Continuity and Rational Functions - Archive ouverte HAL Accéder directement au contenu
Communication Dans Un Congrès Année : 2017

Continuity and Rational Functions

Résumé

A word-to-word function is continuous for a class of languages V if its inverse maps V_languages to V. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. Previous algebraic studies of transducers have focused on the structure of the underlying input automaton, disregarding the output. We propose a comparison of the two algebraic approaches through two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses? 1 Introduction The algebraic theory of regular languages is tightly interwoven with fundamental questions about the computing power of Boolean circuits and logics. The most famous of these braids revolves around A, the class of aperiodic or counter-free languages. Not only is it expressed using the logic FO[<], but it can be seen as the basic building block of AC 0 , the class of languages recognized by circuit families of polynomial size and constant depth, this class being in turn expressed by the logic FO[arb] (see [18] for a lovely account). This pervasive interaction naturally prompts to lift this study to the functional level, hence to rational functions. This was started in [4], where it was shown that a subsequential (i.e., input-deterministic) transducer computes an AC 0 function iff it preserves the regular languages of AC 0 by inverse image. Buoyed by this clean, semantic characterization, we wish to further investigate this latter property for different classes: say that a function f : A * → B * is V_continuous, for a class of languages V, if for every language L ⊆ B * of V, the language f −1 (L) is also a language of V. Our main focus will be on deciding V_continuity for rational functions; before listing our main results, we emphasize two additional motivations. First, there has been some historical progression towards this goal. Noting, in [9], that inverse rational functions provide a uniform and compelling view of a wealth of natural operations on regular languages, Pin and Sakarovitch initiated in [10] a study of regular-continuous functions. It was already known at the time, by a result of Choffrut (see [3, * The first and third authors are partly funded by the DFG Emmy Noether program (KR 4042/2); the second author is funded by the DeLTA project (ANR-16-CE40-0007).
Fichier principal
Vignette du fichier
LIPIcs-ICALP-2017-115.pdf (547.35 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte
Loading...

Dates et versions

hal-01587614 , version 1 (14-09-2017)

Identifiants

Citer

Michaël Cadilhac, Olivier Carton, Charles Paperman. Continuity and Rational Functions. The 44th International Colloquium on Automata, Languages, and Programming ICALP 2017, 2017, Varsovie, Poland. pp.1 - 11514, ⟨10.4230/LIPIcs.ICALP.2017.115⟩. ⟨hal-01587614⟩
97 Consultations
44 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More