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Non-uniform Crossover in Non-uniform Crossover in Genetic Algorithm Methods to Genetic Algorithm Methods to

Speed up the Generation of Test Speed up the Generation of Test Patterns for Sequential CircuitsPatterns for Sequential Circuits

Non-uniform Crossover in Non-uniform Crossover in Genetic Algorithm Methods to Genetic Algorithm Methods to

Speed up the Generation of Test Speed up the Generation of Test Patterns for Sequential CircuitsPatterns for Sequential Circuits Michael Dimopoulos - Panagiotis

Linardis

Department of InformaticsAristotle University of Thessaloniki

Greece

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Digital CircuitsDigital Circuits

CombinationalLogic: :

inputs outputs

: :M

emory

Sequential Circuit

output = f (inputs,time)

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Test GenerationTest GenerationTest GenerationTest 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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Test Problem FormulationTest Problem Formulation

FSM good M=(I,O,S,δ,λ)

Problem Formulation

FSM faulty Mf=(I,Of,Sf,δf,λf)

For a given list of stuck-at faults : F={f1,f2,…,fn}

Find a sequence of input vectors V

(Test Sequence) that detects the faults in F.

outp

uts

CombinationalLogicin

put

s

: :

Memory

: :

?

outp

uts

CombinationalLogicin

put

s

: :

Memory

: :

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ATPG MethodsATPG 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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A Simple GA for ATPGA Simple GA for ATPGA Simple GA for ATPGA Simple GA for ATPG

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

Test

seq.

Test seq.

Crossover0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 1

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

1 0 11

01

01

01

00

10

11

11

10

10

11

01

10

01

01

01

01

01

00

10

11

11

10

10

11

01

10

10

11

Mutation

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

Random Population

Initial

(A)(B)

(E)

End

Ngen <

MAX_GENERATIONS

Ngen = Ngen+1

1 1 0 1

If (Ngen%2) == 0

Expand seq.

(F)

Age = Ngen

(C)

(D)

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0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

0 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 00 1 0 10 1 0 01 0 1 11 1 1 01 0 1 10 1 1 0

Population

Encoding of the IndividualsEncoding of the IndividualsEncoding of the IndividualsEncoding of the Individuals

0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0n-input vector

Sequence of m vectors

n x m bit string

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Crossover:Crossover: Effect on Sequ Circuits Effect on Sequ Circuits

Crossover operation degrades to mutationCrossover operation degrades to mutation

k-th vector

1st vector

LV vector

Detecting properties

arepreserved

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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Biased CrossoverBiased Crossover

k-th vector

1st vector

LV vector

Detecting properties

arepreserved

Detecting properties

may be completely

lostparents offsprings

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GA Test Generation Policy

Slowly increase test sequ sizeSlowly increase test sequ size• Gradually expand candidate test sequences

• Append one new vector every {three} generations

Direct crossover to tail of test sequDirect 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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Proposed Distribution(NonUni)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vector selection (norm)

P

Square probability distribution (normalized) for crossover selection

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GATPG AlgorithmGATPG AlgorithmGATPG AlgorithmGATPG 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<ncross; j +=2, i++) /**** crossover ****/ { cross_over(Individual[j], Individual[j+1], child1, child2) Evaluate_ fsimulation (child1) Evaluate_ fsimulation (child2) } for (i=0; i<nmut; i +=2) /**** mutation ****/ { mutation(Individual[0], child1) mutation(Individual[1], child2) Evaluate_ fsimulation (child1) Evaluate_ fsimulation (child2) } Sort_population If ( (ngen % 3) == 0 ) { Expand_sequence(EXPAND_STEP) Evaluate_fsim(Individual[0]) /* check best */ } ngen++ } while (ngen<MAX_GENERATIONS)

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Fitness FunctionFitness FunctionFitness FunctionFitness Function

fitnessfitness = if (ngen < 0.25*MAX_GENERATIONS) f1 else f2 = 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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POPULATION = 32 MAX_GENERATIONS = 300PCROSSOVER = 0.6PMUTATION = 0.2 EXPAND_STEP = 1

