Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T07:09:43.978Z Has data issue: false hasContentIssue false
  • English
  • Français

Implicative algebras: a new foundation for realizability and forcing

Published online by Cambridge University Press:  03 July 2020

Alexandre Miquel Open the ORCID record for Alexandre Miquel [Opens in a new window]
Alexandre Miquel*
Affiliation:
Instituto de Matemática y Estadística Prof. Ing. Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, MontevideoC.P.11300, Uruguay
*
*Corresponding author. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.

Keywords

RealizabilityForcingCategorical Logic
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 5 , May 2020 , pp. 458 - 510
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

This work was partly supported by the Uruguayan National Research & Innovation Agency (ANII) under the project “Realizability, Forcing and Quantum Computing,” FCE_1_2014_1_104800.

References

Barendregt, H. (1984). The Lambda Calculus: Its Syntax and Semantics, Studies in Logic and The Foundations of Mathematics, vol. 103, North-Holland, Elsevier.Google Scholar
Barendregt, H., Coppo, M. and Dezani-Ciancaglini, M. (1983). A filter lambda model and the completeness of type assignment. Journal of Symbolic Logic 48 (4) 931–940.CrossRefGoogle Scholar
Cohen, P. J. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America 50 (6) 1143–1148.CrossRefGoogle ScholarPubMed
Cohen, P. J. (1964). The independence of the continuum hypothesis II. Proceedings of the National Academy of Sciences of the United States of America 51 (1) 105–110.CrossRefGoogle ScholarPubMed
Coppo, M., Dezani-Ciancaglini, M. and Venneri, B. (1980). Principal type schemes and lambda-calculus semantics. In: Hindley, R. and Seldin, G. (eds.) To H. B. Curry. Essays on Combinatory Logic, Lambda-Calculus and Formalism, London, Academic Press, 480–490.Google Scholar
Ferrer, W. and Malherbe, O. (2017). The category of implicative algebras and realizability. ArXiv e-prints.Google Scholar
Ferrer Santos, W., Frey, J., Guillermo, M., Malherbe, O. and Miquel, A. (2017). Ordered combinatory algebras and realizability. Mathematical Structures in Computer Science 27 (3) 428–458.CrossRefGoogle Scholar
Friedman, H. (1973). Some applications of Kleene's methods for intuitionistic systems. In: Mathias, A. R. D. and Rogers, H. (eds.), Cambridge Summer School in Mathematical Logic, Lecture Notes in Mathematics, vol. 337, Berlin-Heidelberg-New York, Springer-Verlag, 113–170.CrossRefGoogle Scholar
Girard, J. (1987). Linear logic. Theoretical Computer Science 50, 1–102.CrossRefGoogle Scholar
Girard, J.-Y. (1972). Interprétation fonctionnelle et élimination des coupures de l'arithmétique d'ordre supérieur. Doctorat d'État, Université Paris VII.Google Scholar
Girard, J.-Y., Lafont, Y. and Taylor, P. (1989). Proofs and Types. Cambridge, Cambridge University Press.Google Scholar
Griffin, T. (1990). A formulae-as-types notion of control. In: Principles of Programming Languages (POPL'90), 47–58.CrossRefGoogle Scholar
Guillermo, M. and Miquel, A. (2015). Specifying Peirce's law. Mathematical Structures in Computer Science 26 (7) 1269–1303.CrossRefGoogle Scholar
Hyland, J. M. E., Johnstone, P. T. and Pitts, A. M. (1980). Tripos theory. Mathematical Proceedings of the Cambridge Philosophical Society 88 205–232.CrossRefGoogle Scholar
Jech, T. (2002). Set Theory, Third Millennium Edition (Revised and Expanded). Berlin-Heidelberg-New York, Springer.Google Scholar
Kleene, S. C. (1945). On the interpretation of intuitionistic number theory. Journal of Symbolic Logic 10 109–124.CrossRefGoogle Scholar
