Approximate NFA Universality and Related Problems Motivated by Information Theory

TL;DR

This paper introduces a notion of approximate universality for NFAs and provides randomized algorithms to test this property, addressing the computational hardness of exact universality in coding and information theory.

Contribution

It proposes the concept of $(1- ext{epsilon})$-universal automata and develops polynomial randomized algorithms for approximate universality testing.

Findings

Randomized algorithms effectively test approximate universality.

Approximate universality testing is computationally hard for fixed epsilon.

Randomization is essential for practical approximate universality testing.

Abstract

In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is `approximately' maximal, which then boils down to whether a given NFA is `approximately' universal. Here we introduce the notion of a $(1-\epsilon)$-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable $\epsilon$.