Provably Optimal Test Cube Generation using Quantified ... · 01X:Approximativeformulation for Don’t Cares 01.24.13 Sven Reimer – Provably Optimal Test Cube Generation using Quantiﬁed

Provably Optimal Test Cube Generation usingQuantified Boolean Formula SolvingASP-DAC 2013

Albert-Ludwigs-Universität Freiburg

Matthias Sauer, Sven Reimer, Ilia Polian, Tobias Schubert, Bernd BeckerChair of Computer Architecture01.24.13

Motivation – Test pattern relaxation

Test cube:Parts of the pattern are unspecified (Don’t Care)Test requirements still hold

Used for:RefillingMinimizing power consumptionCompaction (e.g., Embedded Deterministic Test)

All known techniques are approximativeOur approach:

Test cube generation with maximum number of Don’t Cares⇒ Optimal test cube

Measure the quality of heuristic methods

01.24.13 Sven Reimer – Provably Optimal Test Cube Generation using Quantified Boolean Formula Solving 2 / 17

Outline

1 Motivation2 Preliminaries

Circuit encodingUnspecified valuesSensitizable paths + small delay faults

3 Optimal test cube generation4 Experimental results5 Conclusion


Circuit encoding

A

C

B

E

D

G

F

D

Boolean satisfiability (SAT) formulation in CNF:Tseitin encoding [Tseitin ’68]Additional variables for each gateLinear in circuit size


Circuit encoding

A

C

B

E

D

G

F

D



Circuit encoding

A

BE



Circuit encoding

A

BE

E ↔ ¬(A ∧B)



Circuit encoding

A

BE

E ↔ ¬(A ∧B)(A ∨ E) ∧

Clause︷︸︸︷(B ∨ E)∧(

Literal︷︸︸︷¬A ∨¬B ∨ ¬E)



Unspecified values – 01X logic [Jain et al. ’00]

A = 101X

C = 101X

B = X01X

E = X01X

D = X01X

G = 101X

F = X01X

D = X01X

Three-valued logic:001X (logic 0), 101X (logic 1), X01X (unknown)01X in SAT: 001X = (0,1), 101X = (1,0), X01X = (0,0)SAT encoding for 01X doubles size of the formulaIn example: Output F is unknown if input B is unspecified


Unspecified values – Exact formulation

A = 1

B

C = 1D

E

F

G

= 0= 1

= 1= 0

= 0= 1

= 1= 1

= 1= 1

D= 0= 1

Simulation for B= 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}︸︷︷︸

Prefix

. ϕ(A, . . . ,G)︸︷︷︸Tseitin encoding

∧(A)∧ (C)∧ (F )︸︷︷︸property

QBF: reconvergent paths are resolved by formulation01X : reconvergent paths may block propagation of values



A = 1

B = ?

C = 1D

E

F = 1

G

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}︸︷︷︸

Prefix






A = 1

B

C = 1D

E

F

G

= 0= 1

= 1= 0

= 0= 1

= 1= 1

= 1= 1

D= 0= 1

Simulation for B= 0= 1


Prefix



QBF: reconvergent paths are resolved by formulation

01X : reconvergent paths may block propagation of values



A = 101X

C = 101X

B = X01X

E = X01X

D = X01X

G = 101X

F = X01X

D = X01X


Prefix






A = 101X

C = 101X

B = X01X

E = X01X

D = X01X

G = 101X

F = X01X

D = X01X


Prefix



QBF: Exact formulation for Don’t Cares01X : Approximative formulation for Don’t Cares


Sensitizable paths + small delay faults

FF1

FF2

FF3

FF4

Sensitizable path: Transition from input to outputLength of a path according to sum of gate delays

Small delay faults: Assume additional delay for one gateOutput transition too late for clockTwo-pattern delay testThe longer the path the higher the detection quality



02

4

4

1

Clock

22

6

7Length 6Length 2

FF1

FF2

FF3

FF4

Sensitizable path: Transition from input to outputLength of a path according to sum of gate delays

Small delay faults: Assume additional delay for one gateOutput transition too late for clockTwo-pattern delay testThe longer the path the higher the detection quality



02

4

4

1

Clock

22

6

7Length 6Length 2

+δ+δ

+δ+δ

+δ

(erroneous behavior)(works fine)

Additional delay of δ = 2

FF1

FF2

FF3

FF4

Sensitizable path: Transition from input to outputLength of a path according to sum of gate delaysSmall delay faults: Assume additional delay for one gateOutput transition too late for clockTwo-pattern delay testThe longer the path the higher the detection quality


Optimal test cube generation

Primary inputs

Two-pattern delay test

Fli

p-Fl

ops

Fli

p-F

lops

p1 ... pipi+1 ... pm

x

x

x

x

Small delay faults over two timeframesTest cube with maximum number of unspecified inputs using QBFQuantify unspecified inputs universally, specified ones existentiallyIf path for small delay fault is sensitizable:Universally quantified inputs: excluded from test cubeExistential quantified inputs: test cubeBut: The quantifier of a variable cannot be changed in QBFUnspecified inputs are unidentified a-prioriWhich inputs have to be quantified universally?


