Automatic Randomness Tests

TL;DR

This paper introduces automatic randomness tests (ART) using automata theory to characterize measure-theoretic typicalness of infinite binary sequences, providing a new framework for understanding automatic randomness.

Contribution

It defines ART within automata theory, characterizes automatic random sequences combinatorially, and compares different types of automatic randomness tests.

Findings

ARTs are equivalent to deterministic Büchi automata recognizing measure zero ω-languages

A collection of ARTs induces a notion of automatic randomness for sequences

Provides a combinatorial characterization of automatic random sequences

Abstract

In this paper we define a notion of automatic randomness tests (ART) which capture measure theoretic typicalness of infinite binary sequences within the framework of automata theory. An individual ART is found to be equivalent to a deterministic B\"{u}chi automaton recognizing $\omega$-language of (Lebesgue) measure zero. A collection of ART's induce a notion of automatic random sequence. We provide a purely combinatorial characterization of an automatic random sequence in the form of a disjunctive property for sequences. At last, we compare two kinds of automatic randomness tests presented in this paper.