Randomness of formal languages via automatic martingales

TL;DR

This paper introduces a new notion of randomness for formal languages based on automatic martingales, showing how randomness depends on the type of text and the language class.

Contribution

It defines a formal framework for language randomness using automatic martingales and analyzes how this randomness varies with language class and text type.

Findings

Nonregular languages are random under general texts when considered individually.

Collections of languages are nonrandom under very general texts.

Computable languages can be nonrandom under certain dynamic texts.

Abstract

We define a notion of randomness for individual and collections of formal languages based on automatic martingales acting on sequences of words from some underlying domain. An automatic martingale bets if the incoming word belongs to the target language or not. Then randomness of both single languages and collections of languages is defined as a failure of automatic martingale to gain an unbounded capital by betting on the target language according to an incoming sequence of words, or a text. The randomness of formal languages turned out to be heavily dependent on the text. For very general classes of texts, any nonregular language happens to be random when considered individually. As for collections of languages, very general classes of texts permits nonrandomness of automatic families of languages only. On the other hand, an arbitrary computable language is be shown to be nonrandom under certain dynamic texts.