Mnacho Echenim

On Variable-inactivity and Polynomial T-Satisfiability Procedures

Verification problems require to reason in theories of data structures and fragments of arithmetic. Thus, decision procedures for such theories are needed, to be embedded in, or interfaced with, proof assistants or software model checkers. Such decision procedures ought to be sound and complete, to avoid false negatives and false positives, efficient, to handle large problems, and easy to combine, because most problems involve multiple theories. The rewrite-based approach to decision procedures aims at addressing these sometimes conflicting issues in a uniform way, by harnessing the power of general first-order theorem proving. In this article, we generalize the rewrite-based approach from deciding the satisfiability of sets of ground literals to deciding that of arbitrary ground formulae in the theory. Next, we present polynomial rewrite-based satisfiability procedures for the theories of records with extensionality and integer offsets. The generalization of the rewrite-based approach to arbitrary ground formulae and the polynomial satisfiability procedure for the theory of records with extensionality use the same key property—termed variable-inactivity—that allows one to combine theories in a simple way in the rewrite-based approach.

Theory decision by decomposition

The topic of this article is decision procedures for satisfiability modulo theories (SMT) of arbitrary quantifier-free formulae. We propose an approach that decomposes the formula in such a way that its definitional part, including the theory, can be compiled by a rewrite-based first-order theorem prover, and the residual problem can be decided by an SMT-solver, based on the Davis-Putnam-Logemann-Loveland procedure. The resulting decision by stages mechanism may unite the complementary strengths of first-order provers and SMT-solvers. We demonstrate its practicality by giving decision procedures for the theories of records, integer offsets and arrays, with or without extensionality, and for combinations including such theories.

Rewrite-Based Satisfiability Procedures for Recursive Data Structures

If a rewrite-based inference system is guaranteed to terminate on the axioms of a theory T and any set of ground literals, then any theorem-proving strategy based on that inference system is a rewrite-based decision procedure for T-satisfiability. In this paper, we consider the class of theories defining recursive data structures, that might appear out of reach for this approach, because they are defined by an infinite set of axioms. We overcome this obstacle by designing a problem reduction that allows us to prove a general termination result for all these theories. We also show that the theorem-proving strategy decides satisfiability problems in any combination of these theories with other theories decided by the rewrite-based approach.

Rewrite-Based Decision Procedures

The rewrite-based approach to satisfiability modulo theories consists of using generic theorem-proving strategies for first-order logic with equality. If one can prove that an inference system generates finitely many clauses from the presentation T of a theory and a finite set of ground unit clauses, then any fair strategy based on that system can be used as a T -satisfiability procedure. In this paper, we introduce a set of sufficient conditions to generalize the entire framework of rewrite-based T-satisfiability procedures to rewrite-based T-decision procedures. These conditions, collectively termed subterm-inactivity, will allow us to obtain rewrite-based T-decision procedures for several theories, namely those of equality with uninterpreted functions, arrays with or without extensionality and two of its extensions, finite sets with extensionality and recursive data structures.

A Resolution Calculus for First-order Schemata

We devise a resolution calculus that tests the satisfiability of infinite families of clause sets, called clause set schemata. For schemata of propositional clause sets, we prove that this calculus is sound, refutationally complete, and terminating. The calculus is extended to first-order clauses, for which termination is lost, since the satisfiability problem is not semi-decidable for nonpropositional schemata. The expressive power of the considered logic is strictly greater than the one considered in our previous work.

An Instantiation Scheme for Satisfiability Modulo Theories

State-of-the-art theory solvers generally rely on an instantiation of the axioms of the theory, and depending on the solvers, this instantiation is more or less explicit. This paper introduces a generic instantiation scheme for solving SMT problems, along with syntactic criteria to identify the classes of clauses for which it is complete. The instantiation scheme itself is simple to implement, and we have produced an implementation of the syntactic criteria that guarantee a given set of clauses can be safely instantiated. We used our implementation to test the completeness of our scheme for several theories of interest in the SMT community, some of which are listed in the last section of this paper.

A calculus for generating ground explanations

We present a modification of the superposition calculus that is meant to generate explanations why a set of clauses is satisfiable. This process is related to abductive reasoning, and the explanations generated are clauses constructed over so-called abductive constants. We prove the correctness and completeness of the calculus in the presence of redundancy elimination rules, and develop a sufficient condition guaranteeing its termination; this sufficient condition is then used to prove that all possible explanations can be generated in finite time for several classes of clause sets, including many of interest to the SMT community. We propose a procedure that generates a set of explanations that should be useful to a human user and conclude by suggesting several extensions to this novel approach.

Permutative rewriting and unification

Permutative rewriting provides a way of analyzing deduction modulo a theory defined by leaf-permutative equations. Our analysis naturally leads to the definition of the class of unify-stable axiom sets, in order to enforce a simple reduction strategy. We then give a uniform unification algorithm modulo theories E axiomatized this way. We prove that it computes complete sets of unifiers of simply exponential cardinality, and that the E-unification decision problem belongs to NP.

Overlapping leaf permutative equations

Leaf permutative equations often appear in equational reasoning, and lead to dumb repetitions through rather trivial though profuse variants of clauses. These variants can be compacted by performing inferences modulo a theory E of leaf permutative equations, using group-theoretic constructions, as proposed in [1]. However, this requires some tasks that happen to be NP-complete (see [7]), unless restrictions are imposed on E. A natural restriction is orthogonality, which allows the use of powerful group-theoretic algorithms to solve some of the required tasks. If sufficient, this restriction is however not necessary. We therefore investigate what kind of overlapping can be allowed in E while retaining the complexity obtained under orthogonality.

NP-Completeness Results for Deductive Problems on Stratified Terms

In [1] Avenhaus and Plaisted proposed the notion of stratified terms, in order to represent concisely the sets of consequences of clauses under leaf permutative theories. These theories contain variable-permuting equations, so that the consequences appear as simple “permuted” variants of each other. Deducing directly with stratified terms can reduce exponentially the search space, but we show that the problems involved (e.g. unifiability) are NP-complete. We use computational group theory to show membership in NP, while NP-hardness is obtained through an interesting problem in group theory.