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Boolean Abstraction for Temporal Logic Satisfiability A. Cimatti 1 , M. Roveri 1 , V. Schuppan 1 , S. Tonetta 2 1 FBK-irst, Trento, Italy 2 University of Lugano, Faculty of Informatics, Lugano, Switzerland CAV07, July 37, 2007, Berlin,

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SLIDE 1

Boolean Abstraction for Temporal Logic Satisfiability

  • A. Cimatti1, M. Roveri1, V. Schuppan1, S. Tonetta2

1FBK-irst, Trento, Italy 2University of Lugano, Faculty of Informatics, Lugano, Switzerland

CAV’07, July 3–7, 2007, Berlin, Germany

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SLIDE 2

Motivation

2

⇒ Property-based system design (PROSYD): work at the level of requirements. ⇒ In model checking, focus is on dealing with complexity in the model. ⇒ Satisfiability of large temporal formulas can be hard. (e.g., [Rozier, Vardi (SPIN’07)])

c 2007 V. Schuppan

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SLIDE 3

Contents

3

  • 1. Boolean Abstraction
  • 2. Pure Literal Simplification
  • 3. Extracting Unsatisfiable Cores
  • 4. Experiments

c 2007 V. Schuppan

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SLIDE 4

Boolean Abstraction

4

(well-known in SMT community)

∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

U

⇒ ∧

combination Boolean temporal formula

c 2007 V. Schuppan

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SLIDE 5

Boolean Abstraction

5

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula Boolean formula abstract

c 2007 V. Schuppan

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SLIDE 6

Boolean Abstraction

6

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula Boolean formula abstract false? = yes unsatisfiable

c 2007 V. Schuppan

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SLIDE 7

Boolean Abstraction

7

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula A1 A3 ¬A 4 Boolean formula abstract false? = no yes unsatisfiable extract prime implicant prime implicant Boolean

c 2007 V. Schuppan

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SLIDE 8

Boolean Abstraction

8

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula A1 A3

F G

∧

U

⇒ ∧

X

∨

G

¬A 4 Boolean formula abstract false? = no yes unsatisfiable extract prime implicant prime implicant Boolean concretize temporal prime implicant ¬

c 2007 V. Schuppan

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SLIDE 9

Boolean Abstraction

9

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula A1 A3

F G

∧

U

⇒ ∧

X

∨

G

¬A 4 Boolean formula abstract false? = no yes unsatisfiable extract prime implicant check satisfiability prime implicant Boolean concretize temporal prime implicant ¬

c 2007 V. Schuppan

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SLIDE 10

Boolean Abstraction

10

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula A1 A3

F G

∧

U

⇒ ∧

X

∨

G

¬A 4 SAT? yes satisfiable Boolean formula abstract false? = no yes unsatisfiable extract prime implicant check satisfiability prime implicant Boolean concretize temporal prime implicant ¬

c 2007 V. Schuppan

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SLIDE 11

Boolean Abstraction

11

(well-known in SMT community)

A1 A2 A3 A4

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

U

⇒ ∧

X

∨

G X

¬

temporal formula

∧ ∨ ¬ ⇔ ⇒ ∨

A2

U

⇒ ∧

combination Boolean fresh proposition temporal formula A1 A3

F G

∧

U

⇒ ∧

X

∨

G

¬A 4 prime implicant remove SAT? no yes satisfiable Boolean formula abstract false? = no yes unsatisfiable extract prime implicant check satisfiability prime implicant Boolean concretize temporal prime implicant ¬

c 2007 V. Schuppan

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SLIDE 12

Contents

12

  • 1. Boolean Abstraction
  • 2. Pure Literal Simplification
  • 3. Extracting Unsatisfiable Cores
  • 4. Experiments

c 2007 V. Schuppan

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SLIDE 13

Pure Literal Simplification — Propositional Logic

13

[Davis, Putnam (1960); Dunham, Fridshal, Sward (1959)]

Assume a propositional formula φ in CNF: (l1,1 ∨...∨l1,n1 ∨ p)∧...∧(lk,1 ∨...∨lk,nk ∨ p)

  • φ1 : p occurs only positively

∧ φ2

  • no occurrence of p

Then: φ is satisfiable iff p∧φ is satisfiable. (And similarly if p occurs only negatively in φ1.)

c 2007 V. Schuppan

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SLIDE 14

Pure Literal Simplification — PSL

14

Extend notion of pure literal to PSL (see paper). Let φ be a PSL formula such that p is pure positive in φ. Then: φ is satisfiable iff (Gp)∧φ is satisfiable. (And similarly if p is pure negative in φ.) (Modal logic K : [Pan, Sattler, Vardi (J. Applied Non-Classical Logics 2006)])

c 2007 V. Schuppan

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SLIDE 15

Boolean Abstraction and Pure Literal Simplification

15

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

F G

∧

U

⇒ ∧

U

⇒ ∧

X

∨

G X

∨

G

A1 A1 A2 A3 A3 A4 ¬A 4 ¬

¬

X

temporal formula Boolean formula Boolean prime implicant temporal prime implicant abstract yes unsatisfiable false? no = extract prime implicant concretize check satisfiability satisfiable yes SAT? no remove prime implicant

