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Morrill and Valentin in the paper "Computational coverage of TLG:Nonlinearity" considered an extension of the Lambek calculus enriched by aso-called "exponential" modality. This modality behaves in the "relevant"style, that is, it allows contraction and permutation, but not weakening.Morrill and Valentin stated an open problem whether this system is decidable.Here we show its undecidability. Our result remains valid if we consider thefragment where all division operations have one direction. We also show thatthe derivability problem in a restricted case, where the modality can beapplied only to variables (primitive types), is decidable and belongs to the NPclass.

Undecidability of the Lambek calculus with a relevant modality