Journal of Logic, Language and Information

, Volume 17, Issue 1, pp 109–129 | Cite as

On combinations of propositional dynamic logic and doxastic modal logics

  • Renate A. Schmidt
  • Dmitry Tishkovsky
Original Article


We prove completeness and decidability results for a family of combinations of propositional dynamic logic and unimodal doxastic logics in which the modalities may interact. The kind of interactions we consider include three forms of commuting axioms, namely, axioms similar to the axiom of perfect recall and the axiom of no learning from temporal logic, and a Church–Rosser axiom. We investigate the influence of the substitution rule on the properties of these logics and propose a new semantics for the test operator to avoid unwanted side effects caused by the interaction of the classic test operator with the extra interaction axioms.


Combinations of modal logics Dynamic logic Doxastic logic Epistemic logic Reasoning about actions Belief and knowledge 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Blackburn P., de Rijke M., Venema Y. (2001). Modal logic, Vol. 53 of Cambridge tracts in theoretical computer science. Cambridge, Cambridge Univ. PressGoogle Scholar
  2. Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (1995). Reasoning about knowledge. MIT Press.Google Scholar
  3. Fischer M.J., Ladner R.E. (1979). Propositional dynamic logic of regular programs. Journal of computer and System Sciences 18(2): 194–211CrossRefGoogle Scholar
  4. Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (2003). Many-dimensional modal logics: Theory and applications, Vol. 148 of studies in logic and the foundations of mathematics. North-Holland.Google Scholar
  5. Gabbay D.M., Shehtman V. (1998). Products of modal logics, Part 1. Logic Journal of the IGPL 6(1): 73–146CrossRefGoogle Scholar
  6. Harel D., Kozen D., Tiuryn J. (2000). Dynamic logic, Foundations of Computing. MIT Press.Google Scholar
  7. Herzig A., Longin D. (2000). Belief dynamics in cooperative dialogues. Journal of Semantics 17(2): 91–118CrossRefGoogle Scholar
  8. Kracht M., Wolter F. (1991). Properties of independently axiomatizable bimodal logics. Journal of Symbolic Logic 56(4): 1469–1485CrossRefGoogle Scholar
  9. Kracht M., Wolter F. (1997). Simulation and transfer results in modal logic—a survey. Studia Logica 59(2): 149–177CrossRefGoogle Scholar
  10. Meyer J.J.C., van der Hoek W., van Linder B. (1999). A logical approach to the dynamics of commitments. Artificial Intelligence 113(1–2): 1–40CrossRefGoogle Scholar
  11. Ohlbach, H. J. (1996). scan—elimination of predicate quantifiers: System description. In M. A. McRobbie & J. K. Slaney (Eds.), Proceedings of the 13th conference on automated deduction (CADE-13), Vol. 1104 of Lecture notes in artificial intelligence (pp. 161–165). Springer.Google Scholar
  12. Plaza, J. A. (1989). Logic 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 (ISMIS’89), pp. 201–216.Google Scholar
  13. Rao, A. S., & Georgieff, M. P. (1991). Modeling rational agents within a BDI-architecture. In R. E. Fikes & E. Sandewall (Eds.), Proceedings of the 2nd international conference on the principle of knowledge representation and reasoning (KR’91), pp. 473–484.Google Scholar
  14. Sahlqvist, H. (1975). Completeness and correspondence in the first and second order semantics for modal logic. In S. Kanger (Ed.), Proceedings of the 3rd scandinavian logic symposium, pp. 110–143.Google Scholar
  15. Schmidt R.A., Tishkovsky D. (2002). On axiomatic products of PDL and S5: substitution, tests and knowledge. Bulletin of the Section of Logic 31(1): 27–36Google Scholar
  16. Schmidt, R. A. & Tishkovsky, D. (2003). Combining dynamic logic with doxastic modal logics. In P. Balbiani, N.-Y. Suzuki, F. Wolter, & M. Zakharyaschev (Eds.), Advances in modal logic, (Vol. 4, pp. 371–392). King’s College Publ.Google Scholar
  17. Schmidt, R. A. & Tishkovsky, D. (2004). Multi-agent dynamic logics with informational test. Annals of Mathematics and Artificial Intelligence, 42(1–3), 5–36. Special issue on Computational Logic in Multi-Agent Systems.Google Scholar
  18. van der Hoek, W. (2001). Logical foundations of agent-based computing. In M. Luck, V. Marík, O. Stepánková, & R. Trappl (Eds.), Multi-agent systems and applications, Vol. 2086 of Lecture notes in artificial intelligence, (pp. 50–73). Springer.Google Scholar
  19. Wolter F. (2000). The product of converse PDL and polymodal K. Journal of Logic and Computation 10(2): 223–251CrossRefGoogle Scholar
  20. Zakharyaschev, M., Wolter, F., & Chagrov, A. (2001). Advanced modal logic. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 3, pp. 83–266). Kluwer (2nd ed.).Google Scholar

Copyright information

© Springer Science+Business Media 2007

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of ManchesterManchesterUK
  2. 2.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

Personalised recommendations