Abstract. A number of different techniques have been introduced in the last few decades to create ɛ-free automata representing regular expressions such as the Glushkov automata, follow automata, or Antimirov automata. This paper presents a simple and unified view of all these construction methods both for unweighted and weighted regular expressions. It describes simpler algorithms with time complexities at least as favorable as that of the best previously known techniques, and provides a concise proof of their correctness. Our algorithms are all based on two standard automata operations: epsilon-removal and minimization. This contrasts with the multitude of complicated and special-purpose techniques previously described in the literature, and makes it straightforward to generalize these algorithms to the weighted case. In particular, we extend the definition and construction of follow automata to the case of weighted regular expressions over a closed semiring and present the first algorithm to compute weighted Antimirov automata. 1

Our aim is to study the set of K-rational expressions describing rational series. More precisely we are concerned with the definition of quotients of this set by coarser and coarser congruences which lead to an extension – in the case of multiplicities – of some classical results stated in the Boolean case. In particular, multiplicity analogues of the well known theorems of Brzozowski and Antimirov are provided.

The partial derivative automaton (Apd) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (Apos). By estimating the number of regular expressions that have ε as a partial derivative, we compute a lower bound of the average number of mergings of states in Apos and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k’s its limit approaches half the number of states in Apos. The lower bound corresponds to consider the Apd automaton for the marked version ofthe regularexpression, i.e.where allitsletters are made different. Experimental results suggest that the average number of states of this automaton, and of the Apd automaton for the unmarked regular expression, are very close to each other.

Abstract. This paper presents some features of the Vaucanson platform. We describe some original algorithms on weighted automata and transducers (computation of the quotient, conversion of a regular expression into a weighted automaton, and composition). We explain how complex declarations due to the generic programming are masked from the user and finally we present a proposal for an XML format that allows implicit descriptions for simple types of automata. 1

Two classical non-deterministic automata recognize the language denoted by a regular expression: the position automaton which deduces from the position sets defined by Glushkov and McNaughton-Yamada, and the equation automaton which can be computed via Mirkin’s prebases or Antimirov’s partial derivatives. Let |E | be the size of the expression and �E � be its alphabetic width, i.e. the number of symbol occurrences. The number of states in the equation automaton is less than or equal to the number of states in the position automaton, which is equal to �E�+1. On the other hand, the worst-case time complexity of Antimirov algorithm is O(�E � 3 · |E | 2), while it is only O(�E � · |E|) for the most efficient implementations yielding the position automaton (Brüggemann-Klein, Chang and Paige, Champarnaud et al.). We present an O(|E | 2) space and time algorithm to compute the equation automaton. It is based on the notion of canonical derivative which makes it possible to efficiently handle sets of word derivatives. By the way, canonical derivatives also lead to a new O(|E | 2) space and time algorithm to construct the position automaton. 1

Abstract. We correct a mistake made in a previous paper in the construction of an automaton from a rational expression. We used there the definition of derivation of expression given by Antimirov, and this definition has to be further adapted for our purpose. 1991 Mathematics Subject Classification. 68Q45, 68Q70. A disturbing example In [7], we were considering the following problem: Is it possible to build an algorithm Ω such that for any rational expression E computed from an automaton A — i.e. E =Φ(A) where Φ is the state elimination method for instance — the following holds: A =Ω(E) ? We did not solve the problem completely, but we have identified two constructions that are good candidates to be the core components of such an algorithm Ω. The first one is the construction of the automaton of derived terms ∆(E) from an expression E, the second one is the computation of the minimal co-quotient Υ(B) of an automaton B. Our construction of the automaton of derived terms of E in [7] is a modification of the automaton of partial derivatives of [2] by the introduction of an initial derivation that gives rises to initial derived terms. We shall recall all these definitions below, but, to make a not so long story even shorter, our key statement (Theorem 3.5), that says that the ‘automaton of derived terms of an expression computed from a co-deterministic automaton is co-deterministic’, is untrue, as shown by the following example. Let A1 be the automaton of Figure 1 (a), and let E1 =(a+b)[(a + b)(a + b)] ∗ be the expression computed from A1, whichwewrite

Acknowledgement I would like to thank my tutor Aarne Ranta, who introduced me to the subject of transducers and has always been there for me throughout the work when I had some questions or if I needed help to get out of a general state of confusion. I would also like to thank my friend Bj"orn Nyberg, who was invaluable in the final step of this work. Finally, I would like to direct a general thanks to all the wonderful people in my life, none mentioned, none forgotten.

Abstract. We propose an algebra of regular hedge expressions built on top of regular hedge grammars as a framework for the analysis and manipulation of hedge languages. We show how linear systems of hedge language equations (LS for short) can be used as an intermediate representation on which to perform the computation of quotient, intersection, product derivative, and factor matrix of regular hedge languages. Regular hedge grammars and LSs are shown to be formalisms of same expressive power for the representation of hedge languages, and we give algorithms to convert between these two formalisms. 1

Abstract. Quotients and factors of regular languages are important notions in the design of computational procedures in the algebra of regular expressions and the analysis of their logical properties. We present algorithms for the computation of quotients and factors of languages specified by regular expressions. 1

Abstract. We propose linear systems of hedge language equations (LSH) as a formalism to represent regular hedge languages. These linear systems are suitable for several computations in the algebra of regular hedge languages. We indicate algorithms to translate between representations by hedge automata and LSH, and for the computation of LSH for the intersection, quotient, left and right factors of regular hedge languages. 1