Regular transducer expressions for regular transformations

Dave, Vrunda; Gastin, Paul; Krishna, Shankara Narayanan

doi:10.1016/j.ic.2020.104655

Cited by 2 publications

(8 citation statements)

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“…The main contribution of [3] was to provide a set of combinators, analogous to the operators used in regular expressions, which help in forming combinator expressions computing the output of the ACRA over finite words. The result of [3] was generalized to infinite words in [5]. The proof technique in [5] is completely different from [3], and, being algebraic, allows a uniform treatment of the result for transductions over finite and infinite words.…”

Section: Supported By Irl Relaxmentioning

confidence: 99%

“…The result of [3] was generalized to infinite words in [5]. The proof technique in [5] 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: Supported By Irl Relaxmentioning

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 [5]. 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: Supported By Irl Relaxmentioning

confidence: 99%

“…A natural operation on functions is that of composition. The composition operation can be used in place of the chained-sum operator of [3], and also in place of the unambiguous 2-chained iteration of [5], preserving expressiveness. In yet another recent paper, [12] proposes simple functions like copy, duplicate and reverse along with function composition to capture regular word transductions.…”

Section: As An Example Consider the Sdrtesmentioning

confidence: 99%

“…The fundamental correspondence between machines and logic at the language level has been generalized to transformations by Engelfreit and Hoogeboom [1], where regular transformations are defined by two-way transducers (2DFTs) as well as by the MSO transductions of Courcelle [2]. A generalization of Kleene's theorem to transformations can be found in [3], [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

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SD-Regular Transducer Expressions for Aperiodic Transformations

Dartois

Gastin

Krishna

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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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 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 SDregular 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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