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A filter lambda model and the completeness of type assignment1

  • Henk Barendregt (a1), Mario Coppo (a2) and Mariangiola Dezani-Ciancaglini
Abstract

In [6, p. 317] Curry described a formal system assigning types to terms of the type-free λ-calculus. In [11] Scott gave a natural semantics for this type assignment and asked whether a completeness result holds.

Inspired by [4] and [5] we extend the syntax and semantics of the Curry types in such a way that filters in the resulting type structure form a domain in the sense of Scott [12]. We will show that it is possible to turn the domain of types into a λ-model, among other reasons because all λ-terms possess a type. This model gives the completeness result for the extended system. By a conservativity result the completeness for Curry's system follows.

Independently Hindley [8], [9] has proved both completeness results using term models. His method of proof is in some sense dual to ours.

For λ-calculus notation see [1].

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1

Research partially supported by the Italian C.N.R. Grant 80.01917.97.

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[5] M. Coppo , Functional characters of solvable terms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 4558.

[6] H. B. Curry and R. Feys , Combinatory logic. I, North-Holland, Amsterdam, 1958.

[7] R. Hindley and G. Longo , Lambda calculus models and extensionality, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 26 (1980), pp. 289310.

[9] R. Hindley , A semantics for Coppo-Dezani type assignment, International symposium on programming ( M. Dezani and H. Montanari , Editors), Lecture Notes in Computer Science, vol. 137, Springer-Verlag, Berlin and New York, 1982, pp. 212226.

[12] D. Scott , Lectures on a mathematical theory of computation, University of Oxford, 1980.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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