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Minimizing N -Detect Tests for Combinational Circuits Master’s Defense Kalyana R. Kantipudi

Minimizing N -Detect Tests for Combinational Circuits Master’s Defense Kalyana R. Kantipudi. Thesis Advisor: Dr. Vishwani D. Agrawal Thesis Committee: Dr. Charles E. Stroud and Dr. Victor P. Nelson Dept. of ECE, Auburn University. Outline. Background Problem Statement Contributions

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Minimizing N -Detect Tests for Combinational Circuits Master’s Defense Kalyana R. Kantipudi

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  1. Minimizing N-Detect Tests for Combinational CircuitsMaster’s DefenseKalyana R. Kantipudi Thesis Advisor:Dr. Vishwani D. Agrawal Thesis Committee:Dr. Charles E. Stroud and Dr. Victor P. Nelson Dept. of ECE, Auburn University MS Thesis Defense

  2. Outline • Background • Problem Statement • Contributions • Theoretical Minimum for N-Detect Tests • ILP Based N-Detect Test Minimization • Relaxed LP based methods • The New Recursive Rounding Approach • Conclusions • Future work MS Thesis Defense

  3. Background • Defects are modeled as faults • Single stuck-at faults ease the test generation process • Bridging faults emulate the defects more accurately • Test sets with greater than 95% fault coverage can produce only 33% coverage of node-to-node bridging faults (Krishnaswamy et al. ITC’01) • About 80% of all bridges occur between a node and Vcc or Vss MS Thesis Defense

  4. N-Detect Tests • Some applications need much lower DPM • New test strategy which can be easily assimilated into the normal test generation process • The problem with N-detect tests is their size • There is no accurate way to achieve a minimal N-detect set • There is no proven lower bound on the size of the N-detect vectors MS Thesis Defense

  5. Problem Statement • To find a lower bound on the size of N-detect tests • To find an exact method for minimizing a given N-detect test set • To derive a polynomial time heuristic algorithm for the N-detect test minimization problem MS Thesis Defense

  6. The Independence Graph • Independence graph: Nodes are faults and edges represent pair-wise independence relationships • A clique is a fully connected sub-graph Example: c17 A. S. Doshi, “Independence Fault Collapsing and Concurrent Test Generation,” Master’s thesis, Auburn University, May 2006. MS Thesis Defense

  7. 1 4 2 5 Lower Bound on Single-Detection Tests • The Independent Fault Set (IFS) is a maximal clique in the graph • Theorem 1: The size of the IFS is a lower bound on the single detection test set size (Akers et al., ITC-87) So, the lower bound for the single detection test set of c17 is ‘4’. MS Thesis Defense

  8. N test Vecs N test Vecs 1 N test Vecs 5 2 1 4 2 5 N test Vecs 4 Theoretical Minimum of an N-Detect Test Set • Theorem 2: The lower bound on the size of the N-detect test set is N times the size of the largest clique in the independence graph (Original Contribution) So, at least 4N vectors are needed to detect each fault ‘N’ times. MS Thesis Defense

  9. Minimized N-Detect Vectors for 74181 ALU MS Thesis Defense

  10. ILP Based N-Detect Test Minimization • Use any N-detect test generation approach to obtain a set of k vectors which detect every fault at least N times. • Use diagnostic fault simulation to get the vector subset Tj for each fault j. • Assign integer variable ti to ith vector such that, • ti = 1 if ith vector is included in the minimal set. • ti = 0 if ith vector is not included. MS Thesis Defense

  11. Objective and Constraints of ILP Njis the multiplicity of detection for the jth fault. Nj can be selected for individual faults based on some criticality criteria or on the capability of the initial vector set. Theorem 3: When the minimization is performed over an exhaustive set of vectors, an ILP solution that satisfies the above expressions is a minimum N-detect test. MS Thesis Defense

