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1 Non-uniform Crossover in Genetic Algorithm Methods to Speed up the Generation of Test Patterns for Sequential Circuits Michael Dimopoulos - Panagiotis Linardis Department of Informatics Aristotle University of Thessaloniki Greece

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2 Digital Circuits Combinational Logic : inputsoutputs : Memory Sequential Circuit output = f (inputs,time)

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3 Test Generation Apply a sequence of inputs to a circuit. Observe the output response and compare the response with a precomputed or expected response. Any discrepancy is said to constitute an error, the cause of which is a physical fault. TEST GENERATION TESTING In the faulty circuit, a single line/wire is S-a-0 or S-a-1. STUCK-AT Fault Model

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4 Test Problem Formulation FSM goodM=(I,O,S, δ, λ ) Problem Formulation FSM faultyM f =(I,O f,S f, δ f, λ f ) For a given list of stuck-at faults : F={f 1,f 2,…,f n } Find a sequence of input vectors V (Test Sequence) that detects the faults in F. outputs Combinational Logic inputs :::: Memory :::: ? outputs Combinational Logic inputs :::: Memory ::::

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5 ATPG Methods Stuck-at Fault Model ATPG Methods for Sequential Circuits Deterministic Simulation-based (random) Automatic Test Pattern Generation (ATPG) Genetic Algorithms Optimum Test Set: NP-Complete problem.

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6 A Simple GA for ATPG Test seq. Test seq. Crossover Mutation Random Population Initial (A) (B) (E) End Ngen < MAX_GENERATIONS Ngen = Ngen If (Ngen%2) == 0 Expand seq. (F) Age = Ngen (C) (D)

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Population Encoding of the Individuals n-input vector Sequence of m vectors n x m bit string

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8 Crossover: Effect on Sequ Circuits Crossover operation degrades to mutation k-th vector 1st vector L V vector Detecting properties are preserved Detecting properties may be completely lost parents offsprings Vectors after the k-th, strongly depend on those before the k-th

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9 Biased Crossover k-th vector 1st vector L V vector Detecting properties are preserved Detecting properties may be completely lost parents offsprings

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10 GA Test Generation Policy Slowly increase test sequ size Slowly increase test sequ size Gradually expand candidate test sequences Append one new vector every {three} generations Direct crossover to tail of test sequ Direct crossover to tail of test sequ Try to optimize newly appended vectors Use non Uniform selection probability with emphasis on tail

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11 Proposed Distribution(NonUni) Square probability distribution (normalized) for crossover selection

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12 GATPG Algorithm Create_random_population For each individual Evaluate_fsimulation(individual) Sort_population/* with descending fit. value */ ngen=0/* generation num. */ do { for (j=0, i=0; i

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13 Fitness Function fitness = if (ngen < 0.25*MAX_GENERATIONS) f1 else f2 where: f1 = 20. R1 + R2. R3 f2 = 20. R1 + R3 + R2. R4. R5 and R1 = fdetected R2 = (sequ_length – eff_length) / sequ_length R3 = factivated / (fremaining+1) R4 = (faults propagated to FFs) / (num_FF. factive. seq_length) R5 = (faults propagated to outputs) / (num_ouputs. factive. sequ_length)

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14 POPULATION = 32 MAX_GENERATIONS = 300 PCROSSOVER = 0.6 PMUTATION = 0.2 EXPAND_STEP = 1 POPULATION = 32 MAX_GENERATIONS = 300 PCROSSOVER = 0.6 PMUTATION = 0.2 EXPAND_STEP = 1 Experimental Results ISCAS89 Benchmark Circuits GATPG parameters:

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15 Uniform Sqr (GATPG) Crossover Probability Distribution GATPG vs Uniform

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16 Experimental Results Comparison with other methods Comparison with other methods

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17 Experimental Results (cont)

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18 Experimental Results (cont) Sequence Lengths GATPGHITECRudnick

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19 Hybrid Methods (b) circuit s400(a) circuit s386

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20 Conclusion Non uniform (biased) probability distribution for cut-point selection in crossover operator Non uniform (biased) probability distribution for cut-point selection in crossover operator Crossover operation degrades to mutation Crossover operation degrades to mutation GA for ATPG of Sequential Circuits: GATPG: Slowly increase test sequ size Slowly increase test sequ size Direct crossover to tail of test sequ Direct crossover to tail of test sequ

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