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Higher order functions and Brouwer’s thesis

Part of: Robert Harper Festschrift Collection

Published online by Cambridge University Press:  19 May 2021

JONATHAN STERLING Open the ORCID record for JONATHAN STERLING [Opens in a new window]
JONATHAN STERLING*
Affiliation:
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: jmsterli@cs.cmu.edu)
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Abstract

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Extending Martín Escardó’s effectful forcing technique, we give a new proof of a well-known result: Brouwer’s monotone bar theorem holds for any bar that can be realized by a functional of type (ℕ→ℕ)→ℕ in Gödel’s System T. Effectful forcing is an elementary alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads, the algebra interpretation of call-by-name λ-calculus, and logical relations.

Our argument proceeds by interpreting System T programs as well-founded dialogue trees whose nodes branch on a query to an oracle of type ℕ→ℕ, lifted to higher type along a call-by-name translation. To connect this interpretation to the bar theorem, we then show that Brouwer’s famous “mental constructions” of barhood constitute an invariant form of these dialogue trees in which queries to the oracle are made maximally and in order.

Type
Research Article
Information
Journal of Functional Programming , Volume 31 , 2021 , e11
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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© The Author(s) 2021. Published by Cambridge University Press

References

Ackermann, W. (1928) Zum Hilbertschen Aufbau der reellen Zahlen. Mathematische Annalen 99(1), 118–133.CrossRefGoogle Scholar
Anel, M. & Joyal, A. (2019) Topo-logie. Preprint.Google Scholar
Awodey, S., Gambino, N., Lumsdaine, P. L. & Warren, M. A. (2009) Lawvere–Tierney sheaves in algebraic set theory. J. Symb. Logic. 74(Sept.), 861–890.CrossRefGoogle Scholar
Bauer, A. (2006) Sometimes all Functions are Continuous. Blog post.Google Scholar
Bickford, M., Cohen, L., Constable, R. L. & Rahli, V. (2018) Computability beyond church-turing via choice sequences. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. pp. 245–254. ACM.CrossRefGoogle Scholar
Brouwer, L. E. J. (1981) Brouwer’s Cambridge Lectures on Intuitionism. Cambridge University Press.Google Scholar
Capretta, V. & Uustalu, T. (2016) A coalgebraic view of bar recursion and bar induction. In Foundations of Software Science and Computation Structures: 19th International Conference, FOSSACS 2016, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016, Eindhoven, The Netherlands, April 2–8, 2016, Proceedings, Jacobs, B. & Löding, C. (eds). Springer Berlin Heidelberg, pp. 91–106.CrossRefGoogle Scholar
Constable, R. L., Allen, S. F., Bromley, H. M., Cleaveland, W. R., Cremer, J. F., Harper, R. W., Howe, D. J., Knoblock, T. B., Mendler, N. P., Panangaden, P., Sasaki, J. T. & Smith, S. F. (1986) Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Inc.Google Scholar
Constable, R. (2014) Virtual Evidence: A Constructive Semantics for Classical Logics. Google Scholar
Constable, R. and Bickford, M. (2014) Intuitionistic completeness of first-order logic. Ann. Pure Appl. Logic 165(1), 164–198. The Constructive in Logic and Applications.CrossRefGoogle Scholar
Coquand, T., Mannaa, B. & Ruch, F. (2017) Stack semantics of type theory. In 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 1–11.CrossRefGoogle Scholar
Coquand, T. & Jaber, G. (2012) A computational interpretation of forcing in type theory. In Epistemology versus Ontology, Dybjer, P., Lindström, S., Palmgren, E. and Sundholm, G. (eds), Logic, Epistemology, and the Unity of Science, vol. 27. Springer Netherlands, pp. 203–213.CrossRefGoogle Scholar
Coquand, T. & Mannaa, B. (2016) The independence of Markov’s principle in type theory. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016), Kesner, D. and Pientka, B. (eds), Leibniz International Proceedings in Informatics (LIPIcs), vol. 52. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 17:1:–17:18.Google Scholar
Coquand, T., Sambin, G., Smith, J. & Valentini, S. (2003) Inductively generated formal topologies. Ann. Pure Appl. Logic 124(1), 71–106.CrossRefGoogle Scholar
