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)


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”.


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

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