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Constructive Dynamic Logic of Relation Changers

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Dynamic Logic. New Trends and Applications (DaLi 2020)

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Abstract

This paper proposes an intuitionistic generalization of van Benthem and Liu’s dynamic logic of relation changers, where relation changers are dynamic operators which rewrite each agent’s accessibility relation. We employ Nishimura’s Kripke semantics for a constructive propositional dynamic logic to define the semantics of relation changers. A sound and complete axiomatization of the constructive dynamic logic of relation changers is provided. Moreover, we follow Hatano et al.’s approach to provide a different semantics for dynamic logic of relation changers, where relation changers are regarded as bounded morphisms. This alternative semantics leads us to a semantic completeness proof of the axiomatization for the original semantics, which does not require a reduction strategy based on recursion axioms.

We would like to thank anonymous reviewers for their helpful comments and suggestions on our manuscript. Both authors were partially supported by JSPS Core-to-Core Program (A. Advanced Research Networks). The work of the second author was also partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Grant Number 19K12113 and JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 17H02258.

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Notes

  1. 1.

    As a part of the study to show the Craig interpolation theorem for \(\mathbf {PDL}\), Leivant mentioned a proof system of “simplified” constructive \(\mathbf {PDL}\) which is based on logical connectives \(\rightarrow \), \(\lnot \) and \([\alpha ]\) and program constructors of \(\mathbf {PDL}\). We owe this point to Malvin Gattinger.

  2. 2.

    We may follow [28] to require that R satisfies \(\leqslant ; R ; \leqslant \subseteq R\), which is equivalent to \(\leqslant ; R ; \leqslant = R\). It is noted that this is a stronger condition than Nishimura [18]’s one. But, this is not a crucial difference.

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Authors and Affiliations

  1. Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, Japan

    Ryo Hatano

  2. Hokkaido University, Nishi 7 Chome, Kita 10 Jo, Kita-ku, Sapporo, Hokkaido, Japan

    Katsuhiko Sano

Authors
  1. Ryo Hatano
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  2. Katsuhiko Sano
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Correspondence to Ryo Hatano .

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Editors and Affiliations

  1. Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Manuel A. Martins

  2. Institute of Computer Science, Czech Academy of Sciences, Praha 8, Czech Republic

    Igor Sedlár

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Hatano, R., Sano, K. (2020). Constructive Dynamic Logic of Relation Changers. In: Martins, M.A., Sedlár, I. (eds) Dynamic Logic. New Trends and Applications. DaLi 2020. Lecture Notes in Computer Science(), vol 12569. Springer, Cham. https://doi.org/10.1007/978-3-030-65840-3_9

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  • DOI: https://doi.org/10.1007/978-3-030-65840-3_9

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