A Circuit Complexity Approach to Transductions

Cadilhac, Michaël; Krebs, Andreas; Ludwig, Michael; Paperman, Charles

doi:10.1007/978-3-662-48057-1_11

Cited by 6 publications

(6 citation statements)

References 20 publications

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“…In [18], the authors use the term "preserving" in the more general context of functions from monoids to monoids. In our study, we focus on word to word functions, in which the natural topological context provides a solid basis for the use of "continuous," as used in [16,5].…”

Section: Preliminariesmentioning

confidence: 99%

“…This pervasive interaction naturally suggests lifting this study to the functional level, hence to rational functions. This was started in [5], where it was shown that a subsequential (i.e., inputdeterministic) 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.…”

Section: Introductionmentioning

confidence: 99%

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Continuity of Functional Transducers: A Profinite Study of Rational Functions

Cadilhac,

Carton,

Paperman

2018

Preprint

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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. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages.Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Continuity of Functional Transducers: A Profinite Study of Rational Functions

Cadilhac,

Carton,

Paperman

2018

Preprint

Self Cite

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“…For word functions, a recent work (Cadilhac et al 2015) shows that it is decidable whether a deterministic rational transduction is definable in the circuit class AC 0 (resp. ACC 0 ), also using an algebraic approach.…”

Section: Related Workmentioning

confidence: 99%

First-order definability of rational transductions

Filiot

Gauwin

Lhote

2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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The algebraic theory of rational languages has provided powerful decidability results. Among them, one of the most fundamental is the definability of a rational language in the class of aperiodic languages, i.e., languages recognized by finite automata whose transition relation defines an aperiodic congruence. An important corollary of this result is the first-order definability of monadic second-order formulas over finite words. Our goal is to extend these results to rational transductions, i.e. word functions realized by finite transducers. We take an algebraic approach and consider definability problems of rational transductions in a given variety of congruences (or monoids). The strength of the algebraic theory of rational languages relies on the existence of a congruence canonically attached to every language, the syntactic congruence. In a similar spirit, Reutenauer and Schützenberger have defined a canonical device for rational transductions, that we extend to establish our main contribution: an effective characterization of V-transductions, i.e. rational transductions realizable by transducers whose transition relation defines a congruence in a (decidable) variety V. In particular, it provides an algorithm to decide the definability of a rational transduction by an aperiodic finite transducer. Using those results, we show that the FO-definability of a rational transduction is decidable, where FO-definable means definable in a first-order restriction of logical transducersà la Courcelle.

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“…In contrast to considering restrictions of the transition congruence of the underlying automata of transducers, a different way of defining subclasses of rational transductions, taking into account the outputs, has been proposed in [CKLP15] and [CCP17]. It is based on the natural notion of continuity, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…In [CKLP15], the authors were able to effectively characterize sequential transductions definable by AC 0 circuits. In [CCP17] the authors show how to decide if a transduction is continuous with respect to many usual varieties and they also compare the two notions of continuity and realizability.…”

Section: Introductionmentioning

confidence: 99%

Logical and Algebraic Characterizations of Rational Transductions

Filiot,

Gauwin,

Lhote

2017

Preprint

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Rational word languages can be defined by several equivalent means: finite state automata, rational expressions, finite congruences, or monadic second-order (MSO) logic. The robust subclass of aperiodic languages is defined by: counter-free automata, star-free expressions, aperiodic (finite) congruences, or first-order (FO) logic. In particular, their algebraic characterization by aperiodic congruences allows to decide whether a regular language is aperiodic.We lift this decidability result to rational transductions, i.e. word-to-word functions defined by finite state transducers. In this context, logical and algebraic characterizations have also been proposed. Our main result is that one can decide if a rational transduction (given as a transducer) is in a given decidable congruence class. We also establish a transfer result from logic-algebra equivalences over languages to equivalences over transductions. As a consequence, it is decidable if a rational transduction is first-order definable, and we show that this problem is PSPACE-complete.

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A Circuit Complexity Approach to Transductions

Cited by 6 publications

References 20 publications

Continuity of Functional Transducers: A Profinite Study of Rational Functions

Continuity of Functional Transducers: A Profinite Study of Rational Functions

First-order definability of rational transductions

Logical and Algebraic Characterizations of Rational Transductions

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