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On the Expressive Power of GF(2)-Grammars

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SOFSEM 2019: Theory and Practice of Computer Science (SOFSEM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11376))

Abstract

GF(2)-grammars, recently introduced by Bakinova et al. (“Formal languages over GF(2)”, LATA 2018), are a variant of ordinary context-free grammars, in which the disjunction is replaced by exclusive OR, whereas the classical concatenation is replaced by a new operation called GF(2)-concatenation: \(K \odot L\) is the set of all strings with an odd number of partitions into a concatenation of a string in K and a string in L. This paper establishes several results on the family of languages defined by these grammars. Over the unary alphabet, GF(2)-grammars define exactly the 2-automatic sets. No language of the form \(\{a^n b^{f(n)} \,\mid \, n \geqslant 1\}\), with uniformly superlinear f, can be described by any GF(2)-grammar. The family is not closed under union, intersection, classical concatenation and Kleene star, non-erasing homomorphisms. On the other hand, this family is closed under injective nondeterministic finite transductions, and contains a hardest language under reductions by homomorphisms.

Supported by Russian Science Foundation, project 18-11-00100.

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Authors and Affiliations

  1. St. Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg, 199034, Russia

    Vladislav Makarov & Alexander Okhotin

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  1. Vladislav Makarov
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  2. Alexander Okhotin
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Correspondence to Vladislav Makarov .

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Editors and Affiliations

  1. University of Genoa, Genoa, Italy

    Barbara Catania

  2. Comenius University, Bratislava, Slovakia

    Rastislav Královič

  3. Poznań University of Technology, Poznań, Poland

    Jerzy Nawrocki

  4. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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Makarov, V., Okhotin, A. (2019). On the Expressive Power of GF(2)-Grammars. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_25

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  • DOI: https://doi.org/10.1007/978-3-030-10801-4_25

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10800-7

  • Online ISBN: 978-3-030-10801-4

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