SMT Solvers - GitHub Pages · 2019. 8. 30. · Linear Integer Arithmetic ΣP = {≤}, ΣF = {0,1,+,−}. Let MLIA be the standard model of integers. Then T LIA is deﬁned to be the

SMT Solvers

Theory & Practice

Leonardo de Moura

[email protected]

Microsoft Research

FMCAD 2006 – p.1/75

Credits

Slides inspired by previous presentations by:

Clark Barrett, Harald Ruess, Natarajan Shankar, Cesare

Tinelli, Ashish Tiwari

Special thanks to:

Clark Barrett, Cesare Tinelli (for contributing some of the

material) and the FMCAD PC (for the invitation).

FMCAD 2006 – p.2/75

Introduction

Industry tools rely on powerful verification engines.

Boolean satisfiability (SAT) solvers.

Binary decision diagrams (BDDs).

Satisfiability Modulo Theories (SMT)

The next generation of verification engines.

SAT solvers + Theories

Arithmetic

Arrays

Uninterpreted Functions

Some problems are more naturally expressed in SMT.

More automation.

FMCAD 2006 – p.3/75

Applications

Extended Static Checking.

Microsoft Spec# and ESP.

ESC/Java

Predicate Abstraction.

Microsoft SLAM/SDV (device driver verification).

Bounded Model Checking (BMC) & k-induction.

Test-case generation.

Microsoft MUTT.

Symbolic Simulation.

Planning & Scheduling.

Equivalence checking.

FMCAD 2006 – p.4/75

SMT-Solvers & SMT-Lib & SMT-Comp

SMT-Solves:

Ario, Barcelogic, CVC, CVC Lite, CVC3, ExtSAT, Harvey,

HTP, ICS (SRI), Jat, MathSAT, Sateen, Simplify, STeP, STP,

SVC, TSAT, UCLID, Yices (SRI), Zap (Microsoft),

Z3 (Microsoft)

SMT-Lib: library of benchmarks

http://goedel.cs.uiowa.edu/smtlib/

SMT-Comp: annual SMT-Solver competition.

FMCAD 2006 – p.5/75

http://goedel.cs.uiowa.edu/smtlib/

Roadmap

Background

Theories

Combination of Theories

SAT + Theories

Decision Procedures for Specific Theories

Applications

FMCAD 2006 – p.6/75

Language: Signatures

A signature Σ is a finite set of:

Function symbols: ΣF = {f, g, . . .}.

Predicate symbols: ΣP = {P,Q, . . .}.

and an arity function: Σ 7→ N

Function symbols with arity 0 are called constants.

A countable set V of variables disjoint of Σ.

FMCAD 2006 – p.7/75

Language: Terms

The set T (Σ,V) of terms is the smallest set such that:

V ⊂ T (Σ,V)

f(t1, . . . , tn) ∈ T (Σ,V) whenever

f ∈ ΣF , t1, . . . , tn ∈ T (Σ,V) and arity(f) = n.

The set of ground terms is defined as T (Σ, ∅).

FMCAD 2006 – p.8/75

Language: Atomic Formulas

P (t1, . . . , tn) is an atomic formula whenever

P ∈ ΣP , arity(P ) = n, and t1, . . . , tn ∈ T (Σ,V).

true and false are atomic formulas.

If t1, . . . , tn are ground terms, then P (t1, . . . , tn) is called a

ground (atomic) formula.

We assume that the binary predicate = is present in ΣP .

A literal is an atomic formula or its negation.

FMCAD 2006 – p.9/75

Language: Quantifier Free Formulas

The set QFF(Σ,V) of quantifier free formulas is the smallest set

such that:

Every atomic formulas is in QFF(Σ,V).

If φ ∈ QFF(Σ,V), then ¬φ ∈ QFF(Σ,V).

If φ1, φ2 ∈ QFF(Σ,V), then

φ1 ∧ φ2 ∈ QFF(Σ,V)

φ1 ∨ φ2 ∈ QFF(Σ,V)

φ1 ⇒ φ2 ∈ QFF(Σ,V)

φ1 ⇔ φ2 ∈ QFF(Σ,V)

FMCAD 2006 – p.10/75

Language: Formulas

The set of first-order formulas is the closure of QFF(Σ,V) under

existential (∃) and universal (∀) quantification.

Free (occurrences) of variables in a formula are those not bound by

a quantifier.

A sentence is a first-order formula with no free variables.

FMCAD 2006 – p.11/75

Theories

A (first-order) theory T (over a signature Σ) is a set of (deductively

closed) sentences (over Σ and V ).

Let DC(Γ) be the deductive closure of a set of sentences Γ.

For every theory T , DC(T ) = T .

A theory T is consistent if false 6∈ T .

We can view a (first-order) theory T as the class of all models of

T (due to completeness of first-order logic).

FMCAD 2006 – p.12/75

Models (Semantics)

A model M is defined as:

Domain S: set of elements.

Interpretation fM : Sn 7→ S for each f ∈ ΣF with

arity(f) = n.

Interpretation PM ⊆ Sn for each P ∈ ΣP with arity(P ) = n.

Assignment xM ∈ S for every variable x ∈ V .

A formula φ is true in a model M if it evaluates to true under the

given interpretations over the domain S.

M is a model for the theory T if all sentences of T are true in M .

