Local proof transformations for flexible interpolation and proof - - PowerPoint PPT Presentation

▶
local proof transformations for flexible interpolation
SMART_READER_LITE
LIVE PREVIEW

Local proof transformations for flexible interpolation and proof - - PowerPoint PPT Presentation

Oct 22, 2023 342 likes •2.21k views

Local proof transformations for flexible interpolation and proof reduction N. Sharygina Formal Verification and Security Group University of Lugano June 21, 2011 Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 1 / 72

slide-1
SLIDE 1

Local proof transformations for flexible interpolation and proof reduction

  • N. Sharygina

Formal Verification and Security Group University of Lugano

June 21, 2011

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 1 / 72

slide-2
SLIDE 2

Outline

1 Background

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 2 / 72

slide-3
SLIDE 3

Outline

1 Background 2 Motivation and Related Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 2 / 72

slide-4
SLIDE 4

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 2 / 72

slide-5
SLIDE 5

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 2 / 72

slide-6
SLIDE 6

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 3 / 72

slide-7
SLIDE 7

Background

Formal Verification in Lugano, Switzerland

  • Program Verification

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 4 / 72

slide-8
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 Transformation June 21, 2011 4 / 72

slide-9
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 Transformation June 21, 2011 4 / 72

slide-10
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 Transformation June 21, 2011 4 / 72

slide-11
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 Transformation June 21, 2011 5 / 72

slide-12
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 Transformation June 21, 2011 5 / 72

slide-13
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 Transformation June 21, 2011 6 / 72

slide-14
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 Transformation June 21, 2011 6 / 72

slide-15
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 Transformation June 21, 2011 6 / 72

slide-16
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 Transformation June 21, 2011 7 / 72

slide-17
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 Transformation June 21, 2011 7 / 72

slide-18
SLIDE 18

Background

Formal Verification in Lugano, Switzerland

  • Boolean and Theory Reasoning (SMT)
  • Generation of interpolants (for QF EUF, RDL)
  • Proof manipulation for interpolation [S.F. Rollini, R. Bruttomesso, N.

Sharygina, A. Tsitovich, ICCAD’10]

  • Resolution proof reduction [S.F. Rollini, R. Bruttomesso, N. Sharygina,

HVC’10]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 8 / 72

slide-19
SLIDE 19

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 9 / 72

slide-20
SLIDE 20

Proof Transformation and Reduction

Motivation

  • Resolution proofs find application in several ambits

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 10 / 72

slide-21
SLIDE 21

Proof Transformation and 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 Transformation June 21, 2011 10 / 72

slide-22
SLIDE 22

Proof Transformation and 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 Transformation June 21, 2011 10 / 72

slide-23
SLIDE 23

Proof Transformation and 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
  • Clean structure of proofs is required for interpolation generation

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 10 / 72

slide-24
SLIDE 24

Proof Transformation and 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
  • Clean structure of proofs is required for interpolation generation
  • Size affects efficiency
  • Size can be exponential w.r.t. input formula

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 10 / 72

slide-25
SLIDE 25

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-26
SLIDE 26

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-27
SLIDE 27

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable
  • I defined over common symbols of A and B

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-28
SLIDE 28

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable
  • I defined over common symbols of A and B
  • I as over-approximation A conflicting with B

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-29
SLIDE 29

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable
  • I defined over common symbols of A and B
  • I as over-approximation A conflicting with B
  • Example

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-30
SLIDE 30

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable
  • I defined over common symbols of A and B
  • I as over-approximation A conflicting with B
  • Example
  • A (p ∨ q) ∧ (p ∨ q)

B (q ∨ r) ∧ (q ∨ r)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-31
SLIDE 31

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable
  • I defined over common symbols of A and B
  • I as over-approximation A conflicting with B
  • Example
  • A (p ∨ q) ∧ (p ∨ q)

B (q ∨ r) ∧ (q ∨ r)

  • Interpolant q

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-32
SLIDE 32

Notation

Interpolation

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • A ⇒ I, I ∧ B unsatisfiable
  • I defined over common symbols of A and B
  • I as over-approximation A conflicting with B
  • Example
  • A (p ∨ q) ∧ (p ∨ q)

B (q ∨ r) ∧ (q ∨ r)

