Gödel numberings, principal morphisms, combinatory algebras

A category-theoretic characterization of functional completeness
  • G. Longo
  • E. Moggi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 176)


Functional languages are based on the notion of application: programs may be applied to data or programs. By application one may define algebraic functions and a programming language is functionally complete when any algebraic function f(x1,...,xn) is representable (i.e. there is a constant a such that f(x1,...,xn) = ax1· ... ·xn). Combinatory Logic (C.L.) is the simplest type-free language which is functionally complete.

In a sound category-theoretic framework the constant a above may be considered an "abstract gödel-number" for f, as gödel-numberings are generalized to "principal morphisms". By this, models of C.L. are categorically characterized and their relation is given to λ-calculus models within Cartesian Closed Categories.

Finally, the partial recursive functionals in any finite higher type are shown to yield models of C.L..


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  1. Barendregt H. (1983) "Lambda-calculus and its models" in Proceedings of Logic Colloquium 82 (Lolli, Longo, Marcja eds), North-Holland.Google Scholar
  2. Barendregt H. (1984) The lambda-calculus: its syntax and semantics, revised and expanded edition, North-Holland.Google Scholar
  3. Berry G. (1979) "Some Syntactic and Categorial Constructions of λ-calculus models" INRIA, Valbonne.Google Scholar
  4. Hindley R., Longo G. (1980) "Lambda-calculus models and extensionality" Zeit. Math. Logik 26 (289–310).Google Scholar
  5. Hindley R., Seldin J. (198?) Introductory book on lambda-calculus and CL (in preparation).Google Scholar
  6. Longo G. (1982) "Hereditary Partial Effective Functionals in any finite type" (Preliminary note), Forschungsinst. Math. ETH, Zürich.Google Scholar
  7. Longo, G., Moggi E. (1983) "The Hereditary Partial Effective Functionals and Recursion Theory in Higher Types" J. Symb. Logic (to appear).Google Scholar
  8. Longo G., Moggi E. (1984) "Carlesian Closed Categories for effective type structures" Part I Symp. Semantics Data Types, LNCS, Springer-Verlag (to appear).Google Scholar
  9. Meyer A. (1982) "What is a model of lambda-calculus?" Info. Contr. 52, 1 (87–122).Google Scholar
  10. Plotkin G. (1978) "Tω as a universal domains" JCSS 17, 2 (209–236).Google Scholar
  11. Scott D.S. (1980) "Relating theories of lambda-calculus" in To H.B. Curry: essays... (Hindley, Seldin eds.), Academic Press (403–450).Google Scholar
  12. Scott D.S. (1982) "Some ordered sets in Computer Science" Ordered sets (Rival ed.), Reidel.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • G. Longo
    • 1
  • E. Moggi
    • 1
  1. 1.Dip. InformaticaUniversità di PisaPisa

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