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Star-Freeness, First-Order Definability and Aperiodicity of Structured Context-Free Languages

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Theoretical Aspects of Computing – ICTAC 2020 (ICTAC 2020)

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Abstract

A classic result in formal language theory is the equivalence among aperiodic finite automata, star-free regular expressions, and first-order logic on words. Extending these results to structured subclasses of context-free languages, such as tree languages, did not work as smoothly: there are star-free tree languages that are counting. We argue that investigating the same properties within the family of operator precedence languages (OPLs) by going back to string languages rather than tree languages may lead to equivalences that perfectly match those on regular languages. We define operator precedence expressions; we show that they define exactly the class of OPLs and that, when restricted to the star-free subclass, coincide with first-order definable OPLs and are aperiodic.

Since operator precedence languages strictly include other classes of structured languages such as visibly pushdown languages, the same results given in this paper hold as trivial corollary for that family too.

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Notes

  1. 1.

    This is a minor deviation from [12], where it was assumed that grammars have only one axiom.

  2. 2.

    See Fig. 3.

  3. 3.

    Note that the G produced by the construction is BD if so are \(G_1\) and \(G_2\), but it could be not necessarily BDR; however, if a BDR OPG has a counting derivation, any equivalent BD grammar has also a counting derivation.

  4. 4.

    \(\Xi := \{ (x,y) \mid A \rightarrow x B y \in P \} \,\cup \, \{ z \mid A \rightarrow z \in P \} \).

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Acknowledgments

We are grateful to the reviewers for their careful reading and suggestions.

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Authors and Affiliations

  1. Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133, Milano, Italy

    Dino Mandrioli, Matteo Pradella & Stefano Crespi Reghizzi

  2. IEIIT, Consiglio Nazionale delle Ricerche, via Ponzio 34/5, 20133, Milano, Italy

    Matteo Pradella & Stefano Crespi Reghizzi

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  1. Dino Mandrioli
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Correspondence to Matteo Pradella .

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Editors and Affiliations

  1. Western Norway University of Applied Sciences, Bergen, Norway

    Violet Ka I Pun

  2. Western Norway University of Applied Sciences, Bergen, Norway

    Volker Stolz

  3. Federal University of Sao Carlos, Sao Carlos, Brazil

    Adenilso Simao

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Mandrioli, D., Pradella, M., Crespi Reghizzi, S. (2020). Star-Freeness, First-Order Definability and Aperiodicity of Structured Context-Free Languages. In: Pun, V.K.I., Stolz, V., Simao, A. (eds) Theoretical Aspects of Computing – ICTAC 2020. ICTAC 2020. Lecture Notes in Computer Science(), vol 12545. Springer, Cham. https://doi.org/10.1007/978-3-030-64276-1_9

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