DPLL-based Checkers for Satisfiability Modulo Theories

DPLL-based Checkers for Satisfiability Modulo Theories Cesare Tinelli Department of Computer Science The University of Iowa Joint work with Jed Hagen (Iowa) Robert Nieuwenhuis, Albert Oliveras (TUC) CMU, Oct 4

Talk Plan • Propositional satisfiability and the DPLL procedure • A calculus for DPLL • Satisfiability modulo theories • The DPLL(T1,…Tn) calculus • Implementing the DPLL(T1,…Tn) calculus • Some experimental results CMU, Oct 4

Background • Propositional satisfiability (SAT) is one of the most fundamental problems in AI and CS. • SAT is decidable, but it is NP-complete. • An effective method for SAT was pioneered by Davis, Putman, Logemann, and Loveland (DPLL). • The best modern SAT solvers (Chaff, Berkmin, Siege, …) are based on DPLL. CMU, Oct 4

Context (partial assignment) The DPLL Procedure as a Calculus CMU, Oct 4

The DPLL Calculus CMU, Oct 4

The DPLL Calculus (cont.) Lits() = { all atoms in  and their negation } CMU, Oct 4

Correctness of DPLL  = { propositional clauses } Prop. Every derivation tree of  |-  is finite. Prop.  is satisfiable iff  |-  has a derivation tree with a successful branch. CMU, Oct 4

Satisfiability Modulo Theories • DPLL-based methods can also decide the satisfiability of quantifier-free, first-order formulas • Often, however, one is interested in the satisfiability of a quantifier-free formula in a given FO theory T • Example: Is R(a,b) R(b,c)  R(a,c) satisfiable? What if R stands for a transitive relation? • Applications: planning, scheduling, verification, compiler optimization, … CMU, Oct 4

Checking Satisfiability Modulo Theories Current approaches: • Eager translation into SAT • Encode problem and theory into an equisatisfiable propositional formula • Feed formula to a SAT-solver • Lazy “translation” into SAT • Couple a SAT solver with a decision procedure for the theory CMU, Oct 4

Idea of Lazy Approach Fact: Many theories of interest have (efficient) decision procedures for sets of literals. Problem: in practice, dealing with Boolean combinations of literals is as hard as in the propositional case. Solution: use propositional satisfiability technology for the Boolean part. CMU, Oct 4

Previous Lazy Approaches [Ar00,Aud02,Ba02,deM02,…] CMU, Oct 4

Our Approach [Tin02, ON03] Embed decision procedures into the SAT solver. More abstractly: Embed decision procedures into the DPLL Calculus. CMU, Oct 4

Checked by decision procedure for T The DPLL(T) Calculus iff every model of T that satisfies  also satisfies l CMU, Oct 4

The DPLL(T) Calculus CMU, Oct 4

The DPLL(T) Calculus All the DPLL rules plus: CMU, Oct 4

Correctness of DPLL(T)  = {quantifier-free clauses in T’s signature} Prop. Every derivation tree of  |-  is finite. Prop.  is T-satisfiable iff  |-  has a derivation tree with a successful branch. CMU, Oct 4

Satisfiability Modulo Multiple Theories • Let T1,…, Tn be distinct theories with resp. decision procedures P1,…,Pn • How can we reason over all of them with DPLL(T)? • Quick Solution: • Combine P1,…,Pn with the Nelson-Oppen method into a decision procedure for T1  …  Tn • Use DPLL(T) with T =T1  …  Tn CMU, Oct 4

Satisfiability Modulo Multiple Theories • Let T1,…, Tn be distinct theories with resp. decision procedures P1,…,Pn • How can we reason over all of them with DPLL(T)? • Better Solution: • Embed the Nelson-Oppen method directly into the calculus • Turn DPLL(T) into DPLL(T1,…,Tn ) CMU, Oct 4

The DPLL(T1,…,Tn) Calculus: Preliminaries • Let n=2, for simplicity • Let Ti be a theory of signature i for i=1,2, with 1 2 =  • Assume wlog that each input literal has signature 1 or 2 (no mixed literals) CMU, Oct 4

The DPLL(T1,…,Tn) Calculus: Notation • Lits(, i) = {i-atoms in  and their negation} • i = { i-literals of  } • s = { x = y | x, y  vars(1)  vars(2) } CMU, Oct 4

