20100822 opensmt bruttomesso

opensmt

The OpenSMT Solver

Roberto Bruttomesso

Edgar Pek, Natasha Sharygina, Aliaksei Tsitovich

University of Lugano, Switzerland(Universita della Svizzera Italiana)

September 18, 2010

Roberto Bruttomesso (USI) The OpenSMT Solver September 18, 2010 1 / 40

opensmt

Outline

1 Introduction

2 Architecture

3 A Variable Elimination Techique for SMTDP + FM = DPFMA crazy benchmark suiteRelated Work

4 Extending and Using OpenSMTExtending OpenSMT

5 Conclusion

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Introduction

• Satisfiability Modulo Theory (SMT) Solvers are key engines ofseveral verification approaches

• Efficient solvers however are proprietary(Z3, Yices, Barcelogic, MathSAT, . . . )

• OpenSMT is an effort of providing a simple, extensible, and efficientinfrastructure for the development of customized decision procedures

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Introduction

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Introduction

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Introduction

• Satisfiability Modulo Theories combines the efficiencyof SAT and theory-specific decision procedures

a ∧ ((x + y ≤ 0) ∨ ¬a) ∧ ((x = 1) ∨ b)

opensmt

Introduction

a ∧ ((x + y ≤ 0) ∨ ¬a ) ∧ ((x = 1) ∨ b )

opensmt

Introduction

a ∧ ( (x + y ≤ 0) ∨ ¬a) ∧ ( (x = 1) ∨ b)

opensmt

Introduction

a ∧ ( (x + y ≤ 0)︸︷︷︸c

∨¬a) ∧ ( (x = 1)︸︷︷︸d

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Introduction

• We need to reason about Boolean combinations ofatoms in a theory T (LRA for instance)

SAT (Boolean)(e.g. DPLL) (e.g. Simplex)

Linear Arithmetic

a1x1 + . . . + anxn + b ≤ 0

opensmt

Introduction

SAT (Boolean)(e.g. DPLL)

(e.g. Simplex)Linear Arithmetic

a1x1 + . . . + anxn + b ≤ 0

opensmt

Introduction

SAT (Boolean)(e.g. DPLL) (e.g. Simplex)

Linear Arithmetic

a1x1 + . . . + anxn + b ≤ 0

opensmt

Introduction

• DPLL + LRA ⇒ DPLL(LRA)

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

DPLL(LRA)

opensmt

Introduction

• DPLL(LRA) seems easy to achieve• Let DPLL enumerate Boolean models• Check LRA constraints with Simplex

• However a lot more has to be done to make itefficient

• Don’t wait for complete Boolean model• Theory Propagation• Preprocessing• Conversion to CNF• Theory Layering• . . .

opensmt

Introduction

• DPLL(LRA) seems easy to achieve• Let DPLL enumerate Boolean models• Check LRA constraints with Simplex

• However a lot more has to be done to make itefficient

• Don’t wait for complete Boolean model• Theory Propagation• Preprocessing• Conversion to CNF• Theory Layering• . . .

opensmt

Introduction

e(DPLL(T)) = e(DPLL) + e(T) + e(COMM)

DPLL T

opensmt

Introduction

e(DPLL(T)) = e(DPLL) + e(T) + e(COMM)

opensmt

Introduction

e(DPLL(T)) ≈ e(T)

opensmt

Outline

1 Introduction

2 Architecture

5 Conclusion

opensmt

Architecture

DL BVLRA

EUFPreprocessor

SMT-Level

SAT-Solver T -solvers

Enumerator

Boolean

Simplifications

Parser

Preprocessor

sat (model) / unsat (proof)

opensmt

Architecture

DL BVLRA

EUFPreprocessor

SMT-Level

Enumerator

Boolean

Simplifications

Parser

Preprocessor

opensmt

Architecture

DL BVLRA

EUFPreprocessor

SMT-Level

Enumerator

Boolean

Simplifications

Parser

Preprocessor

opensmt

Outline

1 Introduction

2 Architecture

5 Conclusion

opensmt

A Generic Template for Variable Elimination Procedures

Variable Types: T1, T2, . . .

Resolution Rules: R1, R2, . . .

