Undecidability of the Lambek calculus with subexponential and bracket modalities
This work resolves a key theoretical question in computational linguistics by establishing undecidability for an extended logical formalism, impacting researchers in formal grammar and logic.
The paper proves that the derivability problem in the Lambek calculus extended with subexponential and bracket modalities is undecidable, addressing a limitation in modeling natural language syntax beyond context-free phenomena. It also shows that restricted fragments of this calculus are in the NP complexity class.
The Lambek calculus is a well-known logical formalism for modelling natural language syntax. The original calculus covered a substantial number of intricate natural language phenomena, but only those restricted to the context-free setting. In order to address more subtle linguistic issues, the Lambek calculus has been extended in various ways. In particular, Morrill and Valentin (2015) introduce an extension with so-called exponential and bracket modalities. Their extension is based on a non-standard contraction rule for the exponential that interacts with the bracket structure in an intricate way. The standard contraction rule is not admissible in this calculus. In this paper we prove undecidability of the derivability problem in their calculus. We also investigate restricted decidable fragments considered by Morrill and Valentin and we show that these fragments belong to the NP class.