Tight Complexity for Modal Extensions of Lec with K, 4, T

17 pages•Published: October 23, 2026

Abstract

Substructural logics are defined by dropping some of the structural properties of intuitionistic logic such as commutativity (exchange), contraction, or weakening. Substructural logics with contraction and without weakening can have very high complexity: in particular, FLec, the Full Lambek calculus with exchange and contraction, is complete for the Ackermannian complexity class. Here, `Full' refers to the presence of additive (lattice) connectives; these are used to simulate zero tests in the lower bounding argument. Indeed, Schmitz (2016) showed that its multiplicative fragment Lec has much lower complexity: provability and deduction are 2EXPTIME-complete. In this paper, we investigate the effect of adding the standard modal rules K, 4, and T to this multiplicative setting.
We show that provability in Lec+K and Lec+4 remains in 2EXPTIME by extending Schmitz’s expansive BVASS argument. We also obtain Ackermannian lower bounds for Lec+K deducibility, by constructing a reduction from an expansive counter machine; the main point here is that the K rule can be used to simulate zero tests. Some of these results hold even in the presence of the T rule. Combined with known results, this yields a tight classification: Lec+K(+T) and Lec+4 provability is 2EXPTIME-complete, Lec+K(+T) deducibility is Ackermannian-complete.

Keyphrases: branching vector addition systems, computational complexity, infinite state systems, modal logic, multiplicative substructural logic, reachability

In: Erika Ábrahám, Nikolaj Bjørner and Michael Rawson (editors). Proceedings of the 26th Conference on Logic for Programming, Artificial Intelligence, and Reasoning, vol 112, pages 364-380.

Links:	https://easychair.org/publications/paper/hK2D
	https://doi.org/10.29007/n4zl

BibTeX entry

@inproceedings{LPAR-26:Tight_Complexity_Modal_Extensions,
  author    = {Wouter Leferink and Revantha Ramanayake},
  title     = {Tight Complexity for Modal Extensions of Lec with K, 4, T},
  booktitle = {Proceedings of the 26th Conference on Logic for Programming, Artificial Intelligence, and Reasoning},
  editor    = {Erika Ábrahám and Nikolaj Bjørner and Michael Rawson},
  series    = {EPiC Series in Computing},
  volume    = {112},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/hK2D},
  doi       = {10.29007/n4zl},
  pages     = {364-380},
  year      = {2026}}

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