Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving

Transcription

1 Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving ASP-DAC 2013 Albert-Ludwigs-Universität Freiburg Matthias Sauer, Sven Reimer, Ilia Polian, Tobias Schubert, Bernd Becker Chair of Computer Architecture

2 Motivation Test pattern relaxation Test cube: Parts of the pattern are unspecified (Don t Care) Test requirements still hold Used for: Refilling Minimizing power consumption Compaction (e.g., Embedded Deterministic Test) All known techniques are approximative Our approach: Test cube generation with maximum number of Don t Cares Optimal test cube Measure the quality of heuristic methods Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 2 / 17

3 Outline 1 Motivation 2 Preliminaries Circuit encoding Unspecified values Sensitizable paths + small delay faults 3 Optimal test cube generation 4 Experimental results 5 Conclusion Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 3 / 17

4 Circuit encoding G A B C E D F D Boolean satisfiability (SAT) formulation in CNF: Tseitin encoding [Tseitin 68] Additional variables for each gate Linear in circuit size Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 4 / 17

5 Circuit encoding G A B C E D F D Boolean satisfiability (SAT) formulation in CNF: Tseitin encoding [Tseitin 68] Additional variables for each gate Linear in circuit size Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 4 / 17

6 Circuit encoding A B E Boolean satisfiability (SAT) formulation in CNF: Tseitin encoding [Tseitin 68] Additional variables for each gate Linear in circuit size Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 4 / 17

7 Circuit encoding E (A B) A B E Boolean satisfiability (SAT) formulation in CNF: Tseitin encoding [Tseitin 68] Additional variables for each gate Linear in circuit size Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 4 / 17

8 Circuit encoding Clause {}}{ (A E) (B E) ( Literal E (A B) {}}{ A B E) A B E Boolean satisfiability (SAT) formulation in CNF: Tseitin encoding [Tseitin 68] Additional variables for each gate Linear in circuit size Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 4 / 17

9 Unspecified values 01X logic [Jain et al. 00] G = 1 01X A = 1 01X E = X 01X B = X 01X C = 1 01X D = X 01X F = X 01X D = X 01X Three-valued logic: 0 01X (logic 0), 1 01X (logic 1), X 01X (unknown) 01X in SAT: 0 01X = (0,1), 1 01X = (1,0), X 01X = (0,0) SAT encoding for 01X doubles size of the formula In example: Output F is unknown if input B is unspecified Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 5 / 17

10 Unspecified values Exact formulation Simulation for B = 0 = 1 A = 1 B = 0 = 1 C = 1 E = 1 = 0 D = 0 = 1 G = 1 = 1 F = 1 = 1 D = 0 = 1 But: F can be set to 1, even if B is unspecified Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 6 / 17

11 Unspecified values Exact formulation A = 1 B =? C = 1 E D G F = 1 But: F can be set to 1, even if B is unspecified QBF: Universally quantified variables for unknown values {A,C} {B} {D,E,F,G}. ϕ(a,...,g) (A) (C) (F) }{{}}{{}}{{} Prefix Tseitin encoding property Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 6 / 17

12 Unspecified values Exact formulation Simulation for B = 0 = 1 A = 1 B = 0 = 1 C = 1 E = 1 = 0 D = 0 = 1 G = 1 = 1 F = 1 = 1 D = 0 = 1 But: F can be set to 1, even if B is unspecified QBF: Universally quantified variables for unknown values {A,C} {B} {D,E,F,G}. ϕ(a,...,g) (A) (C) (F) }{{}}{{}}{{} Prefix Tseitin encoding property QBF: reconvergent paths are resolved by formulation Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 6 / 17

13 Unspecified values Exact formulation G = 1 01X A = 1 01X E = X 01X B = X 01X C = 1 01X D = X 01X F = X 01X D = X 01X But: F can be set to 1, even if B is unspecified QBF: Universally quantified variables for unknown values {A,C} {B} {D,E,F,G}. ϕ(a,...,g) (A) (C) (F) }{{}}{{}}{{} Prefix Tseitin encoding property QBF: reconvergent paths are resolved by formulation 01X: reconvergent paths may block propagation of values Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 6 / 17

