We consider relational periods, where the relation is acompatibility relation on words induced by a relation on letters. Weprove a variant of the theorem of Fine and Wilf for a (pure) periodand a relational period.

Codd defined the relational algebra [E.F. Codd, Communications of the ACM13 (1970) 377–387;E.F. Codd, Relational completeness of data base sublanguages, in Data Base Systems, R. Rustin, Ed., Prentice-Hall (1972) 65–98] as the algebra with operations projection, join, restriction, union and difference. His projection operator can drop, permute and repeat columns of a relation. This permuting and repeating of columns does not really add expressive power to the relational algebra. Indeed, using the join operation, one can rewrite any relational algebra expression into an equivalent expression where no projection operator permutes or repeats columns. The fragment of the relational algebra known as the semijoin algebra, however, lacks a full join operation. Nevertheless, we show that any semijoin algebra expression can still be simulated in a natural way by a set of expressions where no projection operator permutes or repeats columns.

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.

We investigate automatic presentations of ω-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSO-interpretability in algebraic trees, and the decidability of the MSO theory of morphic words.Refining their techniques we observe that the lexicographicpresentation of a (morphic) word is in a certain sense canonical.We then generalize our techniques to a hierarchy of classes of ω-words enjoying the above mentioned definability and decidability properties.We introduce k-lexicographic presentations, and morphisms oflevel k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic orlevel k morphic words. We prove that these presentations arealso canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classesunder MSO-definable recolorings, e.g. closure under deterministic sequential mappings.The classes of k-lexicographic words are shown to constitute an infinite hierarchy.

The necessary and sufficient conditions are extracted for periodicity of bi-ideals. They cover infinitely and finitely generated bi-ideals.

The complexity of infinite words is considered from the point of view of a transformation with a Mealy machine that is the simplest model of a finite automaton transducer. We are mostly interested in algebraic properties of the underlying partially ordered set. Results considered with the existence of supremum, infimum, antichains, chains and density aspects are investigated.

We introduce natural generalizations of two well-known dynamical systems, the Sand Piles Model and the Brylawski'smodel. We describe their order structure, their reachableconfiguration's characterization, their fixed points and theirmaximal and minimal length's chains. Finally, we present aninduced model generating the set of unimodal sequences which amongst other corollaries, implies that this set is equipped with a lattice structure.

A compatibility relation on letters induces a reflexive andsymmetric relation on words of equal length. We consider these wordrelations with respect to the theory of variable length codes andfree monoids. We define an (R,S)-code and an (R,S)-free monoidfor arbitrary word relations R and S. ModifiedSardinas-Patterson algorithm is presented for testing whether finitesets of words are (R,S)-codes. Coding capabilities of relationalcodes are measured algorithmically by finding minimal and maximalrelations. We generalize the stability criterion of Schützenbergerand Tilson's closure result for (R,S)-free monoids. The(R,S)-free hull of a set of words is introduced and we show how itcan be computed. We prove a defect theorem for (R,S)-free hulls.In addition, a defect theorem of partial words is proved as acorollary.

We investigate the intersection of two finitely generated submonoidsof the free monoid on a finite alphabet. To this purpose, weconsider automata that recognize such submonoids and we study theproduct automata recognizing their intersection. By using automatamethods we obtain a new proof of a result of Karhumäki on thecharacterization of the intersection of two submonoids ofrank two, in the case of prefix (or suffix) generators. In a moregeneral setting, for an arbitrary number of generators, we provethat if H and K are two finitely generated submonoids generatedby prefix sets such that the product automaton associated to $H \capK$ has a given special property then $\widetilde{rk}(H \cap K) \leq\widetilde{rk}(H) \widetilde{rk}(K)$ where $\widetilde{rk}(L)=\max(0,rk(L)-1)$ for any submonoid L.

Right (left, two-sided) extendable part of a language consists of all words having infinitely many right (resp. left, two-sided) extensions within the language. We prove that for an arbitrary factorial language each of these parts has the same growth rate of complexity as the language itself. On the other hand, we exhibit a factorial language which grows superpolynomially, while its two-sided extendable part grows only linearly.

