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Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1

Published online by Cambridge University Press:  25 October 2013

FEDERICO ASCHIERI
FEDERICO ASCHIERI*
Affiliation:
Laboratoire de l'Informatique du Parallélisme (UMR 5668), équipe Plume, École Normale Supérieure de Lyon – Université de Lyon, 46 Allée d'Italie, 69007 Lyon, France Email: federico.aschieri@ens-lyon.fr
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Abstract

We introduce a realizability semantics based on interactive learning for full second-order Heyting arithmetic with excluded middle and Skolem axioms over Σ10-formulas. Realizers are written in a classical version of Girard's System $\mathsf{F}$ and can be viewed as programs that learn by interacting with the environment. We show that the realizers of any Π20-formula represent terminating learning processes whose outcomes are numerical witnesses for the existential quantifier of the formula.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 24 , Issue 6 , December 2014 , e240601
Copyright
Copyright © Cambridge University Press 2013 

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References

Aschieri, F. (2011a) Learning, Realizability and Games in Classical Arithmetic, Ph.D. Thesis, Queen Mary, University of London and University of Turin. (Available at http://arxiv.org/abs/1012.4992.Google Scholar
Aschieri, F. (2011b) Transfinite Update Procedures for Predicative Systems of Analysis. In: Bezem, M. (ed.) Computer Science Logic (CSL'11) – 25th International Workshop/20th Annual Conference of the EACSL, Leibniz International Proceedings in Informatics 12 20–34.Google Scholar
Aschieri, F. (2012) A Constructive Analysis of Learning in Peano Arithmetic. Annals of Pure and Applied Logic 163 (11)1448–1470.CrossRefGoogle Scholar
Aschieri, F. (2013) Learning Based Realizability for HA + EM1 and 1-Backtracking Games: Soundness and Completeness. Annals of Pure and Applied Logic 164 (6)591–617.CrossRefGoogle Scholar
Aschieri, F. and Berardi, S. (2010) Interactive Learning-Based Realizability for Heyting Arithmetic with EM1. Logical Methods in Computer Science 6 (3)19.Google Scholar
Aschieri, F. and Berardi, S. (2012) A New Use of Friedman's Translation: Interactive Realizability. In: Berger, U., Diener, H., Schuster, P. and Seisenberger, M. (eds.) Logic, Construction, Computation, Ontos-Verlag Series in Mathematical Logic 11–49.CrossRefGoogle Scholar
Avigad, J. (2000) A Realizability Interpretation for Classical Arithmetic. In: Buss, S. R., Hájek, P. and Pudlák, P. (eds.) Logic Colloquium '98, Lecture Notes in Logic 13, AK Peters 57–90.Google Scholar
Avigad, J. (2002) Update Procedures and 1-Consistency of Arithmetic. Mathematical Logic Quarterly 48 (1)3–13.3.0.CO;2-6>CrossRefGoogle Scholar
Avigad, J. (2003) Eliminating Definitions and Skolem functions in First-Order Logic. ACM Transactions on Computational Logic 4 (3).CrossRefGoogle Scholar
Berardi, S. and de'Liguoro, U. (2008) A Calculus of Realizers for EM1 Arithmetic. In: Kaminski, M. and Martini, S. (eds.) Computer Science Logic: Proceedings 22nd International Workshop, CSL 2008, at 17th Annual Conference of the EACSL. Springer-Verlag Lecture Notes in Computer Science 5213215–229.CrossRefGoogle Scholar
Berardi, S., Bezem, M. and Coquand, T. (1998) On the Computational Content of the Axiom of Choice. Journal of Symbolic Logic 63 (2)600–622.CrossRefGoogle Scholar
Berger, U. (2005) Strong Normalization for Applied Lambda Calculi. Logical Methods in Computer Science 1 (2–3)1–14.Google Scholar
Coquand, T. (1991) A Semantics of Evidence for Classical Arithmetic. In: Informal Proceedings of the Second Workshop on Logical Frameworks 87–99.Google Scholar
Coquand, T. (1995) A Semantics of Evidence for Classical Arithmetic. Journal of Symbolic Logic 60 325–337.CrossRefGoogle Scholar
Friedman, H. (1978) Classically and Intuitionistically Provable Recursive Functions. Springer-Verlag Lecture Notes in Mathematics 669 21–27.CrossRefGoogle Scholar
Gentzen, G. (1935a) Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen 112 493–565.CrossRefGoogle Scholar
Gentzen, G. (1935b) Untersuchungen fiber das logische Schliessen. Mathematische Zeitschrift 39 176–210, 405–431.CrossRefGoogle Scholar
Girard, J.-Y. (1989) Proofs and Types, Cambridge University Press.Google Scholar
Gödel, K. (1958) Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12 (3–4)280–287.CrossRefGoogle Scholar
Kohlenbach, U. (2008) Applied Proof Theory, Springer-Verlag.Google Scholar
Kreisel, G. (1951) On the interpretation of non-finitist proofs, part I. Journal of Symbolic Logic 16 241–267.Google Scholar
Kreisel, G. (1959) Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, North-Holland101–128.Google Scholar
Krivine, J.-L. (1990) Lambda-calcul, types et modèles, Studies in Logic and Foundations of Mathematics, Masson 1–176.Google Scholar
Krivine, J.-L. (2009) Realizability in Classical Logic. In: Curien, P.-L., Herbelin, H., Krivine, J.-L. and Melliès, P.-A. (authors) Interactive models of computation and program behavior, Panoramas et synthèses 27, Socièté Mathématique de France 197–229.Google Scholar
Mints, G., Tupailo, S. and Bucholz, W. (1996) Epsilon Substitution Method for Elementary Analysis. Archive for Mathematical Logic 35 (2)103–130.CrossRefGoogle Scholar
Miquel, A. (2009) Relating classical realizability and negative translation for existential witness extraction. In: Curien, P.-L. (ed.) Typed Lambda Calculi and Applications: Proceedings 9th International Conference, TLCA 2009. Springer-Verlag Lecture Notes in Computer Science 5608188–202.CrossRefGoogle Scholar
Oliva, P. and Striecher, T. (2008) On Krivine Realizability Interpretation of Second-Order Classical Arithmetic. Fundamenta Informaticae 84 (2)207–220.Google Scholar
Schwichtenberg, H. and Troelstra, A. (1996) Basic Proof Theory, Cambridge University Press.Google Scholar
Sorensen, M. H. and Urzyczyn, P. (2006) Lectures on the Curry–Howard isomorphism, Studies in Logic and the Foundations of Mathematics 149, Elsevier.Google Scholar
Spector, C. (1962) Provably Recursive Functionals of Analysis: a Consistency Proof of Analysis by an Extension of Principles in Current Intuitionistic Mathematics. In: Dekker, J. (ed.) Recursive Function Theory: Proceedings of Symposia in Pure Mathematics 5, AMS 1–27.Google Scholar
Troelstra, A. (1973) Metamathematical Investigations of Intuitionistic Arithmetic and Analysis. Springer-Verlag Lecture Notes in Mathematics 344.CrossRefGoogle Scholar
Troelstra, A. and van Dalen, D. (1988) Constructivism in Mathematics, vol. I, North-Holland.Google Scholar