  • Interpolant q
  • A ⇒ q

q ∧ B unsatisfiable

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

slide-33
SLIDE 33

Interpolation

Background

  • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B

[Craig57]

  • I as over-approximation A conflicting with B

A I B

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 12 / 72

slide-34
SLIDE 34

Interpolation

Background

  • Applications in symbolic model checking

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-35
SLIDE 35

Interpolation

Background

  • Applications in symbolic model checking
  • Bounded model checking: approximate cheaper reachability set

computation [McMillan03]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-36
SLIDE 36

Interpolation

Background

  • Applications in symbolic model checking
  • Bounded model checking: approximate cheaper reachability set

computation [McMillan03]

  • Predicate abstraction refinement based on spurious behaviors

[Henzinger04]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-37
SLIDE 37

Interpolation

Background

  • Applications in symbolic model checking
  • Bounded model checking: approximate cheaper reachability set

computation [McMillan03]

  • Predicate abstraction refinement based on spurious behaviors

[Henzinger04]

  • Property-based transition relation approximation [Jhala05]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-38
SLIDE 38

Interpolation

Background

  • Applications in symbolic model checking
  • Bounded model checking: approximate cheaper reachability set

computation [McMillan03]

  • Predicate abstraction refinement based on spurious behaviors

[Henzinger04]

  • Property-based transition relation approximation [Jhala05]
  • Forementioned applications involve

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-39
SLIDE 39

Interpolation

Background

  • Applications in symbolic model checking
  • Bounded model checking: approximate cheaper reachability set

computation [McMillan03]

  • Predicate abstraction refinement based on spurious behaviors

[Henzinger04]

  • Property-based transition relation approximation [Jhala05]
  • Forementioned applications involve
  • Problem encoding into logic (SAT, SMT)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-40
SLIDE 40

Interpolation

Background

  • Applications in symbolic model checking
  • Bounded model checking: approximate cheaper reachability set

computation [McMillan03]

  • Predicate abstraction refinement based on spurious behaviors

[Henzinger04]

  • Property-based transition relation approximation [Jhala05]
  • Forementioned applications involve
  • Problem encoding into logic (SAT, SMT)
  • Problem solving by means of resolution based engines (SAT solvers,

SMT solvers)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

slide-41
SLIDE 41

SAT and SMT

Background

  • Satisfiability (SAT)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

slide-42
SLIDE 42

SAT and SMT

Background

  • Satisfiability (SAT)
  • Example

A (p ∨ q) ∧ (p ∨ q) B (q ∨ r) ∧ (q ∨ r)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

slide-43
SLIDE 43

SAT and SMT

Background

  • Satisfiability (SAT)
  • Example

A (p ∨ q) ∧ (p ∨ q) B (q ∨ r) ∧ (q ∨ r)

  • Satisfiability Modulo Theories (SMT): more expressivity than boolean

logic

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

slide-44
SLIDE 44

SAT and SMT

Background

  • Satisfiability (SAT)
  • Example

A (p ∨ q) ∧ (p ∨ q) B (q ∨ r) ∧ (q ∨ r)

  • Satisfiability Modulo Theories (SMT): more expressivity than boolean

logic

  • Timed automata, hybrid systems, . . .

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

slide-45
SLIDE 45

SAT and SMT

Background

  • Satisfiability (SAT)
  • Example

A (p ∨ q) ∧ (p ∨ q) B (q ∨ r) ∧ (q ∨ r)

  • Satisfiability Modulo Theories (SMT): more expressivity than boolean

logic

  • Timed automata, hybrid systems, . . .
  • Arbitrary precision arithmetic, data structures . . .