POPULATION = 32 MAX_GENERATIONS = 300PCROSSOVER = 0.6PMUTATION = 0.2 EXPAND_STEP = 1

Experimental ResultsExperimental ResultsExperimental ResultsExperimental Results

ISCAS’89 Benchmark CircuitsISCAS’89 Benchmark Circuits

a/a circuit i / o / ff / gates Total Faults

1 s298 3 / 6 / 14 / 119 265

2 s344 9 / 11 / 15 / 160 329

3 s349 9 / 11 / 15 / 161 335

4 s382 3 / 6 / 21 / 158 364

5 s386 7 / 7 / 6 / 159 314

6 s400 3 / 6 / 21 / 164 384

7 s444 3 / 6 / 21 / 181 424

a/a circuit i / o / ff / gates Total Faults

1 s298 3 / 6 / 14 / 119 265

2 s344 9 / 11 / 15 / 160 329

3 s349 9 / 11 / 15 / 161 335

4 s382 3 / 6 / 21 / 158 364

5 s386 7 / 7 / 6 / 159 314

6 s400 3 / 6 / 21 / 164 384

7 s444 3 / 6 / 21 / 181 424

GATGATPGPG parameters: parameters:

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Uniform Sqr (GATPG)

Crossover Probability DistributionCrossover Probability Distribution

Circuit Det. Vec. Time (gen)s298 265 89 282s344 329 61 225s349 335 59 287s382 332 91 272s386 286 52 233s400 347 79 227s444 383 76 272

Sum: 2277 507 1798

Circuit Det. Vec. Time (gen)s298 264 79 242s344 327 56 266s349 332 51 295s382 316 88 284s386 254 39 284s400 329 86 293s444 360 91 258

Sum: 2182 490 1922

GATPG vs UniformGATPG vs UniformGATPG vs UniformGATPG vs Uniform

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Experimental ResultsExperimental ResultsExperimental ResultsExperimental Results

Comparison with other methodsComparison with other methods

Circuit f.c Vec. f.c Vec. f.c Vec. f.c Vec. f.c Vec.

s298 1,000 89 0,955 476 1,000 161 1,000 232 1,000 306s344 1,000 61 0,997 115 1,000 95 1,000 120 0,997 142s349 1,000 59 1,000 95 1,000 137

s382 0,912 91 0,753 246 0,953 281 0,997 2047 0,997 4931

s386 0,911 52 0,930 1230 0,939 154 1,000 311s400 0,904 79 0,930 758 0,951 280 0,995 2162 0,997 4309s444 0,903 76 0,936 519 0,958 275 1,000 1970 0,976 2240

Average:

fault cov. 0,947 0,917 0,972 0,998 0,995

test seq. 72 557 192 1306 1768

GATPG CRIS IGATERudnick HITEC

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Circuit GATPG HITEC Rudnick95

P-133 Sun Sparc SLC Sun Sparc II

s298 12,6m 266,4m 6,05ms344 11,5m 79,8m 5,85ms349 14,0m 52,2m 5,83ms382 15,0m 720m 8,91ms386 16,8m 1,03m 3,45m

s400 13,0m 726m 9,45ms444 14,0m 966m 10,5m

Time (CPU)

Circuit GATPG HITEC Rudnick95

P-133 Sun Sparc SLC Sun Sparc II

s298 12,6m 266,4m 6,05ms344 11,5m 79,8m 5,85ms349 14,0m 52,2m 5,83ms382 15,0m 720m 8,91ms386 16,8m 1,03m 3,45m

s400 13,0m 726m 9,45ms444 14,0m 966m 10,5m

Time (CPU)

Experimental Results Experimental Results (con’t)(con’t)Experimental Results Experimental Results (con’t)(con’t)

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Experimental Results Experimental Results (con’t)(con’t)Experimental Results Experimental Results (con’t)(con’t)

Sequence LengthsSequence Lengths

89306

161 61142

95 59137

9591

4931

281

52

311154

79

4309

280

76

2240

275

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

s298 s344 s349 s382 s386 s400 s444

GATPG HITEC Rudnick

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Hybrid Methods

HYBRID GATPG

0

50

100

150

200

250

300

350

0 100 200 300 400 500

Time (sec)

Faul

ts

HYBRID

GATPG

0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500

Time (sec.)Fa

ults

(b) circuit s400 (a) circuit s386

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Conclusion

Non uniform (biased) probability distribution for cut-point Non uniform (biased) probability distribution for cut-point

selection in crossover operatorselection in crossover operator

Crossover operation degrades to mutationCrossover operation degrades to mutation

GGAA for ATPG of Sequential for ATPG of Sequential CCircuitsircuits::

GATPG:GATPG:

Slowly increase test sequ sizeSlowly increase test sequ size

Direct crossover to tail of test sequDirect crossover to tail of test sequ