20100822 opensmt bruttomesso

The OpenSMT Solver

Roberto Bruttomesso

Edgar Pek, Natasha Sharygina, Aliaksei Tsitovich

University of Lugano, Switzerland
(Universit` della Svizzera Italiana)
a

September 18, 2010

o es
p nmt
Roberto Bruttomesso (USI) The OpenSMT Solver September 18, 2010 1 / 40

Outline

1 Introduction

2 Architecture

3 A Variable Elimination Techique for SMT
DP + FM = DPFM
A crazy benchmark suite
Related Work

4 Extending and Using OpenSMT
Extending OpenSMT

5 Conclusion

o es
p nmt

Introduction

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

o es
p nmt

Introduction


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

o es
p nmt

Introduction


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

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

o es
p nmt

Introduction

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

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

o es
p nmt

Introduction


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

o es
p nmt

Introduction


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

o es
p nmt

Introduction


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

o es
p nmt

Introduction

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

o es
p nmt

Introduction

SAT (Boolean)
(e.g. DPLL)

o es
p nmt

Introduction

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

a1x1 + . . . + an xn + b ≤ 0
≤0
≤0
≤0
≤0

o es
p nmt

Introduction

• DPLL + LRA ⇒ DPLL(LRA)

DPLL(LRA)

o es
p nmt

Introduction

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

o es
p nmt

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

o es
p nmt

Introduction

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

DPLL T

o es
p nmt

Introduction

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

DPLL

o es
p nmt

Introduction

e(DPLL(T)) ≈ e(T)

DPLL

o es
p nmt

Outline

1 Introduction

2 Architecture

DP + FM = DPFM
Related Work

Extending OpenSMT

5 Conclusion

o es
p nmt

Architecture

ϕ

Preprocessor SAT-Solver T -solvers

SMT-Level
Parser Preprocessor
EUF

Simpliﬁcations Boolean
Enumerator DL LRA BV

sat (model) / unsat (proof)

o es
p nmt

Outline

1 Introduction

2 Architecture

DP + FM = DPFM
Related Work

Extending OpenSMT

5 Conclusion

o es
p nmt

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

o es
p nmt

The Davis-Putnam Procedure [DP60]

Variable Types:
Resolution Rules:
Algorithm:
Input:
Repeat
Choose a variable X of type to eliminate
Combine positive and negative occurrences of X, using

o es
p nmt


Variable Types: Bool
Resolution Rules:
Algorithm:
Input:
Repeat

o es
p nmt


Resolution Rules: Boolean Resolution (BR)
Algorithm:
Input:
Repeat

o es
p nmt


Algorithm:
Input: a set of Boolean clauses
Repeat

o es
p nmt


Algorithm:
Repeat
Choose a variable X of type Bool to eliminate

o es
p nmt


Algorithm:
Repeat
Choose a variable X of type Bool to eliminate
Combine positive and negative occurrences of X, using BR

o es
p nmt

Boolean Resolution

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

o es
p nmt

Boolean Resolution

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

o es
p nmt

Boolean Resolution


Boolean Resolution for two clauses
(C1 ∨ a ∨ C2 ) ⊗a (D1 ∨ ¬a ∨ D2 ) := (C1 ∨ C2 ∨ D1 ∨ D2 )

o es
p nmt

Boolean Resolution


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

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

o es
p nmt

Boolean Resolution


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

occurrences of a

Boolean Resolution for sets of clauses
Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa , C2 ∈ S¬a }

o es
p nmt

Boolean Resolution

occurrences of a

Boolean Resolution for sets of clauses
Sa ⊗a S¬a := {C1 ⊗a C2 | C1 ∈ Sa , C2 ∈ S¬a }

Theorem [DP60]
Sa ∪ S¬a is equisatisﬁable with Sa ⊗a S¬a

o es
p nmt

DP - Example (on var a)

