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 $\frac{1}{5}$ .

In this paper, we define the notion of biRFSA which is a residual finate stateautomaton (RFSA) whose the reverse is also an RFSA. The languages recognized bysuch automata are called biRFSA languages. We prove that the canonical RFSA of abiRFSA language is a minimal NFA for this language and that each minimalNFA for this language is a sub-automaton of the canonical RFSA. This leadsto a characterization of the family of biRFSA languages.In the second part of this paper, we define the family of biseparable automata. We prove that every biseparable NFA is uniquely minimal among all NFAs recognizinga same language, improving the result of H. Tamm and E. Ukkonen for bideterministic automata.