Skip to main content
×
×
Home

Realizability in ordered combinatory algebras with adjunction

  • WALTER FERRER SANTOS (a1) (a2), MAURICIO GUILLERMO (a3) and OCTAVIO MALHERBE (a1) (a3)
Abstract

In this work, we continue our consideration of the constructions presented in the paper Krivine's Classical Realizability from a Categorical Perspective by Thomas Streicher. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented, mutatis mutandis, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.

Copyright
References
Hide All
Birkhoff, G. (1995). Lattice Theory, Colloquium Publications, vol. 25, 3rd edition, American Mathematical Society.
Borceux, F. (2008). Handbook of Categorical Algebra Volume 1. Basic Category Theory, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge Univ. Press, Cambridge.
Borceux, F. (2008). Handbook of Categorical Algebra Volume 2. Categories and Structures, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge Univ. Press, Cambridge.
Curry, H. and Feys, R. (1958). Combinatory Logic, vol. 1, North Holland Publishing Co., Amsterdam.
Ferrer Santos, W., Guillermo, M. and Malherbe, O. (2013). A Report on Realizability, arXiv:1309.0706v2 [math.LO], 1–25.
Ferrer Santos, W., Frey, J., Guillermo, M., Malherbe, O. and Miquel, A. (2015). Ordered Combinatory Algebras and Realizability, Mathematical Structures in Computer Science, Camb. Univ. Press, 131.
Griffin, T. G. (1990). A Formulæ-as-Types Notion of Control, In: Proceedings of the Conference Record of the 17th Annual ACM Symposium on Principles of Programming Languages.
Hofstra, P. and van Oosten, J. (2004). Ordered partial combinatory algebras. Mathematical Proceedings of the Cambridge Philosophical Society 134 (3) 445463.
Hofstra, P. (2006). All realizability is relative. Mathematical Proceedings of the Cambridge Philosophical Society 141 (2) 239264.
Hyland, J. M. E. (1982). The effective topos. In: Proceedings of The L.E.J. Brouwer Centenary Symposium (Noordwijkerhout 1981), North Holland, 165–216.
Hyland, J. M. E., Johnstone, P. T. and Pitts, A. M. (1980). Tripos theory. Mathematical Proceedings of the Cambridge Philosophical Society 88 205232.
Krivine, J.-L. (2001). Typed lambda-calculus in classical Zermelo-Fraenkel set theory. Archive for Mathematical Logic 40 (3) 189205.
Krivine, J.-L. (2003). Dependent choice, quote and the clock. Theoretical Computer Science 308 259276.
Krivine, J.-L. (2004). Realizability in Classical Logic, Lessons in Marseille-Lumini, (revised in 2005), 1–29. Available at https://hal-univ-diderot.archives-ouvertes.fr/hal-00154500/document
Krivine, J.-L. (2008). Structures de réalisabilité, RAM et ultrafiltre sur ℕ, Available at http://www.pps.jussieu.fr/krivine/Ultrafiltre.pdf.
Krivine, J.-L. (2009). Realizability in classical logic. In: Interactive Models of Computation and Program Behaviour, Panoramas et synthèses, vol. 27, SMF.
Krivine, J.-L. Realizability Algebras: A Program to Well Order R, Logical Methods in Computer Science, vol. 7, Issue 3, 2.
Krivine, J.-L. (2012). Realizability algebras II : New models of ZF + DC, Logical Methods in Computer Science, vol. 8, Issue 1, 10.
Krivine, J.-L. (2016). Realizability Algebras III: Some Examples, Mathematical Structures in Computer Science, Cambridge University Press 132.
MacLane, S. (1997). Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edition, Springer-Verlag, New York.
Miquel, A. (2014). Implicative Algebras for Noncommutative Forcing, Available at http://smc2014.univ-lyon1.fr/lib/exe/fetch.php?media=miquel.pdf?
Miquel, A. (2016). Implicative Algebras: A New Foundation for Forcing and Realizability, Available at https://www.pedrot.fr/montevideo2016/miquel-slides.pdf
Paré, R. and Schumacher, D. (1978). Abstract Families and the Adjoint Functor Theorems. Lectures Notes in Mathematics, vol. 661.
Streicher, T. (2013). Krivine's Classical Realizability From a Categorical Perspective, Mathematical Structures in Computer Science 23 (6) 12341256.
van Oosten, J. (2008). Realizability, an Introduction to its Categorical Side, Elsevier.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 9 *
Loading metrics...

Abstract views

Total abstract views: 112 *
Loading metrics...

* Views captured on Cambridge Core between 26th April 2018 - 18th August 2018. This data will be updated every 24 hours.