OLD NEW
(a ∨ b ∨ c)
(a ∨ ¬b ∨ ¬c)
(¬a ∨ ¬b ∨ ¬c)
(¬a ∨ ¬b ∨ c)
S¬a (¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c)
Sa
(a ∨ ¬b ∨ ¬c)
(¬a ∨ ¬b ∨ ¬c)
S¬a
(¬a ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)
Sa
(a ∨ ¬b ∨ ¬c)
(¬a ∨ ¬b ∨ ¬c)
S¬a
(¬a ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)
Sa
(a ∨ ¬b ∨ ¬c) (b ∨ c ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c)
S¬a
(¬a ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)
Sa
(a ∨ ¬b ∨ ¬c) (b ∨ c ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) (¬b ∨ ¬c)
S¬a
(¬a ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)
Sa
(a ∨ ¬b ∨ ¬c) (b ∨ c ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) (¬b ∨ ¬c)
S¬a
(¬a ∨ ¬b ∨ c) (¬b ∨ ¬c ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c) (b ∨ c ∨ ¬b ∨ ¬c)
(a ∨ ¬b ∨ ¬c) (b ∨ c ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) (¬b ∨ ¬c)
(¬a ∨ ¬b ∨ c) (¬b ∨ ¬c ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt


OLD NEW
(a ∨ b ∨ c)
(a ∨ ¬b ∨ ¬c)
(¬a ∨ ¬b ∨ ¬c) (¬b ∨ ¬c)
(¬a ∨ ¬b ∨ c)
(¬a ∨ ¬b ∨ ¬c) o es
p nmt

The Fourier-Motzkin Elimination [Fou26]

Variable Types:
Resolution Rules:
Algorithm:
Input:
Repeat

o es
p nmt


Variable Types: Rational
Resolution Rules:
Algorithm:
Input:
Repeat

o es
p nmt


Resolution Rules: LRA Resolution (RR)
Algorithm:
Input:
Repeat

o es
p nmt


Algorithm:
Input: a set of LRA constraints
Repeat

o es
p nmt


Algorithm:
Repeat
Choose a variable X of type Rational to eliminate

o es
p nmt


Algorithm:
Repeat
Choose a variable X of type Rational to eliminate
Combine positive and negative occurrences of X, using RR

o es
p nmt

LRA Resolution

• LRA constraints are expressions like 3x − 5y + 10z ≤ 15

o es
p nmt

LRA Resolution

• Notice that ≤ is suﬃcient to represent also {=, <} (see [DdM06])

o es
p nmt

LRA Resolution

• LRA constraints can be rewritten in terms of upper bound x ≤ p or
lower bound −x ≤ p for a variable x.

o es
p nmt

LRA Resolution


LRA Resolution for two constraints
(x ≤ p) ⊗x (−x ≤ q) := (−q ≤ p)

o es
p nmt

LRA Resolution


(x ≤ p) ⊗x (−x ≤ q) := (−q ≤ p)

• Let Sx , S−x be the set of upper resp. lower bounds for x

o es
p nmt

LRA Resolution


(x ≤ p) ⊗x (−x ≤ q) := (−q ≤ p)


LRA Resolution for sets of constraints
Sx ⊗x S−x := {(x ≤ p) ⊗x (−x ≤ q) | (x ≤ p) ∈ Sx , (−x ≤ q) ∈ S−x }

o es
p nmt

LRA Resolution


LRA Resolution for sets of constraints
Sx ⊗x S−x := {(x ≤ p) ⊗x (−x ≤ q) | (x ≤ p) ∈ Sx , (−x ≤ q) ∈ S−x }

Theorem [Fou26]
Sx ∪ S−x is equisatisﬁable with Sx ⊗x S−x

o es
p nmt

FM - Example (on var z)

OLD NEW
−x + z ≤ −4
x + z ≤ 18
x −z ≤6
−x − z ≤ −16
y ≤5
−y ≤ −3
S−z −x − z ≤ −16 o es
p nmt


