An Efficient and Flexible Approach to Resolution Proof Reduction N. - - PowerPoint PPT Presentation

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An Efficient and Flexible Approach to Resolution Proof Reduction N. - - PowerPoint PPT Presentation

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An Efficient and Flexible Approach to Resolution Proof Reduction N. Sharygina Formal Verification and Security Group University of Lugano March 9, 2011 Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 1 / 60 Outline 1

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SLIDE 1

An Efficient and Flexible Approach to Resolution Proof Reduction

  • N. Sharygina

Formal Verification and Security Group University of Lugano

March 9, 2011

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 1 / 60

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SLIDE 2

Outline

1 Background

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 2 / 60

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SLIDE 3

Outline

1 Background 2 Motivation and Related Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 2 / 60

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SLIDE 4

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 2 / 60

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SLIDE 5

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 2 / 60

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SLIDE 6

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 3 / 60

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SLIDE 7

Background

Formal Verification in Lugano, Switzerland

  • Program Verification

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 4 / 60

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SLIDE 8

Background

Formal Verification in Lugano, Switzerland

  • Program Verification
  • Model checking code (LoopFrog, Synergy, SatAbs (with Oxford),

FunFrog), ANSI-C

  • Efficient decision procedures as computational engines of verification

(OpenSMT)

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 4 / 60

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SLIDE 9

Background

Formal Verification in Lugano, Switzerland

  • Program Verification
  • Model checking code (LoopFrog, Synergy, SatAbs (with Oxford),

FunFrog), ANSI-C

  • Efficient decision procedures as computational engines of verification

(OpenSMT)

  • Abstractions

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 4 / 60

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SLIDE 10

Background

Formal Verification in Lugano, Switzerland

  • Program Verification
  • Model checking code (LoopFrog, Synergy, SatAbs (with Oxford),

FunFrog), ANSI-C

  • Efficient decision procedures as computational engines of verification

(OpenSMT)

  • Abstractions
  • Program Summarization [ATVA’08], [ASE’09]
  • Avoids fix-point computation by constructing symbolic abstract

transformers instead

  • Performs sound over-approximation of (unbounded) loops
  • Precision is tuned by selection of abstract domains
  • Exploits efficiency of SAT/SMT solvers

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 4 / 60

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SLIDE 11

Background

Formal Verification in Lugano, Switzerland

  • Program Termination [CAV’10, TACAS’11]
  • Integration of Loop Summarization with Termination Analysis
  • Compositional Transition Invariants avoid all paths computation of

termination checks

  • Simple abstract domains are used for termination checks

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 5 / 60

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SLIDE 12

Background

Formal Verification in Lugano, Switzerland

  • Program Termination [CAV’10, TACAS’11]
  • Integration of Loop Summarization with Termination Analysis
  • Compositional Transition Invariants avoid all paths computation of

termination checks

  • Simple abstract domains are used for termination checks
  • Synergy of Abstractions [STTT’10]
  • Interleaves precise and over-approximated abstractions
  • Reduces CEGAR iterations
  • Removes multiple counterexamples within a single refinement step
  • Localizes precise abstraction/refinement to relevant parts of the

program

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 5 / 60

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SLIDE 13

Background

Formal Verification in Lugano, Switzerland

  • Model checking mobile code [IFM’08], [JFAC’10]
  • Specification language for security policies
  • Formalization of mobile code distribution net
  • Location-specific abstractions and model checking of security policies

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 6 / 60

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SLIDE 14

Background

Formal Verification in Lugano, Switzerland

  • Model checking mobile code [IFM’08], [JFAC’10]
  • Specification language for security policies
  • Formalization of mobile code distribution net
  • Location-specific abstractions and model checking of security policies
  • Boolean and Theory Reasoning (SMT)
  • Procedure for bit-vector extraction and concatenation [ICCAD’09]
  • Reduces formulae to the theory of equality to avoid, when possible,

expensive reduction to SAT

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 6 / 60

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SLIDE 15

Background

Formal Verification in Lugano, Switzerland

  • Model checking mobile code [IFM’08], [JFAC’10]
  • Specification language for security policies
  • Formalization of mobile code distribution net
  • Location-specific abstractions and model checking of security policies
  • Boolean and Theory Reasoning (SMT)
  • Procedure for bit-vector extraction and concatenation [ICCAD’09]
  • Reduces formulae to the theory of equality to avoid, when possible,

expensive reduction to SAT

  • Generation of explanations in theory propagation [MEMOCODE’10]
  • Computes explanations on demand by reusing the consistency check

algorithm for a generic theory T.