14 Unspecified values Exact formulation G = 1 01X A = 1 01X E = X 01X B = X 01X C = 1 01X D = X 01X F = X 01X D = X 01X But: F can be set to 1, even if B is unspecified QBF: Universally quantified variables for unknown values {A,C} {B} {D,E,F,G}. ϕ(a,...,g) (A) (C) (F) }{{}}{{}}{{} Prefix Tseitin encoding property QBF: Exact formulation for Don t Cares 01X: Approximative formulation for Don t Cares Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 6 / 17

15 Sensitizable paths + small delay faults F F 3 F F 1 F F 2 F F 4 Sensitizable path: Transition from input to output Length of a path according to sum of gate delays Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 7 / 17

16 Sensitizable paths + small delay faults Length 6 Length F F 3 0 F F 1 F F F F Clock Sensitizable path: Transition from input to output Length of a path according to sum of gate delays Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 7 / 17

17 Sensitizable paths + small delay faults Length 6+δ Length 2+δ (erroneous behavior) (works fine) 4 F F F F 1 F F δ 4 F F 4 6 +δ 2 +δ Additional delay of δ = 2 Clock Sensitizable path: Transition from input to output Length of a path according to sum of gate delays Small delay faults: Assume additional delay for one gate Output transition too late for clock Two-pattern delay test The longer the path the higher the detection quality Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 7 / 17

18 Optimal test cube generation Primary inputs Flip-Flops p x 1... p x p i+1 i... p m x x Flip-Flops Two-pattern delay test Small delay faults over two timeframes Test cube with maximum number of unspecified inputs using QBF Quantify unspecified inputs universally, specified ones existentially If path for small delay fault is sensitizable: Universally quantified inputs: excluded from test cube Existential quantified inputs: test cube But: The quantifier of a variable cannot be changed in QBF Unspecified inputs are unidentified a-priori Which inputs have to be quantified universally? Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 8 / 17

19 Multiplexed inputs S 1 E 1 A MUX 1 Primary inputs S n E n A n 0 1 MUX n Flip-Flops Flip-Flops p x 1... p i p x i+1... p x m x Multiplexed Inputs Two-pattern delay test ψ = {S 1,...,S n,e 1,...,E n } {A 1,...,A n }...ϕ Circuit ϕ Property ϕ MUX Dynamic choice of (un-)specified input with multiplexer Select input S i switches between specified (S i = 0 : E i ) and unspecified (S i = 1 : A i ) for any primary input I i Find the maximum number of select inputs that can be set to Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 9 / 17

20 Maximization SO 1... SO n Bitonic Sorting Network S S n E 1 A MUX 1... E n A n 0 1 MUX n Flip-Flops Primary inputs Flip-Flops p x 1... p i p x i+1... p x m x Maximization Multiplexed Inputs Two-pattern delay test Sort select-inputs S i with Bitonic sorting network [Batcher 68] Circuit size of sorter: O(n logn) Input vector S is sorted by 1 s and 0 s Sorted output vector SO Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 10 / 17

21 Optimal test cube generation SO SO 1 k Bitonic SO 0 k+1 Sorting 0Network 0 0 SO n S 1... S n E 1 A MUX 1... E n A n 0 1 MUX n Flip-Flops Primary inputs Flip-Flops p x 1... p i p x i+1... p x m x Maximization Multiplexed Inputs Two-pattern delay test ψ(j) = {SO 1,...,SO n,s 1,...,S n,e 1,...,E n } {A 1,...,A n }... Binary search over j ϕ Circuit ϕ Property ϕ MUX ϕ Sorter (SO j ) Search for k, such that: path is sensitizable with k unspecified inputs (SO k = 1), but not with k + 1 (SO k+1 = 0) QBF solver returns assignment for outermost existential variables: S 1,...,S n : unspecified inputs; remaining E 1,...,E n : test cube Optimal test cube, i.e., maximum number of Don t Cares Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 11 / 17