OLD NEW
−x + z ≤ −4
Sz
x + z ≤ 18
x −z ≤6
S−z
−x − z ≤ −16
y ≤5 y ≤5
−y ≤ −3 −y ≤ −3
S−z −x − z ≤ −16 o es
p nmt


OLD NEW
−x + z ≤ −4 0≤2
Sz
x + z ≤ 18
x −z ≤6
S−z
−x − z ≤ −16
y ≤5 y ≤5
−y ≤ −3 −y ≤ −3
S−z −x − z ≤ −16 o es
p nmt


OLD NEW
−x + z ≤ −4 0≤2
Sz
x + z ≤ 18 −x ≤ −10
x −z ≤6
S−z
−x − z ≤ −16
y ≤5 y ≤5
−y ≤ −3 −y ≤ −3
S−z −x − z ≤ −16 o es
p nmt


OLD NEW
−x + z ≤ −4 0≤2
Sz
x + z ≤ 18 −x ≤ −10
x −z ≤6 x ≤ 12
S−z
−x − z ≤ −16
y ≤5 y ≤5
−y ≤ −3 −y ≤ −3
S−z −x − z ≤ −16 o es
p nmt


OLD NEW
−x + z ≤ −4 0≤2
Sz
x + z ≤ 18 −x ≤ −10
x −z ≤6 x ≤ 12
S−z
−x − z ≤ −16 0≤2
y ≤5 y ≤5
−y ≤ −3 −y ≤ −3
S−z −x − z ≤ −16 o es
p nmt


OLD NEW
−x + z ≤ −4
x + z ≤ 18 −x ≤ −10
x −z ≤6 x ≤ 12
−x − z ≤ −16
y ≤5 y ≤5
−y ≤ −3 −y ≤ −3

o es
p nmt


o es
p nmt

DP + FM = DPFM

Variable Types:
Resolution Rules:
Algorithm:
Input:
Repeat

o es
p nmt

DP + FM = DPFM

Variable Types: Bool , Rational
Resolution Rules:
Algorithm:
Input:
Repeat

o es
p nmt

DP + FM = DPFM

Resolution Rules: BR , SMT(LRA) Resolution (SR)
Algorithm:
Input:
Repeat

o es
p nmt

DP + FM = DPFM

Algorithm:
Input: a set of SMT(LRA) clauses in OCCF
Repeat

o es
p nmt

DP + FM = DPFM

Algorithm:
Repeat
Choose a variable X of type Bool ( Rational ) to eliminate

o es
p nmt

DP + FM = DPFM

Algorithm:
Repeat
Choose a variable X of type Bool ( Rational ) to eliminate
Combine positive and negative occurrences of X, using BR ( SR )

o es
p nmt

One Constraint per Clause Form (OCCF)

• DPFM requires clauses in OCCF, clauses that contain
at most one LRA constraint

o es
p nmt



• it is easy to transform clauses in OCCF, by means of auxiliary
Boolean variables

o es
p nmt



• it is easy to transform clauses in OCCF, by means of auxiliary
Boolean variables

• E.g. (a ∨ (x ≤ 3) ∨ b ∨ (x + y ≤ 10)) can be rewritten as
(a ∨ (x ≤ 3) ∨ b ∨ c ) and ( ¬c ∨ (x + y ≤ 10))

o es
p nmt

SMT(LRA) Resolution

• negated LRA constr. can be expressed in terms of ≤
• e.g. ¬(x ≤ 10) is equiv. to −x ≤ −10 − δ, (δ > 0) (see [DdM06])

o es
p nmt

SMT(LRA) Resolution

• LRA constraints in SMT(LRA) clauses can be rewritten in terms of
upper bound x ≤ p or lower bound −x ≤ p for a variable x.

