A Note on Strictly Positive Logics and Word Rewriting Systems

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


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.


Strictly positive logics Semi-thue systems Deep inference 



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.


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© 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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