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Constrained Expressions and their Derivatives

Champarnaud J.-M., Mignot L., Nicart F.

Fundamenta Informaticae

2016

Vol. 149, nr 3

311--361

This paper proposes an extension to classical regular expressions by the addition of two operators allowing the inclusion of boolean formulae from the zeroth order logic. These expressions are called constrained expressions. The associated language is defined thanks to the notion of interpretation and of realization. We show that the language associated when both interpretation and realization are fixed is stricly regular and can be not regular otherwise. Furthermore, we use an extension of Antimirov’s partial derivatives in order to solve the membership test in the general case. Finally, we show that once the interpretation is fixed, the membership test of a word in the language denoted by a constrained expression can be undecidable whereas it is always decidable when the interpretation is not fixed.

Two-Sided Derivatives for Regular Expressions and for Hairpin Expressions

Champarnaud J.-M., Dubernard J.-P., Jeanne H., Mignot L.

Fundamenta Informaticae

2015

Vol. 137, nr 4

425--455

The aim of this paper is to design a polynomial construction of a finite recognizer for hairpin completions of regular languages. This is achieved by considering completions as new expression operators and by applying derivation techniques to the associated extended expressions called hairpin expressions. More precisely, we extend partial derivation of regular expressions to two-sided partial derivation of hairpin expressions and we show how to deduce a recognizer for a hairpin expression from its two-sided derived term automaton, providing an alternative proof of the fact that hairpin completions of regular languages are linear context-free.