Dummett, M. (2000) Elements of Intuitionism. 2nd edn. Oxford Logic Guides, vol. 39. The Clarendon Press Oxford University Press.Google Scholar
Escardó, M. (2007) Seemingly Impossible Functional Programs. Guest post on Andrej Bauer’s blog.Google Scholar
Escardó, M. (2013) Continuity of Gödel’s System T definable functionals via effectful forcing. Electron. Not. Theor. Comput. Sci. 298, 119–141. Agda development: http://www.cs.bham.ac.uk/mhe/dialogue/dialogue-lambda.html.CrossRefGoogle Scholar
Fourman, M. P. (1982) Notions of choice sequence. In L.E.J. Brouwer Centenary Symposium, van Dalen, D. and Troelstra, A. (eds). North-Holland, pp. 91–105.Google Scholar
Fourman, M. P. (1984) Continuous Truth I: Non-constructive objects. In Logic Colloquium 1982, Lolli, G., G. L. and Marcja, A. (eds), Studies in Logic and the Foundations of Mathematics, vol. 112. Elsevier, pp. 161–180.Google Scholar
Fourman, M. P. (2013) Continuous truth II: Reflections. In Logic, Language, Information, and Computation: 20th International Workshop, WoLLIC 2013, Darmstadt, Germany, August 20-23, 2013. Proceedings, Libkin, L., Kohlenbach, U. and de Queiroz, R. (eds). Springer Berlin Heidelberg, pp. 153–167.CrossRefGoogle Scholar
Fujiwara, M. & Kawai, T. (2019) Equivalence of bar induction and bar recursion for continuous functions with continuous moduli. Ann. Pure Appl. Logic 170(8), 867–890.CrossRefGoogle Scholar
Gambino, N. & Schuster, P. (2007) Spatiality for formal topologies. Mathematical. Structures in Comp. Sci. 17(1), 65–80.CrossRefGoogle Scholar
Gödel, K. (1958) Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12(3–4), 280–287.CrossRefGoogle Scholar
Harper, R. (2012) Exceptions are Shared Secrets. Blog post.Google Scholar
Harper, R. (2016) Practical Foundations for Programming Languages. 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Heyting, A. (1956) Intuitionism, an Introduction. Studies in Logic and the Foundations of Mathematics. North-Holland. Revised edition, 1966.Google Scholar
Hilbert, D. (1926) Über das Unendliche. Mathematische Annalen 95, 161–190.CrossRefGoogle Scholar
Jaber, G., Tabareau, N. & Sozeau, M. (2012) Extending type theory with forcing. In 2012 27th Annual IEEE Symposium on Logic in Computer Science, pp. 395–404.CrossRefGoogle Scholar
Jaber, G., Lewertowski, G., Pédrot, P.-M., Sozeau, M. & Tabareau, N. (2016) The Definitional Side of the Forcing. Logics in Computer Science. CrossRefGoogle Scholar
Johnstone, P. T. (1982) Stone Spaces . Cambridge Studies in Advanced Mathematics. Cambridge University Press.Google Scholar
Johnstone, P. T. (1983) The point of pointless topology. Bull. (New Ser.) Amer. Math. Soc. 8(1), 41–53.CrossRefGoogle Scholar
Kreisel, G. & Troelstra, A. S. (1970) Formal systems for some branches of intuitionistic analysis. Ann. Math. Logic 1(3), 229–387.CrossRefGoogle Scholar
Levy, P. B. (2006) Call-by-push-value: Decomposing call-by-value and call-by-name. Higher-Order Symb. Comput. 19, 377–414.CrossRefGoogle Scholar
Longley, J. (1999) When is a functional program not a functional program? In Proceedings of the Fourth ACM SIGPLAN International Conference on Functional Programming, ICFP 1999. ACM, pp. 1–7.Google Scholar
Longley, J. & Normann, D. (2015) Higher-Order Computability . Theory and Applications of Computability. Springer.Google Scholar
Mac Lane, S. & Moerdijk, I. (1992) Sheaves in Geometry and Logic: A First Introduction to Topos Theory . Universitext. Springer.Google Scholar
Martin-Löf, P. (1970) Notes on Constructive Mathematics. Almqvist and Wiksell.Google Scholar
Martin-Löf, P. (1979) Constructive mathematics and computer programming. In 6th International Congress for Logic, Methodology and Philosophy of Science, pp. 153–175. Published by North Holland, Amsterdam, 1982.Google Scholar
Norell, U. (2009) Dependently typed programming in Agda. In Proceedings of the 4th International Workshop on Types in Language Design and Implementation, TLDI 2009. ACM, pp. 1–2.CrossRefGoogle Scholar
Oliva, P. & Steila, S. (2018) A direct proof of Schwichtenberg’s bar recursion closure theorem. J. Symb. Logic 83(1), 70–83.CrossRefGoogle Scholar