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

slide-46
SLIDE 46

SAT and SMT

Background

  • Satisfiability (SAT)
  • Example

A (p ∨ q) ∧ (p ∨ q) B (q ∨ r) ∧ (q ∨ r)

  • Satisfiability Modulo Theories (SMT): more expressivity than boolean

logic

  • Timed automata, hybrid systems, . . .
  • Arbitrary precision arithmetic, data structures . . .
  • Example

A (5x −y ≤ 1)∧(y −5x ≤ −1) B (y −5z ≤ 3)∧(5z −y ≤ −2)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

slide-47
SLIDE 47

SAT and SMT

Proofs and Solving Engines

  • A ∧ B unsatisfiable: certificate of unsatisfiability

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

slide-48
SLIDE 48

SAT and SMT

Proofs and Solving Engines

  • A ∧ B unsatisfiable: certificate of unsatisfiability
  • Propositional proof of unsatisfiability
  • Generated by logging steps at solving time

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

slide-49
SLIDE 49

SAT and SMT

Proofs and Solving Engines

  • A ∧ B unsatisfiable: certificate of unsatisfiability
  • Propositional proof of unsatisfiability
  • Generated by logging steps at solving time
  • DPLL SAT solver [Davis60,62]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

slide-50
SLIDE 50

SAT and SMT

Proofs and Solving Engines

  • A ∧ B unsatisfiable: certificate of unsatisfiability
  • Propositional proof of unsatisfiability
  • Generated by logging steps at solving time
  • DPLL SAT solver [Davis60,62]
  • Search space boolean assignments
  • Backtracking

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

slide-51
SLIDE 51

SAT and SMT

Proofs and Solving Engines

  • A ∧ B unsatisfiable: certificate of unsatisfiability
  • Propositional proof of unsatisfiability
  • Generated by logging steps at solving time
  • DPLL SAT solver [Davis60,62]
  • Search space boolean assignments
  • Backtracking
  • SMT solver

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

slide-52
SLIDE 52

SAT and SMT

Proofs and Solving Engines

  • A ∧ B unsatisfiable: certificate of unsatisfiability
  • Propositional proof of unsatisfiability
  • Generated by logging steps at solving time
  • DPLL SAT solver [Davis60,62]
  • Search space boolean assignments
  • Backtracking
  • SMT solver
  • DPLL SAT solver
  • Theory solver

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

slide-53
SLIDE 53

Interpolation

Generation

  • Interpolant I for unsatisfiable conjunction of formulae A ∧ B

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

slide-54
SLIDE 54

Interpolation

Generation

  • Interpolant I for unsatisfiable conjunction of formulae A ∧ B
  • State-of-the-art approach [Pudl´

ak97, McMillan04]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

slide-55
SLIDE 55

Interpolation

Generation

  • Interpolant I for unsatisfiable conjunction of formulae A ∧ B
  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability resolution proof of A ∧ B

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

slide-56
SLIDE 56

Interpolation

Generation

  • Interpolant I for unsatisfiable conjunction of formulae A ∧ B
  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability resolution proof of A ∧ B
  • Computation of I from proof structure in linear time

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

slide-57
SLIDE 57

Resolution System

Background

  • Literal

p p

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

slide-58
SLIDE 58

Resolution System

Background

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

slide-59
SLIDE 59

Resolution System

Background

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥

  • Input formula

(p ∨ q) ∧ (r ∨ p) . . . → {pq, rp}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

slide-60
SLIDE 60

Resolution System

Background

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥

  • Input formula

(p ∨ q) ∧ (r ∨ p) . . . → {pq, rp}

  • Resolution rule

pC pD p CD

Antecedents: pC pD Resolvent: CD Pivot: p

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

slide-61
SLIDE 61

Resolution System

Background

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥

  • Input formula

(p ∨ q) ∧ (r ∨ p) . . . → {pq, rp}

  • Resolution rule

pC pD p CD

Antecedents: pC pD Resolvent: CD Pivot: p

  • Resolution proof of unsatisfiability of a set of clauses S

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

slide-62
SLIDE 62

Resolution System

Background

  • Literal

p p

  • Clause

p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥

  • Input formula

(p ∨ q) ∧ (r ∨ p) . . . → {pq, rp}

  • Resolution rule

pC pD p CD

Antecedents: pC pD Resolvent: CD Pivot: p

  • 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 Transformation June 21, 2011 17 / 72

slide-63
SLIDE 63

Resolution Proofs

SAT

  • A {pq, pq}

B {qr, qr}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 18 / 72

slide-64
SLIDE 64

Resolution Proofs

SAT

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq pq p q qr qr r q q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 18 / 72

slide-65
SLIDE 65

Interpolant Generation

SAT [Pudl´ ak97]

