Finite Orbits of Language Operations

TL;DR

This paper proves that applying a specific set of language operations repeatedly to any language results in a finite, bounded set of languages, extending previous findings on individual operations.

Contribution

It generalizes earlier results by showing the finiteness and boundedness of orbits under a broader set of language operations.

Findings

The orbit of any language under the generated monoid is finite.

The orbit size is bounded independently of the initial language.

Generalizes previous results on complement and closure operations.

Abstract

We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure.