On Combining 01X-Logic and QBF Marc Herbstritt (joint work with - - PowerPoint PPT Presentation

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On Combining 01X-Logic and QBF Marc Herbstritt (joint work with - - PowerPoint PPT Presentation

Oct 14, 2022 443 likes •804 views

On Combining 01X-Logic and QBF Marc Herbstritt (joint work with Bernd Becker) Institute of Computer Science Albert-Ludwigs-University Freiburg im Breisgau, Germany Presentation at EuroCAST 2007 Applied Formal Verification Track Feb 14 2007

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SLIDE 1

On Combining 01X-Logic and QBF Marc Herbstritt

(joint work with Bernd Becker)

Institute of Computer Science Albert-Ludwigs-University Freiburg im Breisgau, Germany

Presentation at EuroCAST 2007 Applied Formal Verification Track Feb 14 2007

www.avacs.org

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SLIDE 2

Background

Formal Verification of Circuits

→ Checking correctness between specification and implementation

Model Checking

→ Specification given by a set of (temporal) properties → Model Checking to prove that circuit model fulfills the properties → Bounded Model Checking to falsify properties

Blackbox Designs

→ describe partial circuit implementations → occur naturally in early design phase → can be used for abstraction

This work:

→ Bounded Model Checking of Blackbox Designs (BB-BMC) → Merge two approaches for BB-BMC: 01X-logic and QBF → Flexibility to counteract computational complexity of QBF

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 2 / 14

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SLIDE 3

Background

Formal Verification of Circuits

→ Checking correctness between specification and implementation

Model Checking

→ Specification given by a set of (temporal) properties → Model Checking to prove that circuit model fulfills the properties → Bounded Model Checking to falsify properties

Blackbox Designs

→ describe partial circuit implementations → occur naturally in early design phase → can be used for abstraction

This work:

→ Bounded Model Checking of Blackbox Designs (BB-BMC) → Merge two approaches for BB-BMC: 01X-logic and QBF → Flexibility to counteract computational complexity of QBF

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 2 / 14

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SLIDE 4

Background

Formal Verification of Circuits

→ Checking correctness between specification and implementation

Model Checking

→ Specification given by a set of (temporal) properties → Model Checking to prove that circuit model fulfills the properties → Bounded Model Checking to falsify properties

Blackbox Designs

→ describe partial circuit implementations → occur naturally in early design phase → can be used for abstraction

This work:

→ Bounded Model Checking of Blackbox Designs (BB-BMC) → Merge two approaches for BB-BMC: 01X-logic and QBF → Flexibility to counteract computational complexity of QBF

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 2 / 14

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SLIDE 5

Background

Formal Verification of Circuits

→ Checking correctness between specification and implementation

Model Checking

→ Specification given by a set of (temporal) properties → Model Checking to prove that circuit model fulfills the properties → Bounded Model Checking to falsify properties

Blackbox Designs

→ describe partial circuit implementations → occur naturally in early design phase → can be used for abstraction

This work:

→ Bounded Model Checking of Blackbox Designs (BB-BMC) → Merge two approaches for BB-BMC: 01X-logic and QBF → Flexibility to counteract computational complexity of QBF

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 2 / 14

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SLIDE 6

Overview

1

Introduction

2

An Example: Equivalence Checking of Blackbox Designs

3

Combining 01X-Logic and QBF

4

Application to Bounded Model Checking of Blackbox Designs

5

Experimental Results

6

Conclusions

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SLIDE 7

Applications of Blackbox Designs: ISCAS c3540

BCD−ADD MUX MUX

A B Shifter

MUX MUX

ALU

C3540: ALU with binary and BCD arithmetic, logic and shift operations.

BCD−SUB (Source: Hansen, Yalcin, Hayes − Unveiling the ISCAS85 Benchmarks, IEEE Design&Test, 1999)

1

Abstraction: Hide components that are not necessary

2

Verification of Partial Designs: E.g. in early design stage

3

Error Diagnosis: Localisation of error

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 4 / 14

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SLIDE 8

Applications of Blackbox Designs: ISCAS c3540

BCD−ADD MUX MUX

A B Shifter

MUX MUX

ALU

BCD−SUB

  • p(A,B,+,bin) =

enc(A,bin) + enc(B,bin) ?

1

Abstraction: Hide components that are not necessary

2

Verification of Partial Designs: E.g. in early design stage

3

Error Diagnosis: Localisation of error

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 4 / 14

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SLIDE 9

Applications of Blackbox Designs: ISCAS c3540

enc(A, ) + enc(B, ) ?

but only on encoding

Blackbox Blackbox

MUX MUX

A B

MUX MUX

ALU Blackbox

Property is not dependent

bin bin bin

  • n BCD−units and Shifter,
  • p(A,B,+, ) =

binary

1

Abstraction: Hide components that are not necessary

2

Verification of Partial Designs: E.g. in early design stage

3

Error Diagnosis: Localisation of error

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 4 / 14

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SLIDE 10

Applications of Blackbox Designs: ISCAS c3540

Blackbox Blackbox

MUX MUX

A B

MUX MUX

ALU

Implementation of Shifter and BCD−SUB unit not finished

BCD−ADD

  • p(A,B,+,bin) =

enc(A,bin) + enc(B,bin) ?