Multiplexed inputs

Primary inputsE1 A1

MUX1S1

...

En An

Sn MUXn..

.

Multiplexed Inputs Two-pattern delay test

0 1

0 1

Flip

-Flo

ps

Flip

-Flo

ps

p1 ... pipi+1 ... pm

x

x

x

x

ψ = ∃{S1, . . . ,Sn,E1, . . . ,En}∀{A1, . . . ,An}∃ . . .ϕCircuit ∧ϕProperty ∧ϕMUX

Dynamic choice of (un-)specified input with multiplexer

Select input Si switches between specified (Si = 0 : ∃Ei ) and unspecified(Si = 1 : ∀Ai ) for any primary input IiFind the maximum number of select inputs that can be set to 1


Maximization

Primary inputsE1 A1

MUX1S1

...

En An

Sn MUXn..

.

...

Multiplexed Inputs Two-pattern delay testMaximization

0 1

0 1

BitonicSortingNetwork

SOn

SO1

Flip

-Flo

ps

Flip

-Flo

ps

p1 ... pipi+1 ... pm

x

x

x

x

0100101

1110000

Sort select-inputs Si 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


Optimal test cube generation

Primary inputsE1 A1

MUX1S1

...

En An

Sn MUXn

...

Multiplexed Inputs Two-pattern delay testMaximization

0 1

0 1


SOn

SO1

Flip

-Flo

ps

Flip

-Flo

ps

p1 ... pipi+1 ... pm

x

x

x

x

1110000

SOkSOk+1

ψ(j) = ∃{SO1, . . . ,SOn,S1, . . . ,Sn,E1, . . . ,En}∀{A1, . . . ,An}∃ . . .ϕCircuit ∧ϕProperty ∧ϕMUX ∧ϕSorter ∧ (SOj )

⇒ Binary search over jSearch for k , such that: path is sensitizable with k unspecified inputs(SOk = 1), but not with k +1 (SOk+1 = 0)QBF solver returns assignment for outermost existential variables:S1, . . . ,Sn: unspecified inputs; remaining E1, . . . ,En: test cubeOptimal test cube, i.e., maximum number of Don’t Cares


01X -Optimal test cube generation

Primary inputs

...

...

...

Trigger X Two-pattern delay testMaximization


SOn

SO1

Flip

-Flo

ps

Flip

-Flo

ps

p1 ... pipi+1 ... pm

x

x

x

x

T1

Tn

ϕ(j) = ϕCircuit︸︷︷︸01X encoding

∧ϕProperty ∧ϕTrigger ∧ϕSorter ∧ (SOj )

⇒ Binary search over jSearch for k , such that: path is sensitizable with k unspecified inputs(SOk = 1), but not with k +1 (SOk+1 = 0)If Ti = 1, corresponding input Ii is set to X01X

SAT solver returns assignment for all variables:T1, . . . ,Tn: unspecified inputs; remaining input variables: test cube01X -Optimal test cube, i.e., optimal for 01X encoding


Experimental setup

Sequential versions of ISCAS 89 and ITC 99 benchmarksSAT-based path generator PHAETON [Sauer et al. ’11]:100 longest broadside testable paths of each circuitIn-house SAT solver antom [Schubert et al. ’10] andQBF solver quantom

Cone-of-influence (COI) reductionAverage percentage of Don’t Cares (DC)


Results for ISCAS 89 & ITC 99 circuits

s000

27

s002

08

s002

98

s003

44

s003

49

s003

82

s003

86

s004

00

s004

20

s004

44

s005

10

s005

26

s006

41

s007

13

s008

20

s008

32

s008

38

s009

53

s011

96

s012

38

s014

23

s014

88

s014

94

s053

78

s132

07

s359

32

s384

17

b01

b02

b03

b04

b05

b06

b07

b08

b09

b10

b11

b12

b13

b15

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Cone-of-Influence Optimal test cube

DC


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


Comparison

0.1

1

10

100

1000

10000

QBF-opt. 01X-opt. Lifting Simulation

aver

age

runt

ime

in s

econ

ds

0%

1%

2%

3%

4%

5%

6%

7%

QBF-opt. 01X-opt. Lifting Simulation

qual

itiy

loss

*

∗Loss(Method) = 1− DCMethod−DCCOIDCOptimal−DCCOI


Conclusion

Novel technique for generation test cubes with QBFFirst approach producing test cubes with maximum numberof Don’t CaresFramework adaptable to any task that maximizes number ofunspecified linesCompare heuristic approaches with true optimumNew and fast method for 01X encoding (01X -optimal)

Future workAdapt framework to other applications and fault modelsIncrease scalability of QBF-solver


SAT + QBF

Satisfiability problem or SAT problem:Given propositional formula ϕ. Is there an assignment to thevariables, such that ϕ is satisfied?ϕ in conjunctive normal form (CNF), e.g.,ϕ(x1, . . . ,xn) = (x1∨ ¬x2︸︷︷︸

literal

)∧ (x2∨x3∨¬x4)︸︷︷︸clause

∧ . . .