FMCAD 2006 – p.13/75

Satisfiability and Validity

A formula φ(~x) is satisfiable in a theory T if there is a model of

DC(T ∪ ∃~x.φ(~x)). That is, there is a model M for T in which

φ(~x) evaluates to true, denoted by,

M |=T φ(~x)

This is also called T -satisfiability.

A formula φ(~x) is valid in a theory T if ∀~x.φ(~x) ∈ T . That is

φ(~x) evaluates to true in every model M of T .

T -validity is denoted by |=T φ(~x).

The quantifier free T -satisfiability problem restricts φ to be

quantifier free.

FMCAD 2006 – p.14/75

Checking validity

Checking the validity of φ in a theory T is:

≡ T -satisfiability of ¬φ

≡ T -satisfiability of ~Q~x.φ1 (PNF of ¬φ)

≡ T -satisfiability of ∀~x.φ1 (Skolemize)

≡ T -satisfiability of φ2 (Instantiate)

≡ T -satisfiability of∨

i ψi (DNF of φ2)

≡ T -satisfiability of every ψi

ψi is a conjunction of literals.

FMCAD 2006 – p.15/75

Roadmap

Background

Theories


SAT + Theories


Applications

FMCAD 2006 – p.16/75

Pure Theory of Equality (EUF)

The theory T E of equality is the theory DC(∅).

The exact set of sentences of T E depends on the signature in

question.

The theory does not restrict the possibles values of the symbols in

its signature in any way. For this reason, it is sometimes called the

theory of equality and uninterpreted functions.

The satisfiability problem for T E is the satisfiability problem for

first-order logic, which is undecidable.

The satisfiability problem for conjunction of literals in T E is

decidable in polynomial time using congruence closure.

FMCAD 2006 – p.17/75

Linear Integer Arithmetic

ΣP = {≤}, ΣF = {0, 1,+,−}.

Let MLIA be the standard model of integers.

Then T LIA is defined to be the set of all Σ sentences true in the

model MLIA.

As showed by Presburger, the general satisfiability problem for

T LIA is decidable, but its complexity is triply-exponential.

The quantifier free satisfiability problem is NP-complete.

Remark: non-linear integer arithmetic is undecidable even for the

quantifier free case.

FMCAD 2006 – p.18/75

Linear Real Arithmetic

The general satisfiability problem for T LRA is decidable, but its

complexity is doubly-exponential.

The quantifier free satisfiability problem is solvable in polynomial

time, though exponential methods (Simplex) tend to perform best in

practice.

FMCAD 2006 – p.19/75

Difference Logic

Difference logic is a fragment of linear arithmetic.

Atoms have the form: x− y ≤ c.

Most linear arithmetic atoms found in hardware and software

verification are in this fragment.

The quantifier free satisfiability problem is solvable in O(nm).

FMCAD 2006 – p.20/75

Theory of Arrays

ΣP = ∅, ΣF = {read,write}.

Non-extensional arrays

Let ΛA be the following axioms:

∀a, i, v. read(write(a, i, v), i) = v

∀a, i, j, v. i 6= j ⇒ read(write(a, i, v), j) = read(a, j)

T A = DC(ΛA)

For extensional arrays, we need the following extra axiom:

∀a, b. (∀i.read(a, i) = read(b, i)) ⇒ a = b

The satisfiability problem for T A is undecidable, the quantifier free

case is NP-complete.

FMCAD 2006 – p.21/75

Other theories

Bit-vectors

Partial orders

Tuples & Records

Algebraic data types

. . .

FMCAD 2006 – p.22/75

Roadmap

Background

Theories


SAT + Theories


Applications

FMCAD 2006 – p.23/75


In practice, we need a combination of theories.

Examples:

x+2 = y ⇒ f(read(write(a, x, 3), y−2)) = f(y−x+1)

f(f(x) − f(y)) 6= f(z), x+ z ≤ y ≤ x⇒ z < 0

Given

Σ = Σ1 ∪ Σ2

T 1, T 2 : theories over Σ1,Σ2

T = DC(T 1 ∪ T 2)

Is T consistent?

Given satisfiability procedures for conjunction of literals of T 1 and

T 2, how to decide the satisfiability of T ?FMCAD 2006 – p.24/75

Preamble

Disjoint signatures: Σ1 ∩ Σ2 = ∅.

Stably-Infinite Theories.

Convex Theories.

FMCAD 2006 – p.25/75

Stably-Infinite Theories

A theory is stably infinite if every satisfiable QFF is satisfiable in an

infinite model.

Example. Theories with only finite models are not stably infinite.

T2 = DC(∀x, y, z. (x = y) ∨ (x = z) ∨ (y = z)).

Is this a problem in practice? (We want to support the “finite types”

found in our programming languages)

FMCAD 2006 – p.26/75

Stably-Infinite Theories

A theory is stably infinite if every satisfiable QFF is satisfiable in an

infinite model.

Example. Theories with only finite models are not stably infinite.

T2 = DC(∀x, y, z. (x = y) ∨ (x = z) ∨ (y = z)).

Is this a problem in practice? (We want to support the “finite types”

found in our programming languages)

Answer: No. T2 is not useful in practice. Add a predicate in2(x)

(intuition: x is an element of the “finite type”).