1

Abstraction: Hide components that are not necessary

2

Verification of Partial Designs: E.g. in early design stage

3

Error Diagnosis: Localisation of error

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 4 / 14

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SLIDE 11

Applications of Blackbox Designs: ISCAS c3540

MUX MUX

A B

MUX MUX

ALU Blackbox

BCD−SUB

Shifter

within the blackbox region Check whether error lies

1

Abstraction: Hide components that are not necessary

2

Verification of Partial Designs: E.g. in early design stage

3

Error Diagnosis: Localisation of error

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 4 / 14

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SLIDE 12

Combinational Equivalence Checking using 01X-Logic

x1 x2x3 x4 f S x1 x2x3 x4 f BB

1

Can f BB

1

be completed such that f S and f BB

1

are equal,

  • r are f S and f BB

1

already different?

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 5 / 14

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SLIDE 13

Combinational Equivalence Checking using 01X-Logic

01X-Logic (Jain et al., VTS’00)

Introduce logical value X, i.e., X means unknown Binary encoding of logical values: 01X-value binary encoding (1,0) 1 (0,1) X (0,0) Adapt synthesis operators:

AND01X ((a0, a1), (b0, b1)) := (a0 + b0, a1 · b1) OR01X ((a0, a1), (b0, b1)) := (a0 · b0, a0 + b0) NOT01X ((a0, a1)) := (a1, a0)

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 5 / 14

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SLIDE 14

Combinational Equivalence Checking using 01X-Logic

XEK XEK ENC−XOR

x1 f

M,0 ENC−OR ENC−AND XEK XEK ENC−XOR ENC−AND ENC−OR ENC−OR ENC−AND

x1

1

x

1

x

1

x x x

1

x

2 2 3 3 4 4

0 0 f

M,1

Build miter for f S and f BB

1

Replace blackbox output by new logical value X Encode miter circuit using AND01X, OR01X, and NOT01X Solve propositional problem ⇒ (0, 0, 1, 0) is a counterexample

[f S(0, 0, 1, 0) = 1 = 0 = f BB

1 (0, 0, 1, 0)]

Propositional problem to solve

∃ x0

1x1 1 x0 2x1 2 x0 3x1 3 x0 4x1 4 :

((¬f M,0) · f M,1)(x0

1, x1 1, x0 2, x1 2, x0 3, x1 3, x0 4, x1 4) = 1.

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 5 / 14

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SLIDE 15

Inaccuracy of 01X-Logic and the Demand for QBF

x1 x2x3 x4 f S

x1x2x3x4 f S f BB,01X

3

0000 X 0001 X 0010 1 1 0011 1 1 0100 1 1 0101 1 1 0110 1 X 0111 1 X 1000 1 1 1001 1 1 1010 1 1 1011 1 1 1100 1 1 1101 1 1 1110 1 1 1111 1 1

x1 x2x3 x4 f BB

3

No counterexample can be found using 01X-logic.

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 6 / 14

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SLIDE 16

Inaccuracy of 01X-Logic and the Demand for QBF

QBF counterexample formulation

Instead of new logical value X, introduce fresh propositional variables Zi for blackbox output i Counterexample (CE): An input assignment such that for all values of the blackbox ouputs Zi, f S and f BB

3

differ.

∃ x1 x2 x3 x4 ∀Z1 : f S(x1, x2, x3, x4) = f BB

3 (x1, x2, x3, x4, Z1)

⇒ (0, 1, 1, 0) is a counterexample.

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 6 / 14

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SLIDE 17

Multiple Blackboxes: Combining 01X-Logic and QBF

x1 x2x3 x4 f S x1 x2x3 x4 f BB

4 BB BB 1 2

QBF formulation finds counterexample

∃ x1 x2 x3 x4 ∀Z1 Z2 : f S(x1, x2, x3, x4) = f BB

4 (x1, x2, x3, x4, Z1, Z2)

(0, 0, 0, 0) is a counterexample.

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 7 / 14

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SLIDE 18

Multiple Blackboxes: Combining 01X-Logic and QBF

f BB

4

f BB

4

x1 x2 x3 x4 f S (Z1, Z2-var) (Z1-var + 01X) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Mixing 01X-Logic and QBF formulation

BB2 can be modelled using 01X-logic ⇒ (0, 0, 0, 0) is CE.