(Pure literal) Simplification

c 2007 V. Schuppan

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SLIDE 16

Contents

16

  • 1. Boolean Abstraction
  • 2. Pure Literal Simplification
  • 3. Extracting Unsatisfiable Cores
  • 4. Experiments

c 2007 V. Schuppan

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SLIDE 17

Unsatisfiable Cores

17

Assume φ ≡ (Gp) ∧ (F¬p) ∧ ((Xp) ∨ (XXp)) Prime implicants: (Gp) ∧ (F¬p) ∧ (Xp) (Gp) ∧ (F¬p) ∧ (XXp) They share unsatisfiable part ⇒ no need to check both! Given {φi | i ∈ I} with V

i∈I φi unsatisfiable,

any {φj | j ∈ J ⊆ I} with V

j∈J φj unsatisfiable is an unsatisfiable core.

c 2007 V. Schuppan

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SLIDE 18

Boolean Abstraction and Unsat Core Extraction

18

∧ ∨ ¬ ⇔ ⇒ ∨ ∧ ∨ ¬ ⇔ ⇒ ∨

F G

∧

F G

∧

U

⇒ ∧

U

⇒ ∧

X

∨

G X

∨

G

A1 A1 A2 A3 A3 A4 ¬A 4 ¬

¬

X

temporal formula Boolean formula Boolean prime implicant temporal prime implicant abstract yes unsatisfiable false? no = extract prime implicant concretize check satisfiability satisfiable yes SAT? no remove prime implicant

Extract/remove unsat core(s)

c 2007 V. Schuppan

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SLIDE 19

Activation Variables

19

Propositional case: [Lynce, Marques-Silva (SAT’04)]

  • 1. Assume prime implicant V

i∈I φi.

  • 2. Introduce one fresh, Boolean activation variable Ai per φi.
  • 3. Build B¨

uchi automaton B for

^

i∈I

(Ai → φi) Let J ⊆ I. B has fair path from some initial state with {Aj | j ∈ J} true iff

V

j∈J φ j is satisfiable.

Independent of how B¨ uchi automaton is constructed!

c 2007 V. Schuppan

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SLIDE 20

Extracting Unsatisfiable Cores with BDD-based Solvers

20

Let B be a B¨ uchi automaton for V

i∈I(Ai → φi).

  • 1. Let S be the set of states in B that are the start of a fair path

(e.g., Emerson-Lei).

  • 2. Restrict S to initial states in B.
  • 3. Project S onto {Ai | i ∈ I}.
  • 4. Complement S.

Now S contains the set of unsatisfiable cores of V

i∈I φi.

(We obtain all unsatisfiable cores.)

c 2007 V. Schuppan

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SLIDE 21

Extracting Unsatisfiable Cores with SAT-based Solvers

21

Let B be a B¨ uchi automaton for V

i∈I(Ai → φi).

  • 1. Let k ← 0.
  • 2. Encode feasibility of loop-free path of length k in B.
  • 3. Check satisfiability assuming {Ai | i ∈ I} is true at time 0.
  • 4. If unsat, obtain conflict in terms of assumptions {Aj | j ∈ J ⊆ I}

at time 0.

  • 5. Otherwise, increase k and repeat.

Now {φj | j ∈ J} contains an unsatisfiable core of V

i∈I φi.

(We obtain one unsatisfiable core.)

c 2007 V. Schuppan

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SLIDE 22

Contents

22

  • 1. Boolean Abstraction
  • 2. Pure Literal Simplification
  • 3. Extracting Unsatisfiable Cores
  • 4. Experiments

c 2007 V. Schuppan

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SLIDE 23

Experiments

23

Benchmarks on PSL satisfiability (Used in [Cimatti, Roveri, Semprini, Tonetta (FMCAD’06); Cimatti, Roveri, Tonetta (TACAS’07)])

  • 1. Fill typical patterns extracted from industrial specifications [Ben-David,

Orni (2005)] with random regular expressions.

  • 2. Generate benchmarks by aggregating patterns from step 1 into the fol-

lowing shapes: – large conjunction, – (large conjunction) implies (large conjunction), – (large conjunction) iff (large conjunction), – random Boolean combination. We’d love to have challenging realistic benchmarks from industry.

c 2007 V. Schuppan

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SLIDE 24

Experiments

24

Implementation – Basis: NuSMV – Translation from PSL to automata: [Cimatti, Roveri, Tonetta (TACAS’07)] – BDD-based solver: backward Emerson-Lei, dynamic reordering baseline for BDD-based approaches – SAT-based solver: incremental and complete SBMC with MiniSat [Heljanko, Junttila, Latvala (CAV’05)] baseline for SAT-based approaches Resources – Time out: 120 seconds – Memory out: 768 MB Download http://sra.itc.it/people/roveri/cav07-bapsl/

c 2007 V. Schuppan

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SLIDE 25

Results

25

Boolean abstraction vs. not

SAT BDD

to 100 10 1 0.1 to 100 10 1 0.1

Baseline [seconds] Boolean abstraction [seconds]

to 100 10 1 0.1 to 100 10 1 0.1

Baseline [seconds] Boolean abstraction [seconds]

unsat sat

c 2007 V. Schuppan

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SLIDE 26

Results

26

Pure literal simplification vs. not (without Boolean abstraction)

