Abstract
We characterize those type preorders which yield complete intersection-type assignment systems for λ-calculi, with respect to the three canonical set-theoretical semantics for intersection-types: the inference semantics, the simple semantics, and the F-semantics. These semantics arise by taking as interpretation of types subsets of applicative structures, as interpretation of the preorder relation, ≤, set-theoretic inclusion, as interpretation of the intersection constructor, ∩, set-theoretic intersection, and by taking the interpretation of the arrow constructor, → à la Scott, with respect to either any possible functionality set, or the largest one, or the least one.These results strengthen and generalize significantly all earlier results in the literature, to our knowledge, in at least three respects. First of all the inference semantics had not been considered before. Second, the characterizations are all given just in terms of simple closure conditions on the preorder relation, ≤, on the types, rather than on the typing judgments themselves. The task of checking the condition is made therefore considerably more tractable. Last, we do not restrict attention just to λ-models, but to arbitrary applicative structures which admit an interpretation function. Thus we allow also for the treatment of models of restricted λ-calculi. Nevertheless the characterizations we give can be tailored just to the case of λ-models.
- Abramsky, S. 1991. Domain theory in logical form. Ann. Pure Appl. Logic 51, 1-2, 1--77.Google Scholar
- Abramsky, S. and Ong, C.-H. L. 1993. Full abstraction in the lazy lambda calculus. Inform. Computat. 105, 2, 159--267. Google Scholar
- Barendregt, H. 1984. The Lambda Calculus: Its Syntax and Semantics, revised ed. North-Holland, Amsterdam, The Netherlands.Google Scholar
- Barendregt, H., Coppo, M., and Dezani-Ciancaglini, M. 1983. A filter lambda model and the completeness of type assignment. J. Symbolic Logic 48, 4, 931--940.Google Scholar
- Barendregt, H. P. et al. In press. Typed λ-Calculus and Applications. North-Holland, Amsterdam, The Netherlands.Google Scholar
- Berline, C. 2000. From computation to foundations via functions and application: The λ-calculus and its webbed models. Theoret. Comput. Sci. 249, 81--161. Google Scholar
- Coppo, M. and Dezani-Ciancaglini, M. 1980. An extension of the basic functionality theory for the λ-calculus. Notre Dame J. Formal Logic 21, 4, 685--693.Google Scholar
- Coppo, M., Dezani-Ciancaglini, M., Honsell, F., and Longo, G. 1984. Extended type structures and filter lambda models. In Logic Colloquium '82, G. Lolli, G. Longo, and A. Marcja, Eds. North-Holland, Amsterdam, The Netherlands, 241--262.Google Scholar
- Coppo, M., Dezani-Ciancaglini, M., and Venneri, B. 1981. Functional characters of solvable terms. Z. Math. Logik Grundlag. Math. 27, 1, 45--58.Google Scholar
- Coppo, M., Dezani-Ciancaglini, M., and Zacchi, M. 1987. Type theories, normal forms, and D&infty;-lambda-models. Inform. Computat. 72, 2, 85--116. Google Scholar
- Dezani-Ciancaglini, M., Honsell, F., and Alessi, F. 2000. A complete characterization of the complete intersection-type theories. In Proceedings in Informatics, J. Rolim, A. Broder, A. Corradini, R. Gorrieri, R. Heckel, J. Hromkovic, U. Vaccaro, and J. Wells, Eds. ITRS'00 Workshop, vol. 8. Carleton-Scientific, Waterloo, Ont., Canada, 287--301.Google Scholar
- Dezani-Ciancaglini, M. and Margaria, I. 1986. A characterization of F-complete type assignments. Theoret. Comput. Sci. 45, 2, 121--157. Google Scholar
- Egidi, L., Honsell, F., and Ronchi della Rocca, S. 1992. Operational, denotational and logical descriptions: A case study. Fund. Inform. 16, 2, 149--169. Google Scholar
- Engeler, E. 1981. Algebras and combinators. Algebra Universalis 13, 3, 389--392.Google Scholar
- Hindley, J. and Seldin, J. 1986. Introduction to Combinators and λ-Calculus. Cambridge University Press, Cambridge, U.K. Google Scholar
- Hindley, J. R. 1982. The simple semantics for Coppo--Dezani--Sallé types. In International Symposium on Programming, M. Dezani-Ciancaglini and U. Montanari, Eds. Lecture Notes in Computer Science, vol. 137. Springer, Berlin, Germany, 212--226. Google Scholar
- Hindley, J. R. 1983a. The completeness theorem for typing λ-terms. Theoret. Comput. Sci. 22, 1--17.Google Scholar
- Hindley, J. R. 1983b. Curry's type-rules are complete with respect to F-semantics too. Theoret. Comput. Sci. 22, 127--133.Google Scholar
- Hindley, J. R. and Longo, G. 1980. Lambda-calculus models and extensionality. Z. Math. Logik Grundlag. Math. 26, 4, 289--310.Google Scholar
- Honsell, F. and Lenisa, M. 1999. Semantical analysis of perpetual strategies in λ-calculus. Theoret. Comput. Sci. 212, 1-2, 183--209. Google Scholar
- Honsell, F. and Ronchi della Rocca, S. 1992. An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus. J. Comput. System Sci. 45, 1, 49--75. Google Scholar
- Mitchell, J. 1988. Polymorphic type inference and containment. Inform. Computat. 76, 2--3, 211--249. Google Scholar
- Park, D. 1976. The Y-combinator in Scott's λ-calculus models (revised version). Theory of Computation Report 13, Department of Computer Science, University of Warwick, Coventry, U.K. Google Scholar
- Plotkin, G. D. 1975. Call-by-name, call-by-value and the λ-calculus. Theoret. Comput. Sci. 1, 2, 125--159.Google Scholar
- Plotkin, G. D. 1993. Set-theoretical and other elementary models of the λ-calculus. Theoret. Comput. Sci. 121, 1-2, 351--409. Google Scholar
- Pottinger, G. 1980. A type assignment for the strongly normalizable λ-terms. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Hindley and J. Seldin, Eds. Academic Press, London, U.K., 561--577.Google Scholar
- Scott, D. 1972. Continuous lattices. In Toposes, Algebraic Geometry and Logic, F. Lawvere, Ed. Lecture Notes in Mathematics, vol. 274. Springer, Berlin, Germany, 97--136.Google Scholar
- Scott, D. 1975. Open problem. In Lambda Calculus and Computer Science Theory, C. Böhm, Ed. Lecture Notes in Computer Science, vol. 37. Springer, Berlin, Germany, 369.Google Scholar
- Scott, D. 1976. Data types as lattices. SIAM J. Comput. 5, 3, 522--587.Google Scholar
- Scott, D. 1980a. Lambda calculus: Some models, some philosophy. In The Kleene Symposium, J. Barwise, H. J. Keisler, and K. Kunen, Eds. North-Holland, Amsterdam, The Netherlands, 223--265.Google Scholar
- Scott, D. S. 1980b. Letter to Albert Meyer. Carnegie Mellon University, Pittsburgh, PA.Google Scholar
- van Bakel, S. 1992. Complete restrictions of the intersection type discipline. Theoret. Comput. Sci. 102, 1, 135--163. Google Scholar
- Vickers, S. 1989. Topology Via Logic. Cambridge University Press, Cambridge. Google Scholar
- Yokouchi, H. 1994. F-semantics for type assignment systems. Theoret. Comput. Sci. 129, 1, 39--77. Google Scholar
Index Terms
- A complete characterization of complete intersection-type preorders
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