T2′ = DC(∀x, y, z. in2(x) ∧ in2(y) ∧ in2(z) ⇒

(x = y) ∨ (x = z) ∨ (y = z))

T2′ is stably infinite.

FMCAD 2006 – p.26/75

Stably-Infinite Theories (cont.)

The union of two consistent, disjoint, stably infinite theories is

consistent.

FMCAD 2006 – p.27/75

Convexity

A theory T is convex iff

for all finite sets Γ of literals and

for all non-empty disjunctions∨

i∈I xi = yi of variables,

Γ |=T

∨i∈I xi = yi iff Γ |=T xi = yi for some i ∈ I .

Every convex theory T with non trivial models (i.e.,

|=T ∃x, y. x 6= y) is stably infinite.

All Horn theories are convex – this includes all (conditional)

equational theories.

Linear rational arithmetic is convex.

FMCAD 2006 – p.28/75

Convexity (cont.)

Many theories are not convex:

Linear integer arithmetic.

1 ≤ x ≤ 3 |= x = 1 ∨ x = 2 ∨ x = 3

Nonlinear arithmetic.

x2 = 1, y = 1, z = −1 |= x = y ∨ x = z

Theory of Bit-vectors.

Theory of Arrays.

v1 = read(write(a, i, v2), j), v3 = read(a, j) |=

v1 = v2 ∨ v1 = v3

FMCAD 2006 – p.29/75

Convexity: Example

Let T = T 1 ∪ T 2, where T 1 is EUF (O(nlog(n))) and T 2 is

IDL (O(nm)).

T 2 is not convex.

Satisfiability is NP-Complete for T = T 1 ∪ T 2.

Reduce 3CNF satisfiability to T -satisfiability.

For each boolean variable pi add the atomic formulas:

0 ≤ xi, xi ≤ 1.

For a clause p1 ∨ ¬p2 ∨ p3 add the atomic formula:

f(x1, x2, x3) 6= f(0, 1, 0)

FMCAD 2006 – p.30/75

Nelson-Oppen Combination

Let T 1 and T 2 be consistent, stably infinite theories over disjoint

(countable) signatures. Assume satisfiability of conjunction of

literals can decided in O(T1(n)) and O(T2(n)) time respectively.

Then,

1. The combined theory T is consistent and stably infinite.

2. Satisfiability of quantifier free conjunction of literals in T can be

decided in O(2n2

× (T1(n) + T2(n)).

3. If T 1 and T 2 are convex, then so is T and satisfiability in T is

in O(n4 × (T1(n) + T2(n))).

FMCAD 2006 – p.31/75

Nelson-Oppen Combination Procedure

The combination procedure:

Initial State: φ is a conjunction of literals over Σ1 ∪ Σ2.

Purification: Preserving satisfiability transform φ into φ1 ∧ φ2,

such that, φi ∈ Σi.

Interaction: Guess a partition of V(φ1) ∩ V(φ2) into disjoint

subsets. Express it as conjunction of literals ψ.

Example. The partition {x1}, {x2, x3}, {x4} is represented

as x1 6= x2, x1 6= x4, x2 6= x4, x2 = x3.

Component Procedures : Use individual procedures to decide

whether φi ∧ ψ is satisfiable.

Return: If both return yes, return yes. No, otherwise.

FMCAD 2006 – p.32/75

Purification

Purification:

φ ∧ P (. . . , s[t], . . .) φ ∧ P (. . . , s[x], . . .) ∧ x = t,

t is not a variable.

Purification is satisfiability preserving and terminating.

FMCAD 2006 – p.33/75

Purification

Purification:

φ ∧ P (. . . , s[t], . . .) φ ∧ P (. . . , s[x], . . .) ∧ x = t,



Example:

f(x− 1) − 1 = x, f(y) + 1 = y

FMCAD 2006 – p.33/75

Purification

Purification:

φ ∧ P (. . . , s[t], . . .) φ ∧ P (. . . , s[x], . . .) ∧ x = t,



Example:

f(x− 1) − 1 = x, f(y) + 1 = y

f(u1) − 1 = x, f(y) + 1 = y, u1 = x− 1

FMCAD 2006 – p.33/75

Purification

Purification:

φ ∧ P (. . . , s[t], . . .) φ ∧ P (. . . , s[x], . . .) ∧ x = t,



Example:

f(x− 1) − 1 = x, f(y) + 1 = y

f(u1) − 1 = x, f(y) + 1 = y, u1 = x− 1

u2 − 1 = x, f(y) + 1 = y, u1 = x− 1, u2 = f(u1)

FMCAD 2006 – p.33/75

Purification

Purification:

φ ∧ P (. . . , s[t], . . .) φ ∧ P (. . . , s[x], . . .) ∧ x = t,



Example:

f(x− 1) − 1 = x, f(y) + 1 = y

f(u1) − 1 = x, f(y) + 1 = y, u1 = x− 1

u2 − 1 = x, f(y) + 1 = y, u1 = x− 1, u2 = f(u1)

u2 − 1 = x, u3 + 1 = y, u1 = x− 1, u2 = f(u1), u3 = f(y)

FMCAD 2006 – p.33/75

Purification

Purification:

φ ∧ P (. . . , s[t], . . .) φ ∧ P (. . . , s[x], . . .) ∧ x = t,



Example:

f(x− 1) − 1 = x, f(y) + 1 = y

f(u1) − 1 = x, f(y) + 1 = y, u1 = x− 1

u2 − 1 = x, f(y) + 1 = y, u1 = x− 1, u2 = f(u1)

u2 − 1 = x, u3 + 1 = y, u1 = x− 1, u2 = f(u1), u3 = f(y)

FMCAD 2006 – p.33/75

Purification (cont.)

