Undecidability of the Lambek calculus with a relevant modality

TL;DR

This paper proves the undecidability of a relevant modality-extended Lambek calculus, while identifying a decidable fragment where the modality applies only to variables, with implications for computational linguistics.

Contribution

It establishes the undecidability of the Lambek calculus with a relevant modality and identifies a decidable fragment with restricted modality application.

Findings

The full system is undecidable.

A fragment with modality only on variables is NP-complete.

The undecidability holds even with one-directional division operations.

Abstract

Morrill and Valentin 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 Valentin 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.