SAT BDD

to 100 10 1 0.1 to 100 10 1 0.1

Baseline [sec] Baseline, pure lit. [sec]

to 100 10 1 0.1 to 100 10 1 0.1

Baseline [sec] Baseline, pure lit. [sec]

unsat sat

c 2007 V. Schuppan

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SLIDE 27

Results

27

Pure literal rule vs. not (with Boolean abstraction)

SAT BDD

to 100 10 1 0.1 to 100 10 1 0.1

  • Bool. abs. [sec]
  • Bool. abs., pure lit. [sec]

to 100 10 1 0.1 to 100 10 1 0.1

  • Bool. abs. [sec]
  • Bool. abs., pure lit. [sec]

unsat sat

c 2007 V. Schuppan

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SLIDE 28

Results

28

Unsat core extraction vs. not

SAT BDD run time

to 100 10 1 0.1 to 100 10 1 0.1

  • Bool. abs. [sec]
  • Bool. abs., core extr. [sec]

to 100 10 1 0.1 to 100 10 1 0.1

  • Bool. abs. [sec]
  • Bool. abs., core extr. [sec]

search space

1000 100 10 1 1000 100 10 1

  • Bool. abs. [# prime impl.]
  • Bool. abs., core extr. [# prime impl.]

1000 100 10 1 1000 100 10 1

  • Bool. abs. [# prime impl.]
  • Bool. abs., core extr. [# prime impl.]

c 2007 V. Schuppan

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SLIDE 29

The End

29

Introduce Boolean abstraction for PSL. ⇒ Very helpful with BDD-based, unclear with SAT-based solvers. ⇒ SAT- and BDD-based approaches complementary. Extend pure literal simplification to PSL. ⇒ Very helpful, more so when applied to prime implicants. Extract unsatisfiable cores from solvers. ⇒ Reduces search space, though at the cost of run time. Much room for improvement: – Optimize extraction of unsatisfiable cores. – Reuse partial results between prime implicants. – Improve prioritization of prime implicants. – ... (and some more) ...

Thanks!

c 2007 V. Schuppan

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SLIDE 30

Backup-Slides

30

K e e p

  • u

t ! B a c k u p s l i d e s

c 2007 V. Schuppan

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SLIDE 31

Pure Literal Simplification — PSL

31

  • 1. p(¬p) is a positive (negative) occurrence of p.
  • 2. A positive occurrence of p in φ,r is a positive (negative) occurrence of

p in Xφ φ∨ψ ψ∨φ φ∧ψ ψ∧φ φ U ψ ψ U φ φ R ψ ψ R φ r ✸ → ψ s ✸ → φ r | → ψ s | → φ (and analogously for a negative occurrence of p). p is pure positive (negative) in φ iff all occurrences of p in φ are positive (negative).

c 2007 V. Schuppan

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SLIDE 32

Complete Simple Bounded Model Checking

32

[Heljanko, Junttila, Latvala (CAV’05)]

1 k ← 0; 2 while true do 3 check for contradiction at length k; 4 if contradiction then return no fair path exists fi 5 check for non-redundant path of length k; 6 if no non-redundant path then return no fair path exists fi 7 check for fair lasso-shaped path of length k; 8 if fair lasso-shaped path then return fair path exists fi 9 k++; 10

  • d

Note: all constraints added in lines 3, 5 for k are present at lines 3, 5, 7 for k′ > k.

c 2007 V. Schuppan

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SLIDE 33

Extracting Unsatisfiable Cores with SAT-based Solvers

33

literals assumed CNF "UNSAT" + conflict "SAT" + assignment SAT solver

c 2007 V. Schuppan

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SLIDE 34

Extracting Unsatisfiable Cores with SAT-based Solvers

34

literals assumed CNF "UNSAT" + conflict "SAT" + assignment

∧

j φ j such that is unsat

∧

i Ai → φi SAT solver {A } at time step 0 j at time step 0 i {A } no contradiction up to time step k

(We obtain one unsatisfiable core.)

c 2007 V. Schuppan

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SLIDE 35

Results

35

Pure literal simplification at top + prime implicant levels vs.

  • nly at top

SAT BDD

to 100 10 1 0.1 to 100 10 1 0.1

top [seconds] top + prime implicant [seconds]

to 100 10 1 0.1 to 100 10 1 0.1

top [seconds] top + prime implicant [seconds]

unsat sat

c 2007 V. Schuppan

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SLIDE 36

Results

36

SAT vs. BDD

without Boolean abstraction with Boolean abstraction

to 100 10 1 0.1 to 100 10 1 0.1

BDD (pure lit.) [seconds] SAT (pure lit.) [seconds]

to 100 10 1 0.1 to 100 10 1 0.1

BDD (BA, pure lit.) [sec.] SAT (BA, pure lit.) [seconds]

unsat sat

c 2007 V. Schuppan