o es
p nmt

SMT(LRA) Resolution


SMT(LRA) Resolution for two clauses in OCCF
(C1 ∨(x ≤ p)∨C2 )⊗x (D1 ∨(−x ≤ q)∨D2 ) := C1 ∨C2 ∨(−q ≤ p)∨D1 ∨D2

o es
p nmt

SMT(LRA) Resolution



• Let Sx , S−x be the set of SMT(LRA) clauses where x appears upper
bounded resp. lower bounded

o es
p nmt

SMT(LRA) Resolution




SMT(LRA) Resolution for sets of clauses in OCCF
Sx ⊗x S−x := {C1 ⊗x C2 | C1 ∈ Sx , C2 ∈ S−x }
o es
p nmt

SMT(LRA) Resolution


SMT(LRA) Resolution for sets of clauses in OCCF
Sx ⊗x S−x := {C1 ⊗x C2 | C1 ∈ Sx , C2 ∈ S−x }

Theorem
Sx ∪ S−x is equisatisﬁable with Sx ⊗x S−x

o es
p nmt

DPFM - Example (on var z)

¬a1 ∨ (−z ≤ −3) a1 ∨ (z ≤ 3 − δ) ∨ a2
¬a1 ∨ (−x ≤ −3) ¬a2 ∨ (x ≤ 3 − δ) ∨ a3
¬a1 ∨ (−y ≤ −3) ¬a3 ∨ (y ≤ 3 − δ) ∨ a4
¬a1 ∨ (y ≤ 5) ¬a4 ∨ (−y ≤ 5 − δ) ∨ a5
¬a1 ∨ (x ≤ 5) ¬a5 ∨ (−x ≤ 5 − δ) ∨ a6
¬a1 ∨ (z ≤ 5) ¬a6 ∨ (−z ≤ 5 − δ)
¬b1 ∨ (−z ≤ −2) b1 ∨ (z ≤ 2 − δ) ∨ b2
¬b1 ∨ (−x ≤ −2) ¬b2 ∨ (x ≤ 2 − δ) ∨ b3
¬b1 ∨ (−y ≤ −2) ¬b3 ∨ (y ≤ 2 − δ) ∨ b4
¬b1 ∨ (y ≤ 4) ¬b4 ∨ (−y ≤ 4 − δ) ∨ b5
¬b1 ∨ (x ≤ 4) ¬b5 ∨ (−x ≤ 4 − δ) ∨ b6
¬b1 ∨ (z ≤ 4) ¬b6 ∨ (−z ≤ 4 − δ)
a1 ∨ b1

o es
p nmt


¬a1 ∨ (−z ≤ −3) a1 ∨ (z ≤ 3 − δ) ∨ a2
¬a1 ∨ (−x ≤ −3) ¬a2 ∨ (x ≤ 3 − δ) ∨ a3
¬a1 ∨ (−y ≤ −3) ¬a3 ∨ (y ≤ 3 − δ) ∨ a4
¬a1 ∨ (y ≤ 5) ¬a4 ∨ (−y ≤ −5 − δ) ∨ a5
¬a1 ∨ (x ≤ 5) ¬a5 ∨ (−x ≤ −5 − δ) ∨ a6
¬a1 ∨ (z ≤ 5) ¬a6 ∨ (−z ≤ −5 − δ)
¬b1 ∨ (−z ≤ −2) b1 ∨ (z ≤ 2 − δ) ∨ b2
¬b1 ∨ (−x ≤ −2) ¬b2 ∨ (x ≤ 2 − δ) ∨ b3
¬b1 ∨ (−y ≤ −2) ¬b3 ∨ (y ≤ 2 − δ) ∨ b4
¬b1 ∨ (y ≤ 4) ¬b4 ∨ (−y ≤ −4 − δ) ∨ b5
¬b1 ∨ (x ≤ 4) ¬b5 ∨ (−x ≤ −4 − δ) ∨ b6
¬b1 ∨ (z ≤ 4) ¬b6 ∨ (−z ≤ −4 − δ)
a1 ∨ b1

