A New Approach to Formal Language Theory by Kolmogorov Complexity

TL;DR

This paper introduces a novel approach to formal language theory using Kolmogorov complexity, providing alternative proofs, new characterizations, and separation methods for various language classes, with broad applicability across the Chomsky hierarchy.

Contribution

It offers a new incompressibility-based framework for analyzing formal languages, replacing traditional tools like pumping lemmas and enabling quantification of nonrecursiveness.

Findings

New characterization for regular languages

Alternative proofs for language properties

Method to distinguish deterministic and nondeterministic context-free languages

Abstract

We present a new approach to formal language theory using Kolmogorov complexity. The main results presented here are an alternative for pumping lemma(s), a new characterization for regular languages, and a new method to separate deterministic context-free languages and nondeterministic context-free languages. The use of the new `incompressibility arguments' is illustrated by many examples. The approach is also successful at the high end of the Chomsky hierarchy since one can quantify nonrecursiveness in terms of Kolmogorov complexity. (This is a preliminary uncorrected version. The final version is the one published in SIAM J. Comput., 24:2(1995), 398-410.)