# On Lambek’s Restriction in the Presence of Exponential Modalities

• Max Kanovich
• Stepan Kuznetsov
• Andre Scedrov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9537)

## Abstract

The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called “Lambek’s restriction,” that is, the antecedent of any provable sequent should be non-empty. In this paper we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek’s restriction. We present several versions of the Lambek calculus extended with exponential modalities and prove that those extensions are undecidable, even if we take only one of the two divisions provided by the Lambek calculus.

## Keywords

Lambek calculus Linear logic Exponential modalities Lambek’s restriction Undecidability

## Notes

### Acknowledgments

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 of Russia (grant NŠ 1423.2014.1). Max Kanovich’s research was partly supported by EPSRC (grant EP/K040049/1).

This research was initiated during the visit by Stepan Kuznetsov to the University of Pennsylvania, which was supported in part by that institution. Further work was done during the stay of Kanovich and Scedrov at the National Research University Higher School of Economics, which was supported in part by that institution. We would like to thank Sergei O. Kuznetsov and Ilya A. Makarov for hosting us.

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## Authors and Affiliations

• Max Kanovich
• 1
• 2
• 3
• Stepan Kuznetsov
• 4
• Andre Scedrov
• 5
• 6
1. 1.Queen Mary, University of LondonLondonUK
2. 2.University College LondonLondonUK
3. 3.National Research University Higher School of EconomicsMoscowRussia
4. 4.Steklov Mathematical Institute, RASMoscowRussia