Koppelberg, S. (1989). Handbook of Boolean Algebras, vol. 1, North-Holland.Google Scholar
Krivine, J. L. (1993). Lambda-Calculus, Types and Models. Ellis Horwood. Out of print, now available at the author's web page at: https://www.irif.univ-paris-diderot.fr/krivine/articles/Lambda.pdf.Google Scholar
Krivine, J.-L. (2001). Typed lambda-calculus in classical Zermelo-Fraenkel set theory. Archive for Mathematical Logic 40 (3) 189–205.CrossRefGoogle Scholar
Krivine, J.-L. (2003). Dependent choice, ‘quote' and the clock. Theoretical Computer Science 308 (1–3) 259–276.CrossRefGoogle Scholar
Krivine, J.-L. (2009). Realizability in classical logic. In: Interactive Models of Computation and Program Behaviour, Panoramas et synthèses, vol. 27, Société Mathématique de France, 197–229.Google Scholar
Krivine, J.-L. (2011). Realizability algebras: a program to well order R. Logical Methods in Computer Science 7 1–47.CrossRefGoogle Scholar
Krivine, J.-L. (2012). Realizability algebras II: new models of ZF + DC. Logical Methods for Computer Science 8 (1:10) 1–28.CrossRefGoogle Scholar
Leivant, D. (1983). Polymorphic type inference. In: Proceedings of the 10th ACM Symposium on Principles of Programming Languages, 88–98.Google Scholar
McCarty, D. (1984). Realizability and Recursive Mathematics. Phd thesis, Oxford University.Google Scholar
Miquel, A. (2000). A model for impredicative type systems, universes, intersection types and subtyping. In: LICS, 18–29.CrossRefGoogle Scholar
Miquel, A. (2011). Existential witness extraction in classical realizability and via a negative translation. Logical Methods for Computer Science 7 (2) 1–47.CrossRefGoogle Scholar
Miquel, A. (2011). Forcing as a program transformation. In: LICS, IEEE Computer Society, 197–206.Google Scholar
Myhill, J. (1973). Some properties of intuitionistic Zermelo-Fraenkel set theory. Lecture Notes in Mathematics 337 206–231.CrossRefGoogle Scholar
Parigot, M. (1997). Proofs of strong normalisation for second order classical natural deduction. Journal of Symbolic Logic 62 (4) 1461–1479.CrossRefGoogle Scholar
Pitts, A. M. (1981). The Theory of Triposes. Phd thesis, University of Cambridge.Google Scholar
Pitts, A. M. (2002). Tripos theory in retrospect. Mathematical Structures in Computer Science 12 (3) 265–279.CrossRefGoogle Scholar
Ronchi della Rocca, S. and Venneri, B. (1984). Principal type schemes for an extended type theory. Theoretical Computer Science 28 151–169.CrossRefGoogle Scholar
Ruyer, F. (2006). Preuves, types et sous-types. Thèse de doctorat, Université Savoie Mont Blanc.Google Scholar
Streicher, T. (2013). Krivine's classical realisability from a categorical perspective. Mathematical Structures in Computer Science 23 (6) 1234–1256.CrossRefGoogle Scholar
Tait, W. (1967). Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic 32 (2), 198–212.CrossRefGoogle Scholar
van Bakel, S., Liquori, L., Ronchi della Rocca, S. and Urzyczyn, P. (1994). Comparing cubes. In: Nerode, A. and Matiyasevich, Y. V. (eds.), Proceedings of LFCS'94. Third International Symposium on Logical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 813, Springer-Verlag, 353–365.Google Scholar
van Oosten, J. (2002). Realizability: a historical essay. Mathematical Structures in Computer Science 12 (3) 239–263.CrossRefGoogle Scholar
van Oosten, J. (2008). Realizability, an Introduction to its Categorical Side. Amsterdam, Elsevier.Google Scholar
Werner, B. (1994). Une théorie des Constructions Inductives. Phd thesis, Université Paris VII.Google Scholar