Abstract
We study a family of modal logics interpreted on tree-like structures, and featuring local quantifiers \(\exists ^{k}p\) that bind the proposition p to worlds that are accessible from the current one in at most k steps. We consider a first-order and a second-order semantics for the quantifiers, which enables us to relate several well-known formalisms, such as hybrid logics, \(\textsf {S5Q}\) and graded modal logic. To better stress these connections, we explore fragments of our logics, called herein round-bounded fragments. Depending on whether first or second-order semantics is considered, these fragments populate the hierarchy \({2\textsc {NExp} \subset 3\textsc {NExp} \subset \cdots }\) or the hierarchy \({2\textsc {AExp}_{pol} \subset 3\textsc {AExp}_{pol} \subset \cdots }\), respectively. For formulae up-to modal depth k, the complexity improves by one exponential.
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References
Andréka, H., Németi, I., van Benthem, J.: Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27(3), 217–274 (1998)
Areces, C., Blackburn, P., Marx, M.: Hybrid logics: characterization, interpolation and complexity. The Journal of Symbolic Logic 66(3), 977–1010 (2001)
Areces, C., ten Cate, B.: Hybrid logics. In: Handbook of Modal Logic, Studies in logic and practical reasoning, vol. 3, pp. 821–868. North-Holland (2007)
Areces, C., Fervari, R., Hoffmann, G.: Relation-changing modal operators. Logic Journal of the IGPL 23(4), 601–627 (2015)
Barnaba, M.F., Caro, F.D.: Graded modalities. Studia Logica 44(2), 197–221 (1985)
Bednarczyk, B., Demri, S.: Why propositional quantification makes modal logics on trees robustly hard? In: Logic in Computer Science. pp. 1–13. IEEE (2019)
Bednarczyk, B., Demri, S., Fervari, R., Mansutti, A.: Modal logics with composition on finite forests: Expressivity and complexity. In: Logic in Computer Science. pp. 167–180. ACM (2020)
van Benthem, J.: An essay on sabotage and obstruction. In: Mechanizing Mathematical Reasoning. LNCS, vol. 2605, pp. 268–276 (2005)
Blackburn, P., Braüner, T., Kofod, J.: Remarks on Hybrid Modal Logic with Propositional Quantifiers, pp. 401–426. No. 4 in Logic and Philosophy of Time (2020)
Blackburn, P., Wolter, F., van Benthem, J. (eds.): Handbook of Modal Logics, Studies in logic and practical reasoning, vol. 3. Elsevier (2006)
Bozzelli, L., Molinari, A., Montanari, A., Peron, A.: On the complexity of model checking for syntactically maximal fragments of the interval temporal logic HS with regular expressions. In: GandALF’17. EPTCS, vol. 256, pp. 31–45 (2017)
Bull, R.A.: On modal logic with propositional quantifiers. The Journal of Symbolic Logic 34(2), 257–263 (1969)
Calcagno, C., Cardelli, L., Gordon, A.: Deciding validity in a spatial logic for trees. In: International Workshop on Types in Languages Design and Implementation. pp. 62–73. ACM (2003)
ten Cate, B., Franceschet, M.: On the complexity of hybrid logics with binders. In: Ong, L. (ed.) Computer Science Logic. pp. 339–354 (2005)
Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the ACM 28(1), 114–133 (1981)
Demri, S., Fervari, R.: The power of modal separation logics. Journal of Logic and Computation 29(8), 1139–1184 (2019)
Ding, Y.: On the logics with propositional quantifiers extending s5\(\varPi \). In: Advances in Modal Logic. pp. 219–235. College Publications (2018)
Fine, K.: Propositional quantifiers in modal logic. Theoria 36, 336–346 (1970)
Fischer, M.J., Ladner, R.E.: Propositional modal logic of programs. In: ACM Symposium on Theory of Computing. p. 286–294 (1977)
Fischer, M.J., Rabin, M.O.: Super-exponential complexity of presburger arithmetic. In: Complexity of Computation, SIAM–AMS Proceedings. pp. 27–41 (1974)
Hannula, M., Kontinen, J., Virtema, J., Vollmer, H.: Complexity of propositional logics in team semantic. ACM Transactions on Computational Logic 19(1), 2:1–2:14 (2018)
Kaplan, D.: S5 with quantifiable propositional variables. The Journal of Symbolic Logic 35(2), 355 (1970)
Mansutti, A.: Reasoning with Separation Logics: Complexity, Expressive Power, Proof Systems. Ph.D. thesis, Université Paris-Saclay (December 2020)
Mansutti, A.: Notes on kAExp(pol) problems for deterministic machines (2021)
Meier, A., Mundhenk, M., Thomas, M., Vollmer, H.: The complexity of satisfiability for fragments of CTL and CTL*. Electronic Notes in Theoretical Computer Science 223, 201–213 (2008)
Prior, A.: Past, Present and Future. Oxford Books (1967)
Rabin, M.: Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society 41, 1–35 (1969)
de Rijke, M.: A note on graded modal logic. Studia Logica 64(2), 271–283 (2000)
Schmitz, S.: Complexity hierarchies beyond elementary. ACM Transactions on Computation Theory 8(1), 3:1–3:36 (2016)
Schneider, T.: The complexity of hybrid logics over restricted frame classes. Ph.D. thesis, Friedrich Schiller University of Jena (2007)
Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. Journal of the ACM 32(3), 733–749 (1985)
Yang, F., Väänänen, J.: Propositional logics of dependence. Annals of Pure and Applied Logic 167(7), 557–589 (2016)
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Fervari, R., Mansutti, A. (2022). Modal Logics and Local Quantifiers: A Zoo in the Elementary Hierarchy. In: Bouyer, P., Schröder, L. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2022. Lecture Notes in Computer Science, vol 13242. Springer, Cham. https://doi.org/10.1007/978-3-030-99253-8_16
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