o es
p nmt


OLD NEW
¬a1 ∨ (z ≤ 5)
¬b1 ∨ (z ≤ 4)
a1 ∨ (z ≤ 3 − δ) ∨ a2
b1 ∨ (z ≤ 2 − δ) ∨ b2
¬a6 ∨ (−z ≤ −5 − δ)
¬b6 ∨ (−z ≤ −4 − δ)
¬a1 ∨ (−z ≤ −3)
¬b1 ∨ (−z ≤ −2)
S−z a1 ∨ (z ≤ 3 − δ) ∨ a2

o es
p nmt


OLD NEW
¬a1 ∨ (z ≤ 5)
¬b1 ∨ (z ≤ 4)
Sz
a1 ∨ (z ≤ 3 − δ) ∨ a2
b1 ∨ (z ≤ 2 − δ) ∨ b2
¬a6 ∨ (−z ≤ −5 − δ)
¬b6 ∨ (−z ≤ −4 − δ)
S−z
¬a1 ∨ (−z ≤ −3)
¬b1 ∨ (−z ≤ −2)
S−z a1 ∨ (z ≤ 3 − δ) ∨ a2

o es
p nmt


OLD NEW
¬a1 ∨ (z ≤ 5)
Sz ¬b1 ∨ (z ≤ 4)
a1 ∨ (z ≤ 3 − δ) ∨ a2
b1 ∨ (z ≤ 2 − δ) ∨ b2
¬a6 ∨ (−z ≤ −5 − δ)
S−z ¬b6 ∨ (−z ≤ −4 − δ)
¬a1 ∨ (−z ≤ −3)
¬b1 ∨ (−z ≤ −2)
S−z a1 ∨ (z ≤ 3 − δ) ∨ a2

o es
p nmt


OLD NEW
¬a1 ∨ (z ≤ 5) ¬a1 ∨ (0 ≤ −δ) ∨ ¬a6
Sz ¬b1 ∨ (z ≤ 4)
a1 ∨ (z ≤ 3 − δ) ∨ a2
b1 ∨ (z ≤ 2 − δ) ∨ b2
¬a6 ∨ (−z ≤ −5 − δ)
S−z ¬b6 ∨ (−z ≤ −4 − δ)
¬a1 ∨ (−z ≤ −3)
¬b1 ∨ (−z ≤ −2)
S−z a1 ∨ (z ≤ 3 − δ) ∨ a2

o es
p nmt


OLD NEW
¬a1 ∨ (z ≤ 5) ¬a1 ∨ (0 ≤ −δ) ∨ ¬a6
¬b1 ∨ (z ≤ 4) ¬a1 ∨ (0 ≤ 1 − δ) ∨ ¬b6
Sz
a1 ∨ (z ≤ 3 − δ) ∨ a2 ¬a1 ∨ (0 ≤ 2)
b1 ∨ (z ≤ 2 − δ) ∨ b2 ¬a1 ∨ (0 ≤ 3) ∨ ¬b1
¬a6 ∨ (−z ≤ −5 − δ) ¬b1 ∨ (0 ≤ −1 − δ) ∨ ¬a6
¬b6 ∨ (−z ≤ −4 − δ) ¬b1 ∨ (0 ≤ −δ) ∨ ¬b6
S−z
¬a1 ∨ (−z ≤ −3) ¬b1 ∨ (0 ≤ 1) ∨ ¬a1
¬b1 ∨ (−z ≤ −2) ¬b1 ∨ (0 ≤ 2)
a1 ∨ (0 ≤ −2 − δ) ∨ a2 ∨ ¬a6
a1 ∨ (0 ≤ −1 − δ) ∨ a2 ∨ ¬b6
a1 ∨ (0 ≤ −δ) ∨ ¬a1 ∨ a2
a1 ∨ (0 ≤ 1 − δ) ∨ ¬b1 ∨ a2
b1 ∨ (0 ≤ −3 − δ) ∨ b2 ∨ ¬a6
b1 ∨ (0 ≤ −2 − δ) ∨ b2 ∨ ¬b6
b1 ∨ (0 ≤ −1 − δ) ∨ ¬a1 ∨ b2
b1 ∨ (0 ≤ −δ) ∨ ¬b1 ∨ b2
o es
p nmt
S−z a1 ∨ (z ≤ 3 − δ) ∨ a2


