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A filter lambda model and the completeness of type assignment1

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

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

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[1]Barendregt, H., The lambda calculus, its syntax and semantics, North-Holland, Amsterdam, 1981.
[2]BEN-Yelles, C. B., Type-assignment in the lambda-calculus; Syntax and semantics, Doctoral Thesis, University College of Swansea, 1979.
[3]Coppo, M., Dezani-Ciancaglini, M. and Sallé, P., Functional characterization of some semantic equalities inside λ-calculus, Automata, languages and programming (Maurer, E., Editor), Lecture Notes in Computer Science, vol. 71, Springer-Verlag, Berlin and New York, 1979, pp. 133–146.
[4]Coppo, M., Dezani-Ciancaglini, M. and Venneri, B., Principal type schemes and λ-calculus semantics, To H.B. Curry, Essays in combinatory logic, lambda-calculus and formalism (Hindley, R. and Seldin, J.P., Editors), Academic Press, New York, 1980, pp. 535560.
[5]Coppo, M., Functional characters of solvable terms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 4558.
[6]Curry, H. B. and Feys, R., Combinatory logic. I, North-Holland, Amsterdam, 1958.
[7]Hindley, R. and Longo, G., Lambda calculus models and extensionality, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 26 (1980), pp. 289310.
[8]Hindley, R., The completeness theorem for typing λ-terms, Theoretical Computer Science (to appear).
[9]Hindley, R., A semantics for Coppo-Dezani type assignment, International symposium on programming (Dezani, M. and Montanari, H., Editors), Lecture Notes in Computer Science, vol. 137, Springer-Verlag, Berlin and New York, 1982, pp. 212226.
[10]Prawitz, D., Natural deduction, a proof-theoretical study, Almquist & Wiksell, Stockholm, 1965.
[11]Scott, D., Open problem n˚ II 4, λ-calculus and computer science theory, (Böhm, C., Editor), Lectures Notes in Computer Science, vol. 37, Springer-Verlag, Berlin and New York, 1975, p. 369.
[12]Scott, D., 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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