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From Propositional SAT to SMTHossein M. Sheini and Karem A. Sakallah

SAT 2006

August 13, 2006

Seattle

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Propositional Satisfiability

• DPLL: systematic backtracking search• Branch

• [Pure literal rule]

• Propagate• Unit propagation Boolean Constraint Propagation (BCP)

• Backtrack

• Modern implementations of DPLL• Conflict analysis

• Clause learning

• Non-chronological backtracking

• Efficient propagation• Two watched literals per clause

• Adaptive branching• VSIDS

• Restarts

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Quantifier-Free First-Order Logic

Boolean combination of atoms from background theories

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Decidable Quantifier-Free FOL Theories

• Empty• QF_UF: uninterpreted functions with equality (aka EUF)

• Linear Real Arithmetic• QF_LRA• QF_RDL: real difference logic• UTVPI

• Linear Integer Arithmetic• QF_LIA• QF_IDL: integer difference logic

• Data Structures• QF_A: arrays• QF_BV: bit vectors• Lists, etc.

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QF_UF Atoms

term ::= const | var | func-symbol(term, ,term)

atom ::= propositional-var | term = term

literal::= atom | atom

f a,b a

f f a,b ,b f a,b

f g x,y ,z ,h f a,b y

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Linear Real Arithmetic Atoms

R1 1 n n i ia x a x ~ b a ,x ,b ,~ , ,QF_LRA:

Rx y d x,y ,dQF_RDL:

R 0 1ax by c x,y ,c ,a,b ,UTVPI:

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Linear Integer Arithmetic Atoms

Z1 1 n n i ia x a x ~ b a ,x ,b ,~ , ,QF_LIA:

Zx y d x,y ,dQF_IDL:

Z 0 1ax by c x,y ,c ,a,b ,IUTVPI:

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Chronology of SAT for Quantifier-Free FOL

• Late 70s to mid 80s• Congruence closure algorithms• Combination strategies for disjoint theories

• Mid 80s to mid 90s• Not much!

• Mid to late 90s• Initial attempts at improved propositional reasoning

• Early 21st Century• Significant activity across many fields

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Reasoning About Equality

• To prove validity of EUF formula • Construct disjunctive normal form (DNF) of ¬• Prove unsatisfiability of each conjunct of ¬ using congruence closure

[Shostak 78][NelsonOppen 80]

Logic of equality with uninterpreted functions and predicates (EUF)

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Congruence Closure Prove f a,b a f f a,b ,b a

f a,b a f f a,b ,b a

a b

f

f

is unsatisfiable is valid

[NelsonOppen 80]

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Reasoning About Integer Arithmetic

• To prove validity of QF Presburger formula • Construct disjunctive normal form (DNF) of ¬• Prove unsatisfiability of each conjunct of ¬ using integer linear

programming (ILP)

Quantifier-Free Presburger Arithmetic

[Shostak 79]

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QF Presburger Arithmetic Example 3 2 1x y x x y

3 2 1x y x x y

Negate

Invalid!

3 1 1 1 1 1x y x x x y y x

“Normalize”

3 1 1 1 1

3 1 1 1 1

x y x x x y

x y x x y x

Convert to DNF

Solve ILPs

1 0x ,y[Shostak 79]

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Deciding Combinations of Theories

• Eliminate UFs and UPs using “Ackermann’s reduction” (adding consistency “axioms”) to get a pure equality formula

• Convert complement to DNF• Solve each conjunct as an integer linear program (ILP)• Formula explodes

[Shostak 79]

Quantifier-Free Presburger Arithmetic + EUF

Add all functional consistency axioms(aka substitutivity axioms of equality)

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QF Presburger Arithmetic + EUF Example

0 1 4 3 2 1 0h z z g y z f g y f z h

1 1 3 2 3 2

0 1 4 3 2 1 0

z h z h g y z f g y f z

h z z g y z f g y f z h

Add functional consistency axioms

1 2 3 4 5

1 3 4 5 2

1 3 2

0 1 4 0

z x x x z x x

x z x z x x x

Eliminate function symbols

[Shostak 79]

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Alternatively …

• Add all functional consistency axioms• Eliminate UFs and UPs using “Ackermann’s reduction”

(adding consistency “axioms”) to get a pure equality formula• Convert complement to DNF• Solve each conjunct as an integer linear program (ILP)• Formula explodes

• Add functional consistency axioms as needed• Ignore functional consistency and solve as before• Check functional consistency for symbols with different

values; if violated, add axiom and repeat• Has the flavor of “learning” on demand• ILPs can be solved incrementally

[Shostak 79]

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Deciding Combinations of Theories

• Convert negation to DNF• Add variables to purify different theory conjuncts• Solve separately• Propagate equalities• Split in case no equalities can be inferred• Supported theories:

• Real numbers under + and leq: Simplex• Arrays under store and select• List structures with car, cdr, cons, and atom: congruence

closure• Equality with UF: congruence closure

[NelsonOppen 79]

