One-unambiguity of regular expressions with numeric occurrence indicators

Regular expressions with numeric occurrence indicators are an extension of traditional regular expressions, which let the required minimum and the allowed maximum number of iterations of subexpressions be described with numeric parameters. We consider the problem of testing whether a given regular expression E with numeric occurrence indicators is 1-unambiguous or not. This condition means, informally, that any prefix of any word accepted by expression E determines a unique path of matching symbol positions in E. One-unambiguity appears as a validity constraint in popular document schema languages such as SGML and XML DTDs (document type definitions) and XML Schema; the last one both includes numeric occurrence indicators and requires one-unambiguity of expressions. Previously published solutions for testing the one-unambiguity of regular expressions with numeric occurrence indicators are either erroneous or require exponential time. The main contribution of this paper is a polynomial-time method for solving this problem, and a formal proof of its correctness.