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 6 / 60

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SLIDE 16

Background

Formal Verification in Lugano, Switzerland

  • Boolean and Theory Reasoning (SMT)
  • Generation of interpolants (for QF EUF, RDL)
  • Proof manipulation for interpolation [ICCAD’10]
  • Proof reduction [HVC’10]

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 7 / 60

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SLIDE 17

Background

Formal Verification in Lugano, Switzerland

  • Boolean and Theory Reasoning (SMT)
  • Generation of interpolants (for QF EUF, RDL)
  • Proof manipulation for interpolation [ICCAD’10]
  • Proof reduction [HVC’10]
  • Solver, OpenSMT, combines MiniSAT2 SAT-Solver with

state-of-the-art decision procedures for QF EUF, LRA, BV, RDL, IDL

  • Extensible: the SAT-to-theory interface facilites design and plug-in of

new decision procedures

  • Incremental: suitable for incremental verification
  • Open-source: available under GPL license
  • Efficient: currently the fastest open-source SMT Solver for QF UF,

IDL, RDL, LRA according to SMT-Comp’10.

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 7 / 60

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SLIDE 18

Background

Formal Verification in Lugano, Switzerland

Figure: Working Hard

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 8 / 60

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SLIDE 19

Background

Formal Verification in Lugano, Switzerland

  • Boolean and Theory Reasoning (SMT)
  • Generation of interpolants (for QF EUF, RDL)
  • Proof manipulation for interpolation [ICCAD’10]
  • Resolution proof reduction [S.F. Rollini, R. Bruttomesso, N. Sharygina,

HVC’10]

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 9 / 60

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SLIDE 20

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 10 / 60

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SLIDE 21

Proof Reduction

Motivation

  • Resolution proofs find application in several ambits

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 11 / 60

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SLIDE 22

Proof Reduction

Motivation

  • Resolution proofs find application in several ambits
  • Interpolation-based model checking
  • Abstraction techniques
  • Unsatisfiable core extraction in SAT/SMT
  • Automatic theorem proving

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 11 / 60

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SLIDE 23

Proof Reduction

Motivation

  • Resolution proofs find application in several ambits
  • Interpolation-based model checking
  • Abstraction techniques
  • Unsatisfiable core extraction in SAT/SMT
  • Automatic theorem proving
  • Problems

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 11 / 60

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SLIDE 24

Proof Reduction

Motivation

  • Resolution proofs find application in several ambits
  • Interpolation-based model checking
  • Abstraction techniques
  • Unsatisfiable core extraction in SAT/SMT
  • Automatic theorem proving
  • Problems
  • Size affects efficiency
  • Size can be exponential w.r.t. input formula

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 11 / 60

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SLIDE 25

Proof Reduction

Motivation

  • Resolution proofs find application in several ambits
  • Interpolation-based model checking
  • Abstraction techniques
  • Unsatisfiable core extraction in SAT/SMT
  • Automatic theorem proving
  • Problems
  • Size affects efficiency
  • Size can be exponential w.r.t. input formula
  • Reduction/compression of resolution proofs is important

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 11 / 60

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SLIDE 26

Related Work

Features

  • Post-processing approach

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 12 / 60

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SLIDE 27

Related Work

Features

  • Post-processing approach
  • SAT/SMT solving framework

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 12 / 60

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SLIDE 28

Related Work

Features

  • Post-processing approach
  • SAT/SMT solving framework

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 12 / 60

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SLIDE 29

Related Work

Features

  • Post-processing approach
  • SAT/SMT solving framework
  • Compression techniques

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 12 / 60

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SLIDE 30

Related Work

Features

  • Post-processing approach
  • SAT/SMT solving framework
  • Compression techniques
  • Clauses subsumption checking [Amjad07]
  • Proof reordering based on literals linking [Amjad07]
  • Proof reordering based on variable splitting [Cotton10]
  • Merging of shared substructures in subproofs [Sinz07]
  • Memoization of shared substructures [Amjad08,Cotton10]
  • Algebraic approach, resolution hypergraphs [Fontaine10]
  • Removal pivots redundancies along paths [Bar-Ilan08]

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 12 / 60

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SLIDE 31

Notation

Resolution System

  • Literal

p p

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

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SLIDE 32

Notation

Resolution System

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . .

