Abstract
At LICS 2013, O. Lahav introduced a technique to uniformly construct cut-free hypersequent calculi for basic modal logics characterised by frames satisfying so-called ‘simple’ first-order conditions. We investigate the generalisation of this technique to modal logics with the master modality (a.k.a. reflexive-transitive closure modality). The (co)inductive nature of this modality is accounted for through the use of non-well-founded proofs, which are made cyclic using focus-style annotations. We show that the peculiarities of hypersequents hinder the usual method of completeness via infinitary proof-search. Instead, we construct countermodels from maximally unprovable hypersequents. We show that this yields completeness for a small (yet infinite) subset of simple frame conditions.
This research has been made possible by a grant from the Dutch Research Council NWO, project nr. 617.001.857.
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References
Afshari, B., Leigh, G.E.: Cut-free completeness for modal mu-calculus. In: Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science. LICS 2017 (2017)
Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Log. Comput. 21(6), 1177–1216 (2011)
Brünnler, K., Lange, M.: Cut-free sequent systems for temporal logic. J. Log. Algebr. Program. 76(2), 216–225 (2008)
Docherty, S., Rowe, R.N.S.: A non-wellfounded, labelled proof system for propositional dynamic logic. In: Cerrito, S., Popescu, A. (eds.) TABLEAUX 2019. LNCS (LNAI), vol. 11714, pp. 335–352. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29026-9_19
Doczkal, C., Smolka, G.: Constructive completeness for modal logic with transitive closure. In: Hawblitzel, C., Miller, D. (eds.) CPP 2012. LNCS, vol. 7679, pp. 224–239. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35308-6_18
Enqvist, S.: A circular proof system for the hybrid mu-calculus. arXiv preprint arXiv:2001.04971 (2020)
Enqvist, S., Hansen, H.H., Kupke, C., Marti, J., Venema, Y.: Completeness for game logic. In: 2019 34th Annual ACM/IEEE Symposium on Logic in computer Science (LICS), pp. 1–13. IEEE (2019)
Indrzejczak, A.: Sequents and Trees: An Introduction to the Theory and Applications of Propositional Sequent Calculi. Springer Nature, Basingstoke (2020)
Kikot, S., Shapirovsky, I., Zolin, E.: Completeness of logics with the transitive closure modality and related logics. arXiv preprint arXiv:2011.02205 (2020)
Lahav, O.: From frame properties to hypersequent rules in modal logics. In: 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 408–417. IEEE (2013)
Lange, M., Stirling, C.: Focus games for satisfiability and completeness of temporal logic. In: Proceedings 16th Annual IEEE Symposium on Logic in Computer Science, pp. 357–365. IEEE (2001)
Shamkanov, D.S.: Circular proofs for the Gödel-Löb provability logic. Math. Notes 96(3), 575–585 (2014)
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Rooduijn, J. (2021). Cyclic Hypersequent Calculi for Some Modal Logics with the Master Modality. 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_21
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DOI: https://doi.org/10.1007/978-3-030-86059-2_21
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