Logic of Proofs for Bounded Arithmetic

  • Evan Goris
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3967)


The logic of proofs is known to be complete for the semantics of proofs in Peano Arithmetic PA. In this paper we present a refinement of this theorem, we will show that we can assure that all the operations on proofs can be realized by feasible, that is PTIME-computable, functions. In particular we will show that the logic of proofs is complete for the semantics of proofs in Buss’ bounded arithmetic S \(^{1}_{2}\) . In view of recent applications of the Logic of Proofs in epistemology this result shows that explicit knowledge in the propositional framework can be made computationally feasible.


Explicit Knowledge Propositional Variable Epistemic Logic Induction Scheme Provability Logic 


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  1. 1.
    Artemov, S.N.: Explicit provability and constructive semantics. Bulletin of Symbolic Logic 7(1), 1–36 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Artemov, S., Nogina, E.: On epistemic logic with justification. In: van der Meyden, R. (ed.) Theoretical Aspects of Rationality and Knowledge Proceedings of the Tenth Conference (TARK 2005), June 10-12, 2005, pp. 279–294. National University of Singapore, Singapore (2005)Google Scholar
  3. 3.
    Artemov, S.: Evidence-Based Common Knowledge. Technical Report TR-2004018, CUNY Ph.D. Program in Computer Science (2004)Google Scholar
  4. 4.
    Artemov, S., Kuznets, R.: Explicit knowledge is not logically omniscient (unpublished, 2005)Google Scholar
  5. 5.
    Artemov, S.N., Beklemishev, L.D.: Provability logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn., vol. 13, pp. 229–403. Kluwer, Dordrecht (2004)Google Scholar
  6. 6.
    Buss, S.R.: Bounded arithmetic. Bibliopolis, Napels (1986) Revision of PhD. thesisGoogle Scholar
  7. 7.
    Krajíček, J.: Bounded arithmetic, propositional logic, and complexity theory. Cambridge University Press, New York (1995)CrossRefMATHGoogle Scholar
  8. 8.
    Buss, S.R.: First-Order Theory of Arithmetic. In: Buss, S.R. (ed.) Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics, vol. 137, pp. 475–546. Elsevier, Amsterdam (1998)CrossRefGoogle Scholar
  9. 9.
    Parikh, R.: Existence and feasability in arithmetic. Journal of Symbolic Logic 36, 494–508 (1971)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Evan Goris
    • 1
  1. 1.The Graduate Center of the City University of New YorkNew YorkUSA

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