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A polynomial time algorithm for the Lambek calculus with brackets of bounded order
Pentus (2010) gave a polynomial-time algorithm for determ- ining provability of bounded depth formulas in the Lambek calculus with empty antecedents allowed.
Undecidability of the Lambek calculus with subexponential 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.
Reconciling Lambek's restriction, cut-elimination, and substitution in the presence of exponential modalities
The system with cut elimination and substitution and without Lambek's restriction is folklore and known to be undecidable.
Undecidability of the Lambek calculus with a relevant modality
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.
