Compactness and Learning of Classes of Unions of Erasing Regular Pattern Languages

Abstract

A regular pattern is a string of constant symbols and distinct variables. A semantics of a set P of regular patterns is a union L(P) of erasing pattern languages generated by patterns in P. The paper deals with the class RP^k of sets of at most k regular patterns, and an efficient learning from positive examples of the language class defined by RP^k. In efficient learning languages, the complexity for the MINL problem to find one of minimal languages containing a given sample is one of very important keys. Arimura et al.[5] introduced a notion of compactness w.r.t. containment for more general framework, called generalization systems, than RP^k of language description which guarantees the equivalency between the semantic containment L(P) ⫅ L(Q) and the syntactic containment P ⊑ Q, where ⊑ is a syntactic subsumption over the generalization systems.

Under the compactness, the MINL problem reduces to finding one of minimal sets in RP ^k for a given sample under the subsumption ⊑. They gave an efficient algorithm to find such minimal sets under the assumption of compactness and some conditions.

We first show that for each k ≥ 1, the class RP ^k has compactness if and only if the number of constant symbols is greater than k+1. Moreover, we prove that for each P ∈, RP ^k, a finite subset S ₂(P) is a characteristic set of L(P) within the class, where S ₂(P) consists of strings obtained from P by substituting strings with length two for each variable. Then our class RP ^k is shown to be polynomial time inferable from positive examples using the efficient algorithm of the MINL problem due to Arimura et al.[5], provided the number of constant symbols is greater than k + 1.