In this paper, we establish the following results: (i) every strongly finite dilator is finitely presentable in the category of endofunctors on the category of ordinals; (ii) a dilator F is strongly finite if and only if F is finitely presentable in the category of dilators.

We introduce a domain-free λµ-calculus of call-by-value as a short-hand for the second order Church-style. Our motivation comes from the observation that in Curry-style polymorphic calculi, control operators such as callcc-operators cannot, in general, handle correctly the terms placed on the control operator's left, so that the Curry-style system can fail to prove the subject reduction property. Following the continuation semantics, we also discuss the notion of values in classical system, and propose an extended form of values. It is proved that the CPS-translation is sound with respect to domain-free λ2 (second-order λ-calculus). As a by-product, we obtain the strong normalization property for the second-order λµ-calculus of call-by-value in domain-free style. We also study the problems of type inference, typability, and type checking for the call-by-value system. Finally, we give a brief comparison with standard ML plus callcc, and discuss a natural way to avoid the unsoundness of ML with callcc.

We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of the same length. Specifically, we prove that there are only five types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizations in terms of the number of solutions of Diophantine equations of certain types. In particular, we show that the linear upper bound on the non-exponential complexities by Karhumäki et al. in [9], is tight. There are several consequences of our study. First, we derive that both of the sets of all finite Sturmian words and of all finite Standard words are expressible by word equations. Second, we characterize the languages of non-exponential complexity which are expressible by two-variable word equations as finite unions of several simple parametric formulae and solutions of a two-variable word equation with a finite graph. Third, we find optimal upper bounds on the solutions of (solvable) two-variable word equations, namely, linear bound for one variable and quadratric for the other. From this, we obtain an $\mathcal{O}(n^6)$ algorithm for testing the solvability of two-variable word equations, improving thus very much Charatonik and Pacholski's $\mathcal{O}(n^100)$ algorithm from [3].

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 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.

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

The problem of synchronizing a network of identical processors that work synchronously at discrete steps is studied. Processors are arranged as an array of m rows and n columns and can exchange each other only one bit of information. We give algorithms which synchronize square arrays of (n × n) processors and give some general constructions to synchronize arrays of (m × n) processors. Algorithms are given to synchronize in time n2, $n \lceil \log n\rceil$, $n\lceil\sqrt n \rceil$ and 2n a square array of (n × n) processors. Our approach is a modular description of synchronizing algorithms in terms of "fragments" of cellular automata that are called signals. Compositional rules to obtain new signals (and new synchronization times) starting from known ones are given for an (m × n) array. Using these compositional rules we construct synchronizations in any "feasible" linear time and in any time expressed by 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 of biinfinite languages: the family of very thin codes (which contains the rational codes) and another family containing the circular codes. We propose the conjecture that all codes are minimal generators.