Pédrot, P. & Tabareau, N. (2017) An effectful way to eliminate addiction to dependence. In Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE Press, pp. 1–12.CrossRefGoogle Scholar
Pédrot, P.-M. & Tabareau, N. (2019) The fire triangle: How to mix substitution, dependent elimination, and effects. Proc. ACM Program. Lang. 4(POPL).CrossRefGoogle Scholar
Petrakis, I. (2010) Brouwer’s Fan Theorem. Master’s Thesis.Google Scholar
Rahli, V. & Bickford, M. (2016) A nominal exploration of Intuitionism. In Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, CPP 2016. ACM, pp. 130–141.CrossRefGoogle Scholar
Rahli, V., Bickford, M. & Constable, R. (2017) Bar induction: The good, the bad, and the ugly. In 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 1–12.CrossRefGoogle Scholar
Sambin, G. (2012) Real and ideal in constructive mathematics. In Epistemology versus Ontology, Dybjer, P., Lindström, S., Palmgren, E. and Sundholm, G. (eds), Logic, Epistemology, and the Unity of Science, vol. 27. Springer Netherlands, pp. 69–85.CrossRefGoogle Scholar
Schwichtenberg, H. (1979) On bar recursion of types 0 and 1. J. Symb. Logic 44(3), 325–329.CrossRefGoogle Scholar
Spector, C. (1962) Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In Recursive Function Theory: Proc. Symposia in Pure Mathematics, Dekker, F. D. E. (ed), vol. 5. American Mathematical Society, pp. 1–27.CrossRefGoogle Scholar
Sterling, J. & Harper, R. (2018) Guarded Computational Type Theory. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. ACM.CrossRefGoogle Scholar
Streicher, T. (2005) Universes in toposes. In From Sets and Types to Topology and Analysis: Towards Practical Foundations for Constructive Mathematics, Crosilla, L. & Schuster, P. (eds), Oxford Logical Guides, vol. 48. Oxford University Press, pp. 78–90.CrossRefGoogle Scholar
Sundholm, G. & van Atten, M. (2008) The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the bar theorem. In One Hundred Years of Intuitionism (1907–2007): The Cerisy Conference, van Atten, M., Boldini, P., Bourdeau, M. & Heinzmann, G. (eds). Birkhäuser Basel, pp. 60–77.CrossRefGoogle Scholar
Tait, W. W. (1967) Intensional Interpretations of Functionals of Finite Type I. J. Symb. Logic 32(2), 198–212.CrossRefGoogle Scholar
Troelstra, A. & van Dalen, D. (1988) Constructivism in Mathematics: An Introduction . Studies in Logic and the Foundations of Mathematics, vol. 1. North-Holland.Google Scholar
van Atten, M., van Dalen, D. & Tieszen, R. (2002) Brouwer and weyl: The phenomenology and mathematics of the intuitive continuumt. Philos. Math. 10(2), 203–226.CrossRefGoogle Scholar
van Atten, M. (2004) On Brouwer . Wadsworth Philosophers Series. Thompson/Wadsworth.Google Scholar
van Dalen, D. (2013) L.E.J. Brouwer: Topologist, Intuitionist, Philosopher: How Mathematics is Rooted in Life. Springer.CrossRefGoogle Scholar
van der Hoeven, G. & Moerdijk, I. (1984) Sheaf models for choice sequences. Ann. Pure Appl. Logic 27(1), 63–107.CrossRefGoogle Scholar
van Heijenoort, J. (2002) From Frege to Gödel : A Source Book in Mathematical Logic, 1879–1931 (Source Books in the History of the Sciences). Harvard University Press.Google Scholar
Vezzosi, A., Mörtberg, A. & Abel, A. (2019) Cubical Agda: A dependently typed programming language with univalence and higher inductive types. In Proceedings of the 24th ACM SIGPLAN International Conference on Functional Programming, ICFP 2019. ACM.CrossRefGoogle Scholar
Vickers, S. (1989) Topology via Logic. Cambridge University Press.Google Scholar
Vickers, S. (2007) Locales and Toposes as Spaces. Springer Netherlands, pp. 429–496.Google Scholar
Xu, C. (2015) A Continuous Computational Interpretation of Type Theories. PhD thesis, University of Birmingham.Google Scholar
Xu, C. (2020a) A Gentzen-Style Monadic Translation of Gödel’s System T. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020), Ariola, Z. M. (ed), Leibniz International Proceedings in Informatics (LIPIcs), vol. 167. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, pp. 25:1–25:17.Google Scholar
Xu, C. (2020b) A syntactic approach to continuity of T-definable functionals. Log. Meth. Comput. Sci. 16(Feb.).Google Scholar
Xu, C. & Escardó, M. (2016) Universes in Sheaf Models. Unpublished note.Google Scholar
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