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Holomorphic models of exponential types in linear logic

  • R. F. Blute
  • Prakash Panangaden
  • R. A. G. Seely
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 802)

Abstract

In this paper we describe models of several fragments of linear logic with the exponential operator ! (called Of course) in categories of linear spaces. We model ! by the Fock space construction in Banach (or Hilbert) spaces, a notion originally introduced in the context of quantum field theory. Several variants of this construction are presented, and the representation of Fock space as a space of holomorphic functions is described. This also suggests that the “non-linear” functions we arrive at via! are not merely continuous, but analytic.

Keywords

Fock space linear logic Banach spaces holomorphic functions quantum field theory 

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References

  1. [Ab91]
    Abrusci, V.M. “Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic”, Journal of Symbolic Logic 56 (1991) 1403–1451.Google Scholar
  2. [Ah66]
    Ahlfors, L.V. Complex Analysis. McGraw-Hill, New York, 1966, second edition. (First edition 1953).Google Scholar
  3. [AM-A80]
    Ashtekar, A. and A. Magnon-Ashtekar “A geometrical approach to external potential problems in quantum field theory”, Journal of General Relativity and Gravitation 12 (1980) 205–223.Google Scholar
  4. [BSZ92]
    Baez, J.C., I.E. Segal and Z. Zhou Introduction to algebraic and constructive quantum field theory. Princeton University Press, Princeton, New Jersey, 1992.Google Scholar
  5. [Ba61]
    Bargmann, V. “On a Hilbert space of analytic functions and an associated integral transform”, Communications in Pure and Applied Mathematics 14 (1961) 187–214.Google Scholar
  6. [B76a]
    Barr, M. “Duality of vector spaces”, Cahiers de Topologie et Géometrie Differentielle 17 (1976) 3–14.Google Scholar
  7. [B76b]
    Barr, M. “Duality of Banach spaces”, Cahiers de Topologie et Géometrie Differentielle 17 (1976) 15–52.Google Scholar
  8. [B79]
    Barr, M. *Autonomous Categories. Lecture Notes in Mathematics 752, Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
  9. [B91]
    Barr, M. “*autonomous categories and linear logic”, Mathematical Structures in Computer Science 1 (1991) 159–178.Google Scholar
  10. [BBPH]
    Benton, B.N., G. Bierman, V. de Paiva, and M. Hyland “Term assignment for intuitionistic linear logic (preliminary report)”, Preprint, University of Cambridge, 1992.Google Scholar
  11. [B191]
    Blute, R.F. “Proof nets and coherence theorems”, in Categories and Computer Science, Paris 1991, Lecture Notes in Computer Science 530, Springer-Verlag, Berlin, Heidelberg, New York, 1991.Google Scholar
  12. [B192]
    Blute, R.F. “Linear Logic, Coherence and Dinaturality”, to appear in Theoretical Computer Science.Google Scholar
  13. [B193a]
    Blute, R. “Linear Topology, Hopf Algebras and *-autonomous Categories”, preprint, McGill University (1993)Google Scholar
  14. [B193b]
    Blute, R. “Braided Proof Nets and Categories”, in preparation, (1993).Google Scholar
  15. [BCST]
    Blute, R.F., J.R.B. Cockett, R.A.G. Seely, and T.H. Trimble “Natural deduction and coherence for weakly distributive categories”, Preprint, McGill University, 1992. (Submitted to JPAA)Google Scholar
  16. [BCS93]
    Blute, R.F., J.R.B. Cockett, and R.A.G. Seely “ ! and ? for weakly distributive categories: Storage as tensorial strength”, preprint, McGill University, in preparation.Google Scholar
  17. [CLM79]
    Cigler, J., V. Losert, P. Michor Banach Modules and Functors on Categories of Banach Spaces. Dekker, 1979Google Scholar
  18. [CS91]
    Cockett, J.R.B. and R.A.G. Seely “Weakly distributive categories”, in M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories to Computer Science, London Mathematical Society Lecture Note Series 177 (1992) 45–65.Google Scholar