  • Computation of interpolant I for A ∧ B from proof structure

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

slide-66
SLIDE 66

Interpolant Generation

SAT [Pudl´ ak97]

  • Computation of interpolant I for A ∧ B from proof structure
  • Partial interpolant for leaf

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

slide-67
SLIDE 67

Interpolant Generation

SAT [Pudl´ ak97]

  • Computation of interpolant I for A ∧ B from proof structure
  • Partial interpolant for leaf
  • Partial interpolant for resolvent
  • Pivot
  • Partial interpolants for antecedents

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

slide-68
SLIDE 68

Interpolant Generation

SAT [Pudl´ ak97]

  • Computation of interpolant I for A ∧ B from proof structure
  • Partial interpolant for leaf
  • Partial interpolant for resolvent
  • Pivot
  • Partial interpolants for antecedents
  • Partial interpolant for ⊥ is I

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

slide-69
SLIDE 69

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq pq p q qr qr r q q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-70
SLIDE 70

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q qr qr r q q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-71
SLIDE 71

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q qr {⊤} qr {⊤} r q q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-72
SLIDE 72

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q {⊥ ∨ ⊥} qr {⊤} qr {⊤} r q q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-73
SLIDE 73

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q {⊥} qr {⊤} qr {⊤} r q q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-74
SLIDE 74

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q {⊥} qr {⊤} qr {⊤} r q {⊤ ∧ ⊤} q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-75
SLIDE 75

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q {⊥} qr {⊤} qr {⊤} r q {⊤} q ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-76
SLIDE 76

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q {⊥} qr {⊤} qr {⊤} r q {⊤} q ⊥ {(⊥ ∨ q) ∧ (⊤ ∨ q)}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-77
SLIDE 77

Interpolant Generation

SAT [Pudl´ ak97]

  • A {pq, pq}

B {qr, qr}

  • Proof of unsatisfiability

pq {⊥} pq {⊥} p q {⊥} qr {⊤} qr {⊤} r q {⊤} q ⊥ {q}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

slide-78
SLIDE 78

Resolution Proofs

SMT

  • A {

p

z }| { (5x − y ≤ 1) ,

q

z }| { (y − 5x ≤ − 1) } B {

r

z }| { (y − 5z ≤ 3) ,

s

z }| { (5z − y ≤ − 2) }

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

slide-79
SLIDE 79

Resolution Proofs

SMT

  • A {

p

z }| { (5x − y ≤ 1) ,

q

z }| { (y − 5x ≤ − 1) } B {

r

z }| { (y − 5z ≤ 3) ,

s

z }| { (5z − y ≤ − 2) }

  • Theory lemmata

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

slide-80
SLIDE 80

Resolution Proofs

SMT

  • A {

p

z }| { (5x − y ≤ 1) ,

q

z }| { (y − 5x ≤ − 1) } B {

r

z }| { (y − 5z ≤ 3) ,

s

z }| { (5z − y ≤ − 2) }

  • Theory lemmata
  • LIA:

t

  • (x − z ≤ 0)

u

  • (x − z ≥ 1)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

slide-81
SLIDE 81

Resolution Proofs

SMT

  • A {

p

z }| { (5x − y ≤ 1) ,

q

z }| { (y − 5x ≤ − 1) } B {

r

z }| { (y − 5z ≤ 3) ,

s

z }| { (5z − y ≤ − 2) }

  • Theory lemmata
  • LIA:

t

  • (x − z ≤ 0)

u

  • (x − z ≥ 1)
  • LRA:

p

  • (5x − y 1)

r

  • (y − 5z 3)

u

  • (x − z 1)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

slide-82
SLIDE 82

Resolution Proofs

SMT

  • A {

p

z }| { (5x − y ≤ 1) ,

q

z }| { (y − 5x ≤ − 1) } B {

r

z }| { (y − 5z ≤ 3) ,

s

z }| { (5z − y ≤ − 2) }

  • Theory lemmata
  • LIA:

t

  • (x − z ≤ 0)

u

  • (x − z ≥ 1)
  • LRA:

p

  • (5x − y 1)

r

  • (y − 5z 3)

u

  • (x − z 1)
  • LRA:

q

  • (y − 5x − 1)

s

  • (5z − y − 2)

