Undecidability of the Lambek Calculus with a Relevant Modality
Morrill and Valentín in the paper “Computational coverage of TLG: Nonlinearity” considered an extension of the Lambek calculus enriched by a so-called “exponential” modality. This modality behaves in the “relevant” style, that is, it allows contraction and permutation, but not weakening. Morrill and Valentín stated an open problem whether this system is decidable. Here we show its undecidability. Our result remains valid if we consider the fragment where all division operations have one direction. We also show that the derivability problem in a restricted case, where the modality can be applied only to variables (primitive types), is decidable and belongs to the NP class.
Stepan Kuznetsov’s research was supported by the Russian Foundation for Basic Research (grants 15-01-09218-a and 14-01-00127-a) and by the Presidential Council for Support of Leading Scientific Schools (grant NŠ-9091.2016.1). Max Kanovich’s research was partially supported by EPSRC. Andre Scedrov’s research was partially supported by ONR.
This research was performed in part during visits of Stepan Kuznetsov and Max Kanovich to the University of Pennsylvania. We greatly appreciate support of the Mathematics Department of the University. A part of the work was also done during the stay of Andre Scedrov at the National Research University Higher School of Economics. We would like to thank S.O. Kuznetsov and I.A. Makarov for hosting there.
The paper was prepared in part within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and was partially supported within the framework of a subsidy by the Russian Academic Excellence Project ‘5–100’.
We are indepted to the participants of the research seminars “Logical Problems in Computer Science” and “Algorithmic Problems in Algebra and Logic” at Moscow (Lomonosov) University, in particular, S.I. Adian, L.D. Beklemishev, V.N. Krupski, I.I. Osipov, F.N. Pakhomov, M.R. Pentus, D.S. Shamkanov, I.B. Shapirovsky, V.B. Shehtman, A.A. Sorokin, T.L. Yavorskaya, and others for fruitful discussions and suggestions that allowed us to improve our presentation significantly.
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