ABSTRACT
In this article we study Gödel's functional interpretation from the perspective of learning. We define the notion of a learning algorithm, and show that intuitive realizers of the functional interpretation of both induction and various comprehension schemas can be given in terms of these algorithms. In the case of arithmetical comprehension, we clarify how our learning realizers compare to those obtained traditionally using bar recursion, demonstrating that bar recursive interpretations of comprehension correspond to 'forgetful' learning algorithms. The main purpose of this work is to gain a deeper insight into the semantics of programs extracted using the functional interpretation. However, in doing so we also aim to better understand how it relates to other interpretations of classical logic for which the notion of learning is inbuilt, such as Hilbert's epsilon calculus or the more recent learning-based realizability interpretations of Aschieri and Berardi.
- F. Aschieri. Learning, Realizability and Games in Classical Arithmetic. PhD thesis, Universita degli Studi di Torino and Queen Mary, University of London, 2011.Google Scholar
- F. Aschieri and S. Berardi. Interactive learning-based realizability for Heyting arithmetic with EM1. Logical Methods in Computer Science, 6(3), 2010.Google Scholar
- J. Avigad. Update procedures and the 1-consistency of arithmetic. Mathematical Logic Quarterly, 48(1):3--13, 2002.Google ScholarCross Ref
- J. Avigad and S. Feferman. Gödel's functional ("Dialectica") interpretation. In S. R. Buss, editor, Handbook of Proof Theory, volume 137 of Studies in Logic and the Foundations of Mathematics, pages 337--405. North Holland, Amsterdam, 1998.Google Scholar
- U. Berger. A computational interpretation of open induction. In F. Titsworth, editor, Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science, pages 326--334. IEEE Computer Society, 2004. Google ScholarDigital Library
- T. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 60:325--337, 1995. Google ScholarDigital Library
- V. de Paiva. The Dialectica Categories. PhD thesis, University of Cambridge, 1991. Published as Technical Report 213, Computer Laboratory, University of Cambridge.Google Scholar
- M. Escardó and P. Oliva. Selection functions, bar recursion and backward induction. Mathematical Structures in Computer Science, 20(2):127--168, 2010. Google ScholarDigital Library
- K. Gödel. Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. dialectica, 12:280--287, 1958.Google Scholar
- U. Kohlenbach. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Monographs in Mathematics. Springer, 2008.Google Scholar
- A. Kreuzer. Proof mining and Combinatorics: Program Extraction for Ramsey's Theorem for Pairs. PhD thesis, TU Darmstadt, 2012.Google Scholar
- D. Normann. The continuous functionals. In S. Abramsky, S. Artemov, R. A. Shore, and A. S. Troelstra, editors, Handbook of Computability Theory, volume 140 of Studies in Logic and the Foundations of Mathematics, pages 251--275. North Holland, Amsterdam, 1999.Google Scholar
- P. Oliva. Understanding and using Spector's bar recursive interpretation of classical analysis. In A. Beckmann, U. Berger, B. Löwe, and J. V. Tucker, editors, Proceedings of CiE'2006, volume 3988 of LNCS, pages 423--234, 2006. Google ScholarDigital Library
- P. Oliva and T. Powell. On Spector's bar recursion. Mathematical Logic Quarterly, 58:356--365, 2012.Google ScholarCross Ref
- P. Oliva and T. Powell. A constructive interpretation of Ramsey's theorem via the product of selection functions. Mathematical Structures in Computer Science, 25(8):1755--1778, 2015.Google ScholarCross Ref
- H. Schwichtenberg. Dialectica interpretation of well-founded induction. Mathematical Logic Quarterly, 54(3):229--239, 2008.Google ScholarCross Ref
- C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In F. D. E. Dekker, editor, Recursive Function Theory: Proc. Symposia in Pure Mathematics, volume 5, pages 1--27. American Mathematical Society, Providence, Rhode Island, 1962.Google ScholarCross Ref
- Gödel's functional interpretation and the concept of learning
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