Complete proof systems for algebraic simply-typed terms
Abstract
We show that reasoning by case analysis (on whether subprograms diverge or converge) is complete for proving PCF observational congruences of algebraic terms. The latter are applicative combinations of first-order variables and a constant Ω denoting a diverging program of base type. A restricted version of the logic is complete for proving equality of algebraic terms in the full continuous type hierarchy (equivalently, observational congruence in PCF with parallel conditional). We show that the provability in the latter logic is in co-NP. We also give complete equational proof systems for a subclass of algebraic terms; provability in these systems is in linear time.
References
[1]
Henk P. Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in Logic. North-Holland, 1981. Revised Edition, }984.
[2]
G. Berry, and P.-L. Curien. SequentiM algorithms on concrete d~t~ $tructurc~. ThcorcticM Computer Science 20, 1982.
[3]
G. Berry, P.-L. Curien, and J.-J. L~vy. Full abstraction for sequential languages- the state of the art. In M. Nivat and J. Reynolds, editors, Algebraic Methods i?l. Semantics. Cambridge Univ. Press, 1985.
[4]
R. Cartwright, and M. Felleisen. Observable sequentiality and full abstraction. Principles of Programming Languages, }992.
[5]
Harvey Friedman. Equality between functionals. In Rohit Parikh, editor, Logic Colloquium '73, volume 453 of Lect. Notes in Math., pages 22-37. Springer-Verlag, 1975.
[6]
Michael J.C. Gordon, Robin Milner, and C.P. Wadsworth. Edinburgh LCF: A Mechanical Logic of Computation, volume 78 of Lect. Notes in Computer Sci. Springer- Verlag, 1979.
[7]
Robin Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Sci., 4:1-22, 1977.
[8]
Gordon D. Plotkin. )~~definability and logical relations. Technical Report SAI-RM-4, University of Edinburgh, School of Artificial Intelligence, 1973.
[9]
Gordon D. Plotkin. LCF considered as a programming language. Theoretical Computer Sci., 5:223-257, 1977.
[10]
Gordon D. Plotkin. Notes on completeness of the full continuous type hierarchy. Unpublished manuscript, MiT, November 1982.
[11]
V.Yu. Sazonov. Expressibility of functions in D. Scott's LCF language. Algebra i Logika, 15:308-330, 1976. (Russian).
[12]
Richard Statman. Completeness, invariance and lambda-definability. J. Symbolic Logic, 47:17-~6, 1982.
[13]
Richard Statman. Equality between fuuctionals revisited. In L.A. Harrington, et al., editor, Harvey Friedman's Research on the Foundations of Mathematics, volume 117 of Studies in Logic, pages 331--338. North- Holland, 1985.
[14]
Ramesh Subrahmanyam, Dan Dougherty. Completeness theorem for the set-theoretic coproduct model. Manuscript, 1993.
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- Complete proof systems for algebraic simply-typed terms
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Publication History
Published: 01 July 1994
Published in SIGPLAN-LISPPOINTERS Volume VII, Issue 3
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