Infinitary action logic with exponentiation

We introduce infinitary action logic with exponentiation-that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allow some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is Pi(1)(1)-complete. Second, we show that the closure ordinal of its derivability operator is omega(CK)(1). In the case where no subexponential allows contraction, we show that complexity is the same as for infinitary action logic itself. Namely, the derivability problem in this case is Pi(0)(1)-complete and the closure ordinal is not greater than omega(omega). (C) 2021 Elsevier B.V. All rights reserved.