Cut Free Labelled Sequent Calculus for Dynamic Logic of Relation Changers

  • Ryo HatanoEmail author
  • Katsuhiko Sano
  • Satoshi Tojo
Conference paper
Part of the Logic in Asia: Studia Logica Library book series (LIAA)


Dynamic epistemic logic (\(\mathbf {DEL}\)) is known as a large family of logics that extend standard epistemic logic with dynamic operators. Such dynamic operators can be regarded as epistemic actions over Kripke semantics (or its variant). Therefore, \(\mathbf {DEL}\) is often used to model changes of agents’ knowledge, belief or preference over Kripke semantics in terms of dynamic operators in many literatures. As a variant of \(\mathbf {DEL}\), (van Benthem and Liu, J Appl Non-Classical Logics 17(2):157–18 2007; Liu, Reasoning about preference dynamics, Springer Science & Business Media, Berlin 2011) proposed dynamic logic of relation changers (\(\mathbf {DLRC}\)). They provided a general framework to capture many dynamic operators in terms of relation changing operation written in programs of propositional dynamic logic, and they also provided a sound and complete Hilbert-style axiomatization for \(\mathbf {DLRC}\). While \(\mathbf {DLRC}\) can cover many dynamic operators in a uniform manner, proof theory for \(\mathbf {DLRC}\) is not well-studied except the Hilbert-style axiomatization. Therefore, we propose a cut-free labelled sequent calculus for \(\mathbf {DLRC}\). We show that our sequent calculus is equipollent with the Hilbert-style axiomatization.


Labelled Sequent Calculus Propositional Dynamic Logic Complete Hilbert-style Axiomatization Kripke Semantics Relation Changing Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank an anonymous reviewer for his/her helpful comments to revise our manuscript. We would be grateful to the audiences and organizing committee of the joint conference of the 3rd Asian workshop on philosophical logic and the 3rd Taiwan philosophical logic colloquium (AWPL-TPLC 2016). During the conference, the first author was supported by travel grants from the organizing committee of AWPL-TPLC 2016. The work of the second author was partially supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid for Young Scientists (B) Grant Number 15K21025. The work of the second and the third authors were also supported by JSPS Core-to-Core Program (A. Advanced Research Networks).


  1. Balbiani, P., Van Ditmarsch, H., Herzig, A., & De Lima, T. (2010). Tableaux for public announcement logic. Journal of Logic and Computation, 20(1), 55–76.CrossRefGoogle Scholar
  2. Balbiani, P., Demange, V., & Galmiche, D. (2014). A sequent calculus with labels for Public Announcement Logic. Advances in Modal Logic (AiML 2014), 6.Google Scholar
  3. Bull, R. (1992). Cut elimination for propositional dynamic logic without *. Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik, 38, 85–100.CrossRefGoogle Scholar
  4. Frittella, S., Greco, G., Kurz, A., Palmigiano, A., & Sikimíc, V. (2014). Multi-type display calculus for dynamic epistemic logic. Journal of Logic and Computation, 26(6), 2017–2065.Google Scholar
  5. Gentzen, G. (1964). Investigations into logical deduction. American Philosophical Quarterly, 1(4), 288–306.Google Scholar
  6. Gerbrandy, J., & Groeneveld, W. (1997). Reasoning about information change. Journal of Logic, Language and Information, 6(2), 147–169.CrossRefGoogle Scholar
  7. Harel, D., Kozen, D., & Tiuryn, J. (2000). Dynamic logic. Cambridge: MIT press.Google Scholar
  8. Hill, B., & Poggiolesi, F. (2010). A contraction-free and cut-free sequent calculus for propositional dynamic logic. Studia Logica, 94(1), 47–72.Google Scholar
  9. Kashima, R. (2009). Mathematical logic. Asakura Publishing Co. Ltd. (in Japanese).Google Scholar
  10. Liu, F. (2011). Reasoning about preference dynamics (Vol. 354). Synthese Library, Berlin: Springer Science & Business Media.Google Scholar
  11. Ma, M., Sano, K., Schwarzentruber, F., & Velázquez-Quesada, F. R. (2015). Tableaux for non normal public announcement logic. Logic and Its Applications (ICLA2015), Lecture Notes in Computer Science, 8923, 132–145.Google Scholar
  12. Maffezioli, P., & Naibo, A. (2013). Proof theory of epistemic logic of programs. Logic and Logical Philosophy, 23(3), 301–328.Google Scholar
  13. Maffezioli, P., & Negri, S. (2010). A Gentzen-style analysis of public announcement logic. Proceedings of the International Workshop on Logic and Philosophy of Knowledge, Communication and Action, 293–313.Google Scholar
  14. Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34(5), 507–544.Google Scholar
  15. Nomura, S., Sano, K., & Tojo, S. (2015). Revising a sequent calculus for public announcement logic. In Structural Analysis of Non-classical Logics: Proceedings of the Second Taiwan Philosophical Logic Colloquium (TPLC-2014), 131–157.Google Scholar
  16. Ono, H., & Komori, Y. (1985). Logics without the contraction rule. The Journal of Symbolic Logic, 50(01), 169–201.CrossRefGoogle Scholar
  17. Plaza, J. (1989). Logics of public communications. In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic & Z. W. Ras (Eds.), Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, 201–216.Google Scholar
  18. van Benthem, J., & Liu, F. (2007). Dynamic logic of preference upgrade. Journal of Applied Non-Classical Logics, 17(2), 157–182.CrossRefGoogle Scholar
  19. van Ditmarsch, H., van der Hoek, W., & Kooi, B. P. (2007). Dynamic epistemic logic (Vol. 337). Synthese Library, Berlin: Springer.Google Scholar
  20. Wang, Y., & Cao, Q. (2013). On axiomatizations of public announcement logic. Synthese, 190(1), 103–134.CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Science and TechnologyTokyo University of ScienceNodaJapan
  2. 2.Graduate School of LettersHokkaido UniversityHokkaidoJapan
  3. 3.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan

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