22 01X-Optimal test cube generation Primary inputs SO 1 T 1... SO n Bitonic Sorting Network... T n... Flip-Flops Flip-Flops p x 1... p i p x i+1... p x m x Maximization Trigger X Two-pattern delay test ϕ(j) = ϕ } Circuit ϕ {{} Property ϕ Trigger ϕ Sorter (SO j ) 01X encoding Binary search over j Search for k, such that: path is sensitizable with k unspecified inputs (SO k = 1), but not with k + 1 (SO k+1 = 0) If T i = 1, corresponding input I i is set to X 01X SAT solver returns assignment for all variables: T 1,...,T n : unspecified inputs; remaining input variables: test cube 01X-Optimal test cube, i.e., optimal for 01X encoding Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 12 / 17

23 Experimental setup Sequential versions of ISCAS 89 and ITC 99 benchmarks SAT-based path generator PHAETON [Sauer et al. 11]: 100 longest broadside testable paths of each circuit In-house SAT solver antom [Schubert et al. 10] and QBF solver quantom Cone-of-influence (COI) reduction Average percentage of Don t Cares (DC) Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 13 / 17

24 Results for ISCAS 89 & ITC 99 circuits 100% 90% Cone-of-Influence Optimal test cube 80% 70% 60% 50% 40% 30% 20% 10% 0% s00027 s00208 s00298 s00344 s00349 s00382 s00386 s00400 s00420 s00444 s00510 s00526 s00641 s00713 s00820 s00832 s00838 s00953 s01196 s01238 s01423 s01488 s01494 s05378 s13207 s35932 s38417 b01 b02 b03 b04 b05 b06 b07 b08 b09 b10 b11 b12 b13 b15 DC Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 14 / 17

25 Comparison of QBF-optimal result Static (initial test pattern needed): 1. Lifting [Ravi, Somenzi 04] (best case QBF-optimal) 2. Simulation (best case 01X-optimal) Average over 100 random initial test patterns Dynamic (find test cube directly with given test requirements): 3. 01X-optimal Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 15 / 17

26 Comparison % % average runtime in seconds qualitiy loss* 5% 4% 3% 2% 1% 0.1 0% QBF-opt. 01X-opt. Lifting Simulation QBF-opt. 01X-opt. Lifting Simulation Loss(Method) = 1 DC Method DC COI DC Optimal DC COI Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 16 / 17

27 Conclusion Novel technique for generation test cubes with QBF First approach producing test cubes with maximum number of Don t Cares Framework adaptable to any task that maximizes number of unspecified lines Compare heuristic approaches with true optimum New and fast method for 01X encoding (01X-optimal) Future work Adapt framework to other applications and fault models Increase scalability of QBF-solver Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 17 / 17

28 SAT + QBF Satisfiability problem or SAT problem: Given propositional formula ϕ. Is there an assignment to the variables, such that ϕ is satisfied? ϕ in conjunctive normal form (CNF), e.g., ϕ(x 1,...,x n ) = (x 1 x }{{} 2 ) (x 2 x 3 x 4 )... }{{} literal clause Notation: ϕ(x 1,...,x n ) = {{x 1, x 2 },{x 2,x 3, x 4 },...} Properties of CNF: Clause is satisfied iff at least one literal is assigned to 1. CNF is satisfied iff all clauses are satisfied. Combinational circuits can be transformed into CNF in linear size of the circuit (Tseitin encoding) Well known NP-complete problem with enormous improvements in the last decades Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 18 / 17

29 SAT + QBF Quantified Boolean formula (QBF) is an extension of SAT: variables are quantified existentially ( ) or universally ( ) Example for a QBF ψ in prenex normal form: ψ(x 1,...,x n ) = {x 1 } {x 2,x 3 } {x 4 }... {x n }.ϕ(x }{{} 1,...,x n ) }{{} prefix matrix (CNF) Semantics (for this example): ψ is satisfied iff there exists one assignment for x 1 such that for every assignment of x 2 and x 3, there exists one assignment for x 4 and so forth, such that ϕ is satisfied. PSPACE-complete problem with increasing interest in the last decade Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 19 / 17

