (0.2) detection dominance collapsed set = 6 (0.1) {Structural equiv. = 38, Structural dom. = 30, Diagnostic equiv.= 26, Detection equiv.= 23}

9. 9 Results: Functional Collapsing Circuit Name All Faults Number of Collapsed Faults (Collapse Ratio) StructuralFunctional 4 Functional Collapsing – New Results Diagnostic Criterion Detection Criterion Equiv. 2 Dom. 3 Equiv.Dom.Equiv.Dom.Equiv.Dom. XOR24 16 (0.67) 13 (0.54) 10 (0.42) 4 (0.17) 10 (0.42) 4 (0.17) 10 (0.42) 4 (0.17) Full Adder 60 38 (0.63) 30 (0.50) 26 (0.43) 14 (0.23) 26 (0.43) 12 (0.20) 23 (0.38) 6 (0.10) 8-bit Adder 466 290 (0.62) 226 (0.49) 194 (0.42) 112 (0.24) 194 (0.42) 96 (0.21) 191 (0.41) 48 (0.10) ALU (74181) 502 301 (0.60) 248 (0.49) -- 253 (0.50) 155 (0.31) 234 (0.47) 92 (0.18) 2 Using Hitec (obtained from Univ. of Illinois at Urbana-Champaign) 3 Using Fastest (obtained from Univ. of Wisconsin at Madison) 4 Agrawal et al. ITC’03

10. 10 Results: Test Vectors Circuit No. of test vectors (no. of target faults) StructuralFunctional – New Results EquivalenceDominance Diagnostic Dominance Detection Dominance Full Adder6 (38)6 (30)7 (12)6 (6) 8-bit Adder33 (290)28 (226)32 (96)28 (48) ALU44 (293)44 (240)39 (147)38 (84) Test vectors obtained using Gentest ATPG 6. 6 W. T. Cheng and T. J. Chakraborty, “Gentest: An Automatic Test Generation System for Sequential Circuits,” Computer, vol. 22, no. 4, pp. 43–49, April 1989.

11. 11 Hierarchical Fault Collapsing Total Faults: Full Adder: 60, 64-bit Adder: 3714, 1024-bit Adder: 59394, c432:1116, c499:2646 Detection collapsing can be used only for those sub- circuits whose outputs are POs at the top-level.

12. 12 CPU time (sec) for hierarchical collapsing Flattened (Hitec) Flattened (Our Program) Hierarchical (two-level) Hierarchical (multi-level)

13. 13 Conclusions  Diagnostic and detection collapsing should be used only with smaller circuits.  Collapse ratios using detection dominance collapsing is about 10-20%.  Hierarchical fault collapsing:  Better (lower) collapse ratios due to functional collapsed library  Order of magnitude reduction in collapse time.  Smaller fault sets:  Fewer test vectors  Reduced fault simulation effort  Easier fault diagnosis.  Use caution when using dominance collapsing!!

14. 14 Future Work  Generate fault collapsing library of standard cells (Mentor Graphics, etc.).  Efficient redundancy detection program.  Customized ATPG to obtain minimal test vector set.