Nominal Anti-Unification

Equation solving between nominal terms has been investigated by several authors, who designed and analyzed algorithms for nominal unification [3, 4, 15, 26], nominal matching [5], equivariant unification [6], permissive nominal unification [8]. However, in contrast to unification, its dual problem, anti-unification, has not been studied for nominal terms previously. In [18], it is referred to as “the as-of-yet undiscovered nominal anti-unification”, which “could form a fundamental component of a refactoring tool” for αProlog [7] programs. Software refactoring is one of possible applications of anti-unification. This method, formulated for different theories, has been successfully used in inductive logic programming [17], cognitive modeling [24], analogy making [13], inductive program synthesis [12], proof generalization [27], mathematical reasoning [10, 11], etc. Nominal anti-unification can play a role in extending some of these applications to the nominal setting. For instance, it can be useful to generalize proofs done in nominal logic, or in doing analogical reasoning in mathematics, or in adapting inductive program synthesis methods to αProlog programs, etc. The anti-unification problem for two terms t1 and t2 is concerned with finding a generalization term t such that t1 and t2 are substitutive instances of t. The interesting generalizations are the least general ones (lgg). Plotkin [20] and Reynolds [22] initiated research on anti-unification in the 1970s, developing algorithms for first-order terms. Since then, anti-unification has been studied in various theories, including some of those with binding constructs: calculus of constructions [19], Mλ [9], second-order lambda calculus with type variables [16], simply-typed lambda calculus where generalizations are higher-order patterns [2], just to name a few. In this paper we address the problem of computing lgg’s for nominal terms-in-context, which are pairs of a freshness context and a nominal term. It turned out that without a restriction, there is no lgg for terms-in-context, in general. Therefore we restrict the set of atoms which are permitted in generalizations to be finite. In this case, there exists a single lgg (modulo ') and we design an algorithm to compute it. Computation of nominal lgg’s requires a solution to the equivariance problem which aims at finding a permutation of atoms π for given terms t1 and t2 such that π applied to t1 is α-equivalent to t2 (under a given freshness context). Various anti-unification techniques, such as first-order, higher-order, or equational antiunification have been used in inductive logic programming, logical and relational learning [21], reasoning by analogy [13], program synthesis [23], program verification [16], etc. Nominal anti-unification can, hopefully, contribute in solving similar problems in nominal setting. The anti-unification algorithm has been implemented and is available from www.risc.jku. at/projects/stout/software/nau.php. The implementation of the equivariance algorithm is also accessible separately from www.risc.jku.at/projects/stout/software/nequiv.php.