  • Empty clause

⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

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SLIDE 33

Notation

Resolution System

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . .

  • Empty clause

⊥

  • Resolution rule

pC pD p CD

Antecedent Resolvent Pivot

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

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SLIDE 34

Notation

Resolution System

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . .

  • Empty clause

⊥

  • Resolution rule

pC pD p CD

Antecedent Resolvent Pivot

  • Resolution proof of unsatisfiability of a set of clauses S

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

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SLIDE 35

Notation

Resolution System

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . .

  • Empty clause

⊥

  • Resolution rule

pC pD p CD

Antecedent Resolvent Pivot

  • Resolution proof of unsatisfiability of a set of clauses S
  • Tree
  • Leaves as clauses of S
  • Intermediate nodes as resolvents
  • Root as unique empty clause

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

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SLIDE 36

Resolution Proofs

Example

  • Set of clauses

{pq, pq, qr, qr}

  • Proof of unsatisfiability

pq pq p q qr qr r q q ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 14 / 60

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SLIDE 37

Pivots Redundancies [Bar-Ilan08]

  • No need to resolve more than once on a pivot in a path leaf-root

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 15 / 60

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SLIDE 38

Pivots Redundancies [Bar-Ilan08]

  • No need to resolve more than once on a pivot in a path leaf-root
  • O.Bar-Ilan, O.Fuhrmann, S.Hoory, O.Shacham and O.Strichman:

RecyclePivots

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 15 / 60

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SLIDE 39

Pivots Redundancies [Bar-Ilan08]

  • No need to resolve more than once on a pivot in a path leaf-root
  • O.Bar-Ilan, O.Fuhrmann, S.Hoory, O.Shacham and O.Strichman:

RecyclePivots

  • Perform DFS from root to leaves
  • Track pivots occurrences along paths
  • In case of multiple occurrences keep the closest one to root
  • Output regular proof

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 15 / 60

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SLIDE 40

RecyclePivots

Example

pq po p qo pq q po qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 16 / 60

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SLIDE 41

RecyclePivots

Example

pq po p qo pq q po qr pq q pr p

  • r
  • o

r {r} r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 17 / 60

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SLIDE 42

RecyclePivots

Example

pq po p qo pq q po qr pq q pr p

  • r {r, o}
  • r {r}

r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 18 / 60

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SLIDE 43

RecyclePivots

Example

pq po p qo pq q po {r, o, p} qr pq q pr p

  • r {r, o}
  • r {r}

r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 19 / 60

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SLIDE 44

RecyclePivots

Example

pq po p qo {r, o, p, q} pq q po {r, o, p} qr pq q pr p

  • r {r, o}
  • r {r}

r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 20 / 60

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SLIDE 45

RecyclePivots

Example

pq po p qo {r, o, p, q} pq q po {r, o, p} qr pq q pr p

  • r {r, o}
  • r {r}

r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 21 / 60

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SLIDE 46

RecyclePivots

Example

pq po p qo {r, o, p, q} pq q po {r, o, p} qr pq q pr p

  • r {r, o}
  • r {r}

r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 22 / 60

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SLIDE 47

RecyclePivots

Example

pq pq q po qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 23 / 60

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SLIDE 48

RecyclePivots

Example

pq pq q p qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 24 / 60

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SLIDE 49

RecyclePivots

Example

pq pq q p qr pq q pr p r

  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 25 / 60

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SLIDE 50

RecyclePivots

Example

pq pq q p qr pq q pr p r

  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 26 / 60

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SLIDE 51

RecyclePivots

Example

pq pq q p qr pq q pr p r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 27 / 60

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SLIDE 52

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 28 / 60

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SLIDE 53

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 29 / 60

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SLIDE 54

Transformation Framework

Features

  • Local rewriting rules

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

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SLIDE 55

Transformation Framework

Features

  • Local rewriting rules
  • B reduction
  • A perturbation

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

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SLIDE 56

Transformation Framework

Features

  • Local rewriting rules
  • B reduction
  • A perturbation
  • Rule context

pqC pD p qCD qE q CDE

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

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SLIDE 57

Transformation Framework

Features

  • Local rewriting rules
  • B reduction
  • A perturbation
  • Rule context

pqC pD p qCD qE q CDE

  • Exhaustiveness up to symmetry

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

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SLIDE 58

Transformation Framework

Local rewriting rules

  • B rules

B1 pqC pqD p qCD pqE q pCDE ⇒ pqC pqE q pCE

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

slide-59
SLIDE 59

Transformation Framework

Local rewriting rules

  • B rules

B1 pqC pqD p qCD pqE q pCDE ⇒ pqC pqE q pCE

  • Redundancy as reintroduction variable after elimination

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

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SLIDE 60

Transformation Framework

Local rewriting rules

  • B rules

B1 pqC pqD p qCD pqE q pCDE ⇒ pqC pqE q pCE

  • Redundancy as reintroduction variable after elimination
  • Subproof simplification

