SD-Regular Transducer Expressions for Aperiodic Transformations

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 semin...

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Published in:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 1 - 13
Main Authors: Dartois, Luc, Gastin, Paul, Krishna, Shankara Narayanan
Format: Conference Proceeding
Language:English
Published: IEEE 29.06.2021
Subjects:
Complexity theory
Computer science
Delays
Synchronization
Time complexity
Transducers
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Abstract 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 Schutzenberger 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 fc-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 Schutzenberger'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 fc-chained star for a 1-star.
AbstractList 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 Schutzenberger 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 fc-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 Schutzenberger'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 fc-chained star for a 1-star.
Author Dartois, Luc
Krishna, Shankara Narayanan
Gastin, Paul
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  givenname: Luc
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  email: luc.dartois@lacl.fr
  organization: Univ Paris Est Creteil,LACL,Creteil,France,F-94010
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  givenname: Paul
  surname: Gastin
  fullname: Gastin, Paul
  email: paul.gastin@lsv.fr
  organization: Université Paris-Saclay,ENS Paris-Saclay, CNRS, LMF,Gif-sur-Yvette,France,91190
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  givenname: Shankara Narayanan
  surname: Krishna
  fullname: Krishna, Shankara Narayanan
  email: krishnas@cse.iitb.ac.in
  organization: IIT Bombay,Dept. Computer Science and Engineering,India
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Snippet FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order...
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StartPage 1
SubjectTerms Complexity theory
Computer science
Delays
Synchronization
Time complexity
Transducers
Title SD-Regular Transducer Expressions for Aperiodic Transformations
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