Abstract
Various type structures, called Intersection Type Structures, have been introduced in the literature in order to define models of λ-calculus and simultaneously to reason in a finitary way about λ-terms. All these systems are only employed as meet-semilattices generated by preorders built on prime types. For this reason these structures are linguistically redundant. Starting from this observation we introduce the category of Type Preorders, which arises naturally when we eliminate redundant types from Intersection Type Structures. We give a Stoneduality type result for Type Preorders, showing that they are equivalent to the category of prime-algebraic complete lattices and Scott continuous functions. Thus we clarify the domain-theoretic description of Intersection Type Structures, which often appears opaque. As an application we give the domain-theoretic reading of the Intersection Union Type Structure.
Work partially supported by Science Project MASK Sci#-CT92-0776, 60%–40% MURST
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References
S. Abramsky: Domain Theory in Logical Form. Annals of Pure and Applied Logic, 51 (1991) 1–77.
S. Abramsky: A The lazy λ-calculus. In D. Turner ed. Research Topics in Functional Programming, Addison-Wesley, Reading, MA.
H.P. Barendregt, M. Coppo, M. Dezani-Ciancaglini: A Filter λ-models and Completeness of Type Assignment. Journal of Symbolic Logic, 48 (1983) 931–940.
F. Barbanera, M. Dezani-Ciancaglini: Intersection and Union Types TACS'91, LNCS, 526 (1991), 651–674.
F. Barbanera, M. Dezani-Ciancaglini, U. de' Liguoro:Intersection and Union Types: Syntax and Semantics. Information and Computation, to appear.
G. Boudol: Lambda-Calculi for (strict) Parallel Functions. INRIA / Sophia-Antipolis Technical Report 1387, (1991) To appear on Information and Computation.
M. Coppo, M. Dezani-Ciancaglini, F. Honsell, G. Longo: Extended Type Structures and Filter λ-models. In G. Lolli, G. Longo, A. Morcja editors, Logic Colloquium '82, North Holland, Amsterdam (1984) 241–262.
M. Coppo, M. Dezani-Ciancaglini, M. Zacchi: Type Theories, Normal Forms and D∞ — λ-models. Information and Computation, 72 (1987) 85–116.
M. Dezani-Ciancaglini, U. de' Liguoro, A. Piperno: Filter λ-models for a Parallel and Non-Deterministic λ-calculus. In A.M. Borzyszkowsky, S. Sokolowsky editors, 18th International Symposium on Mathematical Foundation of Computer Science, Gdansk (Poland) (1993) 403–412. Springer-Verlag LNCS Vol 711.
M. Dezani-Ciancaglini, I. Margaria: A Characterization of F-complete Type Assignments. Theoretical Computer Science, 45 (1986) 121–157.
G.K. Gierz, K.H. Hoffmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott: A Compendium of Continuous Lattices. Springer-Verlag, Berlin, 1980.
L. Egidi, F. Honsell, S. Ronchi della Rocca: Operational, Denotational and Logical Descriptions: a case study. Fundamenta Informaticae 16 (2) (1992) 149–169.
F. Honsell, M. Lenisa: Some Results on the Full Abstraction Problem for Restricted Lambda Calculi. In A.M. Borzyszkowsky, S. Sokolowsky editors, 18th International Symposium on Mathematical Foundation of Computer Science, Gdansk (Poland) (1993) 84–104. Springer-Verlag LNCS Vol 711.
F. Honsell, S. Ronchi della Rocca: An Approximation Theorem for Scott Topological Lambda Models and the Topological Incompleteness of Lambda Calculus. To appear on Journal of Computer and System Science.
P.T. Johnstone: Stone Spaces. Volume 3 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1982.
K.G. Larsen, G. Winskel: Using Information Systems to Solve Recursive Domain Equation Effectively. In D.B MacQueen, G. Khan and G. Plotkin editors, Semantics of Data Types, 109–130, Berlin (1984) Springer-Verlag LNCS Vol. 173.
M. Nielsen, G. Plotkin, G. Winskel:Petri Nets, Event Structures and Domains, part I. Theoretical Computer Science, 13 (1981) 85–108.
D.S. Scott: Domain for Denotational Semantics. In M. Nielson, E.M. Schmidt editors, Automata, Languages and Programming: Proceedings 1982. Berlin (1982) Springer-Verlag LNCS 140.
M. Smyth, G. Plotkin: The Category Theoretic Solution of Recursive Domain Equations. SIAM Journal Computing 11 (1982) 761.
S. Vickers: Topology via Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1988.
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© 1994 Springer-Verlag Berlin Heidelberg
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Alessi, F. (1994). Type Preorders. In: Tison, S. (eds) Trees in Algebra and Programming — CAAP'94. CAAP 1994. Lecture Notes in Computer Science, vol 787. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017472
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DOI: https://doi.org/10.1007/BFb0017472
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