# Properties of language classes with finite elasticity

## Abstract

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 for language classes. 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.

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 **G**_{ e } of max length-bounded EFS classes that define classes of languages with finite elasticity. Further-more, we present theorems characterizing a max length-bounded EFS class in the family **G**_{ e }, 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 **G**_{ e } is sufficiently large.

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