t

  • (x − z 0)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

slide-83
SLIDE 83

Resolution Proofs

SMT

  • A {p, q}

B {r, s} L {tu, pru, qst}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 22 / 72

slide-84
SLIDE 84

Resolution Proofs

SMT

  • A {p, q}

B {r, s} L {tu, pru, qst}

  • Proof of unsatisfiability

p pru p ru r r u tu u t qst t qs q q s s s ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 22 / 72

slide-85
SLIDE 85

Interpolant Generation

SMT

  • A {p, q}

B {r, s} L {tu, pru, qst}

  • Proof of unsatisfiability

p pru p ru r r u tu u t qst t qs q q s s s ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

slide-86
SLIDE 86

Interpolant Generation

SMT

  • A {p, q}

B {r, s} L {tu, pru, qst}

  • Proof of unsatisfiability

p {⊥} pru p ru r r u tu u t qst t qs q {⊥} q s s s ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

slide-87
SLIDE 87

Interpolant Generation

SMT

  • A {p, q}

B {r, s} L {tu, pru, qst}

  • Proof of unsatisfiability

p {⊥} pru p ru r {⊤} r u tu u t qst t qs q {⊥} q s s {⊤} s ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

slide-88
SLIDE 88

Interpolant Generation

SMT

  • A {p, q}

B {r, s} L {tu, pru, qst}

  • Proof of unsatisfiability

p {⊥} pru p ru r {⊤} r u tu ? u t qst t qs q {⊥} q s s {⊤} s ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

slide-89
SLIDE 89

Interpolation

Challenge

  • State-of-the-art approach [Pudl´

ak97, McMillan04]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

slide-90
SLIDE 90

Interpolation

Challenge

  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability proof of A ∧ B
  • Computation of interpolant from proof structure in linear time

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

slide-91
SLIDE 91

Interpolation

Challenge

  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability proof of A ∧ B
  • Computation of interpolant from proof structure in linear time
  • Restriction

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

slide-92
SLIDE 92

Interpolation

Challenge

  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability proof of A ∧ B
  • Computation of interpolant from proof structure in linear time
  • Restriction
  • Need for proof not to contain AB-mixed predicates

A-local B-local AB-common AB-mixed

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

slide-93
SLIDE 93

Interpolation

Challenge

  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability proof of A ∧ B
  • Computation of interpolant from proof structure in linear time
  • Restriction
  • Need for proof not to contain AB-mixed predicates

A-local B-local AB-common AB-mixed A { (5x − y ≤ 1) , . . .} B { (y − 5z ≤ 3) , . . .}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

slide-94
SLIDE 94

Interpolation

Challenge

  • State-of-the-art approach [Pudl´

ak97, McMillan04]

  • Derivation of unsatisfiability proof of A ∧ B
  • Computation of interpolant from proof structure in linear time
  • Restriction
  • Need for proof not to contain AB-mixed predicates

A-local B-local AB-common AB-mixed A { (5x − y ≤ 1) , . . .} B { (y − 5z ≤ 3) , . . .} L { (x − z ≤ 0) , . . .}

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

slide-95
SLIDE 95

Interpolation

Possible Solutions

  • Need for proof not to contain AB-mixed predicates

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 25 / 72

slide-96
SLIDE 96

Interpolation

Possible Solutions

  • Need for proof not to contain AB-mixed predicates
  • Tune solvers to avoid generating AB-mixed predicates

[Cimatti08,Beyer08]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 25 / 72

slide-97
SLIDE 97

Interpolation

Possible Solutions

  • Need for proof not to contain AB-mixed predicates
  • Tune solvers to avoid generating AB-mixed predicates

[Cimatti08,Beyer08]

  • Transform proof to remove AB-mixed predicates

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 25 / 72

slide-98
SLIDE 98

Proof Transformation

Motivation

  • Proof transformation approach

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

slide-99
SLIDE 99

Proof Transformation

Motivation

  • Proof transformation approach
  • Motivation: more flexibility by decoupling SMT solving and

interpolant generation

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

slide-100
SLIDE 100

Proof Transformation

Motivation

  • Proof transformation approach
  • Motivation: more flexibility by decoupling SMT solving and

interpolant generation

  • Motivation: standard SMT techniques can require addition of

AB-mixed predicates

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

slide-101
SLIDE 101

Proof Transformation

Motivation

  • Proof transformation approach
  • Motivation: more flexibility by decoupling SMT solving and

interpolant generation

  • Motivation: standard SMT techniques can require addition of

AB-mixed predicates

  • Theory reduction via Lemma on Demand [DeMoura02, Barrett06]