OLD NEW
¬a1 ∨ (z ≤ 5) ¬a1 ∨ ¬a6
¬b1 ∨ (z ≤ 4)
a1 ∨ (z ≤ 3 − δ) ∨ a2
b1 ∨ (z ≤ 2 − δ) ∨ b2
¬a6 ∨ (−z ≤ −5 − δ) ¬b1 ∨ ¬a6
¬b6 ∨ (−z ≤ −4 − δ) ¬b1 ∨ ¬b6
¬a1 ∨ (−z ≤ −3)
¬b1 ∨ (−z ≤ −2)
a1 ∨ a2 ∨ ¬a6
a1 ∨ a2 ∨ ¬b6

b1 ∨ b2 ∨ ¬a6
b1 ∨ b2 ∨ ¬b6
b1 ∨ ¬a1 ∨ b2
o es
p nmt
S−z a1 ∨ (z ≤ 3 − δ) ∨ a2


o es
p nmt

Variable Elimination - Complexity
• High worst-case complexity

o es
p nmt

• Suppose we have n variables and m initial clauses

o es
p nmt

• Suppose we eliminate a1 : in the worst case |Sa1 | = |S¬a1 | = m
2

o es
p nmt

2
2
• After elim. a1 we have m × m = m clauses
2 2 4

o es
p nmt

2
2
2 2 4
2 2 4
• After elim. a2 we have m × m = m3 clauses
8 8 4

o es
p nmt

2
2
2 2 4
2 2 4
8 8 4
n−1 n−1 n
m2 m2 m2
• After elim. an we have n−1 − 1 ) × n−1 − 1 ) = 42n −1
clauses
4(2 2 4(2 2

o es
p nmt

2
2
2 2 4
2 2 4
8 8 4
n−1 n−1 n
m2 m2 m2
• After elim. an we have n−1 − 1 ) × n−1 − 1 ) = 42n −1
clauses
4(2 2 4(2 2

m=10
1e+14

1e+12

1e+10

1e+08

1e+06

10000

100

1
0 1 2 3 4 5
o es
p nmt

Formula Simpliﬁcation and Preprocessing

• Variable Elimination technique for SMT(LRA)

o es
p nmt


• We apply it for preprocessing, similarly to the SATElite [EB05]
preprocessor
• Variable elimination rule (among other rules) is applied in a controlled
manner so that it does not produce too many clauses

o es
p nmt


preprocessor
• In our case we use two upper bounds for two parameters
concerning Rational variables

o es
p nmt


preprocessor
• Centrality (for x) : number of distinct variables that appear in some
constraint with x

o es
p nmt


preprocessor
• Centrality (for x) : number of distinct variables that appear in some
constraint with x
• Trade-oﬀ (for x) : amount of new clauses that we want to “trade”
for eliminating x

o es
p nmt

Formula Simpliﬁcation - (Centrality 2, Trade-oﬀ 128)

OpenSMT on QF IDL/qlock Benchmarks - Structural Data
P.Time (s) Clauses TAtoms TVars
Bench WO W WO W WO W WO W
Ind 37 1.08 6.57 41137 35299 6129 5285 829 185
Ind 38 1.16 6.62 42265 36244 6299 5423 851 188
Ind 39 1.19 7.02 43381 37150 6467 5562 873 189
Ind 40 1.17 7.05 44457 38114 6619 5702 895 203
Base 18 0.80 1.87 18630 16314 2867 2559 375 137
Base 19 0.82 2.31 19780 17269 3045 2702 397 150
Base 20 0.95 2.47 20914 18246 3215 2851 419 151
Base 21 0.94 2.54 22052 19193 3389 2995 441 155

o es
p nmt

Formula Simpliﬁcation - (Centrality 2, Trade-oﬀ 128)

