Anti-unification in Constraint Logic Programming

Abstract

Anti-unification refers to the process of generalizing two (or more) goals into a single, more general, goal that captures some of the structure that is common to all initial goals. In general one is typically interested in computing what is often called a most specific generalization, that is a generalization that captures a maximal amount of shared structure. In this work we address the problem of anti-unification in CLP, where goals can be seen as unordered sets of atoms and/or constraints. We show that while the concept of a most specific generalization can easily be defined in this context, computing it becomes an NP-complete problem. We subsequently introduce a generalization algorithm that computes a well-defined abstraction whose computation can be bound to a polynomial execution time. Initial experiments show that even a naive implementation of our algorithm produces acceptable generalizations in an efficient way. Under consideration for acceptance in TPLP.

Anti-unification of Unordered Goals

“…Anti-unification techniques that are adapted for CLP(X) have been defined in [32], but its focus is set on a polynomial abstraction procedure for a specific case where terms cannot be generalized (only variables can) and where generalization has to be carried out through injective substitutions. While [32] provides useful insights and results, it lacks a more general and in-depth study of the used generalization operator. In this work we broaden, generalize and complete the latter work by providing a detailed and systematic study of generalization operators and their characteristics in the context of CLP.…”

“…In Section 4 we provide an in-depth examination of several key variations of the anti-unification problem, namely variable generalization (where no terms are allowed to be generalized), injective generalization (where the generalizing substitutions need to be injective) and dataflow optimization (where the number of generalizing variables needs to be minimized) -the latter of which is proved to make the anti-unification statement NP-hard. Finally, addressing this last problem more in depth in Section 5 we revisit a tractable abstraction that was introduced in [32] but we provide for the first time a formal proof of its worst-case complexity, showing that the approximation can effectively be computed in polynomially bounded time.…”

“…In what follows, we introduce an abstraction for the largest common generzalization with respect to ⪯ ι that can be computed in polynomial time. The abstraction was already introduced in [32] but no formal proof of its complexity was given. The abstraction is based on the k-swap stability property, which is in turn defined in terms of pairing generalizations.…”

“…An algorithm has been introduced in [32] that builds up a k-swap stable generalization using the process suggested in Example 31. Its practical performance has been assessed on different test cases.…”

Global Guidance for Local Generalization in Model Checking

“…For example, we normalize all terms, take care of implicit multiplication by 1, and of associativity and commutativity of addition. In the future, it is interesting to explore how advanced anti-unification algorithms, such as [ 8 , 27 ], can be adapted for our purpose.…”