In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and first-order logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is two-fold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward message-passing and using backward message-passing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our message-passing ...

We extend Nelson-Oppen combination procedure to the case of theories which are compatible with respect to a common subtheory in the shared signature. The notion of compatibility relies on model completions and related concepts from classical model theory.

This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java programs with dynamically allocated data structures. Developers write Jahob specifications in classical higher-order logic (HOL); Jahob reduces the verification problem to deciding the validity of HOL formulas. I present a new method for proving HOL formulas by combining automated reasoning techniques. My method consists of 1) splitting formulas into individual HOL conjuncts, 2) soundly approximating each HOL conjunct with a formula in a more tractable fragment and 3) proving the resulting approximation using a decision procedure or a theorem prover. I present three concrete logics; for each logic I show how to use it to approximate HOL formulas, and how to decide the validity of formulas in this logic. First, I present an approximation of HOL based on a translation to first-order logic, which enables the use of existing resolution-based theorem provers. Second, I present an approximation of HOL based on field constraint analysis, a new technique that enables

In this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory T1 and one that decides constraint satisfiability with respect to a constraint theory T2, produces a procedure that (semi-)decides constraint satisfiability with respect to the union of T1 and T2. We provide a number of model-theoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete, with special emphasis on the case in which the signatures of the component theories are non-disjoint. We also describe some general classes of theories to which our combination results apply, and relate our approach to some of the existing combination methods in the field.

. We present a flexible framework for cooperating decision procedures. We describe the properties needed to ensure correctness and show how it can be applied to implement an efficient version of Nelson and Oppen's algorithm for combining decision procedures. We also show how a Shostak style decision procedure can be implemented in the framework in such a way that it can be integrated with the Nelson-Oppen method. 1 Introduction Decision procedures for fragments of first-order or higher-order logic are potentially of great interest because of their versatility. Many practical problems can be reduced to problems in some decidable theory. The availability of robust decision procedures that can solve these problem within reasonable time and memory could save a great deal of effort that would otherwise go into implementing special cases of these procedures. Indeed, there are several publicly distributed prototype implementations of decision procedures, such as Presburger arithmetic...

The Nelson-Oppen combination method can be used to combine decision procedures for the validity of quantifier-free formulae in first-order theories with disjoint signatures, provided that the theories to be combined are stably infinite. We show that, even though equational theories need not satisfy this property, Nelson and Oppen's method can be applied, after some minor modifications, to combine decision procedures for the validity of quantifier-free formulae in equational theories.

Abstract. The problem of deciding the satisfiability of a quantifier-free formula with respect to a background theory, also known as Satisfiability Modulo Theories (SMT), is gaining increasing relevance in verification: representation capabilities beyond propositional logic allow for a natural modeling of real-world problems (e.g., pipeline and RTL circuits verification, proof obligations in software systems). In this paper, we focus on the case where the background theory is the combination T1 £ T2 of two simpler theories. Many SMT procedures combine a boolean model enumeration with a decision procedure for T1 £ T2, where conjunctions of literals can be decided by an integration schema such as Nelson-Oppen, via a structured exchange of interface formulae (e.g., equalities in the case of convex theories, disjunctions of equalities otherwise). We propose a new approach for SMT¤T1 £ T2¥, called Delayed Theory Combination, which does not require a decision procedure for T1 £ T2, but only individual decision procedures for T1 and T2, which are directly integrated into the boolean model enumerator. This approach is much simpler and natural, allows each of the solvers to be implemented and optimized without taking into account the others, and it nicely encompasses the case of non-convex theories. We show the effectiveness of the approach by a thorough experimental comparison. 1

Abstract. Lazy algorithms for Satisfiability Modulo Theories (SMT) combine a generic DPLL-based SAT engine with a theory solver for the given theory T that can decide the T-consistency of conjunctions of ground literals. For many theories of interest, theory solvers need to reason by performing internal case splits. Here we argue that it is more convenient to delegate these case splits to the DPLL engine instead. The delegation can be done on demand for solvers that can encode their internal case splits into one or more clauses, possibly including new constants and literals. This results in drastically simpler theory solvers. We present this idea in an improved version of DPLL(T), a general SMT architecture for the lazy approach, and formalize and prove it correct in an extension of Abstract DPLL Modulo Theories, a framework for modeling and reasoning about lazy algorithms for SMT. A remarkable additional feature of the architecture, also discussed in the paper, is that it naturally includes an efficient Nelson-Oppen-like combination of multiple theories and their solvers. 1

Abstract not found

Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T1 ∪ T2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable. In the second part of the paper we strengthen the Nelson-Oppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either arbitrary or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, complementing recent work on combination of theories in the rewrite-based approach to satisfiability. 1