, Volume 191, Issue 3, pp 371–408 | Cite as

Exploring the tractability border in epistemic tasks



We analyse the computational complexity of comparing informational structures. Intuitively, we study the complexity of deciding queries such as the following: Is Alice’s epistemic information strictly coarser than Bob’s? Do Alice and Bob have the same knowledge about each other’s knowledge? Is it possible to manipulate Alice in a way that she will have the same beliefs as Bob? The results show that these problems lie on both sides of the border between tractability (P) and intractability (NP-hard). In particular, we investigate the impact of assuming information structures to be partition-based (rather than arbitrary relational structures) on the complexity of various problems. We focus on the tractability of concrete epistemic tasks and not on epistemic logics describing them.


Epistemic logic Computational complexity Epistemic reasoning Multi-agent systems 


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  1. Ågotnes T., Balbiani P., Van Ditmarsch H., Seban P. (2010) Group announcement logic. Journal of Applied Logic 8(1): 62–81CrossRefGoogle Scholar
  2. Aucher G. (2010) An internal version of epistemic logic. Studia Logica 94(1): 1–22CrossRefGoogle Scholar
  3. Aumann R. J. (1999) Interactive epistemology I: Knowledge. International Journal of Game Theory 28(3): 263–300CrossRefGoogle Scholar
  4. Balbiani P., Baltag A., Van Ditmarsch H., Herzig A., Hoshi T., de Lima T. (2008) ‘Knowable’ as ‘known after an announcement’. The Review of Symbolic Logic 1(03): 305–334CrossRefGoogle Scholar
  5. Balcázar J. L., Gabarró J., Santha M. (1992) Deciding bisimilarity is P-complete. Formal Aspects of Computing 4(6A): 638–648CrossRefGoogle Scholar
  6. Baltag A., Moss L. S. (2004) Logics for epistemic programs. Synthese 139(2): 165–224CrossRefGoogle Scholar
  7. Baltag, A., Moss, L. S., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In TARK ’98: Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge, San Francisco, CA, USA, 1998 (pp. 43–56). Burlington, MA: Morgan Kaufmann Publishers Inc.Google Scholar
  8. Baral C., Zhang Y. (2005) Knowledge updates: Semantics and complexity issues. Artificial Intelligence 164(1–2): 209–243CrossRefGoogle Scholar
  9. Besold, T. R., Gust, H., Krumnack, U., Abdel-Fattah, A., Schmidt, M., & Kühnberger, K.-U.. (2011). An argument for an analogical perspective on rationality and decision-making. In R. Verbrugge, & J. van Eijck (Eds.), Proceedings of the Workshop on Reasoning About Other Minds: Logical and Cognitive Perspectives (RAOM-2011), Groningen, The Netherland, 11 July 2011, volume 751 of CEUR Workshop Proceedings, (pp. 20–31). Scholar
  10. Blackburn, P., De Rijke, M., & Venema, Y. (2001). Modal logic. Number 53 in Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.Google Scholar
  11. De Nardo L., Ranzato F., Tapparo F. (2009) The subgraph similarity problem. IEEE Transactions on Knowledge and Data Engineering 21(5): 748–749CrossRefGoogle Scholar
  12. De Rijke, M. (1993). Extending Modal Logic. PhD Thesis, ILLC, University of Amsterdam. ILLC Dissertation Series DS-93-04.Google Scholar
  13. Dovier A., Piazza C. (2003) The subgraph bisimulation problem. IEEE Transactions on Knowledge and Data Engineering 15(4): 1055–1056CrossRefGoogle Scholar
  14. Edmonds J. (1965) Paths, trees, and flowers. Canadian Journal of Mathematics 17: 449–467CrossRefGoogle Scholar
  15. Fagin R., Halpern J. Y., Moses Y., Vardi M. Y. (1995) Reasoning about Knowledge. MIT Press, Cambridge, MAGoogle Scholar
  16. Feltovich N. (2000) Reinforcement-based vs. beliefs-based learning in experimental asymmetric- information games. Econometrica 68: 605–641CrossRefGoogle Scholar
  17. French, T., & Van Ditmarsch, H. (2008). Undecidability for arbitrary public announcement logic. In C. Areces & R. Goldblatt (Eds.), Advances in Modal Logic (pp. 23–42). College Publications.Google Scholar
  18. Garey M. R., Johnson D. S. (1990) Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Co., New YorkGoogle Scholar
  19. Gerbrandy, J. (1999). Bisimulations on Planet Kripke. PhD Thesis, ILLC, University of Amsterdam. ILLC Dissertation Series DS-1999-01.Google Scholar
  20. Gierasimczuk, N., & Szymanik, J. (2011a). Invariance properties of quantifiers and multiagent information exchange. In M. Kanazawa, A. Kornai, M. Kracht & H. Seki (Eds.), Proceedings of 12th Meeting on Mathematics of Language, volume 6878 of Lecture Notes in Computer Science (pp. 72–89). Berlin: Springer.Google Scholar
  21. Gierasimczuk, N., & Szymanik, J. (2011b). A note on a generalization of the muddy children puzzle. In Krzysztof R. Apt, editor, Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge (TARK-2011), Groningen, The Netherlands, July 12–14, 2011, pages 257–264. ACMGoogle Scholar
