TOWER-Complete Problems in Contraction-Free Substructural Logics

We investigate the computational complexity of a family of substructural logics with exchange and weakening but without contraction. With the aid of the techniques provided 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 TOWER-complete, where TOWER is one of the non-elementary complexity classes 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).