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A Note on Strictly Positive Logics and Word Rewriting Systems

  • Lev Beklemishev
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 15)

Abstract

We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable finitely axiomatizable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive logics generalizing derivations in word rewriting systems. We also make some observations and formulate open questions related to the theory of modal companions of superintuitionistic logics that was initiated by L.L. Maksimova and V.V. Rybakov.

Keywords

Strictly positive logics Semi-thue systems Deep inference 

Notes

Acknowledgements

We thank Daniyar Shamkanov and the anonymous referees for spotting some errors in the previous version of the paper, and Valentin Shehtman, Michael Zakharyaschev and Stas Kikot for pointing out connections with the previous work.

The investigation presented in this article was supported by Russian Foundation for Basic Research project No. 15–01–09218a and by the Presidential council for support of leading scientific schools.

References

  1. Baader, F. (2003). Resricted role-value-maps in a description logic with existential restrictions and terminological cycles. In D. Calvanese, G. De Giacomo, E. Franconi (Ed.), Proceeding of the 2003 International Workshop on Description Logics (DL 2003), Rome, Italy, September 5–7, 2003, Vol. 81, CEUR Workshop Proceedings, CEUR–WS.org.Google Scholar
  2. Beklemishev, L. D. (2012). Calibrating provability logic: from modal logic to reflection calculus. In T. Bolander, T. Braüner, S. Ghilardi, & L. Moss (Eds.), Advances in Modal Logic (Vol. 9, pp. 89–94). London: College Publications.Google Scholar
  3. Beklemishev, L. D. (2014). Positive provability logic for uniform reflection principles. Annals of Pure and Applied Logic, 165(1), 82–105.CrossRefGoogle Scholar
  4. Blok, W.J. (1976). Varieties of interior algebras, PhD thesis, University of Amsterdam.Google Scholar
  5. Chagrov, A.V. & Shehtman, V.B. (1995). Algorithmic aspects of propositional tense logics, In Lecture Notes in Computer Science (Vol. 933, pp. 442–455).Google Scholar
  6. Chagrov, A. V., & Zakharyaschev, M. (1992). Modal companions of intermediate propositional logics. Studia Logica, 51(1), 49–82.CrossRefGoogle Scholar
  7. Dashkov, E. V. (2012). O positivnom fragmente polimodalnoy logiki dokazuemosti GLP, Matematicheskie Zametki 91(3): 331–346. [Translation: “On the positive fragment of the polymodal provability logic GLP”. Mathematical Notes, 91(3), 318–333.Google Scholar
  8. Davis, M., Sigal, R., & Weyuker, E.J. (1994). Computability, complexity, and languages: fundamentals of theoretical computer science (2nd ed.). Academic Press.Google Scholar
  9. Esakia, L. L. (1976). On modal companions of superintuitionistic logics. In VII Soviet Symposium on Logic. Kiev.Google Scholar
  10. Kikot, S., Kurucz, A., Tanaka, Y., Wolter, F. & Zakharyaschev, M. (2016). On the completeness of EL-equiations: First results. In 11th International Conference on Advances in Modal Logic, Short Papers (Budapest, 30 August - 2 September, 2016), pp. 82–87.Google Scholar
  11. Kurucz, A., Wolter, F. & Zakharyaschev, M. (2010). Islands of tractability for relational constraints: towards dichotomy results for the description logic EL. In Advances in Modal Logic Vol. 8 pp. 271–291. College Publications, London.Google Scholar
  12. Kurucz, A., Tanaka, Y., Wolter, F., & Zakharyaschev, M. (2011). Conservativity of Boolean algebras with operators over semilattices with operators. Proceedings of TACL, 49–52.Google Scholar
  13. Maksimova, L.L. & Rybakov, V.V. (1974). A lattice of normal modal logics. Algebra i Logika, 13(2): 188–216.Google Scholar
  14. Shehtman, V.B. (1982). Undecidable propositional calculi. InProblems of cybernetics. Non-classical logics and their applications, Moscow, pp. 74–116. In Russian.Google Scholar
  15. Svyatlovsky, M. (2014) Positivnye fragmenty modalnykh logik, Manuscript, [Positive fragments of modal logics]. http://www.mi.ras.ru/~bekl/Papers/work_2.pdf.
  16. Wolter, F. & Zakharyaschev, M. (2014). On the Blok–Esakia theorem, in G. Bezhanishvili (ed.) Leo Esakia on duality in modal and intuitionistic logics, Vol. 4 of Outstanding Contributions to Logic, Springer, pp. 99–118.Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRAS, MoscowRussia
  2. 2.Moscow M.V. Lomonosov State UniversityMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia

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