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

slide-61
SLIDE 61

Transformation Framework

Local rewriting rules

  • B rules

B1 pqC pqD p qCD pqE q pCDE ⇒ pqC pqE q pCE

  • Redundancy as reintroduction variable after elimination
  • Subproof simplification
  • Subproof root strengthening

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

slide-62
SLIDE 62

Transformation Framework

Local rewriting rules

  • A rules

A2 pqC pD p qCD qE q CDE ⇒ pqC qE q pCE pD p CDE

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

slide-63
SLIDE 63

Transformation Framework

Local rewriting rules

  • A rules

A2 pqC pD p qCD qE q CDE ⇒ pqC qE q pCE pD p CDE

  • Pivots swapping

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

slide-64
SLIDE 64

Transformation Framework

Local rewriting rules

  • A rules

A2 pqC pD p qCD qE q CDE ⇒ pqC qE q pCE pD p CDE

  • Pivots swapping
  • Topology perturbation

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

slide-65
SLIDE 65

Transformation Framework

Local rewriting rules

  • A rules

A2 pqC pD p qCD qE q CDE ⇒ pqC qE q pCE pD p CDE

  • Pivots swapping
  • Topology perturbation
  • Redundancies exposure

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

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SLIDE 66

Local rewriting rules

A1 pqC pqD p qCD qE q CDE ⇒ pqC qE pCE qE pqD q pDE p CDE A2 pqC pD p qCD qE q CDE ⇒ pqC qE q pCE pD p CDE B1 pqC pqD p qCD pqE q pCDE ⇒ pqC pqE q pCE B2 pqC pD p qDC pqE q pCDE ⇒ pqC pqE q pCE pD p CDE B2′ pqC pD p qDC pqE q pCDE ⇒ pqC pqE q pCE B3 pqC pD p qCD pqE q pCDE ⇒ pD

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 33 / 60

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SLIDE 67

Rule-based Approach

Example

pq po p qo pq q po qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 34 / 60

slide-68
SLIDE 68

Rule-based Approach

Example

pq po p qo pq q po qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 35 / 60

slide-69
SLIDE 69

Rule-based Approach

Example

pq pq q p qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 36 / 60

slide-70
SLIDE 70

Rule-based Approach

Example

pq pq q p qr pq q pr p

  • r
  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 37 / 60

slide-71
SLIDE 71

Rule-based Approach

Example

pq pq q p qr pq q pr p r

  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 38 / 60

slide-72
SLIDE 72

Rule-based Approach

Example

pq pq q p qr pq q pr p r

  • o

r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 39 / 60

slide-73
SLIDE 73

Rule-based Approach

Example

pq pq q p qr pq q pr p r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 40 / 60

slide-74
SLIDE 74

Rule-based Approach

Example

pq pq q p qr pq q pr p r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 41 / 60

slide-75
SLIDE 75

Rule-based Approach

Example

qr pq pq q p pq p q q r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 42 / 60

slide-76
SLIDE 76

Rule-based Approach

Example

qr pq pq q p pq p q q r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 43 / 60

slide-77
SLIDE 77

Rule-based Approach

Example

qr pq pq q p pq p q q r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 44 / 60

slide-78
SLIDE 78

Rule-based Approach

Example

qr pq pq p q q r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 45 / 60

slide-79
SLIDE 79

Rule-based Approach

Example

qr pq pq p q q r r r ⊥

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 46 / 60

slide-80
SLIDE 80

Comparison

  • RecyclePivots

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

slide-81
SLIDE 81

Comparison

  • RecyclePivots
  • Pros

Global information Fast and effective

  • Cons

Cannot expose redundancies

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

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SLIDE 82

Comparison

  • RecyclePivots
  • Pros

Global information Fast and effective

  • Cons

Cannot expose redundancies

  • Rule-based approach

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

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SLIDE 83

Comparison

  • RecyclePivots
  • Pros

Global information Fast and effective

  • Cons

Cannot expose redundancies

  • Rule-based approach
  • Pros

Flexibility in rules application Flexibility in amount of transformation Can expose redundancies