OpenSMT on QF IDL/qlock Benchmarks - Structural Data
P.Time (s) Clauses TAtoms TVars
Bench WO W WO W WO W WO W
Ind 37 1.08 6.57 41137 35299 6129 5285 829 185
Ind 38 1.16 6.62 42265 36244 6299 5423 851 188
Ind 39 1.19 7.02 43381 37150 6467 5562 873 189
Ind 40 1.17 7.05 44457 38114 6619 5702 895 203
Base 18 0.80 1.87 18630 16314 2867 2559 375 137
Base 19 0.82 2.31 19780 17269 3045 2702 397 150
Base 20 0.95 2.47 20914 18246 3215 2851 419 151
Base 21 0.94 2.54 22052 19193 3389 2995 441 155

OpenSMT on QF IDL/qlock Benchmarks - Solving Time
Bench Time WO (s) Time W (s) Bench Time WO (s) Time W (s)
Base 18 61.3 59.0 Ind 37 90.5 18.0
Base 19 146.1 138.4 Ind 38 105.7 54.6
Base 20 > 1800 940.1 Ind 39 64.4 46.7
Base 21 1367.9 765.0 Ind 40 98.3 37.3
o es
p nmt

Mixed Boolean-Theory Static Learning

OpenSMT on QF IDL/job shop/jobshop12-2-6-6-2-4-9.smt
Centr. Trade-Oﬀ VE P.Time Clauses TAtoms BAtoms T.Time (s)
- - 0 0.05 216 612 0 > 1800
12 64 0 0.05 216 612 0 > 1800
12 256 2 0.06 458 832 22 180.0
12 1024 4 0.04 1094 968 42 91.4
12 4096 6 0.09 3076 1032 60 67.2
12 16384 6 0.10 3076 1032 60 67.1
18 64 0 0.02 216 612 0 > 1800
18 256 4 0.02 714 1054 56 192.3
18 1024 8 0.07 2005 1566 109 105.6
18 4096 12 0.15 5702 2254 156 125.6
18 16384 12 0.16 5702 2254 156 125.9
24 64 0 0.02 216 612 0 > 1800
24 256 4 0.03 781 1108 66 193.2
24 1024 8 0.07 1978 1638 117 157.1
24 4096 11 0.19 5005 2198 153 89.4
24 16384 12 0.32 5519 2294 163 92.2
o es
p nmt

Fractal Diamonds

Our preprocessor is effective for those formulæ that are difficult to solve
with the initial fixed set of theory atoms

y1 y2 y3

x1 x2 x3 x4

z1 z2 z3

o es
p nmt

Fractal Diamonds

Our preprocessor is effective for those formulæ that are difficult to solve
with the initial fixed set of theory atoms

o es
p nmt

Fractal Diamonds (Centrality 18, Trade-oﬀ 8192)

B = BarceLogic (SMTCOMP’08 1st place for IDL)
Z = Z3 (SMTCOMP’08 2nd place for IDL)
O = OpenSMT (with DPFM based preprocessor)

Fractal Diamonds - Solving time (s) - TO = 1200 s
1 2 3 4 5
Or. B Z O B Z O B Z O B Z O B Z O

o es
p nmt



1 2 3 4 5
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o es
p nmt



1 2 3 4 5
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 118 13 1 T T 3 T T 7

o es
p nmt



1 2 3 4 5
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 118 13 1 T T 3 T T 7
3 0 0 0 0 T 2 T T 153 M T T T T T

o es
p nmt

Related Work

• O. Strichman: “Deciding Disjunctive Linear Arithmetic with
SAT” [Str04]
• Eager reduction to SAT, using a procedure based on FM

o es
p nmt

Related Work

SAT” [Str04]

• N. E´n, A. Biere: “Eﬀective Preprocessing in SAT Through Variable
e
and Clause Elimination” [EB05]
• SATElite algorithm for SAT preprocessing

o es
p nmt

Related Work

SAT” [Str04]