  12. Derivation of N-Detect Tests • Generate an unoptimized M-detect test set (M N) using an ATPG (e.g., ATALANTA). • Remove repeated vectors. • Perform diagnostic fault simulation of the remaining vectors using a fault simulator (e.g., HOPE). • If |Tj | <N for any fault, obtain additional vectors for that fault. • Generate ILP constraints and use an ILP solver to determine the values of the variables ti that minimize the number of vectors =Σti . MS Thesis Defense

  13. Minimal 3-Detect Test Set for c17 Fault Numbers • ATALANTA is used to generate 4 test sets (M = 4 iterations) and the repeated vectors are removed. • HOPE is used to perform diagnostic fault simulation on the remaining vectors. • The simulation information is used to create constraints for the ILP MS Thesis Defense

  14. Constraint Generation • Fault 1 is detected by the vectors 1, 2, 15, 16, 22, 24. • Fault 2 is detected by the vectors 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 22, 24, 28, 29..... so on .... Now the Objective is: and the constraints are: Constraint for fault 1: t1+t2+t15+t16+t22+t24≥ 3 Constraint for fault 21: t13+t15+t16+t19+t23+t24 ≥ 3 MS Thesis Defense

  15. Minimum Test Sets from ILP • The minimum 3-detect test set size is 13 (lower bound = 12). • Vectors are: 2, 6, 7, 11, 14, 15, 16, 17, 18, 21, 23, 24, 28. Suppose ‘fault 21’ is a critical fault to be detected 5 times: Constraint for fault 21: t13+t15+t16+t19+t23+t24 • The minimum test set given by ILP has 14 vectors. • Vectors are: 2, 6, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 28. For large circuits this change in test size can be quite small. MS Thesis Defense

  16. Results Results on Ultra-5 * Ultra-10 MS Thesis Defense

  17. Results for 15-Detect Tests c499, c1355, c1908 - Type – I C880,c2670,c7552 - Type – II Results on Ultra-5 * Ultra-10 ** Sun Fire 280R [1] Lee, Cobb, Dworak, Grimaila and Mercer, Proc. DATE, 2002 [2] Hamzaoglu and Patel, IEEECAD, 2000. MS Thesis Defense

  18. Minimized Vectors for 15-Detect Tests MS Thesis Defense

  19. CPU Time for Minimizing 15-Detect Tests MS Thesis Defense

  20. Classifying Combinational Circuits TYPE - I: TYPE – II: Output cones have large overlap. Any vector detecting a fault F2 will have high probability of detecting other faults, say fault F3 or F1. Non-overlapping output cones. Any vector detecting a particular fault, will have very low probability of detecting any other fault. c499, c1355, c1908 c880, c2670, c7552 MS Thesis Defense

  21. 1-b 1-b 1-b 1-b Ripple Carry Adders Iterations: Number of times test sets are taken from Atalanta ATPG MS Thesis Defense

  22. Relaxed-LP Approach • Though ILP guarantees an optimal solution, it takes exponential time to generate the solution. • Time bounded ILP solutions deviate from optimality. • LP takes polynomial time (sometimes in linear time) to generate a solution. • Redefining the variables tis as real variables in the range [0.0,1.0] converts the ILP problem into a linear one. • The problem now remains to convert it into an ILP solution. • The optimal value of the relaxed-LP of the ILP minimization problem is a lower bound on the value of the optimal integer solution to the problem. MS Thesis Defense

  23. Previous Solutions (Randomized Rounding) • The real variables are treated as probabilities. • A random number xi uniformly distributed over the range [0.0,1.0] is generated for each variable ti. • If ti≥ xi then ti is rounded to 1, otherwise rounded to 0. • If the rounded variables satisfy the constraints, then the rounded solution is accepted. • Otherwise, rounding is again performed starting from the original LP solution. MS Thesis Defense