  • Cons

Local information

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

slide-84
SLIDE 84

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 48 / 60

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SLIDE 85

Implementation

A Simple Algorithm

  • Based on a sequence of proof traversals (e.g. topological order)

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

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SLIDE 86

Implementation

A Simple Algorithm

  • Based on a sequence of proof traversals (e.g. topological order)
  • Parameterized in number of traversals and time limit

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

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SLIDE 87

Implementation

A Simple Algorithm

  • Based on a sequence of proof traversals (e.g. topological order)
  • Parameterized in number of traversals and time limit
  • Examination non-leaf clauses

Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

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SLIDE 88

Implementation

A Simple Algorithm

  • Based on a sequence of proof traversals (e.g. topological order)
  • Parameterized in number of traversals and time limit
  • Examination non-leaf clauses
  • Pivot in both antecedents → update, match context, apply rule

qC ′D′ qE ′ q CDE ⇒ qC ′D′ qE ′ q C ′D′E ′ ⇒ pqC ′ pD′ p qC ′D′ qE ′ q C ′D′E ′ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

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SLIDE 89

Implementation

A Simple Algorithm

  • Based on a sequence of proof traversals (e.g. topological order)
  • Parameterized in number of traversals and time limit
  • Examination non-leaf clauses
  • Pivot in both antecedents → update, match context, apply rule

qC ′D′ qE ′ q CDE ⇒ qC ′D′ qE ′ q C ′D′E ′ ⇒ pqC ′ pD′ p qC ′D′ qE ′ q C ′D′E ′

  • Pivot not in both antecedents → remove resolution step

C ′D′ qE ′ q CDE ⇒ C ′D′ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

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SLIDE 90

Implementation

A Simple Algorithm

  • Based on a sequence of proof traversals (e.g. topological order)
  • Parameterized in number of traversals and time limit
  • Examination non-leaf clauses
  • Pivot in both antecedents → update, match context, apply rule

qC ′D′ qE ′ q CDE ⇒ qC ′D′ qE ′ q C ′D′E ′ ⇒ pqC ′ pD′ p qC ′D′ qE ′ q C ′D′E ′

  • Pivot not in both antecedents → remove resolution step

C ′D′ qE ′ q CDE ⇒ C ′D′

  • Easy combination with RecyclePivots

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SLIDE 91

Evaluation

Framework and Benchmarks

  • Implemented in C++ and integrated with OpenSMT
  • Available at www.inf.usi.ch/phd/rollini/hvc.html

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SLIDE 92

Evaluation

Framework and Benchmarks

  • Implemented in C++ and integrated with OpenSMT
  • Available at www.inf.usi.ch/phd/rollini/hvc.html
  • Benchmarks

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SLIDE 93

Evaluation

Framework and Benchmarks

  • Implemented in C++ and integrated with OpenSMT
  • Available at www.inf.usi.ch/phd/rollini/hvc.html
  • Benchmarks
  • SMT: SMT-LIB library
  • SAT: SAT competition
  • Academic and industrial problems

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SLIDE 94

Combined Approach Evaluation

Experimental results over SMT: QF UF, QF IDL, QF LRA, QF RDL

# Avgnodes Avgedges Avgcore T(s) Maxnodes Maxedges Maxcore RP 1370 6.7% 7.5% 1.3% 1.7 65.1% 68.9% 39.1% Ratio 0.01 1366 8.9% 10.7% 1.4% 3.4 66.3% 70.2% 45.7% 0.025 1366 9.8% 11.9% 1.5% 3.6 77.2% 79.9% 45.7% 0.05 1366 10.7% 13.0% 1.6% 4.1 78.5% 81.2% 45.7% 0.075 1366 11.4% 13.8% 1.7% 4.5 78.5% 81.2% 45.7% 0.1 1364 11.8% 14.4% 1.7% 5.0 78.8% 83.6% 45.7% 0.25 1359 13.6% 16.6% 1.9% 7.6 79.6% 84.4% 45.7% 0.5 1348 15.0% 18.4% 2.0% 11.5 79.1% 85.2% 45.7% 0.75 1341 16.0% 19.5% 2.1% 15.1 79.9% 86.1% 45.7% 1 1337 16.7% 20.4% 2.2% 18.8 79.9% 86.1% 45.7%

