From Two-Way Transducers to Regular Function Expressions

Baudru, Nicolas; Reynier, Pierre-Alain

doi:10.1142/s0129054120410087

Cited by 5 publications

(12 citation statements)

References 17 publications

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“…The fundamental correspondence between machines and logic at the language level has been generalized to transformations by Engelfreit and Hoogeboom [15], where regular transformations are defined by two way transducers (2DFTs) as well as by the MSO transductions of Courcelle [9]. A generalization of Kleene's theorem to transformations can be found in [3], [4] and [11].…”

Section: Introductionmentioning

confidence: 99%

“…The proof technique in [11] is completely different from [3], and, being algebraic, allows a uniform treatment of the result for transductions over finite and infinite words. Subsequently, an alternative proof for the result of [3] appeared in [4].…”

Section: Introductionmentioning

confidence: 99%

“…The counter part of regular expressions for aperiodic languages are star-free expressions, which are obtained by using the complementation operation instead of Kleene-star. The generalization of this result to transformations has been an open problem, as mentioned in [3], [4] and [11]. One of the main challenges in generalizing this result to transformations is the difficulty in finding an analogue for the complementation operation on sets in the setting of transformations.…”

Section: Introductionmentioning

confidence: 99%

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SD-Regular Transducer Expressions for Aperiodic Transformations

Dartois¹,

Gastin²,

Krishna³

2021

Preprint

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FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the long standing open problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SD-regular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Schützenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by structural induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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SD-Regular Transducer Expressions for Aperiodic Transformations

Dartois¹,

Gastin²,

Krishna³

2021

Preprint

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“…This formalism is based on some operators defining basic transformations that are composed to define the target transformation. An alternative construction of these expressions starting directly from two-way unambiguous transducers has been proposed in [2]. These expressions have also been extended to run on infinite words [7].…”

Section: Introductionmentioning

confidence: 99%

“…III, Sec. 5]) 2. Here and in the following, m1m2=m is the sum over all pairs (m1, m2) ∈ M 2 such that m1m2 = m.…”

mentioning

confidence: 99%

Weighted Automata and Expressions over Pre-Rational Monoids

Baudru¹,

Dando²,

Lhote³

et al. 2021

Preprint

Self Cite

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The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature. Lifting this result to a weighted setting has been widely studied. Moreover, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata.

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Modular Descriptions of Regular Functions

Gastin

2019

Algebraic Informatics

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We discuss various formalisms to describe string-to-string transformations. Many are based on automata and can be seen as operational descriptions, allowing direct implementations when the input scanner is deterministic. Alternatively, one may use more human friendly descriptions based on some simple basic transformations (e.g., copy, duplicate, erase, reverse) and various combinators such as function composition or extensions of regular operations. ACM Subject Classification Theory of computation → Transducers

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From Two-Way Transducers to Regular Function Expressions

Cited by 5 publications

References 17 publications

SD-Regular Transducer Expressions for Aperiodic Transformations

SD-Regular Transducer Expressions for Aperiodic Transformations

Weighted Automata and Expressions over Pre-Rational Monoids

Modular Descriptions of Regular Functions

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