We present an algorithm for computing the prefix of an automaton. Automata considered are non-deterministic, labelled on words, and can have ε-transitions. The prefix automaton of an automaton $\mathcal{A}$ has the following characteristic properties. It has the same graph as $\mathcal{A}$ . Each accepting path has the same label as in $\mathcal{A}$ . For each state q, the longest common prefix of the labels of all paths going from q to an initial or final state is empty. The interest of the computation of the prefix of an automaton is that it is the first step of the minimization of sequential transducers.The algorithm that we describe has the same worst case time complexity as another algorithm due to Mohri but our algorithm allows automata that have empty labelled cycles. If we denote by P(q) the longest common prefix of labels of paths going from q to an initial or final state, it operates in time O((P+1) × |E|) where P is the maximal length of all P(q).

We introduce two new classes of codes, namely adjacent codes and codes with finite interpreting delay. Foreach class, we establish an extension of the defect theorem.

Fine and Wilf's theorem has recently been extended to words having threeperiods. Following the method of the authors we extend it to anarbitrary number of periods and deduce from that a characterization ofgeneralized Arnoux-Rauzy sequences or episturmian infinite words.

Foundations of the notion of quantum Turing machines areinvestigated. According to Deutsch's formulation, the time evolution of a quantum Turing machine is to be determined by the localtransition function. In this paper, the local transition functions are characterized for fully general quantum Turing machines, including multi-tape quantum Turing machines, extending the results due to Bernstein and Vazirani.

We consider the multiparty communication model defined in [4] using the formalism from [8]. First, we correct an inaccuracy in the proof of the fundamental result of [6] providing a lower bound on the nondeterministic communication complexity of a function. Then we construct several very hard functions, i.e., functions such that those as well as their complements have the worst possible nondeterministic communication complexity. The problem to find a particular very hard function was proposed in [7], where it has been shown that almost all functions are very hard. We also prove that combining two very hard functions by the Boolean operation xor gives a very hard function.

We prove that the cutwidth of the r-dimensional mesh of d-ary trees is of order $\Theta(d^{(r-1)n+1})$ , which improves and generalizes previous results.

We present a uniform and easy-to-use technique fordeciding the equivalenceproblem for deterministic monadic linear recursive programs. The key ideais to reduce this problem to the well-known group-theoretic problems byrevealing an algebraic nature of program computations. We show thatthe equivalence problem for monadic linear recursive programs over finiteand fixed alphabets of basic functions and logical conditions is decidablein polynomial time for the semantics based on the free monoids and freecommutative monoids.

For a non-negative integer k, we say that a language L is k-poly-slender if the number of words of length n in Lis of order ${\cal O}(n^k)$ . We give a precise characterization of the k-poly-slender context-free languages. The well-known characterizationof the k-poly-slender regular languages is an immediate consequenceof ours.

In the context of object-oriented systems, algorithms for building classhierarchies are currently receiving much attention. We present here a characterization of several globalalgorithms.A global algorithm is one which starts with only the set of classes (provided with all their properties) and directly buildsthe hierarchy.The algorithms scrutinized were developped each in a different framework.In this survey, they are explained in a single framework, which takesadvantage of a substructure of the Galois lattice associated with the binary relationmapping the classes to their properties.Their characterization allows to figure the results of the algorithms without runningthem in simple cases.This study once again highlights the Galois lattice as a main and intuitive model forclass hierarchies.

The aim of this paper is to evaluate the growth orderof the complexity function (in rectangles)for two-dimensional sequencesgenerated by a linear cellular automatonwith coefficients in $\mathbb{Z}/l \mathbb{Z}$ , and polynomial initial condition.We prove that the complexity functionis quadratic when l is a prime and that it increases with respect to the number of distinct prime factors of l.

The problem of synchronizing a networkof identical processors that work synchronouslyat discrete steps is studied. Processors are arranged as an array ofm rows and n columns and can exchange each other only one bitof information.We give algorithms whichsynchronize square arrays of (n × n) processors and give somegeneral constructions to synchronize arrays of (m × n) processors.Algorithms are given to synchronize in time n 2, $n \lceil \log n\rceil$ , $n\lceil\sqrt n \rceil$ and 2n a square array of (n × n) processors.Our approach is a modular description ofsynchronizing algorithms in terms of "fragments" of cellular automata that are calledsignals.Compositional rules to obtain new signals (and new synchronizationtimes) starting from known ones are given for an (m × n) array.Using these compositional rules we constructsynchronizations in any "feasible" linear time and in any time expressedby a polynomial with nonnegative coefficients.

This paper offers characteristic formula constructions in the real-time logic Lν for several behavioural relations between (states of) timed automata. The behavioural relations studied in this work are timed (bi)similarity, timed ready simulation, faster-than bisimilarity and timed trace inclusion. The characteristic formulae delivered by our constructions have size which is linear in that of the timed automaton they logically describe. This also applies to the characteristic formula for timed bisimulation equivalence, for which an exponential space construction was previously offered by Laroussinie, Larsen and Weise.

In this paper we give two families of codes which are minimal generators ofbiinfinite languages: the family of very thin codes (which contains the rationalcodes) and another family containing the circular codes. We propose the conjecture that all codes are minimal generators.