Michel Latteux

Recognizable picture languages and domino tiling

Abstract In [2], Giammarresi and Restivo define the notion of local picture languages by giving a set of authorized 2 × 2 tiles over ∑ ∪ {#} where # is a boundary symbol which surrounds the pictures. Then they define the class of recognizable picture languages as the set of languages which can be obtained by projection of a local one. This class is of interest since it admits several quite different characterizations [3]. Here, we define the hv-local picture languages where 2 × 2 tiles are replaced by horizontal and vertical dominoes. So the horizontal and the vertical scanning can be done separately. However, we prove that every recognizable picture language can be obtained as a projection of a hv-local language.

Cônes rationnels commutatifs

We study full trios generated by commutative languages. We give, for instance, a necessary and sufficient condition for a commutative language to be erasable. This allows us to show that slip commutative languages are erasable. We prove also the existence, in the family of nonrational commutative languages, of a minimal language for rational transductions.

On the Composition of Morphisms and Inverse Morphisms

In order to study composition of morphisms and inverse morphisms, we introduce starry transductions t which are, by definition, those verifying: e ∃ t(e) and for all words u, v, t(u) t(v) ⊂t(uv). We show that each starry transduction can be factored with two morphisms and two inverse morphisms. Then, we study some particular starry transductions. So, we prove that each rational substitution can be factored into a single morphism and two inverse morphisms and that each decreasing starry transduction can be factored into a single inverse morphism and two morphisms. That permits us to give an answer to a question posed in [5], by showing that for every rational language R, there exist morphisms h1, h2, h3, g1, g2, g3, such that R=h 3 −1 o h2 o h 1 −1 (a)=g3 o g 2 −1 o g1(a*b).

Context-sensitive string languages and recognizable picture languages

Abstract The theorem stating that the family of frontiers of recognizable tree languages is exactly the family of context-free languages (see J. Mezei and J. B. Wright, 1967,Inform. and Comput.11, 3–29), is a basic result in the theory of formal languages. In this article, we prove a similar result: the family of frontiers of recognizable picture languages is exactly the family of context-sensitive languages

Iterated Length-Preserving Rational Transductions

The purpose of this paper is the study of the smallest family of transductions containing length-preserving rational transductions and closed under union, composition and iteration. We give several characterizations of this class using restricted classes of length-preserving transductions, by showing the connections with “context-sensitive transductions” and transductions associated with recognizable picture languages. We also study the class obtained by only using length-preserving rational functions and we show the relations with “deterministic context-sensitive transductions”.

VERY SMALL FAMILIES OF ALGEBRAIC NONRATIONAL LANGUAGES

Publisher Summary One of the purposes of formal language theory is to propose some notions that enable deriving from local properties of words and global properties of languages. This point of view justifies all the studies on regularity in words of rational or algebraic languages. There are more conjectures than results in this area, and there is no known characterization of algebraic languages that are rational in terms of local properties on their words. The problem is closely related to the study of algebraic languages that are rational and to the questions about very small rational cones. The languages that are nearly rational deal with a characterization of rationality of algebraic languages. This chapter discusses some results obtained on the problem of minimal cones and necessary conditions are given by the iteration theorems for rational languages; among these, the most powerful one is the Ogden-like version of the star theorem. Moreover, the study of algebraic languages leads to defining different kinds of iterative pairs.

Commutative One-counter languages are regular

Abstract A new characterization of commutative regular languages is given. Using it, it is proved that every commutative one-counter language is regular.

Substitutions dans les EDT0L systèmes ultralinéaires

en We study EDT0L-systems with terminal alphabet in which all words derivated from the axiom countain a bounded number of occurrences of non-terminals. We define, also, unary operators associated with these systems.

Commutation with codes

The centralizer of a set of words X is the largest set of words C(X)commuting with X: XC(X) = C(X)X. It has been a long standing open question due to [J.H. Conway, Regular Algebra and Finite Machines, Chapman & Hall, London (1971).], whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see [M. Kunc, Proc. of ICALP 2004, Lecture Notes in Computer Science, Vol. 3142, Springer, Berlin, 2004, pp. 870-881.], we prove here that the situation is different for codes: the centralizer of any rational code is rational and if the code is finite, then the centralizer is finitely generated. This result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by [B. Ratoandromanana, RAIRO Inform. Theor. 23(4) (1989) 425-444.]--many of the techniques we use in this paper follow her ideas. We also give in this paper an elementary proof for the prefix case.

On computational power of weighted finite automata

Weighted Finite Automata are automata with multiplicities used to compute real functions by reading infinite words. We study what kind of functions can be computed by level automata, a particular subclass of WFA. Several results concerning the continuity and the smoothness of these functions are shown. In particular, the only smooth functions that can be obtained are the polynomials. This allows to decide whether a function computed by a level automaton is smooth or not.