Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T06:06:45.823Z Has data issue: false hasContentIssue false
  • English
  • Français

A General Notion of Realizability

Published online by Cambridge University Press:  15 January 2014

Lars Birkedal
Lars Birkedal*
Affiliation:
The IT University of Copenhagen, Glentevej 67, DK–2400 Copenhagen NV, DenmarkE-mail: birkedal@itu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We show how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a topos, i.e., a model of impredicative intuitionistic higher-order logic.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 8 , Issue 2 , June 2002 , pp. 266 - 282
Copyright
Copyright © Association for Symbolic Logic 2002

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.)

References

REFERENCES

[1] Bauer, A., Birkedal, L., and Scott, D. S., Equilogical spaces, TCS, (1998), 27 pages submitted.Google Scholar
[2] Birkedal, L., Developing theories of types and computability via realizability, Electronic notes in theoretical computer science, vol. 34, 2000, Book version of PhD-thesis, pp. viii + 282.Google Scholar
[3] Birkedal, L., A general notion of realizability, Proceedings of the 15th annual IEEE symposium on logic in computer science, IEEE Computer Society, 12 2000.Google Scholar
[4] Birkedal, L., Carboni, A., Rosolini, G., and Scott, D. S., Type theory via exact categories, Proceedings of the 13th annual IEEE symposium on logic in computer science (Indianapolis, Indiana), IEEE Computer Society, 06 1998, pp. 188–198.Google Scholar
[5] Carboni, A., Some free constructions in realizability and proof theory, Journal of Pure and Applied Algebra, vol. 103 (1995), pp. 117–148.Google Scholar
[6] Carboni, A., Freyd, P. J., and Scedrov, A., A categorical approach to realizability and polymorphic types, Mathematical foundations of programming semantics, Proceedings, New Orleans, Louisiana (Berlin), Lecture Notes in Computer Science, vol. 298, Springer-Verlag, 1988, pp. 23–42.CrossRefGoogle Scholar
[7] Carboni, A. and Rosolini, G., Locally cartesian closed exact completions, Journal of Pure and Applied Algebra, (1999), Submitted.Google Scholar
[8] Davey, B. A. and Priestley, H. A., Introduction to lattices and order, Cambridge University Press, 1990.Google Scholar
[9] de Marchi, F., Robinson, E. P., and Rosolini, G., An abstract look at realizability, Working Draft, 11 27, 1999.Google Scholar
[10] Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., and Scott, D. S., A compendium of continuous lattices, Springer-Verlag, 1980.Google Scholar
[11] Hyland, J. M. E., The effective topos, The L. E. J. Brouwer centenary symposium (Amsterdam) (Troelstra, A. S. and van Dalen, D., editors), Studies in Logic and The Foundations of Mathematics, vol. 110, North-Holland, 1982, pp. 165–216.Google Scholar
[12] Hyland, J. M. E., Johnstone, P. T., and Pitts, A. M., Tripos theory, Mathematical proceedings of the Cambridge Philosophical Society, vol. 88, 1980.Google Scholar
[13] Jacobs, B., Categorical logic and type theory, Studies in logic and the foundations of mathematics, vol. 141, Elsevier Science Publishers B.V., 1999.Google Scholar
[14] Kleene, S. C., On the interpretation of intuitionistic number theory, The Journal of Symbolic Logic, vol. 10 (1945), pp. 109–124.CrossRefGoogle Scholar
[15] Kleene, S. C., Realizability: a retrospective survey, Cambridge summer school in mathematical logic (Mathias, A. R. D. and Rogers, H., editors), Lecture Notes in Mathematics, vol. 337, Springer-Verlag, 1973, pp. 95–112.CrossRefGoogle Scholar
[16] Lambek, J. and Scott, P. J., Introduction to higher order categorical logic, Cambridge University Press, 1986.Google Scholar
[17] Lietz, P. and Streicher, Th., Impredicativity entails untypedness, Mathematical Structures in Computer Science, (2000), Accepted for publication.Google Scholar
[18] Longley, J., Unifying typed and untyped realizability, Available at http://www.dcs.ed.ac.uk/home/jrl/unifying.txt, 05 1999.Google Scholar
[19] Longley, J. R., Matching typed and untyped realizability, Workshop on realizability semantics and applications (Birkedal, L., van Oosten, J., Rosolini, G., and Scott, D. S., editors), Electronic Notes in Computer Science, vol. 23, Elsevier, 1999.Google Scholar
[20] Lane, S. Mac and Moerdijk, I., Sheaves in geometry and logic. A first introduction to topos theory, Springer, New York, 1992.Google Scholar
[21] Pitts, A. M., The theory of triposes, Ph.D. thesis , University of Cambridge, 1981.Google Scholar
[22] Pitts, A. M., Tripos theory in retrospect, Tutorial workshop on realizability semantics, FLoC'99 (Trento, Italy) (Birkedal, L. and Rosolini, G., editors), Electronic Notes in Theoretical Computer Science, vol. 23, Elsevier, 1999.Google Scholar
[23] Robinson, E. P. and Rosolini, G., Categories of partial maps, Information and Computation, vol. 79 (1988), pp. 95–130.Google Scholar
[24] Robinson, E. P. and Rosolini, G., Colimit completions and the effective topos, The Journal of Symbolic Logic, vol. 55 (1990), pp. 678–699.Google Scholar
[25] Rosolini, G., Continuity and effectiveness in Topoi, Ph.D. thesis , University of Oxford, 1986.Google Scholar
[26] Scott, D. S., Relating theories of the lambda calculus, To H. B. Curry: Essays in combinatory logic (Hindley, J. R. and Seldin, J. P., editors), Academic Press, 1980, pp. 403–450.Google Scholar
[27] Taylor, P., Practical foundations of mathematics, Cambridge studies in advanced mathematics, no. 59, Cambridge University Press, 1999.Google Scholar
[28] Troelstra, A. S., Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, ch. VI: Realizability, pp. 407–473, Studies in Logic and the Foundations of Mathematics, Elsevier Science B.V, 1998, pp. 407–473.CrossRefGoogle Scholar
[29] van Oosten, J., Realizability: A historical essay, Tutorial workshop on realizability semantics, FLoC'99 (Trento, Italy) (Birkedal, L., van Oosten, J., Rosolini, G., and Scott, D. S., editors), Electronic Notes in Theoretical Computer Science, vol. 23, Elsevier, 1999.Google Scholar