Widths of regular and context-free languages

TL;DR

This paper investigates the growth rates of antichains in regular and context-free languages under lexicographic order, providing algorithms for regular languages and showing undecidability for context-free languages, with extensions to tree languages.

Contribution

It establishes a dichotomy for regular languages between polynomial and exponential growth, offers a polynomial-time algorithm to distinguish them, and extends the analysis to tree languages with a trichotomy.

Findings

Polynomial-time algorithm for regular languages to determine growth type

Undecidability of growth distinction in context-free languages

Trichotomy of growth rates in regular tree languages

Abstract

Given a partially-ordered finite alphabet $\Sigma$ and a language $L\subseteq \Sigma^*$, how large can an antichain in $L$ be (where $L$ is given the lexicographic ordering)? More precisely, since $L$ will in general be infinite, we should ask about the rate of growth of maximum antichains consisting of words of length $n$. This fundamental property of partial orders is known as the width, and in a companion work we show that the problem of computing the information leakage permitted by a deterministic interactive system modeled as a finite-state transducer can be reduced to the problem of computing the width of a certain regular language. In this paper, we show that if $L$ is regular then there is a dichotomy between polynomial and exponential antichain growth. We give a polynomial-time algorithm to distinguish the two cases, and to compute the order of polynomial growth, with the language specified as an NFA. For context-free languages we show that there is a similar dichotomy, but now the problem of distinguishing the two cases is undecidable. Finally, we generalise the lexicographic order to tree languages, and show that for regular tree languages there is a trichotomy between polynomial, exponential and doubly exponential antichain growth.