# A PrimalDual Solution to Minimal Test Generation Problem

- Slides: 16

A Primal-Dual Solution to Minimal Test Generation Problem Mohammed Ashfaq Shukoor Vishwani D. Agrawal Auburn University, Department of Electrical and Computer Engineering Auburn, AL 36849, USA 12 th IEEE VLSI Design and Test Symposium, 2008, Bangalore 05 December 2020 VDAT '08 1

Problem Statement To find a minimal set of vectors to cover all stuck-at faults in a combinational circuit 12/5/2020 VDAT '08 2

A Known Method: Test Minimization ILP[1] v is a variable assigned to each Objective: minimize Subject to conditions: j J vectors (1)with the following constantofathe kj is 1 only if the fault k is detectedmeaning: by vector j, else it is 0 • If vj = 1, then vector j is included in the minimized vector set Fault number ( j ) j is not (2) 0, then vector k = 1, 2, . . . , K • If vj =Vector number ( k) 1 2 3 4. . . J included in the minimized vector 1 0 set 1 1 0. . . 1 2 0 0 1 0 . . . 1 3 1 0 0 1 . . . 0 4 0 1 0 0 . . . 0 K is the number of faults in a combinational. . circuit. . . vector. . set. J is the number of vectors in the unoptimized . . . 1 vj integer [0, 1], j = 1, 2, . . . , J K 1 1 0 0 (3) [1] P. Drineas and Y. Makris, “Independent Test Sequence Compaction through Integer Programming, ” Proc. International Conf. Computer Design, 2003, pp. 380– 386. 05 December 2020 VDAT '08 3

Motivation n When test minimization is performed over an exhaustive set of vectors, the ILP solution is the smallest possible test set. n For most circuits exhaustive vector sets are impractical. n We need a method to find a non-exhaustive vector set for which the test minimization ILP will give a minimal test set. 05 December 2020 VDAT '08 4

Definitions Independent Faults [2]: Two faults are independent if and only if they cannot be detected by the same test vector. T(f 1) T(f 2) f 1 and f 2 are independent T(f 1) T(f 2) f 1 and f 2 are not independent Independent Fault Set (IFS) [2]: An IFS contains faults that are pair-wise independent. [2] S. B. Akers, C. Joseph, and B. Krishnamurthy, “On the Role of Independent Fault Sets in the Generation of Minimal Test Sets, ” Proc. International Test Conf. , 1987, pp. 1100– 1107. 05 December 2020 VDAT '08 5

Independence Graph n n n Independence graph: Nodes are faults and an edge between two nodes means that the corresponding faults are independent. Example: c 17[3]. An Independent Fault Set (IFS) is a maximum clique in the graph. Size of IFS is a lower bound on test set size (Akers et al. , ITC-87) 1 2 3 4 5 11 6 7 8 9 10 [3] A. S. Doshi and V. D. Agrawal, “Independence Fault Collapsing, ” Proc. 9 th VLSI Design and Test Symp. , Aug. 2005, pp. 357 -364. 05 December 2020 VDAT '08 6

New Definitions Conditionally Independent Faults: Two faults, detectable by vector set V, are conditionally independent with respect to the vector set V if no vector in the set detects both faults. Conditionally Independent Fault Set (CIFS): For a given vector set, a subset of all detectable faults in which no pair of faults can be detected by the same vector, is called a conditionally independent fault set (CIFS). Conditional Independence Graph: An independence graph in which the independence relations between faults are relative to a vector set is called a conditional independence graph 05 December 2020 VDAT '08 7

Primal and Dual Problems[4] n n An optimization problem in an application may be viewed from either of two perspectives, the primal problem or the dual problem These two problems share a common set of coefficients and constants. If the primal minimizes one objective function of one set of variables then its dual maximizes another objective function of the other set of variables Duality theorem states that if the primal problem has an optimal solution, then the dual also has an optimal solution, and the optimized values of the two objective functions are equal. [4] G. Strang, Linear Algebra and Its Applications, Fort Worth: Harcourt Brace Javanovich College Publishers, third edition, 1988. 12/5/2020 VDAT '08 8

Dual ILP Formulation (4) maximize Subject to, fk integer [0, 1], fk is a variable assigned to each of the K faults with the following meaning, • If fk = 1, then fault k is included in the fault set (5) j = 1, 2, . . . , J • If fk = 0, then fault k is not included in the fault set k = 1, 2, . . . , K (6) Theorem 1: A solution of the dual ILP of 4, 5 and 6 provides a largest conditionally independent fault set (CIFS) with respect to the vector set V. 05 December 2020 VDAT '08 9

