The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As an example, an equational characterization of the pseudovariety corresponding to the class of regular languages in AC0 is given.

We compare various computational complexity classes defined within the framework of membrane systems, a distributed parallel computing device which is inspired from the functioning of the cell, with usual computational complexity classes for Turing machines. In particular, we focus our attention on the comparison among complexity classes for membrane systems with active membranes (where new membranes can be created by division of existing membranes) and the classes PSPACE, EXP, and EXPSPACE.

We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et al. – but for the case of circular redexes , for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms.

The notion of solvability in the call-by-value λ-calculus is defined and completely characterized, both from an operational and a logical point of view. The operational characterization is given through a reduction machine, performing the classical β-reduction, according to an innermost strategy. In fact, it turns out that the call-by-value reduction rule is too weak for capturing the solvability property of terms. The logical characterization is given through an intersection type assignment system, assigning types of a given shape to all and only the call-by-value solvable terms.

We present an on-line linear time and space algorithmto check if an integer array f is the border array of at least one string w built on a boundedor unbounded size alphabet Σ. First of all, we show a bijection between the border array of a string w and the skeleton of the DFA recognizing Σ*ω, called a string matching automaton (SMA).Different strings can have the same border array but the originality of the presented method is that the correspondence between a border array anda skeleton of SMA is independent from the underlying strings. This enables to design algorithms for validating and generating border arrays that outperform existing ones.The validating algorithm lowers the delay (maximal number of comparisons onone element of the array) from O(|w|) to 1 + min{|Σ|,1 + log2|ω|}compared to existing algorithms.We then give results on the numbers of distinct border arrays depending on the alphabet size.We also present an algorithm that checks if a given directed unlabeled graph G is the skeleton of aSMA on an alphabet of size s in linear time.Along the process the algorithm can build one string w for which G is the SMA skeleton.

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.

We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm.

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 L is of order ${\cal O}(n^k)$. We give a precise characterization of the k-poly-slender context-free languages. The well-known characterization of the k-poly-slender regular languages is an immediate consequence of ours.

Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric proof of Yasutomi's characterization of all pairs (α,p) such that sα,p is a fixed point of some non-trivial substitution.

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in Cartesian categories.

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of the following two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is (i) to present general results about freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic.

In this paper we prove the decidability of the HD0L ultimate periodicity problem.

The Rabin index of a rational language of infinite words given by a parity automaton with n states is computable in time O(n2c) where c is the cardinality of the alphabet. The number of values used by a parity acceptance condition is always greater than the Rabin index and conversely, the acceptance condition of a parity automaton can always be replaced by an equivalent acceptance condition whose number of used values is exactly the Rabin index. This new acceptance condition can also be computed in time O(n2c).

We define a kind of cellular automaton called a hexagonal partitioned cellular automaton (HPCA), and study logical universality of a reversible HPCA. We give a specific 64-state reversible HPCA H1, and show that a Fredkin gate can be embedded in this cellular space. Since a Fredkin gate is known to be a universal logic element, logical universality of H1 is concluded. Although the number of states of H1 is greater than those of the previous models of reversible CAs having universality, the size of the configuration realizing a Fredkin gate is greatly reduced, and its local transition function is still simple. Comparison with the previous models, and open problems related to these model are also discussed.

For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by μ-terms. We call the category μ-bicomplete if every μ-term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved through parity games: we associate to each game an algebraic expression and turn the game into a term of a categorical theory. We show that μ-terms and parity games are equivalent, meaning that they define the same property of being μ-bicomplete. Finally, the interpretation of a parity game in the category of sets is shown to be the set of deterministic winning strategies for a chosen player.

An illustration of coinduction in terms of a notion of weak bisimilarity is presented. First, an operational semantics $\mbox{${\cal O}$}$ for while programs is defined in terms of a final automaton. It identifies any two programs that are weakly bisimilar, and induces in a canonical manner a compositional model $\mbox{${\cal D}$}$. Next $\mbox{${\cal O}$}= \mbox{${\cal D}$}$ is proved by coinduction.

We provide an elementary proof of the fixpoint alternation hierarchy in arithmetic, which in turn allows us to simplify the proof of the modal mu-calculus alternation hierarchy. We further show that the alternation hierarchy on the binary tree is strict, resolving a problem of Niwiński.

Some of the most interesting and important results concerning quantum finite automata arethose showing that they can recognize certain languages with (much) less resources thancorresponding classical finite automata. This paper shows three results of such a typethat are stronger in some sense than other ones because (a) they deal with models ofquantum finite automata with very little quantumness (so-called semi-quantum one- andtwo-way finite automata); (b) differences, even comparing with probabilistic classicalautomata, are bigger than expected; (c) a trade-off between the number of classical andquantum basis states needed is demonstrated in one case and (d) languages (or the promiseproblem) used to show main results are very simple and often explored ones in automatatheory or in communication complexity, with seemingly little structure that could beutilized.

The notion of a bisimulation relation is of basic importance in many areas of computation theory and logic. Of late, it has come to take a particular significance in work on the formal analysis and verification of hybrid control systems, where system properties are expressible by formulas of the modal μ-calculus or weaker temporal logics. Our purpose here is to give an analysis of the concept of bisimulation, starting with the observation that the zig-zag conditions are suggestive of some form of continuity. We give a topological characterization of bisimularity for preorders, and then use the topology as a route to examining the algebraic semantics for the µ-calculus, developed in recent work of Kwiatkowska et al., and its relation to the standard set-theoretic semantics. In our setting, μ-calculus sentences evaluate as clopen sets of an Alexandroff topology, rather than as clopens of a (compact, Hausdorff) Stone topology, as arises in the Stone space representation of Boolean algebras (with operators). The paper concludes by applying the topological characterization to obtain the decidability of μ-calculus properties for a class of first-order definable hybrid dynamical systems, slightly extending and considerably simplifying the proof of a recent result of Lafferriere et al.