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Example of Nelson-Oppen Procedure

car cons 0 0x y y x ,x P h x h y P

ListsEUFLRA

2 1 5

1 5

2 3 4 3

5 4

true car cons

false

0

x y P g g g ,x

y x g P g

g g g g h x

g g h y

[NelsonOppen 79]

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Example of Nelson-Oppen Procedure

1 5car consg g ,x

5 0g

1 5g g

5 FP g

1 0g 1y x g

y x x y

x y

h x h y

3g h x 4g h y

2 3 4g g g

2 TP g

3 4g g

2 0g

2 5g g

[NelsonOppen 79]

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Deciding Combinations of Theories

• Generalizes Nelson-Oppen method by eliminating the need for extra variables

• Congruence closure is extended to handle different theories as long as they have “canonizers” and solvers

• Found to be “buggy” and not as general as N-O

[Shostak 84]

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Disadvantages of “Old” Combination Methods

• Need to convert to DNF• Inefficient handling of Boolean structure• Can be viewed as lazy integration with an open

feedback loop between a propositional enumerator and the theory solvers

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Disadvantages of “Old” Combination Methods

SAT

No

SMT Instance

DNFConverter

DNF Instance

Yes

MoreConjuncts?

Conjunction

of

Theory Atoms

Theory Solvers

UNSAT

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Improved Propositional Reasoning

• Convert SMT instance to equi-satisfiable propositional form and apply Boolean reasoning techniques (BDDs or SAT)• Small-domain encoding• Per-constraint encoding

• Add more sophisticated Boolean reasoning, but keep background theories intact• Re-write rules and better Boolean splits• Add full-fledged SAT or BDD solvers to handle the Boolean skeleton

• Very Lazy: theory solver returns a conflict clause to SAT solver

• Lazy: theory solver invoked incrementally but does not propagate

• Eager: theory solver propagates and learns etc.

• Variants

• Layered

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Reasoning About Equality: Revisited

MoreConjuncts?

CongruenceClosure

SAT

No

YesUNSAT

EUF Formula

DNFConverter

DNF of = and ≠ Conjuncts

Conjunction of= and ≠

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Reasoning About Equality: Revisited

SAT/BDDSolver

SATUNSAT

EUF Formula

Reduction

E Formula

Range Analysis&

Boolean Encoding

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Bryant’s “ite” Reduction

1 1

2 1 2

3 1 1 1 2 3

4 3 1 3 2 3 1 3 4

4 3 3 2 1

F f

F x y,f ,f

F F x,f , F y ,f ,f

F F x,f , F y ,f , F F ,f ,f

F F F F y F

ite

ite ite

ite ite ite

★

★

★ ★ ★

★ ★ ★ ★ ★

★ ★ ★ ★ ★

Reasoning About Equality: Revisited F F F x F F x F F x F y y F x

CongruenceClosure

x y

F1

F4

F3

F2

Ackermann’sReduction

1 2

1 1 3

3 1 4

1 2 3

3 2 4

1 3 3 4

4 3 3 2 1

x y f f

x f f f

x f f f

y f f f

y f f f

f f f f

f f f f y f

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Pros/Cons of SMT-to-SAT Conversion

Loss of theory semantics (e.g., arithmetic)

Black-box use of modern SAT solversLeveraging of performance/capacity improvementsin SAT solvers

Increase of instance sizes

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Combine SAT and Theory Solvers

SAT

UNSAT

SMT Instance

SAT Solver

PropositionalAbstraction

SAT Instance

SAT

Theory Atoms

Theory Solvers

AbstractionRefinement

UNSAT

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Propositional Abstraction

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Spectrum of Integration

• Very lazily: theory solver does not feedback any info to SAT solver; no refinement

• Lazily: theory solver returns a small explanation of infeasibility to SAT solver• Eagerly: theory solver participates in value propagation (implications) and in

conflict analysis• Very eagerly: direct encoding of all theory semantics in propositional

formula; no abstraction

How aggressively is refinement done?

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Very Lazy Integration

Boolean Solver

1A 1B

2A 2B

3A 3B

4A 4B

61B

71B

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Very Lazy Integration

Integer Solver

1

2

3

4

61

71

5

6

0

12

1

2

B u w

B v w

B z

B u v

B x z

B y z

5

6

0

12

1

2

u w

v w

z

u v

x z

y z

UNSAT

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Spectrum of Integration

• Very lazily: theory solver does not feedback any info to SAT solver; no refinement

• Lazily: theory solver returns a small explanation of infeasibility to SAT solver• Eagerly: theory solver participates in value propagation (implications) and in

conflict analysis• Very eagerly: direct encoding of all theory semantics in propositional

formula; no abstraction

How aggressively is refinement done?