• N. E´n, A. Biere: “Eﬀective Preprocessing in SAT Through Variable
e
and Clause Elimination” [EB05]
• SATElite algorithm for SAT preprocessing

• K. McMillan et al.: “Generalizing DPLL to Richer Logics” [MKS09]
• “Shadow Rule” similar to our notion of SMT(LRA) resolution: one
application of the shadow rule is equiv. to many applications of
SMT(LRA) resolution

o es
p nmt

Outline

1 Introduction

2 Architecture

DP + FM = DPFM
Related Work

Extending OpenSMT

5 Conclusion

o es
p nmt

Extending OpenSMT

e(DPLL(T)) ≈ e(T)

DPLL

o es
p nmt

Extending OpenSMT

e(DPLL(T)) ≈ e(T)

DPLL T

o es
p nmt

Extending OpenSMT

• To create an empty template for a new theory solver
use script create tsolver.sh

o es
p nmt

Extending OpenSMT

• Creates a new directory with basic class ﬁles

o es
p nmt

Extending OpenSMT

• Creates/Sets up Makeﬁle for compilation

o es
p nmt

Extending OpenSMT

• Adds a new logic

o es
p nmt

Extending OpenSMT

• Integrates the new solver with the core

o es
p nmt

Extending OpenSMT

• Integrates the new solver with the core
• Basically, it creates an incomplete solver

o es
p nmt

Extending OpenSMT

class TSolver
{
void inform ( Enode * );
bool assertLit ( Enode * );
bool check ( bool );
void pushBktPoint ( );
void popBktPoint ( );
bool belongsToT ( Enode * );
void computeModel ( );

vector< Enode * > & explanation;
vector< Enode * > & deductions;
vector< Enode * > & suggestions;
}
o es
p nmt

Other Features

• C API: you can compile a library and call OpenSMT via the C API

o es
p nmt

Other Features


• Interpolation: given two mutually unsatisfiable formulæ A and B, an
interpolant [Cra57] is a formula I such that
• A→I
• B ∧ I is unsatisfiable
• I is defined on the variables that are common to A and B

o es
p nmt

Other Features


• A→I

• Basically I is an “overapproximation” of A, that is still unsatisﬁable
with B

o es
p nmt

Other Features


• A→I

with B
• It has applications in Model Checking [McM04]

o es
p nmt

Other Features


• A→I

with B
• It has applications in Model Checking [McM04]
• OpenSMT can compute interpolants for propositional formulæ and
some arithmetic fragments

o es
p nmt

Conclusion

• OpenSMT website
http://www.verify.inf.unisi.ch/opensmt

• Source repository
http://code.google.com/p/opensmt

• Discussion group
http://groups.google.com/group/opensmt

o es
p nmt

W. Craig.
Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory.
J. Symb. Log., pages 269–285, 1957.

B. Dutertre and L. M. de Moura.
A Fast Linear-Arithmetic Solver for DPLL(T).
In CAV’06, pages 81–94, 2006.

Martin Davis and Hilary Putnam.
A Computing Procedure for Quantification Theory.
J. ACM, 7(3):201–215, 1960.

N. Eń and A. Biere.
e
Effective Preprocessing in SAT Through Variable and Clause Elimination.
In SAT, pages 61–75, 2005.

J.B.J. Fourier.
Solution d’une question particulire du calcul des angalits.
Oevres, II:314–328, 1826.

K. L. McMillan.
Applications of Craig Interpolation to Model Checking.
In CSL, pages 22–23, 2004.

K. L. McMillan, A. Kuehlmann, and M. Sagiv.
Generalizing dpll to richer logics.
In CAV, pages 462–476, 2009.

O. Strichman. o es
p nmt

Deciding Disjunctive Linear Arithmetic with SAT.
CoRR, 2004.

o es
p nmt

20100822 opensmt bruttomesso

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (20)

More from Computer Science Club

More from Computer Science Club (20)

20100822 opensmt bruttomesso