Pascal Caron

Characterization of Glushkov automata

Abstract Glushkovs algorithm computes a nondeterministic finite automaton without e -transitions and with n+1 states from a regular expression having n occurrences of letters. The aim of this paper is to give a set of necessary and sufficient conditions characterizing this automaton. Our characterization theorem is formulated in terms of directed graphs. Moreover these conditions allow us to produce an algorithm of conversion of a Glushkov automaton into a regular expression of small size.

Families of locally testable languages

Kim, McNaughton and McCloskey have produced a polynomial time algorithm in order to test if a deterministic automaton recognizes a locally testable language. We provide a characterization in terms of automata for the strictly locally testable languages and for the strongly locally testable languages, two subclasses of locally testable languages. These two characterizations lead us to polynomial time algorithms for testing these families of languages.

Partial derivatives of an extended regular expression

The notion of expression derivative due to Brzozowski leads to the construction of a deterministic automaton from an extended regular expression, whereas the notion of partial derivative due to Antimirov leads to the construction of a non-deterministic automaton from a simple regular expression. In this paper, we generalize Antimirov partial derivatives to regular expressions extended to complementation and intersection. For a simple regular expression with n symbols, Antimirov automaton has at most n+1 states. As far as an extended regular expression is concerned, we show that the number of states can be exponential.

Glushkov Construction for Multiplicities

We present an extension to multiplicities of a classical algorithm for computing a boolean automaton from a regular expression. The Glushkov construction computes an automaton with n + 1 states from a regular expression with n occurences of letters. We show that the Glushkov algorithm still suits to the multiplicity case. Next, we give three equivalent extended step by step algorithms.

Glushkov Construction For Series: The Non Commutative Case

We present an extension to multiplicities of a classical algorithm for computing a boolean automaton from a regular expression. The Glushkov construction computes an automaton with n +1 states from a regular expression with n occurences of letters. We give an extension of the Glushkov algorithm for the multiplicity case in a non commutative semiring. Next, we give four equivalent extended step by step algorithms.

Acyclic automata and small expressions using multi-tilde-bar operators

A regular expression with n occurrences of symbol can be converted into an equivalent automaton with (n+1) states, the so-called Glushkov automaton of the expression. Conversely, it is possible to decide whether a given (n+1)-state automaton is a Glushkov one and, if so, to convert it back to an equivalent regular expression of width n. Our goal is to extend the class of automata for which such a linear retranslation is possible. We define new regular operators, called multi-tilde-bars, allowing us to simultaneously apply a multi-tilde operator and a multi-bar one to a list of expressions. The main results are that a multi-tilde-bar expression of width n can be converted into an (n+1)-state position-like automaton and that any acyclic n-state automaton can be turned into an extended expression of width O(n).

LANGAGE: A Maple Package for Automaton Characterization of Regular Languages

LANGAGE is a set of procedures for deciding whether or not a language given by its minimal automaton is piecewise testable, locally testable, strictly locally testable, or strongly locally testable. New polynomial algorithms are implemented for the two last properties. This package is written using the symbolic computation system Maple. It works with AG, a set of Maple packages for processing automata and finite semigroups.

A general framework for the derivation of regular expressions

The aim of this paper is to design a theoretical framework that allows us to perform the computation of regular expression derivatives through a space of generic structures. Thanks to this formalism, the main properties of regular expression derivation, such as the finiteness of the set of derivatives, need only be stated and proved one time, at the top level. Moreover, it is shown how to construct an alternating automaton associated with the derivation of a regular expression in this general framework. Finally, Brzozowski’s derivation and Antimirov’s derivation turn out to be a particular case of this general scheme and it is shown how to construct a DFA, a NFA and an AFA for both of these derivations.

Generalized one-unambiguity

Bruggemann-Klein and Wood have introduced a new family of regular languages, the one-unambiguous regular languages, a very important notion in XML DTDs. A regular language L is one-unambiguous if and only if there exists a regular expression E over the operators of sum, catenation and Kleene star such that L(E) = L and the position automaton of E is deterministic. It implies that for a one-unambiguous expression, there exists an equivalent linear-size deterministic recognizer. In this paper, we extend the notion of one-unambiguity to weak one-unambiguity over regular expressions using the complement operator ¬. We show that a DFA with at most (n + 2) states can be computed from a weakly one-unambiguous expression and that it is decidable whether or not a given DFA recognizes a weakly one-unambiguous language.

Multi-tilde-bar derivatives

Multi-tilde-bar operators allow us to extend regular expressions. The associated extended expressions are compatible with the structure of Glushkov automata and they provide a more succinct representation than standard expressions. The aim of this paper is to examine the derivation of multi-tilde-bar expressions. Two types of computation are investigated: Brzozowski derivation and Antimirov derivation, as well as the construction of the associated automata.