Abstract
The problem of giving a computational meaning to classical reasoning lies at the heart of logic. This article surveys three famous solutions to this problem – the epsilon calculus, modified realizability and the Dialectica interpretation – and re-examines them from a modern perspective, with a particular emphasis on connections with algorithms and programming.
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References
Ackermann, W. (1940). Zur widerspruchsfreiheit der zahlentheorie. Mathematische Annalen, 117, 162–194.
Aschieri, F. (2011a). Learning, realizability and games in classical arithmetic. Ph.D. thesis, Universita degli Studi di Torino and Queen Mary, University of London.
Aschieri, F. (2011b). Transfinite update procedures for predicative systems of analysis. In Computer Science Logic (CSL’11) (Volume 12 of Leibniz international proceedings in informatics, pp. 1–15).
Aschieri, F., & Berardi, S. (2010). Interactive learning-based realizability for Heyting arithmetic with EM1. Logical Methods in Computer Science, 6(3), 1–22.
Avigad, J. (2002). Update procedures and the 1-consistency of arithmetic. Mathematical Logic Quarterly, 48(1), 3–13.
Avigad, J., & Feferman, S. (1998). Gödel’s functional (“Dialectica”) interpretation. In S. R. Buss (Ed.), Handbook of proof theory (Volume 137 of studies in logic and the foundations of mathematics, pp. 337–405). Amsterdam: North Holland.
Avigad, J., & Zach, R. (2013). The epsilon calculus. In The Stanford encyclopedia of philosophy. Revised edition Nov 2013 edition. Available online at https://plato.stanford.edu/entries/epsilon-calculus/.
Berardi, S., Bezem, M., & Coquand, T. (1998). On the computational content of the axiom of choice. Journal of Symbolic Logic, 63(2), 600–622.
Berger, U. (2004). A computational interpretation of open induction. In Proceedings of LICS 2004 (pp. 326–334). IEEE Computer Society.
Berger, U., Lawrence, A., Forsberg, F., & Seisenberger, M. (2015). Extracting verified decision procedures: DPLL and resolution. Logical Methods in Computer Science, 11(1:6), 1–18.
Berger, U., Miyamoto, K., & Schwichtenberg, H. (2016). Logic for Gray-code computation. In D. Probst & P. Schuster (Eds.), Concepts of proof in mathematics, philosophy, and computer science. Berlin/Boston: De Gruyter.
Berger, U., & Schwichtenberg, H. (1995). Program extraction from classical proofs. In D. Leivant (Ed.), Logic and Computational Complexity Workshop (LCC’94) (Volume 960 of lecture notes in computer science, pp. 77–97).
Berger, U., Seisenberger, M., & Woods, G. (2014). Extracting imperative programs from proofs: In-place quicksort. In Proceedings of TYPES 2013 (Volume 26 of LIPIcs, pp. 84–106).
Birolo, G. (2012). Interative realizability, monads and witness extraction. Ph.D. thesis, Universitá degli Studi di Torino.
Coquand, T. (1995). A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 60, 325–337.
de Paiva, V. (1991). The dialectica categories. Ph.D. thesis, University of Cambridge. Published as Technical report 213, Computer Laboratory, University of Cambridge.
Ferreira, G., & Oliva, P. (2012). On the relation between various negative translations. In Logic, construction, computation (Volume 3 of Ontos-Verlag mathematical logic, pp. 227–258).
Giese, M., & Ahrendt, W. (1999). Hilbert’s epsilon-terms in automated theorem proving. In TABLEAUX (Volume 1617 of LNCS, pp. 171–185). Springer.
Gödel, K. (1958). Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica, 12, 280–287.
Gödel, K. (1990). Collected works (Vol. II). New York: Oxford University Press.
Kohlenbach, U. (1999). On the no-counterexample interpretation. Journal of Symbolic Logic, 64, 1491–1511.
Kohlenbach, U. (2008). Applied proof theory: Proof interpretations and their use in mathematics (Monographs in mathematics). Berlin/Heidelberg: Springer.
Kohlenbach, U. (2018). Kreisel’s ‘shift of emphasis’ and contemporary proof mining. Chapter for forthcoming book ‘Intuitionism, Computation, and Proof: Selected themes from the research of G. Kreisel’.
Kohlenbach, U., & Safarik, P. (2014). Fluctuations, effective learnability and metastability in analysis. Annals of Pure and Applied Logic, 165, 266–304.
Kreisel, G. (1951). On the interpretation of non-finitist proofs, Part I. Journal of Symbolic Logic, 16, 241–267.
Kreisel, G. (1952). On the interpretation of non-finitist proofs, part II: Interpretation of number theory. Journal of Symbolic Logic, 17, 43–58.