As most of the SMT developers will tell you, the purification step is

not really necessary.

Given a set of mixed (impure) literal Γ, define a shared term to be

any term in Γ which is alien in some literal or sub-term in Γ.

In our examples, these were the terms replaced by constants.

Assume that each satisfiability procedure treats alien terms as

constants.

FMCAD 2006 – p.34/75

NO procedure: soundness

Each step is satisfiability preserving.

Say φ is satisfiable (in the combination).

Purification: φ1 ∧ φ2 is satisfiable.

FMCAD 2006 – p.35/75





Iteration: for some partition ψ, φ1 ∧ φ2 ∧ ψ is satisfiable.

FMCAD 2006 – p.35/75






Component procedures: φ1 ∧ ψ and φ2 ∧ ψ are both

satisfiable in component theories.

FMCAD 2006 – p.35/75






Component procedures: φ1 ∧ ψ and φ2 ∧ ψ are both

satisfiable in component theories.

Therefore, if the procedure return unsatisfiable, then φ is

unsatisfiable.

FMCAD 2006 – p.35/75

NO procedure: correctness

Suppose the procedure returns satisfiable.

Let ψ be the partition and A and B be models of T 1 ∧ φ1 ∧ ψ

and T 2 ∧ φ2 ∧ ψ.

FMCAD 2006 – p.36/75




and T 2 ∧ φ2 ∧ ψ.

The component theories are stably infinite. So, assume the

models are infinite (of same cardinality).

FMCAD 2006 – p.36/75




and T 2 ∧ φ2 ∧ ψ.



Let h be a bijection between SA and SB such that

h(xA) = xB for each shared variable.

FMCAD 2006 – p.36/75




and T 2 ∧ φ2 ∧ ψ.





Extend B to B̄ by interpretations of symbols in Σ1:

f B̄(b1, . . . , bn) = h(fA(h−1(b1), . . . , h−1(bn)))

FMCAD 2006 – p.36/75




and T 2 ∧ φ2 ∧ ψ.





Extend B to B̄ by interpretations of symbols in Σ1:

f B̄(b1, . . . , bn) = h(fA(h−1(b1), . . . , h−1(bn)))

B̄ is a model of:

T 1 ∧ φ1 ∧ T 2 ∧ φ2 ∧ ψ

FMCAD 2006 – p.36/75

NO deterministic procedure

Instead of guessing, we can deduce the equalities to be shared.

Purification: no changes.

Interaction: Deduce an equality x = y:

T 1 ⊢ (φ1 ⇒ x = y)

Update φ2 := φ2 ∧ x = y. And vice-versa. Repeat until no

further changes.

Component Procedures : Use individual procedures to decide

whether φi is satisfiable.

Remark: T i ⊢ (φi ⇒ x = y) iff φi ∧ x 6= y is not satisfiable in

T i.

FMCAD 2006 – p.37/75

NO deterministic procedure: correctness

Assume the theories are convex.

Suppose φi is satisfiable.

FMCAD 2006 – p.38/75




Let E be the set of equalities xj = xk (j 6= k) such that,

T i 6⊢ φi ⇒ xj = xk.

FMCAD 2006 – p.38/75






By convexity, T i 6⊢ φi ⇒∨

E xj = xk.

FMCAD 2006 – p.38/75







E xj = xk.

φi ∧∧

E xj 6= xk is satisfiable.

FMCAD 2006 – p.38/75







E xj = xk.

φi ∧∧

E xj 6= xk is satisfiable.

The proof now is identical to the nondeterministic case.

FMCAD 2006 – p.38/75

NO procedure: example

x + 2 = y ∧ f(read(write(a, x, 3), y − 2)) 6= f(y − x+ 1)

T E T LA T A

Purifying

FMCAD 2006 – p.39/75


f(read(write(a, x, 3 ), y − 2)) 6= f(y − x+ 1)

T E T LA T A

x+ 2 = y

Purifying

FMCAD 2006 – p.39/75


f(read(write(a, x, u1), y − 2 )) 6= f(y − x+ 1)

T E T LA T A

x+ 2 = y

u1 = 3

Purifying

FMCAD 2006 – p.39/75


f(read(write(a, x , u1 ), u2 )) 6= f(y − x+ 1)

T E T LA T A

x+ 2 = y

u1 = 3

u2 = y − 2

Purifying

FMCAD 2006 – p.39/75


f(u3) 6= f(y − x + 1 )

T E T LA T A

x+ 2 = y u3 =

u1 = 3 read(write(a, x, u1), u2)

u2 = y − 2

Purifying

FMCAD 2006 – p.39/75


f(u3) 6= f(u4)

