Abstract
What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds’ relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac Lane-Mints approach to coherence problems in category theory.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Abramsky, J. C. Mitchell, A. Scedrov, P. Wadler. Relators (to appear.).
E.S. Bainbridge, P. Freyd, A. Scedrov, and P. J. Scott. Functorial Polymorphism, Theoretical Computer Science 70 (1990), pp. 35–64.
E.S. Bainbridge, P. Freyd, A. Scedrov, and P. J. Scott. Functorial Polymorphism: Preliminary Report, Logical Foundations of Functional Programming, G. Huet, ed., Addison Wesley (1990), pp. 315–327.
M. Barr. *-Autonomous Categories. Springer LNM 752, 1979.
S. Eilenberg and G. M Kelly. A generalization of the functorial calculus, J. Algebra 3 (1966), pp. 366–375.
A. Felty . A Logic Program for Transforming Sequent Proofs to Natural Deduction Proofs. In: Proc. December 1989 Workshop on Extension of Logic Programming, ed. by P. Schroeder-Heister, Springer LNCS, to appear.
P. Freyd. Structural Polymorphism, Manuscript, Univ. of Pennsylvania (1989), to appear in Th. Comp. Science.
P. Freyd, J-Y. Girard, A. Scedrov, and P. J. Scott. Semantic Parametricity in Polymorphic Lambda Calculus, Proc. 3rd IEEE Symposium on Logic in Computer Science, Edinburgh, 1988.
J-Y. Girard . Normal functors, power series, and A-calculus, Ann. Pure and Applied Logic 37 (1986) pp. 129–177.
J-Y. Girard. The System F of Variables Types, Fifteen Years Later, Theoretical Computer Science, 45, pp. 159–192.
J-Y. Girard . Linear Logic, Theoretical Computer Science, 50, 1987, pp. 1–102.
J-Y. Girard, Y.Lafont, and P. Taylor. Proofs and Types, Cambridge Tracts in Theoretical Computer Science, 7, Cambridge University Press, 1989.
G. M. Kelly. Many-variable functorial calculus.I, Coherence in Categories Springer LNM 281, pp. 66–105.
G. M. Kelly and S. Mac Lane. Coherence in closed categories, J. Pure Appl. Alg. 1 (1971), pp. 97–140.
J. Lambek. Deductive Systems and Categories I, J. Math. Systems Theory 2 (1968), pp. 278–318.
J. Lambek. Deductive Systems and Categories II, Springer LNM 86 (1969), pp. 76–122.
J. Lambek. Multicategories Revisited, Contemp. Math. 92 (1989), pp. 217–239.
J. Lambek . Logic without structural rules, Manuscript. (McGill University), 1990.
J. Lambek and P. J. Scott. Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics 7, Cambridge University Press, 1986.
S. Mac Lane. Categories for the Working Mathematician, Springer Graduate Texts in Mathematics, Springer-Verlag, 1971.
S. Mac Lane. Why commutative diagrams coincide with equivalent proofs, Contemp. Math. 13 (1982), pp. 387–401.
G. E. Mints. Closed categories and the theory of proofs, J. Soviet Math 15 (1981), pp. 45–62.
J. C. Mitchell and P. J. Scott. Typed Lambda Models and Cartesian Closed Categories, Contemp. Math. 92, pp. 301–316.
J.C. Mitchell and R. Harper, The Essence of ML, Proc. 15th Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL), San Diego, 1988, pp. 28–46.
A. Pitts . Polymorphism is set-theoretic, constructively, in: Category Theory and Computer Science, Springer LNCS 283 (D. H. Pitt, ed.) (1987) pp. 12–39.
D. Prawitz. Natural Deduction, Almquist & Wiksell, Stockhom, 1965.
J.C.Reynolds . Types, Abstraction, and Parametric Polymorphism, in: Information Processing ’83, R. E. A. Mason, ed. North-Holland, 1983, pp. 513–523.
R. A. G. Seely. Categorical Semantics for Higher Order Polymorphic Lambda Calculus. J. Symb. Logic 52(1987), pp. 969–989.
R. A. G. Seely. Linear Logic, *-Autonomous Categories, And Cofree Coalgebras. Con-temp. Math. 92(1989), pp. 371–382.
R. Statman. Logical Relations and the Typed Lambda Calculus. Inf. and Control 65(1985), pp. 85–97.
A. S. Troelstra and D. van Dalen. Constructivism in Mathematics, Vols. I and II. North-Holland, 1988.
P. Wadler. Theorems for Free! 4 th International Symposium on Functional Programming Languages and Computer Architecture, Assn. Comp. Machinery, London, Sept. 1989.
J. Zucker. The Correspondence between Cut-Elimination and Normalization, Ann. Math. Logic 7 (1974) pp. 1–112.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag New York, Inc
About this paper
Cite this paper
Girard, JY., Scedrov, A., Scott, P.J. (1992). Normal Forms and Cut-Free Proofs as Natural Transformations. In: Moschovakis, Y.N. (eds) Logic from Computer Science. Mathematical Sciences Research Institute Publications, vol 21. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2822-6_8
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2822-6_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7685-2
Online ISBN: 978-1-4612-2822-6
eBook Packages: Springer Book Archive