Compactness and learning of classes of unions of erasing regular pattern languages

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 RPk of sets of at most k regular patterns, and an efficient learning from positive examples of the language class defined by RPk. 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 RPk of language description which guarantees the equivalency between the semantic containment L(P) subset of or equal to L(Q) and the syntactic containment P subset of or equal to Q, where subset of or equal to is a syntactic subsumption over the generalization systems. Under the compactness, the MINL problem reduces to finding one of minimal sets in RPk for a given sample under the subsumption subset of or equal to. 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 greater than or equal to 1, the class RPk has compactness if and only if the number of constant symbols is greater than k + 1. Moreover, we prove that for each P is an element of RPk, 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 RPk 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.