T E T LA T A

x+ 2 = y u3 =


u2 = y − 2

u4 = y − x+ 1

Purifying

FMCAD 2006 – p.39/75


T E T LA T A

f(u3) 6= f(u4) x + 2 = y u3 =


u2 = y − 2

u4 = y − x+ 1

Solving T LA

FMCAD 2006 – p.39/75


T E T LA T A

f(u3) 6= f(u4) y = x+ 2 u3 =


u2 = x

u4 = 3

Propagating u2 = x

FMCAD 2006 – p.39/75


T E T LA T A

f(u3) 6= f(u4) y = x+ 2 u3 =

u2 = x u1 = 3 read(write(a, x , u1), u2 )

u2 = x u2 = x

u4 = 3

Solving T A

FMCAD 2006 – p.39/75


T E T LA T A

f(u3) 6= f(u4) y = x+ 2 u3 = u1

u2 = x u1 = 3 u2 = x

u2 = x

u4 = 3

Propagating u3 = u1

FMCAD 2006 – p.39/75


T E T LA T A

f(u3) 6= f(u4) y = x+ 2 u3 = u1

u2 = x u1 = 3 u2 = x

u3 = u1 u2 = x

u4 = 3

u3 = u1

Propagating u1 = u4

FMCAD 2006 – p.39/75


T E T LA T A

f (u3 ) 6= f (u4 ) y = x+ 2 u3 = u1

u2 = x u1 = 3 u2 = x

u3 = u1 u2 = x

u4 = u1 u4 = 3

u3 = u1

Congruence u3 = u1 ∧ u4 = u1 ⇒ f(u3) = f(u4)

FMCAD 2006 – p.39/75


T E T LA T A

f (u3 ) 6= f (u4 ) y = x+ 2 u3 = u1

u2 = x u1 = 3 u2 = x

u3 = u1 u2 = x

u4 = u1 u4 = 3

f (u3 ) = f (u4 ) u3 = u1

Unsatisfiable!

FMCAD 2006 – p.39/75

Reduction Functions

A reduction function reduces the satisfiability of a complex theory

to the satisfiability problem of a simpler theory.

Ackerman reduction is used to remove uninterpreted functions.

For each application f(~a) in φ create a fresh variable f~a.

For each pair of applications f(~a), f(~c) in φ add the formula

~a = ~c⇒ f~a = f~c.

It is used in some SMT solvers to reduce T LA ∪ T E to T LA.

FMCAD 2006 – p.40/75

Reduction Functions

Theory of commutative functions.

Deductive closure of: ∀x, y.f(x, y) = f(y, x)

Reduction to T E .

For every f(a, b) in φ, do φ := φ ∧ f(a, b) = f(b, a).

Theory of “lists”.

Deductive closure of:

∀x, y. car(cons(x, y)) = x

∀x, y. cdr(cons(x, y)) = y

Reduction to T E

For each term cons(a, b) in φ, do

φ := φ ∧ car(cons(a, b)) = a ∧ cdr(cons(a, b)) = b.

FMCAD 2006 – p.41/75

Roadmap

Background

Theories


SAT + Theories


Applications

FMCAD 2006 – p.42/75

Breakthrough in SAT solving

Breakthrough in SAT solving influenced the way SMT solvers are

implemented.

Modern SAT solvers are based on the DPLL algorithm.

Modern implementations add several sophisticated search

techniques.

Backjumping

Learning

Restarts

Watched literals

FMCAD 2006 – p.43/75

The Original DPLL Procedure

Tries to build incrementally a satisfying truth assignment M for a

CNF formula F .

M is grown by

deducing the truth value of a literal from M and F , or

guessing a truth value.

If a wrong guess leads to an inconsistency, the procedure

backtracks and tries the opposite one.

FMCAD 2006 – p.44/75

Basic DPLL System – Example

∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 3 4 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 3 4 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 4 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 3 4 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 4 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 3 4 5 6 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

FMCAD 2006 – p.45/75


∅ || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 3 4 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Decide)

1 2 3 4 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (UnitProp)

1 2 3 4 5 6 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Backjump)

1 2 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2

Backjumpwith clause 1 ∨ 5FMCAD 2006 – p.45/75


. . .

1 2 3 4 5 6 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Backjump)

1 2 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2.

FMCAD 2006 – p.46/75


. . .

1 2 3 4 5 6 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Backjump)

1 2 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2.

1 ∨ 5 is implied by the original set of clauses. For instance, by

resolution,

1 ∨ 2 6 ∨ 5 ∨ 2

1 ∨ 6 ∨ 5 5 ∨ 6

1 ∨ 5

Therefore, instead deciding 3, we could have deduced 5.

FMCAD 2006 – p.46/75


. . .

1 2 3 4 5 6 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2 =⇒ (Backjump)

1 2 5 || 1 ∨ 2, 3 ∨ 4, 5 ∨ 6, 6 ∨ 5 ∨ 2.

1 ∨ 5 is implied by the original set of clauses. For instance, by

resolution,

1 ∨ 2 6 ∨ 5 ∨ 2

1 ∨ 6 ∨ 5 5 ∨ 6

1 ∨ 5

Therefore, instead deciding 3, we could have deduced 5.

Clauses like 1 ∨ 5 are computed by navigating the implication graph.

FMCAD 2006 – p.46/75

The Eager Approach

Translate formula into equisatisfiable propositional formula and use

off-the-shelf SAT solver.

Why “eager”?

Search uses all theory information from the beginning.

Can use best available SAT solver.

Sophisticated encodings are need for each theory.

Sometimes translation and/or solving too slow.