  19. [C90]
    Conway, J. A Course in Functional Analysis. Springer Verlag Graduate Texts in Mathematics 96, (1990)Google Scholar
  20. [F64]
    Freyd, P.J. Abelian Categories: An introduction to the theory of functors. Harper and Row, New York, 1964.Google Scholar
  21. [Ge85]
    Geroch, R. Mathematical Physics. University of Chicago Press, 1985.Google Scholar
  22. [G87]
    Girard, J.-Y. “Linear logic”, Theoretical Computer Science 50 (1987) 1–102.Google Scholar
  23. [G89]
    Girard, J.-Y. “Geometry of interaction I: Interpretation of system F”, Proceedings of ASL meeting, Padova, 1988.Google Scholar
  24. [GL87]
    Girard, J.-Y. and Y. Lafont “Linear logic and lazy computation”, in Tapsoft'87, 2, Lecture Notes in Computer Science 250, Springer-Verlag, Berlin, Heidelberg, New York, 1987, 52–66.Google Scholar
  25. [GSS91]
    Girard, J.-Y., A. Scedrov, and P.J. Scott “Bounded linear logic”, Theoretical Computer Science 97 (1992) 1–66.Google Scholar
  26. [Gu90]
    Gunning, R. Introduction to Holomorphic functions of several variables, Volume 1: Function theory. Mathematics Series, Wadsworth and Brooks/Cole, Belmont CA, 1990.Google Scholar
  27. [IZ80]
    Itzykson, J, and J. Zuber Introduction to Quantum Field Theory. McGraw-Hill, New York, 1980.Google Scholar
  28. [KR83]
    Kadison, R.V. and J.R. Ringrose Fundamentals of the Theory of Operator Algebras. Academic Press, New York, 1983Google Scholar
  29. [KL80]
    Kelly, G.M. and M. La Plaza, “Coherence for Compact Closed Categories”, Journal of Pure and Applied Algebra 19 (1980) 193–213.Google Scholar
  30. [L65]
    Lang, S. Algebra. Addison-Wesley Pub. Co., Reading, Menlo Park, London, Don Mills, 1965.Google Scholar
  31. [Le41]
    Lefschetz, S. Algebraic Topology. AMS Colloquium Publications 27, 1941.Google Scholar
  32. [M71]
    Mac Lane, S. Categories for the Working Mathematician. Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1971Google Scholar
  33. [M-OM89]
    Martí-Oliet, N. and J. Meseguer “From Petri nets to linear logic”, in D.H. Pitt et al., eds. Category Theory and Computer Science, Manchester 1989, Lecture Notes in Computer Science 389, Springer-Verlag, Berlin, Heidelberg, New York, 1989.Google Scholar
  34. [dP89]
    de Paiva, V.C.V. “A Dialectica-like model of linear logic”, in D.H. Pitt et al., eds., Category Theory and Computer Science, Lecture Notes in Computer Science 389, Springer-Verlag, Berlin, Heidelberg, New York, 1989.Google Scholar
  35. [P79]
    Panangaden, P. “Positive and negative frequency decompositions in curved spacetimes”, Journal of Mathematical Physics 20 (1979) 2506–2514.Google Scholar
  36. [P80]
    Panangaden, P. “Propagators and Renormalization of Quantum Field Theory in Curved Spacetimes.”, Ph.D. Thesis, Physics Department, University of Wisconsin-Milwaukee, 1980. Available through University Microfilms International, Ann Arbor.Google Scholar
  37. [Se89]
    Seely, R.A.G. “Linear logic, *-autonomous categories and cofree coalgebras”, in J. Gray and A. Scedrov (eds.), Categories in Computer Science and Logic, Contemporary Mathematics 92 (Am. Math. Soc. 1989).Google Scholar
  38. [S62]
    Segal, E. “Mathematical characterization of the physical vacuum for linear Bose-Einstein fields”, Illinois Journal of Mathematics 6 (1962) 500–523.Google Scholar
  39. [T79]
    Takesaki M. Theory of Operator Algebras. Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • R. F. Blute
    • 1
  • Prakash Panangaden
    • 2
  • R. A. G. Seely
    • 3
  1. 1.Department of MathematicsUniversity of OttawaOttawaCanada
  2. 2.School of Computer ScienceMcGill UniversityMontrealCanada
  3. 3.Department of MathematicsMcGill UniversityMontrealCanada

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