Definability of Total Objects in PCF and Related Calculi

  • Dag Normann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)


We let PCF be Plotkin’s [8] calculus based on Scott’s 10, 11 LCF, and we consider the standard case with base types for the natural numbers and for the Booleans. We consider the standard interpretation using algebraic domains. Plotkin [8] showed that a finite object in general will not be definable, and isolated two nondeterministic constants PAR and ∃ ω such that each computable object is definable in PCF + PAR + ∃ ω .


  1. 1.
    Escardó, M. H., PCF extended with real numbers: a domain-theoretic approach to higher-order exact number computation, Thesis, University of London, Imperial College of Science, Technology and medicine (1996).Google Scholar
  2. 2.
    Escardó, M. H., PCF extended with real numbers, Theoretical Computer Science 162(1) pp. 79–115 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Niggl, K.-H., M ω considered as a programming language, Annals of Pure and Applied Logic 99, pp. 73–92 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Normann, D., Computability over the partial continuous functionals, Journal of Symbolic Logic 65, pp. 1133–1142, (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Normann, D., The Cook-Berger Problem. A Guide to the solution, In Spreen, D. (ed.): Electronic Notes in Theoretical Computer Science. 2000-10; 35: 9Google Scholar
  6. 6.
    Normann, D., The continuous functionals of finite types over the reals, To appear in Keimel, Zhang, Liu and Chen (eds.) Domains and Processes Proceedings of the 1st International Symposium on Domain Theory, Luwer Academic PublishersGoogle Scholar
  7. 7.
    Normann, D., Exact real number computations relative to hereditarily total functionals, To appear in Theoretical Computer Science.Google Scholar
  8. 8.
    Plotkin, G., LCF considered as a programming language, Theoretical Computer Science 5 (1977) pp. 223–255.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Rördam, C., A comparison of the simply typed lambda calculi M ω and λPA, Cand. Scient. Thesis, Oslo (2000)Google Scholar
  10. 10.
    Scott, D. S., A theory of computable functionals of higher type, Unpublished notes, University of Oxford, Oxford (1969).Google Scholar
  11. 11.
    Scott, D. S., A type-theoretical alternative to ISWIM, CUCH, OWHY, Theoretical Computer Science 121 pp. 411–440 (1993).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Dag Normann
    • 1
  1. 1.Department of MathematicsUniversity of OsloUSA

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