FMCAD 2006 – p.47/75

Lazy approach: SAT solvers + Theories

This approach was independently developed by several groups:

CVC (Stanford), ICS (SRI), MathSAT (Univ. Trento, Italy), and

Verifun (HP).

It was motivated also by the breakthroughs in SAT solving.

SAT solver “manages” the boolean structure, and assigns truth

values to the atoms in a formula.

Efficient theory solvers is used to validate the (partial) assignment

produced by the SAT solver.

When theory solver detects unsatisfiability → a new clause

(lemma) is created.

FMCAD 2006 – p.48/75

SAT solvers + Theories (cont.)

Example:

Suppose the SAT solver assigns

{x = y → T, y = z → T, f(x) = f(z) → F}.

Theory solver detects the conflict, and a lemma is created

¬(x = y) ∨ ¬(y = z) ∨ f(x) = f(z).

Some theory solvers use the “proof” of the conflict to build the

lemma.

Problems in these tools:

The lemmas are imprecise (not minimal).

The theory solver is “passive”: it just detects conflicts. There is

no propagation step.

Backtracking is expensive, some tools restart from scratch

when a conflict is detected. FMCAD 2006 – p.49/75

Precise Lemmas

Lemma:

{a1 = T, a1 = F, a3 = F}is inconsistent ¬a1 ∨ a2 ∨ a3

An inconsistent A set is redundant if A′ ⊂ A is also inconsistent.

Redundant inconsistent sets Imprecise Lemmas Ineffective

pruning of the search space.

Noise of a redundant set: A \ Amin.

The imprecise lemma is useless in any context (partial assignment)

where an atom in the noise has a different assignment.

Example: suppose a1 is in the noise, then ¬a1 ∨ a2 ∨ a3 is

useless when a1 = F .

FMCAD 2006 – p.50/75

Theory Propagation

The SAT solver is assigning truth values to the atoms in a formula.

The partial assignment produced by the SAT solver may imply the

truth value of unassigned atoms.

Example:

x = y ∧ y = z ∧ (f(x) 6= f(z) ∨ f(x) = f(w))

The partial assignment {x = y → T, y = z → T} implies

f(x) = f(z).

Reduces the number of conflicts and the search space.

FMCAD 2006 – p.51/75

Efficient Backtracking

One of the most important improvements in SAT was efficient

backtracking.

Until recently, backtracking was ignored in the design of theory

solvers.

Extreme (inefficient) approach: restart from scratch on every

conflict.

Other easy (and inefficient solutions):

Functional data-structures.

Backtrackable data-structures (trail-stack).

Backtracking should be included in the design of theory solvers.

Restore to a logically equivalent state.

FMCAD 2006 – p.52/75

The ideal theory solver

Efficient in real benchmarks.

Produces precise lemmas.

Supports Theory Propagation.

Incremental.

Efficient Backtracking.

Produces counterexamples.

FMCAD 2006 – p.53/75

Roadmap

Background

Theories


SAT + Theories


Applications

FMCAD 2006 – p.54/75

Congruence Closure

TE -satisfiability can be decided with a simple algorithm known as

congruence closure

Let G = (V,E) be a directed graph such that for each vertex v in G,

the successors of v are ordered.

Let C be any equivalence relation on V .

The congruence closure C∗ of C is the finest equivalence relation on V

that contains C and satisfies the following property for all vertices v and

w:

Let v and w have successors v1, . . . , vk and w1, . . . , wl

respectively. If k = l and (vi, wi) ∈ C∗ for 1 ≤ i ≤ k, then

(v, w) ∈ C∗.

FMCAD 2006 – p.55/75

Congruence Closure

Often, the vertices are labeled by some labeling function λ. In this case,

the property becomes:

If λ(v) = λ(w) and if k = l and (vi, wi) ∈ C∗ for

1 ≤ i ≤ k, then (v, w) ∈ C∗.

FMCAD 2006 – p.56/75

A Simple Algorithm

Let C0 = C and i = 0.

1. Number the equivalence classes in Ci.

2. Let α assign to each vertex v the number α(v) of the equivalence

class containing v.

3. For each vertex v construct a signature

s(v) = λ(v)(α(v1), . . . , α(vk)), where v1, . . . , vk are the

successors of v.

4. Group the vertices into equivalence classes by signature.

5. Let Ci+1 be the finest equivalence relation on V such that two

vertices equivalent under Ci or having the same signature are

equivalent under Ci+1.

6. If Ci+1 = Ci, let C∗ = Ci; otherwise increment i and repeat.FMCAD 2006 – p.57/75

Congruence Closure and T E

Recall that T E is the empty theory with equality over some signature

Σ(C) containing only function symbols.

If Γ is a set of ground Σ-equalities and ∆ is a set of ground

Σ(C)-disequalities, then the satisfiability of Γ ∪ ∆ can be determined

as follows.

Let G be a graph which corresponds to the abstract syntax trees of

terms in Γ ∪ ∆, and let vt denote the vertex of G associated with

the term t.

Let C be the equivalence relation on the vertices of G induced by

Γ.

Γ ∪ ∆ is satisfiable iff for each s 6= t ∈ ∆, (vs, vt) 6∈ C∗.

FMCAD 2006 – p.58/75

Difference Logic

Graph interpretation:

Variables are nodes.

