Unification in the union of disjoint equational theories: Combining decision procedures

Abstract

Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms that compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms that just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures that works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions-a slight generalization of unification problems with constants-is decidable for these theories. As a consequence of this new method, we can, for example, show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol.Our method can also be used to combine algorithms that compute finite complete sets of unifiers. Manfred Schmidt-Schauß' combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also obtain the new result that unification in the union of disjoint equational theories is finitary, if general unification-i.e., unification of terms with additional free function symbols-is finitary in the single theories.

Combining Intruder Theories

“…We shall introduce the notion of ordered rewriting [13], which is a useful technique that has been utilized (e.g. [3]) for proving the correctness of combination of unification algorithms.…”

“…Since S is satisfiable, following the lines of F. Baader and K. Schulz [3] permits to prove that S 1 and S 2 are satisfiable with a linear constant restriction ≺ chosen such that q ≺ q implies q σ is not a subterm of qσ.…”

“…This procedure relies on the combination of constraint solving algorithm for disjoint intruder theories, provided that solvability of ordered intruder constraints, a slight generalization of intruder constraints, can be decided in each theory. Such combination algorithm already exists for solving E-unification problems [19,3]. We have extended it in order to solve intruder constraints on disjoint signatures.…”

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

“…Work on theorem proving has focused on decomposition for parallel implementations [8,5,15,43] and has followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g., [32,40,3,36,44]). Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition.…”

Protocol Analysis Modulo Combination of Theories: A Case Study in Maude-NPA

“…Thus it is important to understand the behavior of algebraic properties in concert as well as separately. This is especially the case for protocol analysis systems based on unification, where the problem of combining unification algorithms [3,35] for different theories is known to be highly non-deterministic and complex, even when efficient unification algorithms exist for the individual theories, and even when the theories are disjoint (that is, share no symbols in common).…”