Regular Expressions with Counting: Weak versus Strong Determinism

Authors:
Wouter Gelade;Marc Gyssens;Wim Martens
Affiliations:
School for Information Technology, Hasselt University and Transnational University of Limburg,;School for Information Technology, Hasselt University and Transnational University of Limburg,;Technical University of Dortmund,
Venue:
MFCS '09 Proceedings of the 34th International Symposium on Mathematical Foundations of Computer Science 2009
Year:
2009

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Abstract

We study deterministic regular expressions extended with the counting operator. There exist two notions of determinism, strong and weak determinism, which almost coincide for standard regular expressions. This, however, changes dramatically in the presence of counting. In particular, we show that weakly deterministic expressions with counting are exponentially more succinct and strictly more expressive than strongly deterministic ones, even though they still do not capture all regular languages. In addition, we present a finite automaton model with counters, study its properties and investigate the natural extension of the Glushkov construction translating expressions with counting into such counting automata. This translation yields a deterministic automaton if and only if the expression is strongly deterministic. These results then also allow to derive upper bounds for decision problems for strongly deterministic expressions with counting.