TOWER-Complete Problems in Contraction-Free Substructural Logics

01/07/2022

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by   Hiromi Tanaka, et al.

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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 Lazić and Schmitz (2015), we show that the deducibility problem for full Lambek calculus with exchange and weakening (𝐅𝐋_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 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).