Tower-Complete Problems in Contraction-Free Substructural Logics

Author Hiromi Tanaka

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Hiromi Tanaka
  • Keio University, Tokyo, Japan


I would like to thank Koji Mineshima for helpful comments on the introduction section. I am also grateful to the anonymous reviewers for useful suggestions, which improve some proofs and deepen the discussion in Section 7.

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Hiromi Tanaka. Tower-Complete Problems in Contraction-Free Substructural Logics. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We investigate the non-elementary computational complexity of a family of substructural logics without contraction. With the aid of the technique pioneered by Lazić and Schmitz (2015), we show that the deducibility problem for full Lambek calculus with exchange and weakening (FL_{ew}) is not in Elementary (i.e., the class of decision problems that can be decided in time bounded by an elementary recursive function), but is in PR (i.e., the class of decision problems that can be decided in time bounded by a primitive recursive function). More precisely, we show that this problem is complete for Tower, which is a non-elementary complexity class forming a part of the fast-growing complexity hierarchy introduced by Schmitz (2016). The same complexity result holds even for deducibility in BCK-logic, i.e., the implicational fragment of FL_{ew}. We furthermore show the Tower-completeness of the provability problem for elementary affine logic, which was proved to be decidable by Dal Lago and Martini (2004).

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Proof theory
  • substructural logic
  • linear logic
  • full Lambek calculus
  • BCK-logic
  • computational complexity
  • provability
  • deducibility


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