This paper considers properties of language classes with finite elasticity in the viewpoint of set theoretic operations. Finite elasticity was introduced by Wright as a sufficient condition for language classes to be inferable from positive data, and as a property preserved by (not usual) union operation to generate a class of unions of languages. We show that the family of language classes with finite elasticity is closed under not only union but also various operations for language classes such as intersection, concatenation and so on, except complement operation. As a framework defining languages, we introduce restricted elementary formal systems (EFS's for short), called max length-bounded by which any context-sensitive language is definable. We define various operations for EFS's corresponding to usual language operations and also for EFS classes, and investigate closure properties of the family Ge of max length-bounded EFS classes that define classes of languages with finite elasticity. Furthermore, we present theorems characterizing a max length-bounded EFS class in the family Ge, and that for the language class to be inferable from positive data, provided the class is closed under subset operation. From the former, it follows that for any n, a language class definable by max length-bounded EFS's with at most n axioms has finite elasticity. This means that Ge is sufficiently large.

In this paper, we deal with inductive inference of an indexed family of recursive languages. We give two sufficient conditions for inductive inferability of an indexed family from positive data, each of which does not depend on the indexing of the family. We introduce two notions of finite cross property for a class of languages and a pair of finite tell-tales for a language. The former is a generalization of finite elasticity due to Wright and the latter consists of two finite sets of strings one of which is a finite tell-tale introduced by Angluin. The main theorem in this paper is that if any language of a class has a pair of finite tell-tales, then the class is inferable from positive data. Also, it is shown that any language of a class with finite cross property has a pair of finite tell-tales. Hence a class with finite cross property is inferable from positive data. Further-more, it is proved that a language has a finite tell-tale if and only if there does not exist any infinite cross sequence of languages contained in the language.