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The Proof Theory of Common Knowledge

  • Michel Marti
  • Thomas Studer
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 12)

Abstract

Common knowledge of a proposition A can be characterized by the following infinitary conjunction: everybody knows A and everybody knows that everybody knows A and everybody knows that everybody knows that everybody knows A and so on. We present a survey of deductive systems for the logic of common knowledge. In particular, we present two different Hilbert-style axiomatizations and two infinitary cut-free sequent systems. Further we discuss the problem of syntactic cut-elimination for common knowledge. The paper concludes with a list of open problems.

Keywords

Common knowledge Multi-agent systems Proof theory Infinitary deductive systems Cut-elimination 

Notes

Acknowledgements

We would like to thank Rajeev Goré for many helpful comments. This research is supported by the SNSF project 153169.

References

  1. 1.
    Abate P, Goré R, Widmann F (2007) Cut-free single-pass tableaux for the logic of common knowledge. In: Workshop on agents and deduction at TABLEAUX 2007Google Scholar
  2. 2.
    Alberucci L, Jäger G (2005) About cut elimination for logics of common knowledge. Ann Pure Appl Log 133:73–99Google Scholar
  3. 3.
    Antonakos E (2007) Justified and common knowledge: limited conservativity. In: Artemov SN, Nerode A (eds) Logical foundations of computer science, LFCS 2007. LNCS, vol 4514. Springer, pp 1–11Google Scholar
  4. 4.
    Antonakos E (2013) Explicit generic common knowledge. In: Artemov S, Nerode A (eds) Logical foundations of computer science. Lecture Notes in Computer Science, vol 7734. Springer, pp 16–28Google Scholar
  5. 5.
    Antonakos E (2016) Pairing traditional and generic common knowledge. In: Artemov S, Nerode A (eds) Logical foundations of computer science. Lecture Notes in Computer Science, vol 9537. Springer, pp 14–26Google Scholar
  6. 6.
    Artemov S (2001) Explicit provability and constructive semantics. Bull Symb Log 7(1):1–36Google Scholar
  7. 7.
    Artemov S (2006) Justified common knowledge. Theor Comput Sci 357(1):4–22Google Scholar
  8. 8.
    Aumann R (1976) Agreeing to disagree. Ann Stat 4(6):1236–1239Google Scholar
  9. 9.
    Barwise J (1988) Three views of common knowledge. In: Vardi M (ed) Proceedings of theoretical aspects of reasoning about knowledge. Morgan Kaufman, pp 365–379Google Scholar
  10. 10.
    Barwise J (1989) The situation in logic. In: CSLI lecture notes, vol 17Google Scholar
  11. 11.
    Brünnler K (2006) Deep sequent systems for modal logic. In: Governatori G, Hodkinson I, Venema Y (eds) Advances in modal logic, vol 6. College Publications, pp 107–119Google Scholar
  12. 12.
    Brünnler K, Lange M (2008) Cut-free sequent systems for temporal logics. J Log Algeb Program 76:216–225Google Scholar
  13. 13.
    Brünnler K, Studer T (2009) Syntactic cut-elimination for common knowledge. Ann Pure Appl Log 160:82–95.  https://doi.org/10.1016/j.apal.2009.01.014
  14. 14.
    Brünnler K, Studer T (2012) Syntactic cut-elimination for a fragment of the modal mu-calculus. Ann Pure Appl Log 163(12):1838–1853Google Scholar
  15. 15.
    Bucheli S (2012) Justification logics with common knowledge. PhD thesis, Universität BernGoogle Scholar
  16. 16.
    Bucheli S, Kuznets R, Studer T (2010) Two ways to common knowledge. In: Bolander T, Braüner T (eds) Proceedings of the 6th workshop on methods for modalities (M4M–6 2009), Copenhagen, Denmark, 12–14 Nov 2009. Electronic Notes in Theoretical Computer Science. Elsevier, pp 83–98.  https://doi.org/10.1016/j.entcs.2010.04.007
  17. 17.
    Bucheli S, Kuznets R, Studer T (2011) Justifications for common knowledge. J Appl Non-Class Log 21(1):35–60.  https://doi.org/10.3166/JANCL.21.35-60
  18. 18.
    Buchholz W (1981) The \(\Omega _{\mu +1}\)-rule. In: Buchholz W, Feferman S, Pohlers W, Sieg W (eds) Iterated inductive definitions and subsystems of analysis: recent proof theoretic studies. Lecture Notes in Mathematics, vol 897. Springer, pp 189–233.  https://doi.org/10.1007/BFb0091898
  19. 19.
    Dax C, Hofmann M, Lange M (2006) A proof system for the linear time \(\mu \)-calculus. In: Proceedings of the 26th conference on foundations of software technology and theoretical computer science, FSTTCS’06. LNCS, vol 4337. Springer, pp 274–285Google Scholar
  20. 20.
    Fagin R, Halpern JY, Moses Y, Vardi MY (1995) Reasoning about knowledge. MIT PressGoogle Scholar
  21. 21.
    Fontaine G (2008) Continuous fragment of the mu-calculus. In: Kaminski M, Martini S (eds) Computer science logic. LNCS, vol 5213. Springer, pp 139–153.  https://doi.org/10.1007/978-3-540-87531-4-12
  22. 22.