Only change The DPLL(T1,…,Tn) Calculus All the DPLL rules but with new CMU, Oct 4

The DPLL(T1,…,Tn) Calculus (cont.) New theory rules (i=1,2): CMU, Oct 4

Correctness of DPLL(T1,…,Tn) A theory T is stably infinite iff every T-satisfiable qff is satisfiable in an infinite model of T •  = { quantifier-free clauses in • (T1...Tn)’s signature } • Prop. If T1, …, Tn are • stably-infinite and • pairwise signature-disjoint • then  is (T1...Tn)-satisfiable iff •  |-  has a derivation tree with a successful branch. CMU, Oct 4

Making DPLL Efficient • Literal selection strategies • Intelligent backtracking(backjumping) • Learning (lemma generation) CMU, Oct 4

Making DPLL(T1,…,Tn) Efficient • T-based literal selection strategies • T-based intelligent backtracking • T-based learning CMU, Oct 4

Implementing DPLL(T1,…,Tn) • In theory, • it suffices to have decision proceduresPi s.t. • Pi({l1, …, ln}) = true • iff • {l1, …, ln} is Ti-unsatisfiable • In practice, • it is better if to have a solver for each Ti with additional functionalities CMU, Oct 4

Ti-solver Interface type status = Valid | Unsat | Undef type lit = sign * pred Class Solver { attr context : lit list meth status : pred  status meth explanation : pred  lit set meth assert : lit  lit set meth backtrack : int  unit } CMU, Oct 4

Ti-solver Specification type status = Valid | Unsat | Undef type lit = sign * pred Class Solver { attr context : lit list … methstatus (p : pred) : status ensures result = Valid <=> context |=T p & result = Unsat <=> context |=T ~p … } CMU, Oct 4

Ti-solver Specification type status = Valid | Unsat | Undef type lit = sign * pred Class Solver { attr context : lit list … methexplanation (p : pred) : lit set requires status(p) != Undef ensures result  context & status(p) = Valid => result |=T p & status(p) = Unsat => result |=T ~p … } CMU, Oct 4

Ti-solver Specification type status = Valid | Unsat | Undef type lit = sign * pred Class Solver { attr context : lit list … methassert ((sign,p) : lit) : lit set requires status(p) = Undef ensurescontext’ = (sign,p)::context & (sign,p)  result & forall (lit  result) context’ |=T lit & not(context |=T lit) … } CMU, Oct 4

Ti-solver Specification type status = Valid | Unsat | Undef type lit = sign * pred Class Solver { attr context : lit list … methbacktrack (n : int) : unit requires 1 <= n & n <= length(context) ensures context = lit1:: … ::litn::context’ … } CMU, Oct 4

Experimental Results T = theory of successor and predecessor + free symbols p(s(x)) = x s(p(x)) = x p(x) = p(y) => x = y s(x) = s(y) => x = y 0  pn(x) (for each n > 0) 0  sn(x) (for each n > 0) pn(x)  pm(x) (for each m,n with m > n > 0) sn(x)  sm(x) (for each m,n with m > n > 0) Popular in hardware verification CMU, Oct 4

Experimental Results • T-solver based on a novel congruence closure algorithm [ON03] • DPLL(T) engine with: • Chaff-style literal selection function • 2-watched literal unit propagation • conflict set-based backjumping • 1st-UIP lemma generation • C implementation CMU, Oct 4

Experimental Results • Benchmarks produced with UCLID tool [BLJ02] • Comparisons with • SVC • 4 SAT-translation methods by Bryant et al. • Machine: Pentium IV, 2.63 GHz, 512MB RAM • Times in seconds, with timeout at 6000s CMU, Oct 4

See external table CMU, Oct 4

Conclusions • DPLL(T1,…,Tn) is a sound and complete calculus for satisfiability modulo T1…Tn. • General framework for integrating decision procedures into the DPLL method. • Decision procedures drive the derivation instead of just validating solutions. • Major optimization techniques from SAT can be lifted to DPLL(T1,…,Tn). • Initial experimental results are very promising CMU, Oct 4

Further Work (in Progress) • New version of the DPLL(T) calculus, to model more accurately modern SAT engines • Non-clausal version of DPLL(T) • Implementation with more theories • A DPLL(T)-based system for solving integer linear programming (optimization) problems CMU, Oct 4

DPLL-based Checkers for Satisfiability Modulo Theories