Notation: ϕ(x1, . . . ,xn) = {{x1,¬x2},{x2,x3,¬x4}, . . .}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 linearsize of the circuit (Tseitin encoding)Well known NP-complete problem with enormousimprovements in the last decades


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(x1, . . . ,xn) = ∃{x1}∀{x2,x3}∃{x4} . . .∃{xn}︸︷︷︸

prefix

.ϕ(x1, . . . ,xn)︸︷︷︸matrix (CNF)

Semantics (for this example):ψ is satisfied iff there exists one assignment for x1 such thatfor every assignment of x2 and x3, there exists oneassignment for x4 and so forth, such that ϕ is satisfied.PSPACE-complete problem with increasing interest in thelast decade


Circuit encoding

Circuit to propositional formulae in CNF viaTseitin encoding [Tseitin ’68]

Introduces additional Tseitin variablesResulting formula is linear in circuit size


Circuit encoding

A

B

CD

D ↔ B ∧ C{{B,¬D}, {C,¬D}, {¬B,¬C,D}}

Circuit to propositional formulae in CNF viaTseitin encoding [Tseitin ’68]Introduces additional Tseitin variables

Resulting formula is linear in circuit size


Circuit encoding

A

B

CD

D ↔ B ∧ C{{B,¬D}, {C,¬D}, {¬B,¬C,D}}

F ↔ D ∨ E{{¬D,F}, {¬E,F}, {D,E,¬F}}

E

F

G

G ↔ A ∨ E{{¬A,G}, {¬E,G}, {A,E,¬G}}E ↔ ¬(A ∧B)

{{A,E}, {B,E}, {¬A,¬B,¬E}}

Circuit to propositional formulae in CNF viaTseitin encoding [Tseitin ’68]Introduces additional Tseitin variablesResulting formula is linear in circuit size


Encode small delay faults

A

B

CD

D ↔ B ∧ C{{B,¬D}, {C,¬D}, {¬B,¬C,D}}

F ↔ D ∨ E{{¬D,F}, {¬E,F}, {D,E,¬F}}

E

F

G

G ↔ A ∨ E{{¬A,G}, {¬E,G}, {A,E,¬G}}E ↔ ¬(A ∧B)

{{A,E}, {B,E}, {¬A,¬B,¬E}}

Encode both timeframes . . .

. . . and trigger path with unit clauses(in this example: {{¬C1},{C2},{¬D1},{D2},{¬F1},{F2}})



A

B

C[D1, D2]

{{B2,¬D2}, {C2,¬D2}, {¬B2,¬C2, D2}}

{{¬D2, F2}, {¬E2, F2}, {D2, E2,¬F2}}

[F1, F2]

[G1, G2]

{{¬A2, G2}, {¬E2, G2}, {A2, E2,¬G2}}

{{B1,¬D1}, {C1,¬D1}, {¬B1,¬C1, D1}}

{{¬D1, F1}, {¬E1, F1}, {D1, E1,¬F1}}

{{¬A1, G1}, {¬E1, G1}, {A1, E1,¬G1}}

[E1, E2][A1, A2]

[B1, B2]

[C1, C2]





A

B

C[D1, D2]

{{¬D2, F2}, {¬E2, F2}, {D2, E2,¬F2}}

[F1, F2]

{{¬D1, F1}, {¬E1, F1}, {D1, E1,¬F1}}[B1, B2]

[C1, C2]




Literature

[Batcher ’68] K. E. Batcher, “Sorting networks and their applications,” in AFIPS Spring

Joint Computing Conference, pp. 307–314, ACM, 1968.

[Jain et al. ’00] A. Jain, V. Boppana, R. Mukherjee, J. Jain, M. Fujita, and M. S. Hsiao, “Tes-

ting, Verification, and Diagnosis in the Presence of Unknowns,” in VLSI Test

Symp., pp. 263–269, 2000.

[Ravi, Somenzi ’04] K. Ravi and F. Somenzi, “Minimal assignments for bounded model

checking,” in Tools and Algorithms for the Construction and Analysis of Sys-

tems, vol. 2988, pp. 31–45, Springer, 2004.

[Sauer et al. ’11] M. Sauer, A. Czutro, T. Schubert, S. Hillebrecht, I. Polian, and B. Becker,

“SAT-based analysis of sensitisable paths,” in IEEE Design and Diagnostics

of Electronic Circuits and Systems, pp. 93–98, 2011.

[Schubert et al. ’10] T. Schubert, M. Lewis, and B. Becker, “antom — Solver Description,” in SAT

Race, 2010.

[Tseitin ’68] G.S. Tseitin, “On the complexity of derivations in propositional calculus,” in

Studies in Constructive Mathematics and Mathematical Logics, 1968.


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Provably Optimal Test Cube Generation using Quantified ... · 01X:Approximativeformulation for Don’t Cares 01.24.13 Sven Reimer – Provably Optimal Test Cube Generation using Quantiﬁed