Atoms x− y ≤ c are weighted edges: yc

−→ x.

A set of literals is satisfiable iff there is no negative cycle:

x1

c1−→ x2 . . . xncn−→ x1, C = c1 + . . . + cn < 0. That is,

negative cycle implies 0 ≤ C < 0.

Bellman-Ford like algorithm to find such cycles in O(mn).

FMCAD 2006 – p.59/75

Linear arithmetic

Most SMT solvers use algorithms based on Fourier-Motzkin or

Simplex.

Fourier Motzkin:

Variable elimination method.

t1 ≤ ax, bx ≤ t2 bt1 ≤ at2

Polynomial time for difference logic.

Double exponential and consumes a lot of memory.

Simplex:

Very efficient in practice.

Worst-case exponential (I’ve never seen this behavior in real

benchmarks).

FMCAD 2006 – p.60/75

Fast Linear Arithmetic

Simplex General Form.

New algorithm based on the Dual Simplex.


Efficient Theory Propagation.

New approach for solving strict inequalities (t > 0).

Preprocessing step.

It outperforms even solvers using algorithms for the Difference

Logic fragment.

FMCAD 2006 – p.61/75

Fast Linear Arithmetic: General Form

General Form: Ax = 0 and lj ≤ xj ≤ uj

Example:

x ≥ 0 ∧ (x+ y ≤ 2 ∨ x+ 2y ≥ 6) ∧ (x+ y = 2 ∨ x+ 2y > 4)

(s1 = x+ y ∧ s2 = x+ 2y) ∧

(x ≥ 0 ∧ (s1 ≤ 2 ∨ s2 ≥ 6) ∧ (s1 = 2 ∨ s2 > 4))

Only bounds (e.g., s1 ≤ 2) are asserted during the search.

Unconstrained variables can be eliminated before the beginning of

the search.

FMCAD 2006 – p.62/75

Equations + Bounds + Assignment

An assignment β is a mapping from variables to values.

We maintain an assignment that satisfies all equations and bounds.

The assignment of non basic variables implies the assignment of

basic variables.

Equations + Bounds can be used to derive new bounds.

Example: x = y − z, y ≤ 2, z ≥ 3 x ≤ −1.

The new bound may be inconsistent with the already known

bounds.

Example: x ≤ −1, x ≥ 0.

FMCAD 2006 – p.63/75

Roadmap

Background

Theories


SAT + Theories


Applications

FMCAD 2006 – p.64/75

Bounded Model Checking (BMC)

To check whether a program with initial state I and next-state

relation T violates the invariant Inv in the first k steps, one checks:

I(s0)∧T (s0, s1)∧ . . .∧T (sk−1, sk)∧ (¬Inv(s0)∨ . . .∨¬Inv(sk))

This formula is satisfiable if and only if there exists a path of length

at most k from the initial state s0 which violates the invariant k.

Formulas produced in BMC are usually quite big.

The SAL bounded model checker from SRI uses SMT solvers.

http://sal.csl.sri.com

FMCAD 2006 – p.65/75

http://sal.csl.sri.com

MUTT: MSIL Unit Testing Tools

http://research.microsoft.com/projects/mutt

Unit tests are popular, but it is far from trivial to write them.

It is quite laborious to write enough of them to have confidence in

the correctness of an implementation.

Approach: symbolic execution.

Symbolic execution builds a path condition over the input symbols.

A path condition is a mathematical formula that encodes data

constraints that result from executing a given code path.

FMCAD 2006 – p.66/75

http://research.microsoft.com/projects/mutt

MUTT: MSIL Unit Testing Tools

When symbolic execution reaches a if-statement, it will explore two

execution paths:

1. The if-condition is conjoined to the path condition for the

then-path.

2. The negated condition to the path condition of the else-path.

SMT solver must be able to produce models.

SMT solver is also used to test path feasibility.

FMCAD 2006 – p.67/75

Spec#: Extended Static Checking

http://research.microsoft.com/specsharp/

Superset of C#

non-null types

pre- and postconditions

object invariants

Static program verification

Example:

public StringBuilder Append(char[] value, int startIndex,

int charCount);

requires value == null ==> startIndex == 0 && charCount == 0;

requires 0 <= startIndex;

requires 0 <= charCount;

requires value == null ||

startIndex + charCount <= value.Length;

FMCAD 2006 – p.68/75

http://research.microsoft.com/specsharp/

Spec#: Architecture

Verification condition generation:

Spec# compiler: Spec# MSIL (bytecode).

Bytecode translator: MSIL Boogie PL.

V.C. generator: Boogie PL SMT formula.

SMT solver is used to prove the verification conditions.

Counterexamples are traced back to the source code.

The formulas produces by Spec# are not quantifier free.

Heuristic quantifier instantiation is used.

FMCAD 2006 – p.69/75

SLAM: device driver verification

http://research.microsoft.com/slam/

SLAM/SDV is a software model checker.

Application domain: device drivers.

Architecture

c2bp C program boolean program (predicate abstraction).

bebop Model checker for boolean programs.

newton Model refinement (check for path feasibility)

SMT solvers are used to perform predicate abstraction and to

check path feasibility.

c2bp makes several calls to the SMT solver. The formulas are

relatively small.

FMCAD 2006 – p.70/75

http://research.microsoft.com/slam/

Conclusion

SMT is the next generation of verification engines.

