By considering graphs as logical structures, one...

Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1 Grammars : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2. Systems of equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.1 Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Resolution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.3 Linear systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.4 Parikh's theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

Abstract not found

We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series - the semilinear and the bounded series - both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language - the largest set commuting with a language -, of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago - asking whether or not the centralizer of any rational language is rational - in the case of periodic, binary, and ternary sets of words, as well as for rational c-codes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an c-code X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.

Abstract. We consider standard algorithms of finite graph theory, like for instance shortest path algorithms. We present two general methods to polynomially extend these algorithms to infinite graphs generated by deterministic graph grammars. 1

Abstract. We discuss a generalization of Parikh’s Theorem for contextfree languages to classes of many-sorted relational structures which are both definable in Monadic Second Order Logic and which are of bounded patch-width. Patch-width is a generalization of both tree-width and clique-width. This gives a powerful unifying tool to prove that certain classes of graphs are of unbounded width. For R. Parikh, at the occasion of his 70th birthday 1 Generalizing Parikh’s Theorem R. Parikh’s celebrated theorem, first proved in [Par66], counts the number of occurrences of letters in words of a context-free languages L over an alphabet of k letters. For a given word w, the numbers of these occurrences is denoted by a vector n(w) ∈ N k, and the theorem states Theorem 1 (Parikh 1966). For a context-free language L, the set Par(L) = {n(w) ∈ N k: w ∈ L} is semi-linear. A set X ⊆ Ns is linear in Ns iff there is vector ā ∈ Ns and a matrix M ∈ Ns×r such that X = Aā, M ¯ = { ¯b ∈ Ns: there is ū ∈ Nr with ¯b = ā + M · ū}. Singletons are linear sets with M = 0. If M ̸ = 0 the series is nontrivial. X ⊆ Ns is semi-linear in Ns iff X is a finite union of linear sets Ai ⊆ Ns. For s = 1 the semi-linear sets are exactly the ultimately periodic sets. The terminology is from [Par66], and has since become standard terminology in formal language theory. Several alternative proofs of Parikh’s Theorem have appeared since. D. Pilling [Pil73] put it into a more algebraic form, and more recently, L. Aceto, Z. Esik and A. Ingolfsdottir [AÉI02] showed that it depends only on a few equational properties of least pre-fixed points. B. Courcelle [Cou95]. has generalized Theorem 1 further to certain graph grammars, and relational structures which are