Lexicographic Decomposition of k-Valued Transducers

Sakarovitch, Jacques; Souza, Rodrigo de

doi:10.1007/s00224-009-9206-6

Cited by 23 publications

(17 citation statements)

References 19 publications

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“…The multi-sequential ones are the union of all ksequential transducers for all k [16,7]. Finally, the k-valued transducers are the transducers for which any input word has at most k output words [14,19], and the finite-valued ones are all the k-valued transducers for all k [22,23,18]. All these classes, according to the given references, are decidable in PTIME.…”

Section: A Pattern Logic For Transducersmentioning

confidence: 99%

A Pattern Logic for Automata with Outputs

Filiot

Mazzocchi

Raskin

2018

Developments in Language Theory

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We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric, and we provide sufficient conditions on the predicates under which the model-checking problem is decidable. We then consider three particular automata models (finite automata, transducers and automata weighted by integers -sum-automata -) and instantiate the generic logic for each of them. We give tight complexity results for the three logics and the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. Consequently to our complexity results, we directly obtain that these classical properties can be decided in PTIME. IntroductionMotivations An important aspect of automata theory is the definition of automata subclasses with particular properties, of algorithmic interest for instance. As an example, the inclusion problem for non-deterministic finite automata is PSPACE-C but becomes PTIME if the automata are k-ambiguous for a fixed k [21].By automata theory, we mean automata in the general sense of finite state machines processing finite words. This includes what we call automata with outputs, which may also produce output values in a fixed monoid M = (D, ⊕, 0). In such an automaton, the transitions are extended with an (output) value in D, and the value of an accepting path is the sum (for ⊕) of all the values occurring along its transitions. Automata over finite words in Λ * and with outputs in M define subsets of Λ * × D as follows: to any input word w ∈ Λ * , we associate the set of values of all the accepting paths on w. For example, transducers are automata with outputs in a free monoid: they process input words and produce output words and therefore define binary relations of finite words [15].The many decidability properties of finite automata do not carry over to transducers, and many restrictions have been defined in the literature to recover decidability, or just to define subclasses relevant to particular applications. The inclusion problem for transducer is undecidable [13], but decidable for finitevalued transducers [23]. Another well-known subclass is that of the determinisable transducers [5], defining sequential functions of words. Finite-valuedness and determinisability are two properties decidable in PTIME, i.e., it is decidable in PTIME, given a transducer, whether it is finite-valued (resp. determinisable). As a second example of automata with outputs, we also consider sum-automata, i.e. automata with outputs in (Z, +, 0), which defines relations from words to Z. Properties such as functionality, determinisability, and k-valuedness (for a fixed k) are decidable in PTIME for sum-automata [11,10].In our e...

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Section: A Pattern Logic For Transducersmentioning

confidence: 99%

A Pattern Logic for Automata with Outputs

Filiot

Mazzocchi

Raskin

2018

Developments in Language Theory

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“…As mentioned in Theorem 12, finite-valued bottom-up tree transducers can be effectively decomposed into a finite number of single-valued ones of double-exponential order of the size of the original transducers. Whereas, in the word case, k-valued (word) transducers can be effectively decomposed into k single-valued (unambiguous) ones [38], [39] of single-exponential size [40]. Can k-valued tree transducers decomposed into k single-valued ones of single-exponential size?…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Query Rewriting for Nondeterministic Tree Transducers

Miyahara

Hashimoto

Seki

2016

IEICE Trans. Inf. & Syst.

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SUMMARYWe consider the problem of deciding whether a query can be rewritten by a nondeterministic view. It is known that rewriting is decidable if views are given by single-valued non-copying devices such as compositions of single-valued extended linear top-down tree transducers with regular look-ahead, and queries are given by deterministic MSO tree transducers. In this paper, we extend the result to the case that views are given by nondeterministic devices that are not always single-valued. We define two variants of rewriting: universal preservation and existential preservation, and discuss the decidability of them.

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“…This result was already known for finite-state transducers [Eil74, Sch76, EM65] and here we extend it to VPTs with rather simple constructions based on the concept of look-aheads. This characterization of functional finite-state transducers has been generalized to k-valued and k-ambiguous finite-state transducers [Web93] and with a better upper-bound [SdS10] based on lexicographic decomposition of transducers.…”

Section: Chaptermentioning

confidence: 99%

“…The equivalence of functional NFT transduction has been extended to k-valued NFT first in [Web93] with a double exponential procedure and then recently in [SdS10] with a single exponential bound. For VPT, we proved that the equivalence and inclusion problem of functional VPT is decidable in PTime (See Theorem 5.4.7).…”

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confidence: 99%