More automation: it is push-button technology.

SMT solvers are used in different applications.

The breakthrough in SAT solving influenced the new generation of

SMT solvers:

Precise lemmas.

Theory Propagation.

Incrementality.


FMCAD 2006 – p.71/75

References

[Ack54] W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundation of

Mathematics, 1954

[ABC +02] G. Audemard, P. Bertoli, A. Cimatti, A. Kornilowicz, and R. Sebastiani. A SAT based approach

for solving formulas over boolean and linear mathematical propositions. In Proc. of CADE’02, 2002

[BDS00] C. Barrett, D. Dill, and A. Stump. A framework for cooperating decision procedures. In 17th

International Conference on Computer-Aided Deduction, volume 1831 of Lecture Notes in Artificial

Intelligence, pages 79–97. Springer-Verlag, 2000

[BdMS05] C. Barrett, L. de Moura, and A. Stump. SMT-COMP: Satisfiability Modulo Theories Competition.

In Int. Conference on Computer Aided Verification (CAV’05), pages 20–23. Springer, 2005

[BDS02] C. Barrett, D. Dill, and A. Stump. Checking satisfiability of first-order formulas by incremental

translation to SAT. In Ed Brinksma and Kim Guldstrand Larsen, editors, Proceedings of the 14th

International Conference on Computer Aided Verification (CAV ’02), volume 2404 of Lecture Notes in

Computer Science, pages 236–249. Springer-Verlag, July 2002. Copenhagen, Denmark

[BBC +05] M. Bozzano, R. Bruttomesso, A. Cimatti, T. Junttila, P. van Rossum, S. Ranise, and

R. Sebastiani. Efficient satisfiability modulo theories via delayed theory combination. In Int. Conf. on

Computer-Aided Verification (CAV), volume 3576 of LNCS. Springer, 2005

[Chv83] V. Chvatal. Linear Programming. W. H. Freeman, 1983

FMCAD 2006 – p.72/75

References

[CG96] B. Cherkassky and A. Goldberg. Negative-cycle detection algorithms. In European Symposium on

Algorithms, pages 349–363, 1996

[DLL62] M. Davis, G. Logemann, and D. Loveland. A machine program for theorem proving.

Communications of the ACM, 5(7):394–397, July 1962

[DNS03] D. Detlefs, G. Nelson, and J. B. Saxe. Simplify: A theorem prover for program checking. Technical

Report HPL-2003-148, HP Labs, 2003

[DST80] P. J. Downey, R. Sethi, and R. E. Tarjan. Variations on the Common Subexpression Problem.

Journal of the Association for Computing Machinery, 27(4):758–771, 1980

[dMR02] L. de Moura and H. Rueß. Lemmas on demand for satisfiability solvers. In Proceedings of the

Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT 2002).

Cincinnati, Ohio, 2002

[DdM06] B. Dutertre and L. de Moura. Integrating simplex with DPLL(T ). Technical report, CSL, SRI

International, 2006

[GHN+04] H. Ganzinger, G. Hagen, R. Nieuwenhuis, A. Oliveras, and C. Tinelli. DPLL(T): Fast decision

procedures. In R. Alur and D. Peled, editors, Int. Conference on Computer Aided Verification (CAV

04), volume 3114 of LNCS, pages 175–188. Springer, 2004

FMCAD 2006 – p.73/75

References

[MSS96] J. Marques-Silva and K. A. Sakallah. GRASP - A New Search Algorithm for Satisfiability. In Proc.

of ICCAD’96, 1996

[NO79] G. Nelson and D. C. Oppen. Simplification by cooperating decision procedures. ACM Transactions

on Programming Languages and Systems, 1(2):245–257, 1979

[NO05] R. Nieuwenhuis and A. Oliveras. DPLL(T) with exhaustive theory propagation and its application to

difference logic. In Int. Conference on Computer Aided Verification (CAV’05), pages 321–334.

Springer, 2005

[Opp80] D. Oppen. Reasoning about recursively defined data structures. J. ACM, 27(3):403–411, 1980

[PRSS99] A. Pnueli, Y. Rodeh, O. Shtrichman, and M. Siegel. Deciding equality formulas by small

domains instantiations. Lecture Notes in Computer Science, 1633:455–469, 1999

[Pug92] William Pugh. The Omega test: a fast and practical integer programming algorithm for

dependence analysis. In Communications of the ACM, volume 8, pages 102–114, August 1992

[RT03] S. Ranise and C. Tinelli. The smt-lib format: An initial proposal. In Proceedings of the 1st

International Workshop on Pragmatics of Decision Procedures in Automated Reasoning

(PDPAR’03), Miami, Florida, pages 94–111, 2003

FMCAD 2006 – p.74/75

References

[RS01] H. Ruess and N. Shankar. Deconstructing shostak. In 16th Annual IEEE Symposium on Logic in

Computer Science, pages 19–28, June 2001

[SLB03] S. Seshia, S. Lahiri, and R. Bryant. A hybrid SAT-based decision procedure for separation logic

with uninterpreted functions. In Proc. 40th Design Automation Conference, pages 425–430. ACM

Press, 2003

[Sho81] R. Shostak. Deciding linear inequalities by computing loop residues. Journal of the ACM,

28(4):769–779, October 1981

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