Commutative positive varieties of languages

We study the commutative positive varieties of languages closed under various operations: shuffle, renaming and product over one-letter alphabets

A topological approach to transductions

AbstractThis paper is a contribution to the mathematical foundations of the theory of automata. We give a topological characterization of the transductions τ from a monoid M into a monoid N, such that if R is a recognizable subset of N,τ-1(R) is a recognizable subset of M. We impose two conditions on the monoids, which are fullfilled in all cases of practical interest: the monoids must be residually finite and, for every positive integer n, must have only finitely many congruences of index n. Our solution proceeds in two steps. First we show that such a monoid, equipped with the so-called Hall distance, is a metric space whose completion is compact. Next we prove that τ can be lifted to a map τ^ from M into the set of compact subsets of the completion of N. This latter set, equipped with the Hausdorff metric, is again a compact monoid. Finally, our main result states that τ-1 preserves recognizable sets if and only if τ^ is continuous

Advances and applications of automata on words and trees : abstracts collection

From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Advances and applications of automata on words and trees : executive summary

Seminar: 10501 - Advances and Applications of Automata on Words and Trees. The aim of the seminar was to discuss and systematize the recent fast progress in automata theory and to identify important directions for future research. For this, the seminar brought together more than 40 researchers from automata theory and related fields of applications. We had 19 talks of 30 minutes and 5 one-hour lectures leaving ample room for discussions. In the following we describe the topics in more detail

Topologie pp-adique sur les mots

This is a survey article on the combinatorial aspects of the p-adic metric and p-adic topology on words. We give several equivalent definitions of these notions, illustrated by several examples and properties. After giving a detailed description of the open sets, we prove that the p-adic metric is uniformly equivalent with a metric based on the binomial coefficients defined on words. We also give two examples of converging sequences for the p-adic topology. The first example consists of the sequence of the pn powers of a given word, that converges to the empty word. The second one consists of the sequence of prefixes of the Prouhet-Thue-Morse word: for each prime number p, on can extract from this sequence a subsequence converging to the empty word in the p-adic topology. Most of the proofs are omitted, apart from the very short ones

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

We present an extension of Eilenberg's variety theorem, a well-known result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids.The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat Politècnica de València

Languages associated with saturated formations of groups

In a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The first author acknowledges support from MEC. The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat Politècnica de València

Upper set monoids and length preserving morphisms

Length preserving morphisms and inverse of substitutions are two well-studied operations on regular languages. Their connection with varieties generated by power monoids was established independently by Reutenauer and Straubing in 1979. More recently, an ordered version of this theory was proposed by Polák and by the authors. In this paper, we present an improved version of these results and obtain the following consequences. Given a variety of finite ordered monoids V, let P ¿V be the variety of finite ordered monoids generated by the upper set monoids of members of V. Then P ¿(P ¿V)=P ¿V. This contrasts with the known results for the unordered case: the operator PV corresponding to power monoids satisfies P 3V=P 4V, but the varieties V, PV, P 2V and P 3V can be distinct. © 2011 Elsevier B.V.Work supported by the integrated action Picasso 19245ZC and by the AuthoMathA Programme of the European Science Foundation. The first author was supported by the project Tecnicas de Inferencia Gramatical y aplicacion al procesamiento de biosecuencias (TIN2007-60769) supported by the Spanish Ministry of Education and Sciences. The second author was supported by the project ANR 2010 BLAN 0202 02 FREC.Cano Gómez, A.; Pin, J. (2012). Upper set monoids and length preserving morphisms. Journal of Pure and Applied Algebra. 216(5):1178-1183. https://doi.org/10.1016/j.jpaa.2011.10.022S11781183216

Newton series, coinductively

We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shue, (iii) the inltration, and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a nal coalgebra, we use coinduction to prove that a classical operator from dierence calculus in mathematics: the Newton transform, generalises (from innite sequences) to weighted lan- guages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into an inltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them

Newton series, coinductively: a comparative study of composition

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