Parameterized Regular Expressions and their Languages

TL;DR

This paper investigates regular expressions with variables interpreted as alphabet letters, characterizing the complexity of key decision problems and extending results to arbitrary regular language domains.

Contribution

It provides a complete complexity characterization for problems related to parameterized regular expression languages, including extensions to arbitrary regular language domains.

Findings

Languages are regular under both semantics.

Complexity of nonemptiness, universality, containment, membership is characterized.

Languages remain regular with arbitrary regular language domains.

Abstract

We study regular expressions that use variables, or parameters, which are interpreted as alphabet letters. We consider two classes of languages denoted by such expressions: under the possibility semantics, a word belongs to the language if it is denoted by some regular expression obtained by replacing variables with letters; under the certainly semantics, the word must be denoted by every such expression. Such languages are regular, and we show that they naturally arise in several applications such as querying graph databases and program analysis. As the main contribution of the paper, we provide a complete characterization of the complexity of the main computational problems related to such languages: nonemptiness, universality, containment, membership, as well as the problem of constructing NFAs capturing such languages. We also look at the extension when domains of variables could be arbitrary regular languages, and show that under the certainty semantics, languages remain regular and the complexity of the main computational problems does not change.