We prove that for each positive integer n, the finite commutative languageEn = c(a 1 a 2...an ) possesses a test set of size at most 5n. Moreover, it is shown that each test set for E n has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) alph(w) = alph}(L); and (ii) each symbol a ∈ alph}(L) occurs at least twice in w if it occurs at least twice in some word of L: each such L possesses a test set of size 11n, where n = Card(alph(L)). The considerations rest on the analysis of some basic types of word equations.

We present a model in which, due to the quantum nature of the signals controlling the implementation time of successive unitary computational steps, physical irreversibility appears in the execution of a logically reversible computation.

We provide alternative proofs and algorithms for results proved by Sénizergues on rational and recognizable free group languages. We consider two different approaches to the basic problem of deciding recognizability for rational free group languages following two fully independent paths: the symmetrification method (using techniques inspired by the study of inverse automata and inverse monoids) and the right stabilizer method (a general approach generalizable to other classes of groups). Several different algorithmic characterizations of recognizability are obtained, as well as other decidability results.

We define a notion of asynchronous sliding block map that can be realized by transducers labeled in A* × B*. We show that, under some conditions, it is possible to synchronize this transducer by state splitting, in order to get a transducer which defines the same sliding block map and which is labeled in A × Bk, where k is a constant integer. In the case of a transducer with a strongly connected graph, the synchronization process can be considered as an implementation of an algorithm of Frougny and Sakarovitch for synchronization of rational relations of bounded delay. The algorithm can be applied in the case where the transducer has a constant integer transmission rate on cycles and has a strongly connected graph. It keeps the locality of the input automaton of the transducer. We show that the size of the sliding window of the synchronous local map grows linearly during the process, but that the size of the transducer is intrinsically exponential. In the case of non strongly connected graphs, the algorithm of Frougny and Sakarovitch does not keep the locality of the input automaton of the transducer. We give another algorithm to solve this case without losing the good dynamic properties that guaranty the state splitting process.

It is shown that the class of left-to-right regular languages coincides with the class of languages that are accepted by monotone deterministic RL-automata, in this way establishing a close correspondence between a classical parsing algorithm and a certain restricted type of analysis by reduction.

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model$\mbox{IPM}_k$(n), a generalization of the sand pile model$\mbox{SPM}$(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice $\mbox{IPM}_k$(n): this lets us design an algorithm which generates all the ice piles of $\mbox{IPM}_k$(n) in amortized time O(1) and in space O($\sqrt n$).

An alternative (tree-based) semantics for a class of regular expressions is proposed that assigns a central rôle to the + operator and thus to nondeterminism and nondeterministic choice. For the new semantics a consistent and complete axiomatization is obtained from the original axiomatization of regular expressions by Salomaa and by Kozen by dropping the idempotence law for + and the distribution law of • over +.

The characteristic parameters K w and R w of a word w over a finite alphabet are defined as follows: K w is the minimal natural number such that w has no repeated suffix of length K w and R w is the minimal natural number such that w has no right special factor of length R w . In a previous paper, published on this journal, we have studied the distributions of these parameters, as well as the distribution of the maximal length of a repetition, among the words of each length on a given alphabet. In this paper we give the exact values of these distributions in a special case. However, these values give upper bounds to the distributions in the general case. Moreover, we study the most frequent and the average values of the characteristic parameters and of the maximal length of a repetition over the set of all words of length n.

We introduce in this article a new method to calculate all absolute andrelatif primitive invariants of finite groups. This method is inspiredfrom K. Girstmair which calculate an absolute primitive invariant ofminimal degree.Are presented two algorithms, the first one enable us to calculate allprimitive invariants of minimal degree, and the second one calculate allabsolute or relative primitive invariants with distincts coefficients. Thiswork take place in Galois Theory and Invariant Theory.

In this paper we analyze some intrusion detection strategies proposed in the literature and we show that they represent the various facets of a well known formal languages problem: computing the distance between a string x and a language L. In particular, the main differences among the various approaches adopted for building intrusion detection systems can be reduced to the characteristics of the language L and to the notion of distance adopted. As a further contribution we will also show that from the computational point of view all these strategies are equivalent and they are amenable to efficient parallelization.

We show that semigroups representable by triangular matrices over a fixed finite field form a decidable pseudovariety and provide a finite pseudoidentity basis for it.

In this article, we study the complexity of drunken man infinite words. We show that these infinite words, generated by a deterministic and complete countable automaton, or equivalently generated by a substitution over a countable alphabet of constant length, have complexity functions equivalent to n(log2n)2 when n goes to infinity.


Coalgebras have been proposed as formal basis for the semantics of objects in the sense of object-oriented programming. This paper shows that this semantics provides a smooth interpretation for subtyping, a central notion in object-oriented programming. We show that different characterisations of behavioural subtyping found in the literature can conveniently be expressed in coalgebraic terms. We also investigate the subtle difference between behavioural subtyping and refinement.