  22. Halpern J. Y., Moses Y. (1992) A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence 54: 319–379CrossRefGoogle Scholar
  23. Halpern J. Y., Vardi M. Y. (1989) The complexity of reasoning about knowledge and time. I. Lower bounds. Journal of Computer and Systems Science 38(1): 195–237CrossRefGoogle Scholar
  24. Henzinger, M. R., Henzinger, T. A., & Kopke, P. W. (1995). Computing simulations on finite and infinite graphs. In FOCS ’95: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (pp. 453–462). IEEE Computer Society Press.Google Scholar
  25. Hoffmann, C. M. (1982). Group-theoretic algorithms and graph isomorphism, volume 136 of Lecture Notes in Computer Science. Berlin: SpringerGoogle Scholar
  26. Karp R. M. (1972) Reducibility among combinatorial problems. In: Miller R. E., Thatcher J. W. (eds) Complexity of Computer Computations. Plenum Press, New York, pp 85–103CrossRefGoogle Scholar
  27. Köbler J., Schöning U., Torán J. (1993) The Graph Isomorphism Problem: Its Structural Complexity. Birkhauser Verlag, BaselCrossRefGoogle Scholar
  28. Kooi B., Van Benthem J. (2004) Reduction axioms for epistemic actions. In: Schmidt R., Pratt-Hartmann I., Reynolds M., Wansing H. (eds) Advances in Modal Logic 2004. Department of Computer Science, University of Manchester, Manchester, pp 197–211Google Scholar
  29. Kučera, A., & Mayr, R. (2002). Why is simulation harder than bisimulation?. In CONCUR ’02: Proceedings of the 13th International Conference on Concurrency Theory (pp. 594–610), London, UK, 2002. Berlin: Springer.Google Scholar
  30. Meijering B., Van Rijn H., Taatgen N. A., Verbrugge R. (2012) What eye movements can tell about theory of mind in a strategic game. PLoS ONE 7(9): e45961,09CrossRefGoogle Scholar
  31. Osborne M. J., Rubinstein A. (1994) A Course in Game Theory. MIT Press, Cambridge, MAGoogle Scholar
  32. Papadimitriou C. H. (1993) Computational Complexity. Addison Wesley, BostonGoogle Scholar
  33. Papadimitriou C. H., Steiglitz K. (1982) Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall Inc., Upper Saddle River, NJGoogle Scholar
  34. Plaza J. A. (1989) Logics of public communications. In: Emrich M. L., Pfeifer M.S., Hadzikadic M., Ras Z. W. (eds) Proceedings of the Fourth International Symposium on Methodologies for Intelligent Systems: Poster session program. Oak Ridge National Laboratory, Oak Ridge, TN, pp 201–216Google Scholar
  35. Pratt-Hartmann I., Moss L. S. (2009) Logics for the relational syllogistic. The Review of Symbolic Logic 2(04): 647–683CrossRefGoogle Scholar
  36. Van Benthem J. (1983) Modal Logic and Classical Logic. Bibliopolis, NaplesGoogle Scholar
  37. Van Benthem, J. (2010). Modal Logic for Open Minds. Number 199 in CSLI lecture notes. Stanford, CA: Center for the Study of Language and Information.Google Scholar
  38. Van Benthem J. (2011) Logical Dynamics of Information Flow. Cambridge University Press, Cambridge, MACrossRefGoogle Scholar
  39. Van Benthem J., Pacuit E. (2006) The tree of knowledge in action: Towards a common perspective. In: Hodkinson I., Governatori G., Venema Y. (eds) Advances in Modal Logic. College Publications,Google Scholar
  40. Van Ditmarsch, H., & French, T. (2009). Simulation and information: Quantifying over epistemic events. In Knowledge Representation for Agents and Multi-Agent Systems: First International Workshop, KRAMAS 2008, Sydney, Australia, 17 September 2008, Revised Selected Papers (pp. 51–65), Berlin, Heidelberg, 2009. Berlin: Springer.Google Scholar
  41. Van Ditmarsch H., der Hoek W., Kooi B. (2007) Dynamic Epistemic Logic. Springer, DordrechtGoogle Scholar
  42. Van Rooij I. (2008) The tractable cognition thesis. Cognitive Science: A Multidisciplinary Journal 32(6): 939–984CrossRefGoogle Scholar
  43. Szymanik J. (2010) Computational complexity of polyadic lifts of generalized quantifiers in natural language. Linguistics and Philosophy 33: 215–250CrossRefGoogle Scholar
  44. Szymanik J., Zajenkowski M. (2010) Comprehension of simple quantifiers. Empirical evaluation of a computational model. Cognitive Science: A Multidisciplinary Journal 34(3): 521–532CrossRefGoogle Scholar
  45. Verbrugge R. (2009) Logic and social cognition. The facts matter, and so do computational models. Journal of Philosophical Logic 38(6): 649–680CrossRefGoogle Scholar
  46. Wang, Y. (2010). Epistemic Modelling and Protocol Dynamics. PhD Thesis, ILLC, University of Amsterdam. ILLC Dissertation Series DS-2010-06.Google Scholar
  47. Weber R. (2001) Behavior and learning in the “dirty faces” game. Experimental Economics 4: 229–242Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Cédric Dégremont
    • 1
  • Lena Kurzen
    • 2
  • Jakub Szymanik
    • 1
  1. 1.Institute of Artificial IntelligenceUniversity of GroningenGroningenThe Netherlands
  2. 2.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands

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