Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T07:53:36.581Z Has data issue: false hasContentIssue false
  • English
  • Français

THE MODAL LOGIC OF STEPWISE REMOVAL

Part of: General logic

Published online by Cambridge University Press:  21 July 2020

JOHAN VAN BENTHEM ,
KRZYSZTOF MIERZEWSKI  and
FRANCESCA ZAFFORA BLANDO
JOHAN VAN BENTHEM
Affiliation:
STANFORD UNIVERSITY AND LOGICAL DYNAMICS LAB, CSLI STANFORD, CA, USA and ILLC, UNIVERSITY OF AMSTERDAM AMSTERDAM, NETHERLANDS and TSINGHUA UNIVERSITY BEIJING, CHINA E-mail:[email protected] STANFORD UNIVERSITY AND LOGICAL DYNAMICS LAB, CSLI STANFORD, CA, USAE-mail:[email protected]:[email protected]
KRZYSZTOF MIERZEWSKI
Affiliation:
STANFORD UNIVERSITY AND LOGICAL DYNAMICS LAB, CSLI STANFORD, CA, USA and ILLC, UNIVERSITY OF AMSTERDAM AMSTERDAM, NETHERLANDS and TSINGHUA UNIVERSITY BEIJING, CHINA E-mail:[email protected]
FRANCESCA ZAFFORA BLANDO
Affiliation:
STANFORD UNIVERSITY AND LOGICAL DYNAMICS LAB, CSLI STANFORD, CA, USAE-mail:[email protected]:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the modal logic of stepwise removal of objects, both for its intrinsic interest as a logic of quantification without replacement, and as a pilot study to better understand the complexity jumps between dynamic epistemic logics of model transformations and logics of freely chosen graph changes that get registered in a growing memory. After introducing this logic (MLSR) and its corresponding removal modality, we analyze its expressive power and prove a bisimulation characterization theorem. We then provide a complete Hilbert-style axiomatization for the logic of stepwise removal in a hybrid language enriched with nominals and public announcement operators. Next, we show that model-checking for MLSR is PSPACE-complete, while its satisfiability problem is undecidable. Lastly, we consider an issue of fine-structure: the expressive power gained by adding the stepwise removal modality to fragments of first-order logic.