This paper analyses the complexity of model checking fixpoint logic with Chop – an extension of the modal μ-calculus with a sequential composition operator. It uses two known game-based characterisations to derive the following results: the combined model checking complexity as well as the data complexity of FLC are EXPTIME-complete. This is already the case for its alternation-free fragment. The expression complexity of FLC is trivially P-hard and limited from above by the complexity of solving a parity game, i.e. in UP ∩ co-UP. For any fragment of fixed alternation depth, in particular alternation- free formulas it is P-complete.

Taking the view that infinite plays are draws, we study Conwaynon-terminating games and non-losing strategies. These admit asharp coalgebraic presentation, where non-terminating games are seen as afinal coalgebra and game contructors, such as disjunctivesum, as final morphisms. We have shown, in a previous paper,that Conway’s theory of terminating games can be rephrased naturally in terms of game(pre)congruences. Namely, various conceptually independent notions ofequivalence can be defined and shown to coincide on Conway’sterminating games. These are the equivalence induced by the ordering on surrealnumbers, the contextual equivalence determined by observingwhat player has a winning strategy, Joyal’s categoricalequivalence, and, for impartial games, the denotationalequivalence induced by Grundy semantics. In this paper, wediscuss generalizations of such equivalences to non-terminating games andnon-losing strategies. The scenario is even more rich and intriguing inthis case. In particular, we investigate efficient characterizations of the contextualequivalence, and we introduce a category of fair strategies and acategory of fair pairs of strategies, both generalizing Joyal’s categoryof Conway games and winning strategies. Interestingly, the category of fair pairs capturesthe equivalence defined by Berlekamp, Conway, Guy on loopy games.

In this paper, we study the continuity of rational functions realized by Büchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function f has at least one point of continuity and that its continuity set C(f) cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational ${\bf \Pi}^0_2$-subset of Σω for some alphabet Σ is the continuity set C(f) of an ω-rational synchronous function f defined on Σω.

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-language \hbox{$L(\mathcal{A})$}L(𝒜) accepted by a Büchi 1-counter automaton \hbox{$\mathcal{A}$}𝒜. We prove the following surprising result: there exists a 1-counter Büchi automaton \hbox{$\mathcal{A}$}𝒜 such that the cardinality of the complement \hbox{$L(\mathcal{A})^-$}L(𝒜) −  of the ω-language \hbox{$L(\mathcal{A})$}L(𝒜) is not determined by ZFC: (1) There is a model V1 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) −  is countable. (2) There is a model V2 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) −  has cardinal 2ℵ0. (3) There is a model V3 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) −  has cardinal ℵ1 with ℵ0 < ℵ1 < 2ℵ0.

We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton ℬ. As a corollary, this proves that the continuum hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter ω-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ω-language (respectively, infinitary rational relation) is countable is in Σ13\(Π12 ∪ Σ12). This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

Minimizing a deterministic finite automata (DFA) is a very important problem in theory of automata and formal languages.Hopcroft's algorithm represents the fastest known solution to the such a problem. In this paper we analyze the behavior of this algorithm on a family binary automata, called tree-like automata, associated to binary labeled trees constructed by words. We prove that all the executions of the algorithm on tree-like automata associated to trees, constructed by standard words, have running time with the same asymptotic growth rate. In particular, we provide a lower and upper bound for the running time of the algorithm expressed in terms of combinatorial properties of the trees. We consider also tree-like automata associated to trees constructed by de Brujin words,and we prove that a queue implementation of the waiting set gives a Θ(n log n) execution while a stack implementation produces a linear execution. Such a result confirms the conjecture given in [A. Paun, M. Paun and A. Rodríguez-Patón. Theoret. Comput. Sci.410 (2009) 2424–2430.] formulated for a family of unary automata and, in addition, gives a positive answer also for the binary case.

We extend the simply typedλ-calculus with unbind and rebind primitiveconstructs. That is, a value can be a fragment of open code,which in order to be used should be explicitly rebound. Thismechanism nicely coexists with standard static binding. Themotivation is to provide an unifying foundation for mechanisms ofdynamic scoping, where the meaning of a name isdetermined at runtime, rebinding, such as dynamic updatingof resources and exchange of mobile code, and delegation,where an alternative action is taken if a binding is missing.Depending on the application scenario, we consider twoextensions which differ in the way type safety is guaranteed. Theformer relies on a combination of static and dynamic type checking.That is, rebind raises a dynamic error if for some variablethere is no replacing term or it has the wrong type. In the latter,this error is prevented by a purely static type system, at the priceof more sophisticated types.

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).