30 Circuit encoding Circuit to propositional formulae in CNF via Tseitin encoding [Tseitin 68] Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 20 / 17

31 Circuit encoding A B D C D B C {{B, D}, {C, D}, { B, C, D}} Circuit to propositional formulae in CNF via Tseitin encoding [Tseitin 68] Introduces additional Tseitin variables Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 20 / 17

32 Circuit encoding E (A B) {{A, E}, {B, E}, { A, B, E}} A B E D C D B C {{B, D}, {C, D}, { B, C, D}} G A E {{ A, G}, { E, G}, {A, E, G}} G F D E {{ D, F }, { E, F }, {D, E, F }} F Circuit to propositional formulae in CNF via Tseitin encoding [Tseitin 68] Introduces additional Tseitin variables Resulting formula is linear in circuit size Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 20 / 17

33 Encode small delay faults E (A B) {{A, E}, {B, E}, { A, B, E}} A B E D C D B C {{B, D}, {C, D}, { B, C, D}} G A E {{ A, G}, { E, G}, {A, E, G}} G F D E {{ D, F }, { E, F }, {D, E, F }} F Encode both timeframes Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 21 / 17

34 Encode small delay faults {{ A 1, G 1}, { E 1, G 1}, {A 1, E 1, G 1}} {{ A 2, G 2}, { E 2, G 2}, {A 2, E 2, G 2}} [A 1, A 2 ] A [B 1, B 2 ] B [C 1, C 2 ] C [E [G 1, G 2 ] 1, E 2 ] {{ D 1, F 1}, { E 1, F 1}, {D 1, E 1, F 1}} {{ D 2, F 2}, { E 2, F 2}, {D 2, E 2, F 2}} [F 1, F 2 ] [D 1, D 2 ] {{B 1, D 1}, {C 1, D 1}, { B 1, C 1, D 1}} {{B 2, D 2}, {C 2, D 2}, { B 2, C 2, D 2}} Encode both timeframes Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 21 / 17

35 Encode small delay faults A [B 1, B 2 ] B [C 1, C 2 ] C [D 1, D 2 ] {{ D 1, F 1}, { E 1, F 1}, {D 1, E 1, F 1}} {{ D 2, F 2}, { E 2, F 2}, {D 2, E 2, F 2}} [F 1, F 2 ] Encode both timeframes and trigger path with unit clauses (in this example: {{ C 1 },{C 2 },{ D 1 },{D 2 },{ F 1 },{F 2 }}) Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 21 / 17

36 Literature [Batcher 68] [Jain et al. 00] K. E. Batcher, Sorting networks and their applications, in AFIPS Spring Joint Computing Conference, pp , ACM, A. Jain, V. Boppana, R. Mukherjee, J. Jain, M. Fujita, and M. S. Hsiao, Testing, Verification, and Diagnosis in the Presence of Unknowns, in VLSI Test [Ravi, Somenzi 04] Symp., pp , K. Ravi and F. Somenzi, Minimal assignments for bounded model [Sauer et al. 11] checking, in Tools and Algorithms for the Construction and Analysis of Systems, vol. 2988, pp , Springer, M. Sauer, A. Czutro, T. Schubert, S. Hillebrecht, I. Polian, and B. Becker, SAT-based analysis of sensitisable paths, in IEEE Design and Diagnostics [Schubert et al. 10] [Tseitin 68] of Electronic Circuits and Systems, pp , T. Schubert, M. Lewis, and B. Becker, antom Solver Description, in SAT Race, G.S. Tseitin, On the complexity of derivations in propositional calculus, in Studies in Constructive Mathematics and Mathematical Logics, Sven Reimer Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 22 / 17