The Membership Problem for Regular Expressions with Unordered Concatenation and Numerical Constraints

We study the membership problem for regular expressions extended with operators for unordered concatenation and numerical constraints. The unordered concatenation of a set of regular expressions denotes all sequences consisting of exactly one word denoted by each of the expressions. Numerical constraints are an extension of regular expressions used in many applications, e.g. text search (e.g., UNIX grep), document formats (e.g. XML Schema). Regular expressions with unordered concatenation and numerical constraints denote the same languages as the classical regular expressions, but, in certain important cases, exponentially more succinct. We show that the membership problem for regular expressions with unordered concatenation (without numerical constraints) is already NP-hard. We show a polynomial-time algorithm for the membership problem for regular expressions with numerical constraints and unordered concatenation, when restricted to a subclass called strongly 1-unambiguous.