Theorem 2: For a combinational circuit, suppose V 1 and V 2 are two vector sets such that and V 1 detects all detectable faults of the circuit. If CIFS(V 1) and CIFS(V 2) are the largest CIFS with respect to V 1 and V 2, respectively, then |CIFS(V 1)| ≥ |CIFS(V 2)|. Conditional Independence Graph for vector set V 1 1 2 3 4 Conditional Independence Graph for vector set V 2 5 1 2 3 4 5 11 6 7 8 9 11 10 6 8 9 10 |CIFS(V 2)| = 4 |CIFS(V 1)| = 5 05 December 2020 7 VDAT '08 10

Primal Dual ILP Algorithm for Test Minimization 1. 2. 3. 4. 5. 6. Generate an initial vector set to detect all (or most) faults Obtain diagnostic matrix (conditional independence graph) by fault simulation without fault dropping Solve dual ILP to determine CIFS. Go to 6 if CIFS has converged Augment vector set by additional tests for CIFS Go to step 2 Solve primal ILP for final compacted vector set 05 December 2020 VDAT '08 11

Example 1: c 1355 Problem type Iteration number No. of vectors ATPG CPU s Fault sim. CPU s CIFS size Dual 1 2 3 114 507 903 0. 033 0. 085 0. 333 1. 517 2. 683 85 84 84 Primal 903 No. of min. vectors ILP CPU s 0. 24 0. 97 1. 91 84 3. 38 SUN Fire 280 R, 900 MHz Dual Core machine 05 December 2020 VDAT '08 12

Example 2: c 2670 Problem type Iteration number No. of vectors ATPG CPU s Fault sim. CPU s CIFS size Dual 1 2 3 4 5 6 7 8 9 10 11 12 194 684 1039 1424 1738 2111 2479 2836 3192 3537 3870 4200 2. 167 1. 258 1. 176 1. 168 1. 136 1. 128 1. 112 1. 086 1. 073 1. 033 1. 048 1. 033 3. 670 5. 690 6. 895 8. 683 10. 467 12. 333 14. 183 15. 933 17. 717 19. 267 20. 983 22. 600 102 82 79 78 76 76 74 73 72 70 70 70 Primal 4200 No. of min. vectors ILP CPU s 1. 99 3. 22 7. 90 3. 69 5. 89 7. 43 7. 16 8. 45 9. 81 10. 90 12. 02 13. 44 70 316. 52 SUN Fire 280 R, 900 MHz Dual Core machine 05 December 2020 VDAT '08 13

Comparing primal_LP–dual_ILP solution with LP-alone solution Primal-dual minimization [this paper] Circuit Name Lower bound on vectors Unopt. vectors LP CPU s Minimized vectors Unopt. vectors Total CPU s Minimized vectors c 432 27 608 2. 00 36 983 5. 52 31 c 499 52 379 1. 00 52 221 1. 35 52 c 880 13 1023 31. 00 28 1008 227. 21 25 c 1355 84 755 5. 00 84 507 1. 95 84 c 1908 106 1055 8. 00 107 728 2. 50 107 c 2670 44 959 9. 00 84 1039 17. 41 79 c 3540 78 1971 197. 00 105 2042 276. 91 95 c 5315 37 1079 464. 00 72 1117 524. 53 67 c 6288 6 243 78. 00 18 258 218. 9 17 c 7552 65 2165 151. 00 145 2016 71. 21 139 LP-alone minimization [5] SUN Fire 280 R, 900 MHz Dual Core machine [5] K. R. Kantipudi and V. D. Agrawal, “A Reduced Complexity Algorithm for Minimizing N-Detect Tests, ” Proc. 20 th International Conf. VLSI Design, Jan. 2007, pp. 492– 497. 05 December 2020 VDAT '08 14

Conclusion n n A new algorithm based on primal dual ILP is introduced for test optimization. The dual ILP helps in obtaining proper vectors, which then can be optimized by the primal ILP. Future Work n n According to Theorem 2, CIFS must converge to IFS as the vector set approaches the exhaustive set. We should explore strategies for generating vectors for the dual problem in order to have the CIFS quickly converge to IFS before vector set becomes exhaustive. A useful application of the dual ILP and the conditionally independent fault set (CIFS), we believe, is in fault diagnosis. We hope to explore that in the future. 05 December 2020 VDAT '08 15

Thank you … 05 December 2020 VDAT '08 16

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