Advertisement

Coherence Spaces and Uniform Continuity

  • Kei MatsumotoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)

Abstract

We consider a model of classical linear logic based on coherence spaces endowed with a notion of totality. If we restrict ourselves to total objects, each coherence space can be regarded as a uniform space and each linear map as a uniformly continuous function. The linear exponential comonad then assigns to each uniform space \({\varvec{X}}\) the finest uniform space \(\,!\,{\varvec{X}}\) compatible with \({\varvec{X}}\). By a standard realizability construction, it is possible to consider a theory of representations in our model. Each (separable, metrizable) uniform space, such as the real line \(\mathbb {R}\), can then be represented by (a partial surjecive map from) a coherence space with totality. The following holds under certain mild conditions: a function between uniform spaces \(\mathbb {X}\) and \(\mathbb {Y}\) is uniformly continuous if and only if it is realized by a total linear map between the coherence spaces representing \(\mathbb {X}\) and \(\mathbb {Y}\).

Notes

Acknowledgement

The author is greatful to Naohiko Hoshino and Kazushige Terui (RIMS) for useful comments, and to the anonymous referees for their thoughtful reviews that help improve the manuscript.

This work was partly supported by JSPS Core-to-Core Program (A. Advanced Research Networks) and by KAKENHI 25330013.

References

  1. [AHS02]
    Abramsky, S., Haghverdi, E., Scott, P.J.: Geometry of interaction and linear combinatory algebras. Math. Struct. in Comput. Sci. 12(5), 625–665 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AL00]
    Abramsky, S., Lenisa, M.: A fully complete PER model for ML polymorphic types. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 140–155. Springer, Heidelberg (2000). doi: 10.1007/3-540-44622-2_9 CrossRefGoogle Scholar
  3. [AL05]
    Abramsky, S., Lenisa, M.: Linear realizability and full completeness for typed lambda-calculi. Ann. Pure Appl. Logic 134(2–3), 122–168 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [As90]
    Asperti, A.: Stability and computability in coherent domains. Inf. Comput. 86, 115–139 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Ba00]
    Bauer, A.: The Realizability Approach to Computable Analysis and Topology. Ph.D. thesis, School of Computer Science, Carnegie Mellon University (2000)Google Scholar
  6. [Ba02]
    Bauer, A.: A relationship between equilogical spaces and type two effectivity. Math. Logic Q. 48(S1), 1–15 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Ba05]
    Bauer, A.: Realizability as the connection between computable and constructive mathematics. In: Proceedings of CCA , Kyoto, Japan (2005)Google Scholar
  8. [Be93]
    Berger, U.: Total sets and objects in domain theory. Ann. Pure Appl. Logic 60, 91–117 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [BHW08]
    Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 425–491. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. [Bi94]
    Bierman, G.: On intuitionistic linear logic. Ph.D. thesis, University of Cambridge (1994)Google Scholar
  11. [Bl97]
    Blanck, J.: Computability on topological spaces by effective domain representations. Ph.D. thesis, Uppsala University (1997)Google Scholar
  12. [Ehr05]
    Ehrhard, T.: Finiteness spaces. Math. Str. Comput. Sci. 15(4), 615–646 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Es96]
    Escardo, M.H.: PCF extended with real numbers. Theoret. Comput. Sci. 162(1), 79–115 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [ES99]
    Edalat, A., Sunderhauf, P.: A domain-theoretic approach to computability on the real line. Theoret. Comput. Sci. 210(1), 73–98 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [ES14]
    Escardo, M.H., Simpson, A.: Abstract datatypes for real numbers in type theory. In: Proceedings of RTA-TLCA, pp. 208–223 (2014)Google Scholar
  16. [Gi86]
    Girard, J.-Y.: The system F of variable types, fifteen years later. Theoret. Comput. Sci. 45(2), 159–192 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Gi87]
    Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50(1), 1–101 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [HS03]
    Hyland, M., Schalk, A.: Glueing and orthogonality for models of linear logic. Theo. Comput. Sci. 294(1–2), 183–231 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Is64]
    Isbell, J.R.: Uniform Spaces. American Mathematical Society, Providence (1964)CrossRefzbMATHGoogle Scholar
  20. [Ja99]
    Jacobs, B.: Categorical Logic and Type Theory. North Holland, Amsterdam (1999)zbMATHGoogle Scholar
  21. [Ke75]
    Kelley, J.L.: General Topology. Springer Science & Business Media, New York (1975)zbMATHGoogle Scholar
  22. [KN97]
    Kristiansen, L., Normann, D.: Total objects in inductively defined types. Arch. Math. Logic 36, 405–436 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Ko91]
    Ko, K.: Complexity Theory of Real Functions. Birkhäuser, Boston (1991)CrossRefzbMATHGoogle Scholar
  24. [KW85]
    Kreitz, C., Weihrauch, K.: Theory of representations. Theoret. Comput. Sci. 38, 35–53 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Loa94]
    Loader, R.: Linear logic, totality and full completeness. In: Proceedings of the 9th Annual IEEE Symposium on Logic in Computer Science, pp. 292–298 (1994)Google Scholar
  26. [Lon94]
    Longley, J.R.: Realizability Toposes and Language Semantics. Ph.D. thesis, University of Edinburgh (1994)Google Scholar
  27. [LS02]
    Lietz, P., Streicher, T.: Impredicativity entails untypedness. Math. Struct. Comput. Sci. 12(3), 335–347 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Me09]
    Mellies, P.-A.: Categorical semantics of linear logic. Interactive models of computation and program behaviour. In: Panoramas et Syntheses vol. 27. Soc. Math. de France (2009)Google Scholar
  29. [MT16]
    Matsumoto, K., Terui, K.: Coherence spaces for real functions and operators. Submitted (2016). http://www.kurims.kyoto-u.ac.jp/~terui/pub.html
  30. [No90]
    Normann, D.: Formalizing the notion of total information, pp. 67–94. In: Mathematical Logic. Plenum Press (1990)Google Scholar
  31. [Sc02]
    Schröder, M.: Extended admissibility. Theoret. Comput. Sci. 284(2), 519–538 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [SHT08]
    Stoltenberg-Hansen, V., Tucker, J.V.: Computability on topological spaces via domain representations. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 153–194. Springer, New York (2008)CrossRefGoogle Scholar
  33. [Si03]
    Simpson, A.: Towards a category of topological domains. In: Proceedings of the of Thirteenth ALGI Workshop. RIMS, Kyoto University (2003)Google Scholar
  34. [We00]
    Weihrauch, K.: Computable Analysis – An Introduction. Texts in Theoretical Computer Science. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  35. [Wi70]
    Willard, S.: General Topology. Courier Corp., New York (1970)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.RIMS, Kyoto UniversityKyotoJapan

Personalised recommendations