Skip to main content
SpringerLink
Log in

Exploring the tractability border in epistemic tasks

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ågotnes T., Balbiani P., Van Ditmarsch H., Seban P. (2010) Group announcement logic. Journal of Applied Logic 8(1): 62–81

    Article  Google Scholar 

  • Aucher G. (2010) An internal version of epistemic logic. Studia Logica 94(1): 1–22

    Article  Google Scholar 

  • Aumann R. J. (1999) Interactive epistemology I: Knowledge. International Journal of Game Theory 28(3): 263–300

    Article  Google Scholar 

  • 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): 305–334

    Article  Google Scholar 

  • Balcázar J. L., Gabarró J., Santha M. (1992) Deciding bisimilarity is P-complete. Formal Aspects of Computing 4(6A): 638–648

    Article  Google Scholar 

  • Baltag A., Moss L. S. (2004) Logics for epistemic programs. Synthese 139(2): 165–224

    Article  Google Scholar 

  • 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.

  • Baral C., Zhang Y. (2005) Knowledge updates: Semantics and complexity issues. Artificial Intelligence 164(1–2): 209–243

    Article  Google Scholar 

  • 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). CEUR-WS.org.

  • Blackburn, P., De Rijke, M., & Venema, Y. (2001). Modal logic. Number 53 in Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.

  • De Nardo L., Ranzato F., Tapparo F. (2009) The subgraph similarity problem. IEEE Transactions on Knowledge and Data Engineering 21(5): 748–749

    Article  Google Scholar 

  • De Rijke, M. (1993). Extending Modal Logic. PhD Thesis, ILLC, University of Amsterdam. ILLC Dissertation Series DS-93-04.

  • Dovier A., Piazza C. (2003) The subgraph bisimulation problem. IEEE Transactions on Knowledge and Data Engineering 15(4): 1055–1056

    Article  Google Scholar 

  • Edmonds J. (1965) Paths, trees, and flowers. Canadian Journal of Mathematics 17: 449–467

    Article  Google Scholar 

  • Fagin R., Halpern J. Y., Moses Y., Vardi M. Y. (1995) Reasoning about Knowledge. MIT Press, Cambridge, MA

    Google Scholar 

  • Feltovich N. (2000) Reinforcement-based vs. beliefs-based learning in experimental asymmetric- information games. Econometrica 68: 605–641

    Article  Google Scholar 

  • 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.

  • Garey M. R., Johnson D. S. (1990) Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman and Co., New York

    Google Scholar 

  • Gerbrandy, J. (1999). Bisimulations on Planet Kripke. PhD Thesis, ILLC, University of Amsterdam. ILLC Dissertation Series DS-1999-01.

  • 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.

  • 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. ACM

  • Halpern J. Y., Moses Y. (1992) A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence 54: 319–379

    Article  Google Scholar 

  • 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): 195–237

    Article  Google Scholar 

  • 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.

  • Hoffmann, C. M. (1982). Group-theoretic algorithms and graph isomorphism, volume 136 of Lecture Notes in Computer Science. Berlin: Springer

  • 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 85–103

    Chapter  Google Scholar 

  • Köbler J., Schöning U., Torán J. (1993) The Graph Isomorphism Problem: Its Structural Complexity. Birkhauser Verlag, Basel

    Book  Google Scholar 

  • 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 197–211

    Google Scholar 

  • 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.

  • 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): e45961,09

    Article  Google Scholar 

  • Osborne M. J., Rubinstein A. (1994) A Course in Game Theory. MIT Press, Cambridge, MA

    Google Scholar 

  • Papadimitriou C. H. (1993) Computational Complexity. Addison Wesley, Boston

    Google Scholar 

  • Papadimitriou C. H., Steiglitz K. (1982) Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall Inc., Upper Saddle River, NJ

    Google Scholar 

  • 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 201–216

    Google Scholar 

  • Pratt-Hartmann I., Moss L. S. (2009) Logics for the relational syllogistic. The Review of Symbolic Logic 2(04): 647–683

    Article  Google Scholar 

  • Van Benthem J. (1983) Modal Logic and Classical Logic. Bibliopolis, Naples

    Google Scholar 

  • 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.

  • Van Benthem J. (2011) Logical Dynamics of Information Flow. Cambridge University Press, Cambridge, MA

    Book  Google Scholar 

  • 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,

  • 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.

  • Van Ditmarsch H., der Hoek W., Kooi B. (2007) Dynamic Epistemic Logic. Springer, Dordrecht

    Google Scholar 

  • Van Rooij I. (2008) The tractable cognition thesis. Cognitive Science: A Multidisciplinary Journal 32(6): 939–984

    Article  Google Scholar 

  • Szymanik J. (2010) Computational complexity of polyadic lifts of generalized quantifiers in natural language. Linguistics and Philosophy 33: 215–250

    Article  Google Scholar 

  • Szymanik J., Zajenkowski M. (2010) Comprehension of simple quantifiers. Empirical evaluation of a computational model. Cognitive Science: A Multidisciplinary Journal 34(3): 521–532

    Article  Google Scholar 

  • Verbrugge R. (2009) Logic and social cognition. The facts matter, and so do computational models. Journal of Philosophical Logic 38(6): 649–680

    Article  Google Scholar 

  • Wang, Y. (2010). Epistemic Modelling and Protocol Dynamics. PhD Thesis, ILLC, University of Amsterdam. ILLC Dissertation Series DS-2010-06.

  • Weber R. (2001) Behavior and learning in the “dirty faces” game. Experimental Economics 4: 229–242

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Artificial Intelligence, University of Groningen, Groningen, The Netherlands

    Cédric Dégremont & Jakub Szymanik

  2. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Lena Kurzen

Authors
  1. Cédric Dégremont
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lena Kurzen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jakub Szymanik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cédric Dégremont.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dégremont, C., Kurzen, L. & Szymanik, J. Exploring the tractability border in epistemic tasks. Synthese 191, 371–408 (2014). https://doi.org/10.1007/s11229-012-0215-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-012-0215-7

Keywords

Search

Navigation