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.
Keywords
The work was supported by the Russian Science Foundation, in cooperation with the Austrian Science Fund, under grant RSF–FWF 20-41-05002.
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- 1.
Kozen uses a different notation.
- 2.
For accuracy, we distinguish \(A^m\) as a sequence from \(A^{{\bullet } m}\) as one formula. In fact, they are of course equivalent.
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Acknowledgments
The author is grateful to Anupam Das and Stanislav Speranski for fruitful discussions. The author also thanks the reviewers for thorough consideration of the paper and many valuable suggestions.
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Kuznetsov, S.L. (2021). Complexity of a Fragment of Infinitary Action Logic with Exponential via Non-well-founded Proofs. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_19
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