Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

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 finite and infinite models, respectively. We exhibit a theory T 1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T 1 ∪ T 2 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 finite or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, generalizing recent work on combination of theories in the rewrite-based approach to satisfiability.

A Decidability Result for the Model Checking of Infinite-State Systems

“…It turns out that, in general, the constraint satisfiability problem for T 1 ∪ T 2 can be undecidable, even if the intersection of the signatures 1 ∩ 2 contains only the equality predicate (see [4]). A way to gain decidability is given by the following result: 15]) Suppose that the constraint satisf iability problem for both theories T 1 , T 2 (in signatures 1 , 2 ) is decidable; then the constraint satisf iability problem for the ( 1 ∪ 2 )-theory T 1 ∪ T 2 is also decidable as long as T 1 , T 2 are both T 0 -compatible for some universal and effectively locally f inite…”

Combinable Extensions of Abelian Groups

“…It is known (cf. [5]) that such a problem without any other assumption on T 1 and T 2 is undecidable; nevertheless, the following theorem holds: Theorem 1 ( [11]) Consider two theories T 1 , T 2 in signatures Σ 1 , Σ 2 such that:…”

Decision procedures for extensions of the theory of arrays

“…As a preliminary step to applying SP, we need to partially instantiate axioms (18) and (20) with the constants of sort array and bool occurring in L. This is so because SP does not seem to terminate on theories axiomatizing enumerated data types such as the Booleans (see [4] for a discussion on this point).…”