Abstract
It is known that the \(\mu \)-calculus collapses to its alternation-free fragment over transitive frames and to modal logic over equivalence relations. We adapt a proof by D’Agostino and Lenzi to show that the \(\mu \)-calculus collapses to its alternation-free fragment over weakly transitive frames. As a consequence, we show that the \(\mu \)-calculus with derivative topological semantics collapses to its alternation-free fragment. We also study the collapse over frames of \(\mathsf {S4.2}\), \(\mathsf {S4.3}\), \(\mathsf {S4.3.2}\), \(\mathsf {S4.4}\) and \(\textsf{KD45}\), logics important for Epistemic Logic. At last, we use the \(\mu \)-calculus to define degrees of ignorance on Epistemic Logic and study the implications of \(\mu \)-calculus’s collapse over the logics above.
Supported by MEXT, Japan.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The winning condition for general formulas is: \(\textsf{V}\) wins a run when \(\textsf{R}\) can’t make a move or the outermost \(\eta X.\psi \) which appears infinitely often in the run is some \(\nu X.\varphi \).
References
Alberucci, L., Facchini, A.: The modal \(\mu \)-calculus hierarchy over restricted classes of transition systems. J. Symbolic Logic 74(4), 1367–1400 (2009). https://doi.org/10.2178/jsl/1254748696
Alberucci, L., Facchini, A.: On modal \(\mu \)-calculus and gödel-löb logic. Stud. Logica. 91(2), 145–169 (2009). https://doi.org/10.1007/s11225-009-9170-9
Arnold, A.: The \(\mu \)-calculus alternation-depth hierarchy is strict on binary trees. RAIRO-Theor. Inform. Appl. 33(4–5), 329–339 (1999). https://doi.org/10.1051/ita:1999121
Aucher, G.: Principles of knowledge, belief and conditional belief. In: Rebuschi, M., Batt, M., Heinzmann, G., Lihoreau, F., Musiol, M., Trognon, A. (eds.) Interdisciplinary Works in Logic, Epistemology, Psychology and Linguistics. LAR, vol. 3, pp. 97–134. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-03044-9_5
Baltag, A., Bezhanishvili, N., Fernández-Duque, D.: The topological mu-calculus: completeness and decidability. In: 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 1–13 (2021). https://doi.org/10.1109/lics52264.2021.9470560
Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001). cbo9781107050884
Bradfield, J.C.: Simplifying the modal mu-calculus alternation hierarchy. In: Annual Symposium on Theoretical Aspects of Computer Science, pp. 39–49. Springer (1998). https://doi.org/10.1007/bfb0028547
Bradfield, J.C.: The modal mu-calculus alternation hierarchy is strict. Theoret. Comput. Sci. 195(2), 133–153 (1998). https://doi.org/10.1016/s0304-3975(97)00217-x
Bradfield, J., Walukiewicz, I.: The \(\mu \)-calculus and model checking. In: Handbook of Model Checking, pp. 871–919. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10575-8_26
Carrara, M., Chiffi, D., De Florio, C., Pietarinen, A.: We don’t know we don’t know: asserting ignorance. Synthese 198(4), 3565–3580 (2021). https://doi.org/10.1007/s11229-019-02300-y
D’Agostino, G., Lenzi, G.: On the \(\mu \)-calculus over transitive and finite transitive frames. Theoret. Comput. Sci. 411(50), 4273–4290 (2010). https://doi.org/10.1016/j.tcs.2010.09.002
D’Agostino, G., Lenzi, G.: On modal \(\mu \)-calculus over reflexive symmetric graphs. J. Log. Comput. 23(3), 445–455 (2012). https://doi.org/10.1093/logcom/exs028
Esakia, L.: Intuitionistic logic and modality via topology. Ann. Pure Appl. Logic 127(1–3), 155–170 (2004). https://doi.org/10.1016/j.apal.2003.11.013
Fan, J.: A logic for disjunctive ignorance. J. Philos. Log. 50(6), 1293–1312 (2021). https://doi.org/10.1007/s10992-021-09599-4
Fan, J., Wang, Y., van Ditmarsch, H.: Contingency and knowing whether. Rev. Symbolic Logic 8(1), 75–107 (2015). https://doi.org/10.1017/S1755020314000343
Fine, K.: Ignorance of ignorance. Synthese 195(9), 4031–4045 (2017). https://doi.org/10.1007/s11229-017-1406-z
Grädel, E., Thomas, W., Wilke, T. (eds.): Automata Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36387-4
Hetherington, S.: Good knowledge, bad knowledge: on two dogmas of epistemology. Clarendon Press (2001). https://doi.org/10.1093/acprof:oso/9780199247349.001.0001
Lehrer, K., Paxson, T.: Knowledge: undefeated justified true belief. J. Philos. 66(8), 225–237 (1969). https://doi.org/10.2307/2024435
Lenzen, W.: Recent work in epistemic logic. Acta Philos. Fennica 30, 1–219 (1978)
Lenzi, G.: A hierarchy theorem for the \(\mu \)-calculus. In: Meyer, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 87–97. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61440-0_119
Lenzi, G.: Recent results on the modal \(\mu \)-calculus: a survey. Rendiconti dell’Istituto di Matematica dell’Università di Trieste 42, 235–255 (2010)
McKinsey, J.C.C., Tarski, A.: The algebra of topology. Ann. Math. 141–191 (1944). https://doi.org/10.2307/1969080
Olsson, E. J., Proietti, C.: Explicating ignorance and doubt: a possible worlds approach. In: The Epistemic Dimensions of Ignorance, pp. 81–95. Cambridge University Press (2016). https://doi.org/10.1017/9780511820076.005
Peels, R., Blaauw, M.: The Epistemic Dimensions of Ignorance. Cambridge University Press, Cambridge (2016). https://doi.org/10.1017/9780511820076
Ranalli, C., van Woudenberg, R.: Collective ignorance: an information theoretic account. Synthese 198(5), 4731–4750 (2019). https://doi.org/10.1007/s11229-019-02367-7
Schwarz, G., Truszczyński, M.: Modal logic s4f and the minimal knowledge paradigm. In: Proceedings of the 4th Conference on Theoretical Aspects of Reasoning about Knowledge, pp. 184–198. Morgan Kaufmann Publishers Inc. (1992). https://doi.org/10.5555/1029762.1029779
Stalnaker, R.: On logics of knowledge and belief. Philos. Stud. Int. J. Philos. Anal. Tradit. 128(1), 169–199 (2006). https://doi.org/10.1007/s11098-005-4062-y
Van Benthem, J., Bezhanishvili, G.: Modal Logics of Space. In: Handbook of spatial logics, pp. 217–298. Springer (2007). https://doi.org/10.1007/978-1-4020-5587-4_5
Van Der Hoek, W., Lomuscio, A.: A logic for ignorance. Electron. Notes Theor. Comput. Sci. 85(2), 117–133 (2004). https://doi.org/10.1007/978-3-540-25932-9_6
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Pacheco, L., Tanaka, K. (2022). The Alternation Hierarchy of the \(\mu \)-calculus over Weakly Transitive Frames. In: Ciabattoni, A., Pimentel, E., de Queiroz, R.J.G.B. (eds) Logic, Language, Information, and Computation. WoLLIC 2022. Lecture Notes in Computer Science, vol 13468. Springer, Cham. https://doi.org/10.1007/978-3-031-15298-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-031-15298-6_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-15297-9
Online ISBN: 978-3-031-15298-6
eBook Packages: Computer ScienceComputer Science (R0)
