Complexity of a Fragment of Infinitary Action Logic with Exponential via Non-well-founded Proofs

Abstract

Infinitary action logic ($\mathbf {ACT}_\omega$) can be viewed as an extension of the multiplicative-additive Lambek calculus ($\mathbf {MALC}$) with iteration (Kleene star) governed by an omega-rule (Buszkowski, Palka 2007). An alternative formulation utilizes non-well-founded proofs instead of the omega-rule (Das, Pous 2017). Another unary operation commonly added to $\mathbf {MALC}$ is the exponential. We consider a system which has both Kleene star and the exponential. In general, this system is of a very high complexity level: it is $\varPi ^1_1$-complete (Kuznetsov, Speranski 2020), while $\mathbf {ACT}_\omega$ itself is $\varPi ^0_1$-complete. As a reasonable intermediate logic, we consider the fragment where Kleene star is not allowed to appear in the scope of the exponential. For this fragment we manage to construct a formulation based on non-well-founded proofs, with an easily checkable correctness criterion. Using this formulation, we prove that this fragment indeed has strictly intermediate complexity, namely, we prove a $\varPi ^0_2$ lower bound and a $\varDelta ^1_1$ upper bound. We also prove a negative result that this fragment does not enjoy Palka’s *-elimination property, which would have given a $\varPi ^0_2$ upper bound as well.

The work was supported by the Russian Science Foundation, in cooperation with the Austrian Science Fund, under grant RSF–FWF 20-41-05002.