Reduction of AX to EUF Reduction of LIA to LRA Ackermann’s Expansion

  • Theory combination via DTC [Bozzano05]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

slide-102
SLIDE 102

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 27 / 72

slide-103
SLIDE 103

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 28 / 72

slide-104
SLIDE 104

Contribution

Proof Transformation Framework

  • Proof rewriting framework based on local rules

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 29 / 72

slide-105
SLIDE 105

Contribution

Proof Transformation Framework

  • Proof rewriting framework based on local rules
  • Isolation of AB-mixed predicates into subtrees

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 29 / 72

slide-106
SLIDE 106

Contribution

Proof Transformation Framework

  • Proof rewriting framework based on local rules
  • Isolation of AB-mixed predicates into subtrees
  • Removal of AB-mixed subtrees

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 29 / 72

slide-107
SLIDE 107

Contribution

Proof Transformation Framework

  • Proof rewriting framework based on local rules
  • Isolation of AB-mixed predicates into subtrees
  • Removal of AB-mixed subtrees
  • No more AB-mixed predicates, proof still valid

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 29 / 72

slide-108
SLIDE 108

Proof Transformation

Effect

(a) Initial proof: A-local, B-local, AB-common, AB-mixed (b) Transformed proof: AB-mixed predicates isolated into subtrees (c) Final proof: AB-mixed subtrees removed, new leaves are theory lemmata

(a) (b) (c)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 30 / 72

slide-109
SLIDE 109

Proof Transformation

Advantages

  • No more AB-mixed predicates, new leaves are theory lemmata

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 31 / 72

slide-110
SLIDE 110

Proof Transformation

Advantages

  • No more AB-mixed predicates, new leaves are theory lemmata
  • Easy combination of SMT and interpolation techniques

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 31 / 72

slide-111
SLIDE 111

Proof Transformation

Advantages

  • No more AB-mixed predicates, new leaves are theory lemmata
  • Easy combination of SMT and interpolation techniques
  • Theory reduction, theory combination without restrictions

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 31 / 72

slide-112
SLIDE 112

Proof Transformation

Advantages

  • No more AB-mixed predicates, new leaves are theory lemmata
  • Easy combination of SMT and interpolation techniques
  • Theory reduction, theory combination without restrictions
  • Interpolant generation for propositional resolution proofs of

unsatisfiability [Pudl´ ak97]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 31 / 72

slide-113
SLIDE 113

Proof Transformation

Advantages

  • No more AB-mixed predicates, new leaves are theory lemmata
  • Easy combination of SMT and interpolation techniques
  • Theory reduction, theory combination without restrictions
  • Interpolant generation for propositional resolution proofs of

unsatisfiability [Pudl´ ak97]

  • (Partial) interpolant generation for theory (combination) lemmata

[Yorsh05]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 31 / 72

slide-114
SLIDE 114

Proof Transformation Framework

Features

  • Local rewriting rules

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 32 / 72

slide-115
SLIDE 115

Proof Transformation Framework

Features

  • Local rewriting rules
  • Rule context

pqC pD p qCD qE q CDE

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 32 / 72

slide-116
SLIDE 116

Proof Transformation Framework

Features

  • Local rewriting rules
  • Rule context

pqC pD p qCD qE q CDE

  • Exhaustiveness up to symmetry

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 32 / 72

slide-117
SLIDE 117

Proof Transformation Framework

Local Rewriting Rules

  • pqC

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 33 / 72

slide-118
SLIDE 118

Proof Transformation Framework

Local Rewriting Rules

  • pqC

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

  • Pivots swapping

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 33 / 72

slide-119
SLIDE 119

Proof Transformation Framework

Local Rewriting Rules

  • pqC

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

  • Pivots swapping
  • AB-mixed predicates isolation into subtrees

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 33 / 72

slide-120
SLIDE 120

Reduction LIA to LRA

Transformation

  • A {p, q}

B {r, s} L {tu, pru, qst}

  • Proof of unsatisfiability

p pru p ru r r u tu u t qst t qs q q s s s ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 34 / 72