  • Ratio — time threshold as fraction of solving time
  • # — number of benchmarks solved
  • Avgnodes, Avgedges, Avgcore — average reduction in proof size
  • T(s) — average transformation time in seconds
  • Maxnodes, Maxedges, Maxcore — max reduction in proof size

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SLIDE 95

Combined Approach Evaluation

Experimental results over SMT: QF UF, QF IDL, QF LRA, QF RDL

# Avgnodes Avgedges Avgcore T(s) Maxnodes Maxedges Maxcore RP 1370 6.7% 7.5% 1.3% 1.7 65.1% 68.9% 39.1% Ratio 0.01 1366 8.9% 10.7% 1.4% 3.4 66.3% 70.2% 45.7% 0.025 1366 9.8% 11.9% 1.5% 3.6 77.2% 79.9% 45.7% 0.05 1366 10.7% 13.0% 1.6% 4.1 78.5% 81.2% 45.7% 0.075 1366 11.4% 13.8% 1.7% 4.5 78.5% 81.2% 45.7% 0.1 1364 11.8% 14.4% 1.7% 5.0 78.8% 83.6% 45.7% 0.25 1359 13.6% 16.6% 1.9% 7.6 79.6% 84.4% 45.7% 0.5 1348 15.0% 18.4% 2.0% 11.5 79.1% 85.2% 45.7% 0.75 1341 16.0% 19.5% 2.1% 15.1 79.9% 86.1% 45.7% 1 1337 16.7% 20.4% 2.2% 18.8 79.9% 86.1% 45.7%

  • Ratio — time threshold as fraction of solving time
  • # — number of benchmarks solved
  • Avgnodes, Avgedges, Avgcore — average reduction in proof size
  • T(s) — average transformation time in seconds
  • Maxnodes, Maxedges, Maxcore — max reduction in proof size

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SLIDE 96

Combined Approach Evaluation

Experimental results over SMT: QF UF, QF IDL, QF LRA, QF RDL

# Avgnodes Avgedges Avgcore T(s) Maxnodes Maxedges Maxcore RP 1370 6.7% 7.5% 1.3% 1.7 65.1% 68.9% 39.1% Ratio 0.01 1366 8.9% 10.7% 1.4% 3.4 66.3% 70.2% 45.7% 0.025 1366 9.8% 11.9% 1.5% 3.6 77.2% 79.9% 45.7% 0.05 1366 10.7% 13.0% 1.6% 4.1 78.5% 81.2% 45.7% 0.075 1366 11.4% 13.8% 1.7% 4.5 78.5% 81.2% 45.7% 0.1 1364 11.8% 14.4% 1.7% 5.0 78.8% 83.6% 45.7% 0.25 1359 13.6% 16.6% 1.9% 7.6 79.6% 84.4% 45.7% 0.5 1348 15.0% 18.4% 2.0% 11.5 79.1% 85.2% 45.7% 0.75 1341 16.0% 19.5% 2.1% 15.1 79.9% 86.1% 45.7% 1 1337 16.7% 20.4% 2.2% 18.8 79.9% 86.1% 45.7%

  • Ratio — time threshold as fraction of solving time
  • # — number of benchmarks solved
  • Avgnodes, Avgedges, Avgcore — average reduction in proof size
  • T(s) — average transformation time in seconds
  • Maxnodes, Maxedges, Maxcore — max reduction in proof size

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SLIDE 97

Combined Approach Evaluation

Experimental results over SAT

# Avgnodes Avgedges Avgcore T(s) Maxnodes Maxedges Maxcore RP 25 5.9% 6.5% 1.7% 10.8 33.1% 33.4% 30.3% Ratio 0.01 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.025 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.05 25 7.0% 8.2% 1.8% 40.0 34.0% 34.4% 30.5% 0.075 25 7.2% 8.4% 1.8% 49.3 34.7% 35.1% 30.5% 0.1 25 7.3% 8.4% 1.8% 60.2 34.7% 35.1% 30.5% 0.25 25 7.6% 8.8% 1.9% 125.3 39.8% 40.6% 31.7% 0.5 25 7.8% 9.1% 1.9% 243.5 41.0% 41.9% 32.1% 0.75 25 7.9% 9.3% 1.9% 360.0 41.6% 42.6% 32.1% 1 23 8.4% 9.9% 2.1% 175.6 33.1% 33.4% 30.6%

