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Expressive Power and Succinctness of the Positive Calculus of Relations

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Relational and Algebraic Methods in Computer Science (RAMiCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12062))

Abstract

In this paper, we study the expressive power and succinctness of the positive calculus of relations. We show that (1) the calculus has the same expressive power as that of three-variable existential positive (first-order) logic in terms of binary relations, and (2) the calculus is exponentially less succinct than three-variable existential positive logic, namely, there is no polynomial-size translation from three-variable existential positive logic to the calculus, whereas there is a linear-size translation in the converse direction. Additionally, we give a more fine-grained expressive power equivalence between the (full) calculus of relations and three-variable first-order logic in terms of the quantifier alternation hierarchy. It remains open whether the calculus of relations is also exponentially less succinct than three-variable first-order logic.

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Notes

  1. 1.

    Here, z and u in \(\mathrm {TC}_{z,u}(\psi )\) are viewed as bound variables (i.e., \(\mathbf {FV}([\mathrm {TC}_{z,u}(\psi )](x,y))\) is defined by \(\mathbf {FV}([\mathrm {TC}_{z,u}(\psi )](x,y)) :=(\mathbf {FV}(\psi ) \setminus \{z,u\}) \cup \{x,y\}\)). See also [6, Sec. 9].

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Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Yoshiki Nakamura

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  1. Yoshiki Nakamura
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Correspondence to Yoshiki Nakamura .

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Editors and Affiliations

  1. Laboratoire d’informatique (LIX), École Polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Faculty of Mathematics, Chapman University, Orange, CA, USA

    Peter Jipsen

  3. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Michael Winter

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Nakamura, Y. (2020). Expressive Power and Succinctness of the Positive Calculus of Relations. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2020. Lecture Notes in Computer Science(), vol 12062. Springer, Cham. https://doi.org/10.1007/978-3-030-43520-2_13

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  • DOI: https://doi.org/10.1007/978-3-030-43520-2_13

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  • Online ISBN: 978-3-030-43520-2

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