Parameterizations and Fixed-Point Operators on Control Categories

  • Yoshihiko Kakutani
  • Masahito Hasegawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2701)

Abstract

The λµ-calculus features both variables and names, together with their binding mechanisms. This means that constructions on open terms are necessarily parameterized in two different ways for both variables and names. Semantically, such a construction must be modeled by a bi-parameterized family of operators. In this paper, we study these bi-parameterized operators on Selinger’s categorical models of the λµ-calculus called control categories. The overall development is analogous to that of Lambek’s functional completeness of cartesian closed categories via polynomial categories. As a particular and important case, we study parameterizations of uniform fixed-point operators on control categories, and show bijective correspondences between parameterized fixed-point operators and non-parameterized ones under uniformity conditions.

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References

  1. 1.
    S. Bloom and Z. Ésik. Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.Google Scholar
  2. 2.
    C. Führmann. Varieties of effects. In Foundations of Software Science and Computation Structures, volume 2303 of LNCS, pages 144–158, Springer-Verlag, 2002.CrossRefGoogle Scholar
  3. 3.
    M. Hasegawa. Models of Sharing Graphs: A Categorical Semantics of let and letrec. PhD thesis, University of Edinburgh, 1997.Google Scholar
  4. 4.
    M. Hasegawa and Y. Kakutani. Axioms for recursion in call-by-value. Higher-Order and Symbolic Computation, 15(2):235–264, 2002.MATHCrossRefGoogle Scholar
  5. 5.
    B. Jacobs. Parameters and parameterization in specification, using distributive categories. Fundamenta Informaticae, 24(3):209–250, 1995.MATHMathSciNetGoogle Scholar
  6. 6.
    Y. Kakutani. Duality between call-by-name recursion and call-by-value iteration. In Computer Science Logic, volume 2471 of LNCS, pages 506–521, Springer-Verlag, 2002.CrossRefGoogle Scholar
  7. 7.
    J. Lambek. Functional completeness of cartesian categories. Annals of Mathematical Logic, 6:259–292, 1970.CrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Lambek and P.J. Scott. Introduction to Higher-Order Categorical Logic. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1986.Google Scholar
  9. 9.
    M. Parigot. λµ-calculus: an algorithmic interpretation of classical natural deduction. In Logic Programming and Automated Reasoning, volume 624 of LNCS, pages 190–201, Springer-Verlag, 1992.CrossRefGoogle Scholar
  10. 10.
    A.J. Power and E.P. Robinson. Premonoidal categories and notions of computation. Mathematical Structures in Computer Science, 7(5):453–468, 1997.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Selinger. Control categories and duality: on the categorical semantics of the lambda-mu calculus. Mathematical Structures in Computer Science, 11(2):207–260, 2001.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Simpson and G. Plotkin. Complete axioms for categorical fixed-point operators. In Proceedings of 15th Annual Symposium on Logic in Computer Science, pages 30–41, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Yoshihiko Kakutani
    • 1
  • Masahito Hasegawa
    • 1
    • 2
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyoto
  2. 2.PRESTO21Japan Science and Technology CorporationJapan

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