Advertisement

Interactive Learning-Based Realizability Interpretation for Heyting Arithmetic with EM1

  • Federico Aschieri
  • Stefano Berardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)

Abstract

We interpret classical proofs as constructive proofs (with constructive rules for ∨ , ∃) over a suitable structure \({\mathcal N}\) for the language of natural numbers and maps of Gödel’s system \({\mathcal{T}}\). We introduce a new Realization semantics we call “Interactive learning-based Realizability”, for Heyting Arithmetic plus EM 1 (Excluded middle axiom restricted to \(\Sigma^0_1\) formulas). Individuals of \({\mathcal N}\) evolve with time, and realizers may “interact” with them, by influencing their evolution. We build our semantics over Avigad’s fixed point result [1], but the same semantics may be defined over different constructive interpretations of classical arithmetic (in [7], continuations are used). Our notion of realizability extends Kleene’s realizability and differs from it only in the atomic case: we interpret atomic realizers as “learning agents”.

Keywords

Free Variable Atomic Formula Natural Deduction Closed Term Classical Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avigad, J.: Update Procedures and the 1-Consistency of Arithmetic. Math. Log. Q. 48(1), 3–13 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akama, Y., Berardi, S., Hayashi, S., Kohlenbach, U.: An Arithmetical Hierarchy of the Law of Excluded Middle and Related Principles. In: LICS 2004, pp. 192–201 (2004)Google Scholar
  3. 3.
    Aschieri, F., Berardi, S.: An Interactive Realizability... (Full Paper), Tech. Rep., Un. of Turin (2009), http://www.di.unito.it/~stefano/Realizers2009.pdf
  4. 4.
    Berardi, S.: Classical Logic as Limit .... MSCS 15(1), 167–200 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Berardi, S.: Some intuitionistic equivalents of classical principles for degree 2 formulas. Annals of Pure and Applied Logic 139(1-3), 185–200 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berardi, S., Coquand, T., Hayashi, S.: Games with 1-Bactracking. In: GALOP 2005 (2005)Google Scholar
  7. 7.
    Berardi, S., de’Liguoro, U.: A calculus of realizers for EM 1-Arithmetic. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 215–229. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Coquand, T.: A Semantic of Evidence for Classical Arithmetic. Journal of Symbolic Logic 60, 325–337 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dalen, D.v.: Logic and Structure, 3rd edn. Springer-, Heidelberg (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Girard, J.-Y.: Proofs and Types. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  11. 11.
    Gold, E.M.: Limiting Recursion. Journal of Symbolic Logic 30, 28–48 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hayashi, S., Sumitomo, R., Shii, K.: Towards Animation of Proofs -Testing Proofs by Examples. Theoretical Computer Science (2002)Google Scholar
  13. 13.
    Hayashi, S.: Can Proofs be Animated by Games? FI 77(4), 331–343 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hayashi, S.: Mathematics based on incremental learning - Excluded Middle and Inductive Inference. Theoretical Computer Science 350, 125–139 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kleene, S.C.: On the Interpretation of Intuitionistic Number Theory. Journal of Symbolic Logic 10(4), 109–124 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Popper, K.: The Logic of Scientific Discovery. Routledge Classics, Routledge (2002)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Federico Aschieri
    • 1
  • Stefano Berardi
    • 1
  1. 1.C.S. Dept.University of TurinItaly

Personalised recommendations