Abstract
We contribute to the syntactic study of F ≤, a variant of second order λ-calculus F which appears as a paradigmatic kernel language for polymorphism and subtyping. The type system of F ≤ has a maximum type Top and bounded quantification. We endow this language with the familiar β-rules (for terms and types), to which we add extensionality rules: the η-rules (for terms and types), and a rule (top) which equates all terms of type Top. These rules are suggested by the axiomatization of cartesian closed categories. We show that this theory βηtop≤ is decidable, by exhibiting an effectively weakly normalizing and confluent rewriting system for it. Our proof of confluence relies on the confluence of a corresponding system on F 1 (the extension of F with a terminal type), and follows a general pattern that we investigate for itself in a separate paper.
This work was carried on with the partial support of E.E.C., Esprit Basic Research Action 3003 CLICS.
This work was carried on with the partial support of E.E.C., Esprit Basic Research Action 3070 FIDE and of Italian C.N.R., P.F.I. “Sistemi informatici e calcolo parallelo”.
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References
V. Breazu-Tannen, T. Coquand, C. Gunter, A. Scedrov, Inheritance and Explicit Coercion, to appear in Information and Computation, preliminary version in Proc. Logic in Computer Science 89.
K. Bruce, G. Longo, A Modest Model of Records, Inheritance and Bounded Quantification, Information and Computation 87, 1/2, pp.196–240 (1990).
L. Cardelli, S. Martini, J. Mitchell, A. Scedrov, An Extension of System F with Subtyping, in these Proceedings.
L. Cardelli, P. Wegner, On Understanding Types, Data Abstraction and Polymorphism, ACM Computing Surveys 17 (4) (1985).
L. Cardelli, Typeful Programming, DEC SRC Research Report 45 (1989).
L. Cardelli, G. Longo, A Semantic Basis for Quest, DEC SRC Research Report 55, short version in Proc. Conf. on Lisp and Functional Programming 1990, Nice (1990).
P.-L. Curien, Roberto Di Cosmo, Confluence in the typed λ-calculus extended with Surjective Pairing and a Terminal Type, Proc. ICALP 91, to appear in LNCS.
P.-L. Curien, G. Ghelli, Coherence of Subsumption, Mathematical Structures in Computer Science, to appear; short version in Proc. CAAP 90, LNCS 431.
P.-L. Curien, G. Ghelli, On Confluence of Weakly Normalizing Systems, Proc. RTA 91, to appear in LNCS.
P.-L. Curien, G. Ghelli, Subtyping + Extensionality: Confluence of βηtop reduction in F≤, long version, Technical Report, University of Pisa and LIENS, to appear (1991).
D. van Daalen, The Language Theory of Automath, PhD Thesis, Technical University of Eindhoven, 1980.
G. Ghelli, Proof-theoretic Studies about a Minimal Type System Integrating Inclusion and Parametric Polymorphism, PhD Thesis, TD-6/90, Univ. of Pisa, 1990.
J.-L. Krivine, λ-calcul, Types et Modèles, Masson (1990).
A. Salvesen, The Church-Rosser Theorem for LF with βη-reduction, draft (1989).
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© 1991 Springer-Verlag Berlin Heidelberg
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Curien, PL., Ghelli, G. (1991). Subtyping + extensionality: Confluence of βηtop reduction in F≤ . In: Ito, T., Meyer, A.R. (eds) Theoretical Aspects of Computer Software. TACS 1991. Lecture Notes in Computer Science, vol 526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54415-1_72
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DOI: https://doi.org/10.1007/3-540-54415-1_72
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