Kreisel, G. (1959). Interpretation of analysis by means of functionals of finite type. In A. Heyting (Ed.), Constructivity in mathematics (pp. 101–128). Amsterdam: North-Holland.
Krivine, J.-L. (2009). Realizability in classical logic. In Interactive models of computation and program behaviour (Volume 27 of Panoramas et synthèses, pp. 197–229). Société Mathématique de France.
Minlog. (2018). http://www.mathematik.uni-muenchen.de/~logik/minlog/. Official homepage of MINLOG, as of January 2018.
Mints, G. (2008). Cut elimination for a simple formulation of the epsilon calculus. Annals of Pure and Applied Logic, 152(1–3), 148–169.
Miquel, A. (2011). Existential witness extraction in classical realizability and via a negative translation. Logical Methods in Computer Science, 7(2:2), 1–47.
Moggi, E. (1991). Notions of computation and monads. Information and Computation, 93(1), 55–92.
Moser, G. (2006). Ackermann’s substitution method (remixed). Annals of Pure and Applied Logic, 142(1–3), 1–18.
Moser, G., & Zach, R. (2006). The epsilon calculus and Herbrand complexity. Studia Logica, 82(1), 133–155.
Novikoff, P. S. (1943). On the consistency of certain logical calculi. Matematiceskij Sbornik, 12(54), 230–260.
Oliva, P. (2006). Unifying functional interpretations. Notre Dame Journal of Formal Logic, 47(2), 263–290.
Oliva, P., & Powell, T. (2015a). A constructive interpretation of Ramsey’s theorem via the product of selection functions. Mathematical Structures in Computer Science, 25(8), 1755–1778.
Oliva, P., & Powell, T. (2015b). A game-theoretic computational interpretation of proofs in classical analysis. In R. Kahle & M. Rathjen (Eds.), Gentzen’s centenary: The quest for consistency (pp. 501–532). Cham: Springer.
Oliva, P., & Powell, T. (2017). Spector bar recursion over finite partial functions. Annals of Pure and Applied Logic, 168(5), 887–921.
Pedrot, P.-M. (2015). A materialist dialectica. Ph.D. thesis, Université Paris Diderot.
Powell, T. (2012). Applying Gödel’s Dialectica interpretation to obtain a constructive proof of Higman’s lemma. In Proceedings of classical logic and computation’12 (Volume 97 of EPTCS, pp. 49–62).
Powell, T. (2013). On bar recursive interpretations of analysis. Ph.D. thesis, Queen Mary University of London.
Powell, T. (2014). The equivalence of bar recursion and open recursion. Annals of Pure and Applied Logic, 165(11), 1727–1754.
Powell, T. (2015). Parametrised bar recursion: A unifying framework for realizability interpretations of classical dependent choice. Journal of Logic and Computation. (Advance access).
Powell, T. (2016). Gödel’s functional interpretation and the concept of learning. In Proceedings of Logic in Computer Science (LICS 2016) (pp. 136–145). ACM.
Powell, T. (2018a). A functional interpretation with state. In Proceedings of Logic in Computer Science (LICS 2018). ACM.
Powell, T. (2018b, To appear). Well quasi-orders and the functional interpretation. In P. Schuster, M. Seisenberger, & A. Weiermann (Eds.), Well quasi-orders in computation, logic, language and reasoning (Trends in logic). Springer.
Schwichtenberg, H., & Wainer, S. (2011). Proofs and computations (Perspectives in logic). Cambridge: Cambridge University Press.
Smullyan, R. (1978). What is the name of this book? The riddle of Dracula and other logical puzzles. Englewood Cliffs/London: Prentice Hall.
Spector, C. (1962). Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In F. D. E. Dekker (Ed.), Recursive function theory: Proceedings of symposia in pure mathematics (Vol. 5, pp. 1–27). Providence, Rhode Island: American Mathematical Society.
Trifonov, T. (2011). Analysis of methods for extraction of programs from non-constructive proofs. Ph.D. thesis, Ludwig-Maximilians-Universität Munich.
Wadler, P. (1995). Monads for functional programming. In Advanced functional programming (Volume 925 of lecture notes in computer science).
Acknowledgements
I am grateful to the anonymous referee, together with Ulrik Buchholtz, Felix Canavoi and Sam Sanders, whose corrections and suggestions improved this paper considerably.
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Powell, T. (2019). Computational Interpretations of Classical Reasoning: From the Epsilon Calculus to Stateful Programs. In: Centrone, S., Negri, S., Sarikaya, D., Schuster, P.M. (eds) Mathesis Universalis, Computability and Proof. Synthese Library, vol 412. Springer, Cham. https://doi.org/10.1007/978-3-030-20447-1_14
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