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Synthese

, Volume 190, Supplement 1, pp 135–162 | Cite as

Public announcement logic with distributed knowledge: expressivity, completeness and complexity

Article

Abstract

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with distributed knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (\(\mathcal{PAL }\)) with distributed knowledge, in particular their expressivity, axiomatisations and complexity. \(\mathcal{PAL }\) extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on \(\mathcal{PACD }\), the result of adding both common and distributed knowledge to \(\mathcal{PAL }\), which is more expressive than each of its component logics. We introduce an axiomatisation of \(\mathcal{PACD }\), which is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with \(S5\) knowledge, distributed knowledge, common knowledge and public announcements at the same time. We furthermore show that \(\mathcal{PACD }\) is decidable, more precisely that it is \(\textsc {exptime}\)-complete. This result also carries over to \(\mathcal{S 5\mathcal CD }\) with common and distributed knowledge operators for all coalitions (and not only the grand coalition). Finally, we propose a notion of a trans-bisimulation to generalise certain results and give deeper insight into the proofs.

Keywords

Public announcement logic Distributed knowledge Expressivity Completeness Decidability Computational complexity Unravelling Folding Trans-bisimulation 

Notes

Acknowledgments

We thank the attendees at LORI-III in Guangzhou, the anonymous LORI reviewers and Synthese reviewers, and Pål Grønås Drange for helpful remarks and suggestions. Yì Wáng gratefully acknowledges funding support from the Major Project of National Social Science Foundation of China (No. 11&ZD088). Thomas Ågothes is also affiliated with Southwest University, China, and Bergen University College, Norway.

References

  1. Baltag, A., & Moss, L. (2004). Logics for epistemic programs. Synthese, 139, 165–224.CrossRefGoogle Scholar
  2. Baltag, A., Moss, L., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In Procedings of TARK VII (pp. 43–56).Google Scholar
  3. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. In Cambridge tracts in theoretical computer science (Vol. 53).Google Scholar
  4. Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about Knowledge. MIT.Google Scholar
  5. Fagin, R., Halpern, J., & Vardi, M. (1992). What can machines know? On the properties of knowledge in distributed systems. Journal of the ACM, 39(2), 328–376.CrossRefGoogle Scholar
  6. Fischer, M. J., & Ladner, R. E. (1977). Propositional modal logic of programs. In Proceedings of STOC ’77 (pp. 286–294). New York: ACM.Google Scholar
  7. Fischer, M. J., & Ladner, R. E. (1979). Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18(2), 194–211.CrossRefGoogle Scholar
  8. Gerbrandy, J. (1999) Bisimulations on planet kripke. Ph.D. thesis, ILLC.Google Scholar
  9. Göller, S., Lohrey, M., & Lutz, C. (2007). PDL with intersection and converse is 2EXP-complete. In H. Seidl (Ed.), FOSSACS 2007, LNCS (Vol. 4423, pp. 198–212). Berlin: Springer.Google Scholar
  10. Halpern, J., & Moses, Y. (1992). A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54(3), 319–379.CrossRefGoogle Scholar
  11. Kooi, B., & van Benthem, J. (2004). Reduction axioms for epistemic actions. In Proceedings of AiML 2004 (pp. 197–211).Google Scholar
  12. Ladner, R. E. (1977). The computational complexity of provability in systems of modal propositional logic. SIAM Journal on Computing, 6(3), 467–480.CrossRefGoogle Scholar
  13. Lutz, C. (2005). PDL with intersection and converse is decidable. In L. Ong (Ed.), CSL 2005, LNCS (Vol. 3634, pp. 413–427). Berlin: Springer.Google Scholar
  14. Lutz, C. (2006). Complexity and succinctness of public announcement logic. In Proceedings of AAMAS ’06 (pp. 137–143). New York: ACM.Google Scholar
  15. Meyer, J. J., & van der Hoek, W. (1995). Epistemic logic for AI and computer science (Vol. 41). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Plaza, J. (1989). Logics of public communications. In Proceedings of ISMIS ’89 (pp. 201–216).Google Scholar
  17. Pratt, V. R. (1979). Models of program logics. In Proceedings of the 20th annual symposium on the foundations of computer science (FOCS’79) (pp. 115–122).Google Scholar
  18. Roelofsen, F. (2007). Distributed knowledge. Journal of Applied Non-Classical Logics, 17(2), 255–273.CrossRefGoogle Scholar
  19. van Benthem, J. (2006). Open problems in logical dynamics. In D. M. Gabbay, S. S. Goncharov, & M. Zakharyaschev (Eds.), Mathematical problems from applied logic I: Logics for the XXIst century (pp. 137–192). Berlin: Springer.Google Scholar
  20. van Benthem, J. F. A. K. (1999). Update as relativization. ILLC (manuscript).Google Scholar
  21. van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic epistemic logic. In Synthese library (Vol. 337). Springer.Google Scholar
  22. van der Hoek, W., & Meyer, J. J. (1992). Making some issues of implicit knowledge explicit. International Journal of Foundations of Computer Science, 3(2), 193–224.CrossRefGoogle Scholar
  23. van der Hoek, W., & Meyer, J. J. (1997). A complete epistemic logic for multiple agents: Combining distributed and common knowledge. In Epistemic logic and the theory of games and decisions (pp. 35–68). Dordrecht: Kluwer.Google Scholar
  24. Wáng, Y. N., & Ågotnes, T. (2011). Public announcement logic with distributed knowledge. In Proceedings of LORI-III, LNCS (Vol. 6953, pp. 328–341).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Computing, Mathematics and PhysicsBergen University CollegeBergenNorway
  2. 2.Department of Information Science and Media StudiesUniversity of BergenBergenNorway

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