slide-121
SLIDE 121

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 35 / 72

slide-122
SLIDE 122

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 36 / 72

slide-123
SLIDE 123

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 37 / 72

slide-124
SLIDE 124

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 38 / 72

slide-125
SLIDE 125

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 39 / 72

slide-126
SLIDE 126

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 40 / 72

slide-127
SLIDE 127

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 41 / 72

slide-128
SLIDE 128

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 42 / 72

slide-129
SLIDE 129

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 43 / 72

slide-130
SLIDE 130

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 44 / 72

slide-131
SLIDE 131

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 45 / 72

slide-132
SLIDE 132

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 46 / 72

slide-133
SLIDE 133

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 47 / 72

slide-134
SLIDE 134

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 48 / 72

slide-135
SLIDE 135

Reduction LIA to LRA

Transformation

  • Proof of unsatisfiability
  • ⊥

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 49 / 72

slide-136
SLIDE 136

Proof Transformation Framework

Considerations

  • Potential drawbacks

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 50 / 72

slide-137
SLIDE 137

Proof Transformation Framework

Considerations

  • Potential drawbacks
  • Overhead w.r.t. solving time

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 50 / 72

slide-138
SLIDE 138

Proof Transformation Framework

Considerations

  • Potential drawbacks
  • Overhead w.r.t. solving time
  • Increase of proof size

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 50 / 72

slide-139
SLIDE 139

Transformation Framework

Features

  • Local rewriting rules

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 51 / 72

slide-140
SLIDE 140

Transformation Framework

Features

  • Local rewriting rules
  • B reduction
  • A perturbation

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 51 / 72

slide-141
SLIDE 141

Transformation Framework

Features

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

pqC pD p qCD qE q CDE

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 51 / 72

slide-142
SLIDE 142

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 Transformation June 21, 2011 51 / 72

slide-143
SLIDE 143

Transformation Framework

Local rewriting rules

  • B rules

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 52 / 72

slide-144
SLIDE 144

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 Transformation June 21, 2011 52 / 72

slide-145
SLIDE 145

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 Transformation June 21, 2011 52 / 72

slide-146
SLIDE 146

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 Transformation June 21, 2011 52 / 72

slide-147
SLIDE 147

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 Transformation June 21, 2011 53 / 72

slide-148
SLIDE 148

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 Transformation June 21, 2011 53 / 72

slide-149
SLIDE 149

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 Transformation June 21, 2011 53 / 72

slide-150
SLIDE 150

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 Transformation June 21, 2011 53 / 72

slide-151
SLIDE 151

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 Transformation June 21, 2011 54 / 72

slide-152
SLIDE 152

Evaluation

Framework and Benchmarks

  • Natasha Sharygina (USI)

Flexible Proof Transformation June 21, 2011 55 / 72

slide-153
SLIDE 153

Evaluation

Framework and Benchmarks

  • C++ open-source SMT solver developed at USI
  • Fastest open-source solver in SMT-comp 2009, 2010 for various logics

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 55 / 72

slide-154
SLIDE 154

Evaluation

Framework and Benchmarks

  • C++ open-source SMT solver developed at USI
  • Fastest open-source solver in SMT-comp 2009, 2010 for various logics
  • Benchmarks

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 55 / 72

slide-155
SLIDE 155

Evaluation

Framework and Benchmarks

  • C++ open-source SMT solver developed at USI
  • Fastest open-source solver in SMT-comp 2009, 2010 for various logics
  • Benchmarks
  • SMT: SMT-LIB library
  • Academic and industrial problems

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 55 / 72

slide-156
SLIDE 156

Evaluation

Experimental results over QF UFIDL

Group # #AB %time %nodes %edges RDS 2 7 93% 2% 2% EufLaAr 2 103 91% 30% 26% pete 6 4 33% 8% 9% pete2 56 17 59% 27% 32% uclid 8 11 64% 37% 42% Overall 74 17 59% 26% 30%

