Widths of Regular and Context-Free Languages

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingAcademicpeer-review

Abstract

Given a partially-ordered finite alphabet Σ and a language L ⊆ Σ, 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 [10] 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.

Original languageEnglish
Title of host publication39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)
PublisherLeibnitz Int. Proceedings in Informatics
Pages49:1-49:14
Number of pages14
Volume150
ISBN (Electronic)978-3-95977-131-3
DOIs
Publication statusPublished - 2019
Externally publishedYes

Fingerprint

Dive into the research topics of 'Widths of Regular and Context-Free Languages'. Together they form a unique fingerprint.

Cite this