A Modular Equational Generalization Algorithm

Abstract

Abstract. This paper presents a modular equational generalization algorithm, where function symbols can have any combination of associativity, commutativity, and identity axioms (including the empty set). This is suitable for dealing with functions that obey algebraic laws, and are typically mechanized by means of equational atributes in rule-based languages such as ASF+SDF, Elan, OBJ, Cafe-OBJ, and Maude. The algorithm computes a complete set of least general generalizations modulo the given equational axioms, and is specified by a set of inference rules that we prove correct. This work provides a missing connection between least general generalization and computing modulo equational theories, and opens up new applications of generalization to rule-based languages, theorem provers and program manipulation tools such as partial evaluators, test case generators, and machine learning techniques, where function symbols obey algebraic axioms. A Web tool which implements the algorithm has been developed which is publicly available.

Anti-unification for Unranked Terms and Hedges

“…By the definition of Rgeneralization, the same holds for Xσ ,s, andq because R(top(s), top(q)) = ∅. 2 ) ins 2 such thatq is an R-generalization ofs 1 ands 2 . By the induction hypothesis, replacing an occurrence of X 2 inq with X 1 gives a hedge that is again an R-generalization ofs 1 ands 2 .…”

“…The rules for associative anti-unification with the unit element (AU anti-unification) considered in Alpuente et al [2] can be directly simulated in the minimal and complete unranked anti-unification algorithm. Conversely, to simulate unranked anti-unification rules in AU anti-unification, one needs to introduce sorts to distinguish between term variables and hedge variables (and their instantiations).…”

“…It may compute up to 3 n generalizations, 2 where n is the size of the input. (For instance, 5 ) it computes 11,685 generalizations, most of them several times, until it selects a single one, e.g., f (x 1 , x 2 , x 3 , x 4 , x 5 ), on the minimization step.)…”

“…Associative Anti-Unif ication with Unit Anti-unification with an associative function symbol (A) and a unit element (U) has been studied in Alpuente et al [2]. AU anti-unification with constants (where terms are built over associative function symbol, unit element, and free constants) can be directly modeled as unranked antiunification, using the algorithm G to compute generalizations.…”

ACUOS: A System for Modular ACU Generalization with Subtyping and Inheritance

“…Experimental results given in Section 3 show that ACUOS performs efficiently in practice. For a discussion of the related literature, we refer to [2,3,1,4,10] 2 Use Case: Extracting Analogies…”