Tower-Complete Problems in Contraction-Free Substructural Logics

TL;DR

This paper proves that deducibility in certain contraction-free substructural logics, including full Lambek calculus with exchange and weakening, is Tower-complete, indicating extremely high non-elementary computational complexity.

Contribution

It establishes the Tower-completeness of deducibility problems in full Lambek calculus with exchange and weakening, and in elementary affine logic, revealing their non-elementary complexity.

Findings

Deducibility in $ extbf{FL}_{ extbf{ew}}$ is Tower-complete.

Deducibility in BCK-logic is also Tower-complete.

Provability in elementary affine logic is Tower-complete.

Abstract

We investigate the non-elementary computational complexity of a family of substructural logics without contraction. With the aid of the technique pioneered by Lazi\'c and Schmitz (2015), we show that the deducibility problem for full Lambek calculus with exchange and weakening ($\mathbf{FL}_{\mathbf{ew}}$) 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 $\mathbf{FL}_{\mathbf{ew}}$. 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).