∃ x1 x2 x3 x4 ∀Z1 : f S(x1, x2, x3, x4) = f BB

4 (x1, x2, x3, x4, Z1, X)

But how to combine 01X-logic and QBF?

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 7 / 14

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SLIDE 19

Multiple Blackboxes: Combining 01X-Logic and QBF

A first shot for mixing 01X-Logic and QBF

Apply the same quantifier to encoding variables: ∃ x0

1x1 1 x0 2x1 2 x0 3x1 3 x0 4x1 4 ∀Z0 1Z1 1 Z0 2Z1 2 : ((¬f M,0) · f M,1) = 1.

This is incorrect, since ... ∀Z0

i Z1 i checks (0, 0), (0, 1), (1, 0), (1, 1), i.e., X01X, 101X, 001X, and

(1, 1) is not an 01X-value! Zi is a propositional variable, not a 01X-variable!

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 7 / 14

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SLIDE 20

Multiple Blackboxes: Combining 01X-Logic and QBF

Correct Transformation for 01X-Logic and QBF

We know: 001X = (1, 0) 101X = (0, 1) The transformation consists of two steps:

1

∀Zi ⇒ ∀Z0

i ∃Z1 i

2

Now, when Z0

i is assigned, we have to force either 001X or 101X by

adding two clauses: (Z0

i → (¬Z1 i ))

to force 001X ((¬Z0

i ) → Z1 i )

to force 101X

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 7 / 14

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SLIDE 21

Multiple Blackboxes: Combining 01X-Logic and QBF

QBF for CEC of f S and f BB

4

∃ x0

1x1 1 x0 2x1 2 x0 3x1 3 x0 4x1 4 ∀Z0 1 ∃Z1 1 ∃T :

(((¬f M,0) · f M,1) = 1)CNF· (Z0

1 → (¬Z1 1)) · ((¬Z0 1) → Z1 1).

Pros and cons for mixing 01X-logic and QBF

Pros: Eliminating universal quantified variables Cons: Binary encoding increases number of existential quantified variables

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 7 / 14

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SLIDE 22

BMC of Blackbox Designs (QBF-BB-BMC)

Basic Modelling

Use propositional variable Z(i,j) for output j of blackbox BBi Counterexample has to be valid for all possible blackbox behaviours ⇒ variables Z(i,j) are universally quantified (∀) Counterexample states the existence of a series of input assignments leading to a state that violates the property ⇒ primary inputs x0, x1, . . . , xn are existentially quantified (∃)

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 8 / 14

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SLIDE 23

QBF-BB-BMC: Input-Output-Consistency

IOC(β, d) is a predicate that assures that timed instantiations of all combinational blackboxes behave uniform within different time frames (for β-many blackboxes and unfolding depth d).