Keywords

dynamic logicshybrid logiclogics for graph gamescomplexity and decidability

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03B42: Logics of knowledge and belief (including belief change)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 1 , March 2022 , pp. 36 - 63
Check for updates
Copyright
© 2020, Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andréka, H., Bezhanishvili, N., van Benthem, J., & Németi, I. (2016). Changing a semantics: Opportunism or courage? In Alonso, E., Manzano, M., and Sain, I., editors. The Life and Work of Leon Henkin. Basel: Birkhaueser Verlag, pp. 307–337.Google Scholar
Areces, C., Fervari, R., & Hoffmann, G. (2015). Relation-changing modal operators. Logic Journal of the IGPL, 23(4), 601–627.CrossRefGoogle Scholar
Areces, C., Figueira, D., Figueira, S., & Mera, S. (2008). Expressive power and decidability for memory logics. In Hodges, W., and de Queiroz, R., editors. Logic, Language, Information and Computation, WoLLIC 2008. Lecture Notes in Computer Science, Vol. 5110. Berlin, Heidelberg/Germany: Springer, pp. 56–68.Google Scholar
Areces, C., & ten Cate, B. (2006). Hybrid logics. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Amsterdam, Netherlands: Elsevier Science, pp. 821–868.Google Scholar
Aucher, G., van Benthem, J., & Grossi, D. (2018). Modal logics of sabotage revisited. Journal of Logic and Computation, 28(2), 269–303,CrossRefGoogle Scholar
van Benthem, J. (2005). An essay on sabotage and obstruction. In Hutter, D., & Stephan, W., editors. Mechanizing Mathematical Reasoning: Essays in Honor of Jörg H. Siekmann on the Occasion of His 60th Birthday. Lecture Notes in Artificial Intelligence, Vol. 2605. Berlin: Springer-Verlag, pp. 268–276.CrossRefGoogle Scholar
van Benthem, J. (2011). Logical Dynamics of Information and Interaction. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
van Benthem, J., & Liu, F. (2020). Graph games and logic design. In Liu, F., Ono, H., and Yu, J., editors. Knowledge, Proof and Dynamics. Singapore, Asia: Springer, pp. 125–146.CrossRefGoogle Scholar
van Benthem, J., Gerbrandy, J., Hoshi, T., & Pacuit, E. (2009). Merging frameworks for interaction. Journal of Philosophical Logic, 38(5), 491–526.CrossRefGoogle Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2011). Modal Logic. Cambridge: Cambridge University Press.Google Scholar
Gabbay, D. (2013). Introducing reactive Kripke semantics and arc accessibility. In Gabbay, D., editor. Reactive Kripke Semantics. Dordrecht, Netherlands: Springer Science, pp. 29–76.CrossRefGoogle Scholar
Hansen, J. U. (2011). A hybrid public announcement logic with distributed knowledge. In Bolander, T., and Braüner, T., editors. International Workshop on Hybrid Logic and Applications 2010. Electronic Notes in Theoretical Computer Science, Vol. 273. Amsterdam, Netherlands: Elsevier, pp. 33–50.Google Scholar
Hintikka, J., & Sandu, G. (1997). Game-theoretic semantics. In ter Meulen, A., and van Benthem, J., editors. Handbook of Logic and Language. Amsterdam, Netherlands: Elsevier Science, pp. 361–410.CrossRefGoogle Scholar
Immerman, N. (1999). Descriptive Complexity. Dordrecht, Netherlands: Springer Science Publishers.CrossRefGoogle Scholar
Kanellakis, P., & Smolka, S. (1983). CCS expressions, finite state processes, and three problems of equivalence. Proceedings of the 2nd ACM Symposium on Principles of Distributed Computing. Dordrecht: Springer Science, pp. 228–240.CrossRefGoogle Scholar
Kracht, M., & Wolter, F. (1999). Normal monomodal logics can simulate all others. Journal of Symbolic Logic, 64(1), 99–138.CrossRefGoogle Scholar
Li, D. (2020). Losing connection: The modal logic of definable link deletion. Journal of Logic and Computation, 30(3), 715–743.CrossRefGoogle Scholar
Löding, C., & Rohde, P. (2003). Solving the Sabotage Game is PSPACE-Hard. In Rovan, B., and Vojtas, P, editors. Mathematical Foundations of Computer Science 2003. Lecture Notes in Computer Science, Vol. 2747. Berlin: Springer-Verlag, pp. 531–540.CrossRefGoogle Scholar
Marx, M. (2006). Complexity of modal logic. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Amsterdam, Netherlands: Elsevier Science, pp. 139–179.Google Scholar
de Lavalette, G. R. (2001). A logic of modification and creation. In Condoravdi, C., and de Lavalette, G. R., editors. Logical Perspectives on Language and Information. Stanford, CA: CSLI Publications, pp. 197–219.Google Scholar
Rohde, P. (2005). On games and logics over dynamically changing structures. Ph.D. Dissertation, RWTH Aachen University (Germany), pp. 1–216.Google Scholar
Stockmeyer, L. J., & Meyer, A. R. (1973). Word problems requiring exponential time. Proceedings of the 5th ACM Symposium on Theory of Computing, STOC ’73, pp. 1–9.Google Scholar
Thompson, D. (2020). Local fact change logic. In Liu, F., Ono, H., and Yu, J., editors. Knowledge, Proof and Dynamics. Singapore: Springer, pp. 73–96.CrossRefGoogle Scholar
Zaffora Blando, F., Mierzewski, K., & Areces, C. (2020). The modal logics of the poison game. In Liu, F., Ono, H., and Yu, J., editors. Knowledge, Proof and Dynamics. Singapore: Springer, pp. 3–23.CrossRefGoogle Scholar