  24. Limitations of Randomized Rounding • Consider three faults f1,f2 and f3, and three vectors. • We assign a real variable ti to vector i. • Now the single detection problem is specified as: • Minimize t1 + t2 + t3 • Subject to constraints, • f1 : t1 + t2≥ 1 • f2 : t2 + t3 ≥ 1 • f3 : t3 + t1 ≥ 1 • The number of tests is much larger than the size of the minimal test set. • The randomized rounding becomes a random search. MS Thesis Defense

  25. Recursive Rounding (New Method) • Step 1: Obtain an LP solution. Stop if each ti is either 0.0 or 1.0 • Step 2: Round the largest ti and fix its value to 1.0 If several ti’s have the largest value, arbitrarily set only one to 1.0. Go to Step 1. • Maximum number of LP runs is bounded by the final minimized test set size. • Final set is guaranteed to cover all faults. • This method takes polynomial time even in the worst case. • LP provides a lower bound on solution. Lower Bound ≤ exact ILP solution ≤ recursive LP solution Absolute optimality is not guaranteed. MS Thesis Defense

  26. The 3V3F Example • Step 1: LP gives t1 = t2 = t3 = 0.5 • Step 2: We arbitrarily set t1 = 1.0 • Step 1: Gives t2 = 1, t3 = 0 ■ or t2 = 0, t3 = 1 ■ or t2 = t3 = 0.5 • Step 2: (last case) We arbitrarily set t2 = 1.0 • Step 1: Gives t3 = 0 MS Thesis Defense

  27. Minimal Tests for Array Multipliers • There exists a huge difference between its theoretical lower bound of six and its practically achieved test set of size 12. • A 15 x 16 matrix of full-adders (FA) and half-adders (HA). • To make use of its recursive structure and apply linear programming techniques. MS Thesis Defense

  28. Tests for c6288: 16-Bit Multiplier • Known results (Hamzaoglu and Patel, IEEE-TCAD, 2000): • Theoretical lower bound = 6 vectors • Smallest known set = 12 vectors, 306 CPU s • Our results: • Up to four-bit multipliers need six vectors • Five-bit multiplier requires seven vectors • c6288 • 900 vectors constructed from optimized vector sets of smaller multipliers • ILP, 10 vectors in two days of CPU time • Recursive LP, lower bound = 7, optimized set = 12, in 301 CPU s MS Thesis Defense

  29. Comparison of ILP and Recursive LP method MS Thesis Defense

  30. Sizes of 5-Detect Tests for ISCAS85 Circuits MS Thesis Defense

  31. Time Taken for 5-Detect Tests MS Thesis Defense

  32. Optimized 15-Detect Tests [1] Lee, Cobb, Dworak, Grimaila and Mercer, Proc. DATE, 2002 MS Thesis Defense

  33. Conclusion • A Lower Bound for N-Detect tests is derived. • An N-Detect test minimization method based on ILP is formulated which always guarantees optimality. • A polynomial time consuming recursive rounding LP, which can give close to optimal solutions for single and N-detect tests is presented. • A smallest ever, 10 vector set derived for c6288 signifies the shortcomings of present test minimization techniques. • The new recursive rounding LP method has numerous other applications where ILP is traditionally used and is found to be expensive. MS Thesis Defense

  34. Future Work • The dual problem of the test minimization problem looks promising. • The dual problem: • The Duality Theorem: If m is the minimum value of the primal problem and M is the maximum value of the dual problem, then m = M. MS Thesis Defense

  35. The Previous c17 Example • The primal problem gave a solution of 4 vectors. • The dual problem also gave a solution of 4, selecting faults 1, 10, 16 and 18. • It is observed that these four faults are independent of each other. • So the dual problem yielded an IFS of the circuit. • In cases where relaxed-LP gives non-integer solutions for the dual problem, rounding techniques can be used. • This new approach has the potential of generating much tighter lower bound compared to theIFS. MS Thesis Defense

  36. Thank You . .. MS Thesis Defense

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