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Lazy Integration

1A 1B

2A 2B

3A 3B

4A 4B

61B

71B

Boolean Solver

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Lazy Integration

11u v 0 1

5u w 1B

6v w 2B

12u v

1

2

3

4

61

71

5

6

0

12

1

2

B u w

B v w

B z

B u v

B x z

B y z

Integer Solver

3B 0z

Create conflict clause

and return to Boolean solver

1 2 4B B B 61B

71B

4B

1B

2B

4B

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Spectrum of Integration

• Very lazily: theory solver does not feedback any info to SAT solver; no refinement

• Lazily: theory solver returns a small explanation of infeasibility to SAT solver• Eagerly: theory solver participates in value propagation (implications) and in

conflict analysis• Very eagerly: direct encoding of all theory semantics in propositional

formula; no abstraction

How aggressively is refinement done?

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Eager Integration: Incremental Propagation

1A 1B

2A 2B

3A 3B

4A 4B

11u v

0 1

5u w

6v w

12u v

0z

1B

2B

4B

1 2 4B B B

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Eager Integration: Incremental Propagation

1A 1B

2A 2B

3A 4B

11u v

5u w

6v w

3B

0z

4A

61B

71B

1x z

2y z

1x

2y

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Integration Trade-offs

• Must balance cost of generating new “facts” against utility of such facts in pruning the search space• E.g., a priori generation of transitivity constraints for all

possible equalities is overkill• Suggests an “on-demand” learning strategy analogous to

conflict analysis in modern SAT solvers

• On-demand learning requires incremental backtrackable theory solvers that maintain state

• Cost of propagation for various theories and sub-theories:• EUF: congruence closure is O(n log n) • Difference constraints: negative cycle detection is O(nm)• UTVPI: transitive closure is O(n3)• Real arithmetic: incremental Simplex

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Offline Integration of LRA Solver

1A 1B

2A 2B

3A 4B

11u v

5u w

6v w

3B

0z

4A

61B

71B

1x z

2y z

1x

2y

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Offline Integration of LRA Solver

1A 1B

2A 2B

3A 4B

11u v

5u w

6v w

3B

0z

4A

61B

71B

1x z

2y z

1x

2y

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Offline Integration of LRA Solver

5

6

0

1

2

11

1

2

4 4 0

u w

v w

z

x z

y z

u

u v x

v

y

x

y

11

1

2

4 4 0

u v

x

y

u v x y

5

6

0

1

2

1

2

11

4 4 0

u

u w

v w

z

x z

y z

x

v

u v y

y

x

11

1

4 4

5

0

2

0

6

1

2

u v

x

u w

v w

z

x z

y z

y

u v x y

11

1

2

4 4

5

0

0

6

1

2

u v

x

u

y

u v x y

w

v w

z

x z

y z

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Learning Strategies

• CNF clauses: disjunctions of existing atoms (in terms of their indicator variables)

• Introduction of new theory atoms: cutting planes

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Introduction of New Atoms

1A 1B

2A 2B11u v

5u w

6v w

4 4 0u v x y

2x y

8BNEW

1 2 8B B B 1 2 8B B B

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Offline Integration of LRA Solver

1A 1B

2A 2B

3A 4B

11u v

5u w

6v w

3B

0z

4A

61B 1x z

1x

8B 2x y

1y

71B 72B 73B

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DPLL(T) Framework

• Declarative “calculus” for tight integration of a solver for theory T within a propositional DPLL solver

• Inspired by CLP(X)

• Defines SolverT as an abstract data type with the following methods:• Initialize(L: Literal set)• SetTrue(l: L-literal): L-literal set• IsTrue?(l: L-literal): Boolean• Backtrack(n: Natural)• Explanation(l: L-literal): L-literal set

[Tinelli 02][Nieuwenhuis-Oliveras 03]

GETFOL1993

KSAT1996

MATH-SAT2001

MATH-SAT2002

MATH-SAT2005

LPSAT1999

CVC Lite2004

Nelson-Oppen1979

Shostak1984

SVC1996

CVC2002/2004

Verifun2003

Simplify1998?

ICS2001

ICS+Chaff2002

Simplics2005

DPLL(T)2002

Sammy2005

BarcelogicTools2005

Yices2006

Ario2005

Nelson-Oppen1979

MATH-SAT2001

KSAT1996

GETFOL1993

MATH-SAT2002

MATH-SAT2005

LPSAT1999

SVC1996

CVC2002/2004

CVC Lite2004

Shostak1984

Simplify1998?

Verifun2003

ICS2001

Simplics2005

ICS+Chaff2002

DPLL(T)2002

Sammy2005

BarcelogicTools2005

Yices2006

Ario2005

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2005 Competition Results: QF_UF

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2005 Competition Results: QF_RDL

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2005 Competition Results: QF_IDL

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2005 Competition Results: QF_UFIDL

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2005 Competition Results: QF_LRA

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2005 Competition Results: QF_LIA

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2005 Competition Results: QF_AUFLIA

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Conclusions

• SAT does it again!• Modern SAT technology critical enabler of SMT solvers

• Clear winner: tight integration of SAT and Theory solvers• Incremental propagation• Incremental conflict analysis and learning• Careful tuning

• SMT is bringing different communities together (SAT, CP, AI, OR)

• Competition is good