  • Ratio — time threshold as fraction of solving time
  • # — number of benchmarks solved
  • Avgnodes, Avgedges, Avgcore — average reduction in proof size
  • T(s) — average transformation time in seconds
  • Maxnodes, Maxedges, Maxcore — max reduction in proof size

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SLIDE 98

Combined Approach Evaluation

Experimental results over SAT

# Avgnodes Avgedges Avgcore T(s) Maxnodes Maxedges Maxcore RP 25 5.9% 6.5% 1.7% 10.8 33.1% 33.4% 30.3% Ratio 0.01 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.025 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.05 25 7.0% 8.2% 1.8% 40.0 34.0% 34.4% 30.5% 0.075 25 7.2% 8.4% 1.8% 49.3 34.7% 35.1% 30.5% 0.1 25 7.3% 8.4% 1.8% 60.2 34.7% 35.1% 30.5% 0.25 25 7.6% 8.8% 1.9% 125.3 39.8% 40.6% 31.7% 0.5 25 7.8% 9.1% 1.9% 243.5 41.0% 41.9% 32.1% 0.75 25 7.9% 9.3% 1.9% 360.0 41.6% 42.6% 32.1% 1 23 8.4% 9.9% 2.1% 175.6 33.1% 33.4% 30.6%

  • Ratio — time threshold as fraction of solving time
  • # — number of benchmarks solved
  • Avgnodes, Avgedges, Avgcore — average reduction in proof size
  • T(s) — average transformation time in seconds
  • Maxnodes, Maxedges, Maxcore — max reduction in proof size

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SLIDE 99

Combined Approach Evaluation

Experimental results over SAT

# Avgnodes Avgedges Avgcore T(s) Maxnodes Maxedges Maxcore RP 25 5.9% 6.5% 1.7% 10.8 33.1% 33.4% 30.3% Ratio 0.01 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.025 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.05 25 7.0% 8.2% 1.8% 40.0 34.0% 34.4% 30.5% 0.075 25 7.2% 8.4% 1.8% 49.3 34.7% 35.1% 30.5% 0.1 25 7.3% 8.4% 1.8% 60.2 34.7% 35.1% 30.5% 0.25 25 7.6% 8.8% 1.9% 125.3 39.8% 40.6% 31.7% 0.5 25 7.8% 9.1% 1.9% 243.5 41.0% 41.9% 32.1% 0.75 25 7.9% 9.3% 1.9% 360.0 41.6% 42.6% 32.1% 1 23 8.4% 9.9% 2.1% 175.6 33.1% 33.4% 30.6%

  • Ratio — time threshold as fraction of solving time
  • # — number of benchmarks solved
  • Avgnodes, Avgedges, Avgcore — average reduction in proof size
  • T(s) — average transformation time in seconds
  • Maxnodes, Maxedges, Maxcore — max reduction in proof size

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SLIDE 100

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Reduction Framework Implementation and Evaluation

4 Summary and Future Work

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SLIDE 101

Summary and Future Work

  • Summary

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SLIDE 102

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework

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SLIDE 103

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework
  • Pivots redundancies

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SLIDE 104

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework
  • Pivots redundancies
  • Comparison and evaluation

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SLIDE 105

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework
  • Pivots redundancies
  • Comparison and evaluation
  • Future Work

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SLIDE 106

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework
  • Pivots redundancies
  • Comparison and evaluation
  • Future Work
  • Exploitation of DPLL proof structure

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SLIDE 107

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework
  • Pivots redundancies
  • Comparison and evaluation
  • Future Work
  • Exploitation of DPLL proof structure
  • Evaluation on concrete applications (e.g. interpolation)

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SLIDE 108

Summary and Future Work

  • Summary
  • Rule-based proof reduction framework
  • Pivots redundancies
  • Comparison and evaluation
  • Future Work
  • Exploitation of DPLL proof structure
  • Evaluation on concrete applications (e.g. interpolation)
  • Rule-based control of interpolants’ strength

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SLIDE 109

Publications

  • Proof reduction

S.F. Rollini, R. Bruttomesso and N. Sharygina An Efficient and Flexible Approach to Resolution Proof Reduction. HVC 2010.

  • Proof manipulation for interpolation
  • R. Bruttomesso, S.F. Rollini, N. Sharygina and A. Tsitovich

Flexible Interpolation with Local Proof Transformations. ICCAD 2010

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SLIDE 110

Thank you for your attention!

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