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
(FLew) 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
FLew. 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).Comment: The full version of the paper accepted to CSL 202