Algorithm:

Input: a set of constraints

Repeat

Choose a variable X of type Ti to eliminate

Combine positive and negative occurrences of X, using Ri

opensmt

The Davis-Putnam Procedure [DP60]

Variable Types:

Resolution Rules:

Boolean Resolution (BR)

Algorithm:

Input:

a set of Boolean clauses

Repeat

Choose a variable X of type

to eliminate

Combine positive and negative occurrences of X, using

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Variable Types: Bool

Resolution Rules:

Boolean Resolution (BR)

Algorithm:

Input:

Repeat

to eliminate

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Resolution Rules: Boolean Resolution (BR)

Algorithm:

Input:

Repeat

to eliminate

opensmt

Algorithm:

Input: a set of Boolean clauses

Repeat

to eliminate

opensmt

Algorithm:

Repeat

Choose a variable X of type Bool to eliminate

opensmt

Algorithm:

Repeat

Choose a variable X of type Bool to eliminate

Combine positive and negative occurrences of X, using BR

opensmt

Boolean Resolution

• Clauses are expressions like (a ∨ ¬b ∨ c), i.e., disjunctions of literals(Boolean variables or negated Boolean variables)

• In the following C1, C2, D1, D2 are disjunctions of literals

Boolean Resolution for two clauses

(C1 ∨ a ∨ C2)⊗a (D1 ∨ ¬a ∨ D2) := (C1 ∨ C2 ∨ D1 ∨ D2)

• Let Sa, S¬a be the set of clauses with positive resp. negativeoccurrences of a

Boolean Resolution for sets of clauses

Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa,C2 ∈ S¬a }

opensmt

Boolean Resolution

(C1 ∨ a ∨ C2)⊗a (D1 ∨ ¬a ∨ D2) := (C1 ∨ C2 ∨ D1 ∨ D2)

Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa,C2 ∈ S¬a }

opensmt

Boolean Resolution

(C1 ∨ a ∨ C2)⊗a (D1 ∨ ¬a ∨ D2) := (C1 ∨ C2 ∨ D1 ∨ D2)

Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa,C2 ∈ S¬a }

opensmt

Boolean Resolution

(C1 ∨ a ∨ C2)⊗a (D1 ∨ ¬a ∨ D2) := (C1 ∨ C2 ∨ D1 ∨ D2)

Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa,C2 ∈ S¬a }

opensmt

Boolean Resolution

(C1 ∨ a ∨ C2)⊗a (D1 ∨ ¬a ∨ D2) := (C1 ∨ C2 ∨ D1 ∨ D2)

Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa,C2 ∈ S¬a }

opensmt

Boolean Resolution

Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa,C2 ∈ S¬a }

Theorem [DP60]

Sa ∪ S¬a is equisatisfiable with Sa ⊗a S¬a

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DP - Example (on var a)

OLD NEW

(a ∨ b ∨ c)

(a ∨ ¬b ∨ ¬c)

(¬a ∨ ¬b ∨ ¬c)

(¬a ∨ ¬b ∨ c)

S¬a (¬a ∨ ¬b ∨ ¬c)Roberto Bruttomesso (USI) The OpenSMT Solver September 18, 2010 16 / 40

opensmt

OLD NEW

Sa(a ∨ b ∨ c)

(a ∨ ¬b ∨ ¬c)

S¬a(¬a ∨ ¬b ∨ ¬c)

(¬a ∨ ¬b ∨ c)

(¬a ∨ ¬b ∨ ¬c)Roberto Bruttomesso (USI) The OpenSMT Solver September 18, 2010 16 / 40

opensmt

OLD NEW

Sa(a ∨ b ∨ c)

(a ∨ ¬b ∨ ¬c)

S¬a(¬a ∨ ¬b ∨ ¬c)

(¬a ∨ ¬b ∨ c)

opensmt

OLD NEW

Sa(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)

(a ∨ ¬b ∨ ¬c)

S¬a(¬a ∨ ¬b ∨ ¬c)

(¬a ∨ ¬b ∨ c)

opensmt

OLD NEW

Sa(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)

(a ∨ ¬b ∨ ¬c)

S¬a(¬a ∨ ¬b ∨ ¬c)

(¬a ∨ ¬b ∨ c)

opensmt

OLD NEW

Sa(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)