1 1 1 1 . . . . . . . . . . . . BB xi

n−1

xi

1

xi si

k−1

si

0 si 1

BB xj

n−1

xj

1

sj

0 sj 1

sj

k−1

xj

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 9 / 14

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SLIDE 24

QBF-BB-BMC: Final QBF Formula

Simulation-driven counterexample formulation ϕCE

d

:= ∃x0 ∃s0 ∃χ0

0 ∀γ0 0 . . . ∃χ0 β−1 ∀γ0 β−1

∃x1 ∃s1 ∃χ1

0 ∀γ1 0 . . . ∃χ1 β−1 ∀γ1 β−1

. . . ∃xd−1 ∃sd−1 ∃χd−1 ∀γd−1 . . . ∃χd−1

β−1 ∀γd−1 β−1

∃sd : IOC(β, d) →

  • I(s0) · TB(s0, sd−1) · (¬P(sd))
  • c

Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 10 / 14

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SLIDE 25

QBF-BB-BMC: Final QBF Formula

Sequence of input assignments ϕCE

d

:= ∃x0 ∃s0 ∃χ0

0 ∀γ0 0 . . . ∃χ0 β−1 ∀γ0 β−1

∃x1 ∃s1 ∃χ1

0 ∀γ1 0 . . . ∃χ1 β−1 ∀γ1 β−1

. . . ∃xd−1 ∃sd−1 ∃χd−1 ∀γd−1 . . . ∃χd−1

β−1 ∀γd−1 β−1

∃sd : IOC(β, d) →

  • I(s0) · TB(s0, sd−1) · (¬P(sd))
  • c

Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 10 / 14

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SLIDE 26

QBF-BB-BMC: Final QBF Formula

Sequence of states ϕCE

d

:= ∃x0 ∃s0 ∃χ0

0 ∀γ0 0 . . . ∃χ0 β−1 ∀γ0 β−1

∃x1 ∃s1 ∃χ1

0 ∀γ1 0 . . . ∃χ1 β−1 ∀γ1 β−1

. . . ∃xd−1 ∃sd−1 ∃χd−1 ∀γd−1 . . . ∃χd−1

β−1 ∀γd−1 β−1

∃sd : IOC(β, d) →

  • I(s0) · TB(s0, sd−1) · (¬P(sd))
  • c

Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 10 / 14

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SLIDE 27

QBF-BB-BMC: Final QBF Formula

Blackbox input assignments (dependent on current state and primary inputs) ϕCE

d

:= ∃x0 ∃s0 ∃χ0

0 ∀γ0 0 . . . ∃χ0 β−1 ∀γ0 β−1

∃x1 ∃s1 ∃χ1

0 ∀γ1 0 . . . ∃χ1 β−1 ∀γ1 β−1

. . . ∃xd−1 ∃sd−1 ∃χd−1 ∀γd−1 . . . ∃χd−1

β−1 ∀γd−1 β−1

∃sd : IOC(β, d) →

  • I(s0) · TB(s0, sd−1) · (¬P(sd))
  • c

Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 10 / 14

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SLIDE 28

QBF-BB-BMC: Final QBF Formula

Universal quantification of blackbox outputs (due to falsification of realizability) ϕCE

d

:= ∃x0 ∃s0 ∃χ0

0 ∀γ0 0 . . . ∃χ0 β−1 ∀γ0 β−1

∃x1 ∃s1 ∃χ1

0 ∀γ1 0 . . . ∃χ1 β−1 ∀γ1 β−1

. . . ∃xd−1 ∃sd−1 ∃χd−1 ∀γd−1 . . . ∃χd−1

β−1 ∀γd−1 β−1

∃sd : IOC(β, d) →

  • I(s0) · TB(s0, sd−1) · (¬P(sd))
  • c

Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 10 / 14

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SLIDE 29

Experimental Results

PicoJava/biu from VIS benchmark suite introduced single defects (e.g., IPF = Input Phase Fault) added 3 blackboxes “around” the error for finite unfoldings wrt. BMC, we generated QBFs with different modelling of multiple blackboxes:

all BBs with 01X 1 BB with Zi, 2 BBs with 01X 2 BBs with Zi, 1 BB with 01X 3 BBs with Zi 3 BBs with Zi, but no encoding (MTV’06)

applied combination of QBF pre-processor qubepp and QBF solver qube3.0 (versions as of 2007-01-02 and 2006-11-23, resp.)

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 11 / 14

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SLIDE 30

Experimental Results

10 20 30 40 50 60 70 5 10 15 20

Time [s] Unfolding depth k

biu.mv.xl_ao.bb-b003-p020-IPF06-c01

3 BBs with Zi 2 BBs with Zi 1 BB with Zi pure Zi (Zi,Zi,Zi) (01X,Zi,Zi) (Zi,01X,Zi) (Zi,Zi,01X) (01X,01X,Zi) (01X,Zi,01X) (Zi,01X,01X) (01X,01X,01X)

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 11 / 14

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SLIDE 31

Experimental Results

0.01 0.1 1 10 100 1 2 3 4 5 6 7 8

Time [s] (logscale) Unfolding depth k

biu.mv.xl_ao.bb-b003-p020-IPF01-c05

3 BBs with Zi 2 BBs with Zi 1 BB with Zi pure Zi (Zi,Zi,Zi) (01X,Zi,Zi) (Zi,01X,Zi) (Zi,Zi,01X) (01X,01X,Zi) (01X,Zi,01X) (Zi,01X,01X) (01X,01X,01X)

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 11 / 14

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SLIDE 32

Conclusions and Future Work

Conclusions

We have combined 01X-logic and QBF in the context of blackbox designs. This combination enables flexible modelling of multiple blackboxes. Experimental results for BMC of blackbox designs showed the feasibility.

Future Work

Submission of selected QBF examples to QBFEVAL’07. More experimental results will be available in forthcoming technical report. ...

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 12 / 14

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SLIDE 33

Questions ⇒ Answers

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 13 / 14

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SLIDE 34

Acknowledgements and References

Acknowledgements

Christoph Scholl, Stefan Disch, and Tobias Nopper for fruitful discussions

References

Jain et al., “Testing, Verification, and Diagnosis in the Presence of Unknowns”, VTS’00 Kuehlmann et al., “Robust Boolean Reasoning for Equivalence Checking and Functional Property Verification”, TCAD’02 Scholl, Becker, “Checking Equivalence for Partial Implementations”, DAC’01 Herbstritt, Becker, “On SAT-based Bounded Invariant Checking of Blackbox Designs”, MTV’05 Herbstritt, Becker, Scholl, “Advanced SAT-Techniques for Bounded Model Checking of Blackbox Designs”, MTV’06

c Marc Herbstritt (University Freiburg) On Combining 01X-Logic and QBF AFV@EuroCAST’07 14 / 14