In formal language theory, many families of languages are defined using either grammars or finite acceptors. For instance, context-sensitive languages are the languages generated by growing grammars, or equivalently those accepted by Turing machines whose work tape's size is proportional to that of their input. A few years ago, a new characterisation of context-sensitive languages as the sets of traces, or path labels, of rational graphs (infinite graphs defined by sets of finite-state transducers) was established. We investigate a similar characterisation in the more general framework of graphs defined by term transducers. In particular, we show that the languages of term-automatic graphs between regular sets of vertices coincide with the languages accepted by alternating linearly bounded Turing machines.
As a technical tool, we also introduce an arborescent variant of tiling systems, which provides yet another characterisation of these languages.

A set T ⊆ L is a Parikh test set of L ifc(T) is a test set of c(L).We give a characterization of Parikh test sets for arbitrary language in terms of its Parikh basis, and the coincidence graph of letters.

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(log2 n)2 when n goes to infinity.


It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. The main tool are ζ-semigroups, considered as recognition structures for sofic subshifts. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, bydetermining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts and aperiodic subshifts. The algebraic invariant is compared with other robust conjugacy invariants.

A language L ⊆A*is literally idempotent in case thatua2v ∈ L if and only if uav ∈ L, for each u,v ∈ A*, a ∈ A.Varieties of literally idempotent languages result naturally by takingall literally idempotent languages in a classical (positive) varietyor by considering a certain closure operator on classes of languages.We initiate the systematic study of such varieties. Various classes ofliterally idempotent languages can be characterized using syntactic methods.A starting example is the classof all finite unions of $B^*_1 B^*_2\dots B^*_k$ where B1,...,Bk aresubsets of a given alphabet A.

We prove two cases of a strong version of Dejean's conjectureinvolving extremal letter frequencies. The results are that thereexist an infinite $\left({\frac{5}{4}^+}\right)$ -free word over a 5 letteralphabet with letter frequency $\frac{1}{6}$ and an infinite $\left({\frac{6}{5}^+}\right)$ -free word over a 6 letter alphabet withletter frequency $\frac{1}{5}$ .

We present a higher-order call-by-need lambda calculus enriched with constructors, case expressions, recursive letrec expressions, a seq operator for sequential evaluation and a non-deterministic operator amb that is locally bottom-avoiding. We use a small-step operational semantics in the form of a single-step rewriting system that defines a (non-deterministic) normal-order reduction. This strategy can be made fair by adding resources for book-keeping. As equational theory, we use contextual equivalence (that is, terms are equal if, when plugged into any program context, their termination behaviour is the same), in which we use a combination of may- and must-convergence, which is appropriate for non-deterministic computations. We show that we can drop the fairness condition for equational reasoning, since the valid equations with respect to normal-order reduction are the same as for fair normal-order reduction. We develop a number of proof tools for proving correctness of program transformations. In particular, we prove a context lemma for both may- and must- convergence that restricts the number of contexts that need to be examined for proving contextual equivalence. Combining this with so-called complete sets of commuting and forking diagrams, we show that all the deterministic reduction rules and some additional transformations preserve contextual equivalence. We also prove a standardisation theorem for fair normal-order reduction. The structure of the ordering ≤c is also analysed, and we show that Ω is not a least element and ≤c already implies contextual equivalence with respect to may-convergence.

Ontologies play a key role in the advent of the Semantic Web. An important problem when dealing with ontologies is the modification of an existing ontology in response to a certain need for change. This problem is a complex and multifaceted one, because it can take several different forms and includes several related subproblems, like heterogeneity resolution or keeping track of ontology versions. As a result, it is being addressed by several different, but closely related and often overlapping research disciplines. Unfortunately, the boundaries of each such discipline are not clear, as the same term is often used with different meanings in the relevant literature, creating a certain amount of confusion. The purpose of this paper is to identify the exact relationships between these research areas and to determine the boundaries of each field, by performing a broad review of the relevant literature.

Computational social choice is a new discipline currently emerging at the interface of social choice theory and computer science. It is concerned with the application of computational techniques to the study of social choice mechanisms, and with the integration of social choice paradigms into computing. The first international workshop specifically dedicated to this topic took place in December 2006 in Amsterdam, attracting a mix of computer scientists, people working in artificial intelligence and multiagent systems, economists, game and social choice theorists, logicians, mathematicians, philosophers, and psychologists as participants.