Hierarchical Combination of Unification Algorithms

International audienceA critical question in unification theory is how to obtain a unification algorithm for the combination of non-disjoint equational theories when there exists unification algorithms for the constituent theories. The problem is known to be difficult and can easily be seen to be undecidable in the general case. Therefore, previous work has focused on identifying specific conditions and methods in which the problem is decidable. We continue the investigation in this paper, building on previous combination results. We are able to develop a novel approach to the non-disjoint combination problem. The approach is based on a new set of restrictions and combination method such that if the restrictions are satisfied the method produces an algorithm for the unification problem in the union of non-disjoint equational theories

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

Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which 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 which compute finite complete sets of unifiers. Manfred Schmidt-Schauß\u27 combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get 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

Implementing a Unification Algorithm for Protocol Analysis with XOR

In this paper, we propose a unification algorithm for the theory $E$ which combines unification algorithms for E\_{\std} and E\_{\ACUN} (ACUN properties, like XOR) but compared to the more general combination methods uses specific properties of the equational theories for further optimizations. Our optimizations drastically reduce the number of non-deterministic choices, in particular those for variable identification and linear orderings. This is important for reducing both the runtime of the unification algorithm and the number of unifiers in the complete set of unifiers. We emphasize that obtaining a ``small'' set of unifiers is essential for the efficiency of the constraint solving procedure within which the unification algorithm is used. The method is implemented in the CL-Atse tool for security protocol analysis

Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455

Unification theory

The purpose of this paper is not to give an overview of the state of art in unification theory. It is intended to be a short introduction into the area of equational unification which should give the reader a feeling for what unification theory might be about. The basic notions such as complete and minimal complete sets of unifiers, and unification types of equational theories are introduced and illustrated by examples. Then we shall describe the original motivations for considering unification (in the empty theory) in resolution theorem proving and term rewriting. Starting with Robinson\u27s first unification algorithm it will be sketched how more efficient unification algorithms can be derived. We shall then explain the reasons which lead to the introduction of unification in non-empty theories into the above mentioned areas theorem proving and term rewriting. For theory unification it makes a difference whether single equations or systems of equations are considered. In addition, one has to be careful with regard to the signature over which the terms of the unification problems can be built. This leads to the distinction between elementary unification, unification with constants, and general unification (where arbitrary free function symbols may occur). Going from elementary unification to general unification is an instance of the so-called combination problem for equational theories which can be formulated as follows: Let E, F be equational theories over disjoint signatures. How can unification algorithms for E, F be combined to a unification algorithm for the theory E cup F

Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

AbstractThis paper addresses the problem of systematically building a matching algorithm for the union of two disjoint theoriesE1∪E2provided that matching algorithms are known in both theoriesE1andE2. In general, the blind use of combination techniques introduces unification. Two different restrictions are considered in order to reduce this unification to matching. First, we show that combining matching algorithms (with linear constant restriction) is always sufficient for solving a pure fragment of combined matching problems. Second, the investigated method is complete for the largest class of theories where unification is not needed, including regular collapse-free theories and linear theories. Syntactic conditions are given to define this class of theories in which solving the combined matching problem is performed in a modular way

Three Optimisations for Sharing

In order to improve precision and efficiency sharing analysis should track both freeness and linearity. The abstract unification algorithms for these combined domains are suboptimal, hence there is scope for improving precision. This paper proposes three optimisations for tracing sharing in combination with freeness and linearity. A novel connection between equations and sharing abstractions is used to establish correctness of these optimisations even in the presence of rational trees. A method for pruning intermediate sharing abstractions to improve efficiency is also proposed. The optimisations are lightweight and therefore some, if not all, of these optimisations will be of interest to the implementor.Comment: To appear in Theiry and Practice of Logic Programmin

Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of Logic Programming (TPLP