  • # — number of benchmarks solved
  • #AB — average number of AB-mixed predicates in proof
  • %time — average time overhead
  • %nodes, %edges — average difference in proof size

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 56 / 72

slide-157
SLIDE 157

Comparison

  • RecyclePivots (closest related work) [Strichman’08]

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 57 / 72

slide-158
SLIDE 158

Comparison

  • RecyclePivots (closest related work) [Strichman’08]
  • Pros

Global information Fast and effective

  • Cons

Cannot expose redundancies

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 57 / 72

slide-159
SLIDE 159

Comparison

  • RecyclePivots (closest related work) [Strichman’08]
  • Pros

Global information Fast and effective

  • Cons

Cannot expose redundancies

  • Rule-based approach

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 57 / 72

slide-160
SLIDE 160

Comparison

  • RecyclePivots (closest related work) [Strichman’08]
  • 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 Transformation June 21, 2011 57 / 72

slide-161
SLIDE 161

Implementation

Reduction Algorithm

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 58 / 72

slide-162
SLIDE 162

Implementation

Reduction 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 Transformation June 21, 2011 58 / 72

slide-163
SLIDE 163

Implementation

Reduction 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 Transformation June 21, 2011 58 / 72

slide-164
SLIDE 164

Implementation

Reduction 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 Transformation June 21, 2011 58 / 72

slide-165
SLIDE 165

Implementation

Reduction 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 Transformation June 21, 2011 58 / 72

slide-166
SLIDE 166

Implementation

Reduction 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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 58 / 72

slide-167
SLIDE 167

Evaluation

Framework and Benchmarks

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 59 / 72

slide-168
SLIDE 168

Evaluation

Framework and Benchmarks

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 59 / 72

slide-169
SLIDE 169

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 59 / 72

slide-170
SLIDE 170

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 60 / 72

slide-171
SLIDE 171

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 61 / 72

slide-172
SLIDE 172

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 62 / 72

slide-173
SLIDE 173

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 63 / 72

slide-174
SLIDE 174

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 64 / 72

slide-175
SLIDE 175

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 65 / 72

slide-176
SLIDE 176

Outline

1 Background 2 Motivation and Related Work 3 Contribution

Proof Transformation for Interpolation and Reduction

4 Summary and Future Work

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 66 / 72

slide-177
SLIDE 177

Summary

  • Proof transformation

1 Interpolation, SMT, AB-mixed predicates

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 67 / 72

slide-178
SLIDE 178

Summary

  • Proof transformation

1 Interpolation, SMT, AB-mixed predicates 2 Proof transformation framework for AB-mixed predicates removal

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 67 / 72

slide-179
SLIDE 179

Summary

  • Proof transformation

1 Interpolation, SMT, AB-mixed predicates 2 Proof transformation framework for AB-mixed predicates removal 3 Easy combination:

  • Standard SMTs
  • State-of-the art interpolant generation procedures

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 67 / 72

slide-180
SLIDE 180

Summary

  • Proof transformation

1 Interpolation, SMT, AB-mixed predicates 2 Proof transformation framework for AB-mixed predicates removal 3 Easy combination:

  • Standard SMTs
  • State-of-the art interpolant generation procedures
  • Rule-based proof reduction

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 67 / 72

slide-181
SLIDE 181

Summary

  • Proof transformation

1 Interpolation, SMT, AB-mixed predicates 2 Proof transformation framework for AB-mixed predicates removal 3 Easy combination:

  • Standard SMTs
  • State-of-the art interpolant generation procedures
  • Rule-based proof reduction
  • Pivots redundancies detection and removal

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 67 / 72

slide-182
SLIDE 182

Future Work

  • Exploitation of DPLL proof structure

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 68 / 72

slide-183
SLIDE 183

Future Work

  • Exploitation of DPLL proof structure
  • Evaluation on concrete applications (e.g. interpolation)

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 68 / 72

slide-184
SLIDE 184

Future Work

  • Exploitation of DPLL proof structure
  • Evaluation on concrete applications (e.g. interpolation)
  • Rule-based control of interpolants’ strength

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 68 / 72

slide-185
SLIDE 185

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

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 69 / 72

slide-186
SLIDE 186

Thanks for your attention!

http://www.verify.inf.usi.ch/

Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 70 / 72