Undecidability of the Lambek Calculus with a Relevant Modality

Morrill and Valent́ın 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 Valent́ın 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. 1 The Lambek Calculus Extended by a Relevant Modality We will introduce !L—an extension of the Lambek calculus [9] with one modality, denoted by !. We consider the version of the Lambek calculus L that allows empty left-hand sides of sequents (introduced in [10]). Formulae of !L are built from a set of variables (Var = {p, q, r, . . .}) using two binary connectives, / (right division) and \ (left division), and additionally one unary connective, !. Capital Latin letters denote formulae; capital Greek letters denote finite (possibly empty) linearly ordered sequences of formulae. Following the linguistic tradition, formulae of the Lambek calculus (and its extensions) are also called types. In this terminology, variables are called primitive types.