An abstract notion of application

  • Pietro Di Gianantonio
  • Furio Honsell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 664)


Many concrete notions of function application, suitable for interpreting typed lambda calculi with recursive types, have been introduced in the literature. These arise in different fields such as set theory, multiset theory, type theory and functor theory and are apparently unrelated. In this paper we introduce the general concept of applicative exponential structure and show that it subsumes all these notions. Our approach is based on a generalization of the notion of intersection type. We construe all these structures in a finitary way, so as to be able to utilize uniformly a general form of type assignment system for defining the interpretation function. Applicative exponential structures are just combinatory algebras, in general. Our approach suggests a wide variety of entirely new concrete notions of function application: e.g. in connection with boolean sets. Applicative exponential structures can be used for modeling various forms of non-deterministic operators.


Boolean Algebra Combinatory Structure Heyting Algebra Filter Model Denotational Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S.Abramsky: Domain Theory in Logical Form. Annals of Pure and Applied Logic, (1991)Google Scholar
  2. 2.
    H.Barendregt: Lambda Calculus: its Syntax and Semantics revised version. Studies in Logic. Amsterdam: North Holland 1984Google Scholar
  3. 3.
    H.Barendregt, M.Coppo, M.Dezani, Ciancaglini: A Filter Lambda Model and the Completeness of Type Assignment, Journal to Symbolic Logic, 48, 4 (1983)Google Scholar
  4. 4.
    M.Coppo, M.Dezani Ciancaglini, F.Honsell, G.Longo: Extended Type Structures and Filter Lambda Models. In: G.Longo et al. (eds.): Logic Colloquium'82. Amsterdam: North Holland 1983Google Scholar
  5. 5.
    M.Coppo, M.Dezani Ciancaglini, B.Venneri: Principal Type Schemes and Lambda Calculus Semantics. In: J.Seldin et al. (eds): To H.B.Curry: Essays. Academic Press 1980Google Scholar
  6. 6.
    P.Di Gianantonio, F.Honsell: A General Type Assignment System for an Abstract Notion of Domain. Talks given at the 4th and 6th Meetings of the Jumelage Typed Lambda Calculus. Edinburgh, October 1989 and Paris, January 1991Google Scholar
  7. 7.
    E.Engeler: Algebras and Combinators. Berichte des Instituts f. Informatik 32, ETH, Zurich 1979Google Scholar
  8. 8.
    J.Y.Girard: Normal Functors Power Series and Lambda Calculus. Annals of Pure and Applied Logic, 37, 2 (1988)Google Scholar
  9. 9.
    R.Hindley, G.Longo: Lambda Calculus Models and Extensionalily. Zeit. f. Math. Logik u. Grund. d. Math., 26 (1980)Google Scholar
  10. 10.
    F.Honsell, S.Ronchi della Rocca: Reasoning about interpretations in qualitative Lambda Models. In: M.Broy et al. (eds.) Programming Concepts and Methods. 1990Google Scholar
  11. 11.
    F.Lamarche: Quantitative Domains and Infinitary Algebras. Unpublished manuscript, 1990Google Scholar
  12. 12.
    Ch.-E.Ore: Introducing Girard's quantitative domains. PhD Thesis, Research Report 113. University of Oslo 1988Google Scholar
  13. 13.
    G.Plotkin: A set-theoretical definition of application. Memorandum MIP-R-95, School of Artificial Intelligence, University of Edinburgh, 1972Google Scholar
  14. 14.
    G.Plotkin: Domains for Denotational Semantics, course notes, Stanford 1985Google Scholar
  15. 15.
    D.Scott: Some philosophical issues concerning theories of combinators, lambda calculus and computer science theory. In LNCS 37, Springer Verlag, 1975Google Scholar
  16. 16.
    D.Scott: Data Types as Lattices. SIAM Journal of computing, 5 (1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Pietro Di Gianantonio
    • 1
  • Furio Honsell
    • 1
  1. 1.Dipartimento di Matematica e InfonnaticaUniversità di UdineUdineItaly

Personalised recommendations