    Goré R (2014) And-or tableaux for fixpoint logics with converse: LTL, CTL, PDL and CPDL. In: Demri S, Kapur D Weidenbach C (eds) Automated reasoning: 7th International Joint Conference, IJCAR 2014. Proceedings. Springer, pp 26–45.  https://doi.org/10.1007/978-3-319-08587-6-3
  23. 23.
    Grädel E, Thomas W, Wilke T (eds) (2002) Automata logics, and infinite games: a guide to current research. Springer. ISBN 3-540-00388-6Google Scholar
  24. 24.
    Gudzhinskas E (1982) Syntactical proof of the elimination theorem for von Wrights temporal logic. Math Logika Primenen 2:113–130Google Scholar
  25. 25.
    Halpern JY, Moses Y (1990) Knowledge and common knowledge in a distributed environment. J ACM 37(3):549–587Google Scholar
  26. 26.
    Hintikka J (1962). Knowledge and belief: an introduction to the logic of the two notions. Cornell University PressGoogle Scholar
  27. 27.
    Jäger G, Marti M (2015) Intuitionistic common knowledge or beliefGoogle Scholar
  28. 28.
    Jäger G, Studer T (2011) A Buchholz rule for modal fixed point logics. Logica Universalis 5:1–19.  https://doi.org/10.1007/s11787-010-0022-1
  29. 29.
    Jäger G, Kretz M, Studer T (2007) Cut-free common knowledge. J Appl Log 5(4):681–689Google Scholar
  30. 30.
    Kashima R (1994) Cut-free sequent calculi for some tense logics. Studia Logica 53(1):119–136Google Scholar
  31. 31.
    Kokkinis I, Studer T (2016) Cyclic proofs for linear temporal logic. In: Probst D, Schuster P (eds) Concepts of proof in mathematics, philosophy, and computer science. Ontos mathematical logic, vol 6. De GruyterGoogle Scholar
  32. 32.
    Kretz M, Studer T (2006) Deduction chains for common knowledge. J Appl Log 4:331–357Google Scholar
  33. 33.
    Kuznets R, Studer T (2012) Justifications, ontology, and conservativity. In: Bolander T, Braüner T, Ghilardi S, Moss L (eds) AiML 9. College Publications, pp 437–458Google Scholar
  34. 34.
    Kuznets R, Studer T (2013) Update as evidence: belief expansion. In: Artemov SN, Nerode A (eds) Logical foundations of computer science LFCS 13. LNCS, vol 7734. Springer, pp 266–279Google Scholar
  35. 35.
    Lange M, Stirling C (2001) Focus games for satisfiability and completeness of temporal logic. In: Proceedings of LICSGoogle Scholar
  36. 36.
    Leivant D (1981) A proof theoretic methodology for propositional dynamic logic. In: Proceedings of the international colloquium on formalization of programming concepts. LNCS. Springer, pp 356–373Google Scholar
  37. 37.
    Lewis D (1969) Convention: a philosophical studyGoogle Scholar
  38. 38.
    Martin D (1975) Borel determinacy. Ann Math 102:363–371Google Scholar
  39. 39.
    McCarty J, Sato M, Hayashi T, Igarishi S (1978) On the model theory of knowledge. Technical report STAN-CS-78-657. Stanford UniversityGoogle Scholar
  40. 40.
    Meyer J-J, van der Hoek W (1995) Epistemic logic for AI and computer science. Cambridge University PressGoogle Scholar
  41. 41.
    Mints G, Studer T (2012) Cut-elimination for the mu-calculus with one variable. In: Fixed points in computer science 2012. EPTCS, vol 77. Open Publishing Association, pp 47–54Google Scholar
  42. 42.
    Niwinski D, Walukiewicz I (1996) Games for the mu-calculus. Theor Comput Sci 163(1&2):99–116Google Scholar
  43. 43.
    Paech B (1989) Gentzen-systems for propositional temporal logics. In: Börger E, Büning HK, Richter MM (eds) CSL ’88: 2nd Workshop on Computer Science Logic, Proceedings. Springer, pp 240–253.  https://doi.org/10.1007/BFb0026305
  44. 44.
    Poggiolesi F, Hill B (2015) Common knowledge: a finitary calculus with a syntactic cut-elimination procedure. Logique et Analyse 58(230):136–159Google Scholar
  45. 45.
    Pohlers W (1989) Proof theory—an introduction. SpringerGoogle Scholar
  46. 46.
    Pohlers W (1998) Subsystems of set theory and second order number theory. In: Buss S (ed) Handbook of proof theory. Elsevier, pp 209–335Google Scholar
  47. 47.
    Schelling T (1960) The strategy of conflictGoogle Scholar
  48. 48.
    Schiffer S (1972) MeaningGoogle Scholar
  49. 49.
    Schütte K (1977) Proof theory. SpringerGoogle Scholar
  50. 50.
    Streett RS, Emerson EA (1989) An automata theoretic decision procedure for the propositional modal mu-calculus. Inf Comput 81:249–264Google Scholar
  51. 51.
    Studer T (2009) Common knowledge does not have the Beth property. Inf Process Lett 109:611–614Google Scholar
  52. 52.
    van Benthem J, van Eijck J, Kooi B (2005) Common knowledge in update logics. In: Proceedings of the 10th conference on theoretical aspects of rationality and knowledge, TARK ’05, pp 253–261. ISBN 981-05-3412-4Google Scholar
  53. 53.
    Wehbe R (2010) Annotated systems for common knowledge. PhD thesis, Universität BernGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.University of BernBernSwitzerland

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