Robert Nieuwenhuis

Solving SAT and SAT Modulo Theories: From an abstract Davis--Putnam--Logemann--Loveland procedure to DPLL( T )

We first introduce Abstract DPLL, a rule-based formulation of the Davis--Putnam--Logemann--Loveland (DPLL) procedure for propositional satisfiability. This abstract framework allows one to cleanly express practical DPLL algorithms and to formally reason about them in a simple way. Its properties, such as soundness, completeness or termination, immediately carry over to the modern DPLL implementations with features such as backjumping or clause learning.We then extend the framework to Satisfiability Modulo background Theories (SMT) and use it to model several variants of the so-called lazy approach for SMT. In particular, we use it to introduce a few variants of a new, efficient and modular approach for SMT based on a general DPLL(X) engine, whose parameter X can be instantiated with a specialized solver SolverT for a given theory T, thus producing a DPLL(T) system. We describe the high-level design of DPLL(X) and its cooperation with SolverT, discuss the role of theory propagation, and describe different DPLL(T) strategies for some theories arising in industrial applications.Our extensive experimental evidence, summarized in this article, shows that DPLL(T) systems can significantly outperform the other state-of-the-art tools, frequently even in orders of magnitude, and have better scaling properties.

DPLL(T): Fast Decision Procedures

The logic of equality with uninterpreted functions (EUF) and its extensions have been widely applied to processor verification, by means of a large variety of progressively more sophisticated (lazy or eager) translations into propositional SAT. Here we propose a new approach, namely a general DPLL(X) engine, whose parameter X can be instantiated with a specialized solver Solver T for a given theory T, thus producing a system DPLL(T). We describe this DPLL(T) scheme, the interface between DPLL(X) and Solver T , the architecture of DPLL(X), and our solver for EUF, which includes incremental and backtrackable congruence closure algorithms for dealing with the built-in equality and the integer successor and predecessor symbols. Experiments with a first implementation indicate that our technique already outperforms the previous methods on most benchmarks, and scales up very well.

Theorem proving with ordering and equality constrained clauses

Abstract Deduction methods for first-order constrained clauses with equality are described within an abstract framework: constraint strategies , consisting of an inference system, a constraint inheritance strategy and redundancy criteria for clauses and inferences. We give simple conditions for such a constraint strategy to be complete (refutationally and in the sense of Knuth-Bendix-like completion). This allows to prove in a uniform way the completeness of several instantiations of the framework with concrete strategies. For example, strategies in which equality constraints are inherited are basic : no inferences are needed on subterms introduced by unifiers of previous inferences. Ordering constraints reduce the search space by inheriting the ordering restrictions of previous inferences and increase the expressive power of the logic.

Cardinality Networks: a theoretical and empirical study

We introduce Cardinality Networks, a new CNF encoding of cardinality constraints. It improves upon the previously existing encodings such as the sorting networks of Eén and Sörensson (JSAT 2:1–26, 2006) in that it requires much less clauses and auxiliary variables, while arc consistency is still preserved: e.g., for a constraint x1 + ... + xn ≤ k, as soon as k variables among the xi’s become true, unit propagation sets all other xi’s to false. Our encoding also still admits incremental strengthening: this constraint for any smaller k is obtained without adding any new clauses, by setting a single variable to false. Here we give precise recursive definitions of the clause sets that are needed and give detailed proofs of the required properties. We demonstrate the practical impact of this new encoding by careful experiments comparing it with previous encodings on real-world instances.

Abstract DPLL and Abstract DPLL Modulo Theories

We introduce Abstract DPLL, a general and simple abstract rule-based formulation of the Davis-Putnam-Logemann-Loveland (DPLL) procedure. Its properties, such as soundness, completeness or termination, immediately carry over to the modern DPLL implementations with features such as non-chronological backtracking or clause learning. This allows one to formally reason about practical DPLL algorithms in a simple way. In the second part of this paper we extend the framework to Abstract DPLL modulo theories. This allows us to express—and formally reason about—state-of-the-art concrete DPLL-based techniques for satisfiability modulo background theories, such as the different lazy approaches, or our DPLL(T) framework.

The Barcelogic SMT Solver

This is the first system description of the Barcelogic SMT solver, which implements all techniques that our group has been developing over the last four years as well as state-of-the-art features developed by other research groups. We pay special attention to the theory solvers and to functionalities that are not common in SMT solvers.

Simple LPO constraint solving methods

Abstract We present simple techniques for deciding the satisfiability of lexicographic path ordering constraints under two different semantics: solutions built over the given signature and solutions in extended signatures. For both cases we give the first NP algorithms, which is optimal as we prove the problems to be NP-complete. We discuss the efficient applicability of the techniques in practice, where, as far as we know, their simply exponential bound improves upon the existing methods, and describe some optimizations.

Basic superposition is complete

We define equality constrained equations and clauses and use them to prove the completeness of what we have called basic superposition: a restricted form of superposition in which only the subterms not created in previous inferences is superposed upon. We first apply our results to the equational case and define an unfailing Knuth-Bendix completion procedure that uses basic superposition as inference rule. Second, we extend the techniques to completion of full first-order clauses with equality. Moreover, we prove the refutational completeness of a new simple inference system.

Proof-producing congruence closure

Many applications of congruence closure nowadays require the ability of recovering, among the thousands of input equations, the small subset that caused the equivalence of a given pair of terms. For this purpose, here we introduce an incremental congruence closure algorithm that has an additional

Theorem Proving with Ordering Constrained Clauses

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