Abstract
It is shown, that ordinary isomorphisms (associativity and commutativity of tensor, isomorphisms for tensor unit and currying) provide a complete axiom system for the isomorphism of types in Symmetric Monoidal Closed categories. This gives also a decision algorithm. The problem originally arises from computer science, as the isomorphism of types in SMC categories corresponds to the isomorphism of types in linear lambda calculus, and could be used for search in databases.
supported by CLICS grant (on leave from S.Petersburg Institute for Informatics, Russian Academy of Sciences)
Preview
Unable to display preview. Download preview PDF.
References
M.Rittri. Retrieving library functions by unifying types modulo linear isomorphism. Proceedings of Conference on Lisp and Functional Programming, 1992.
S.V.Soloviev The category of finite sets and cartesian closed categories. Zapiski Nauchnych Seminarov Leningradskogo Otdelenya Matematicheskogo Instituta im.V.A.Steklova AN SSSR, 105:174–194, 1981 (English translation in: Journal of Soviet Mathematics, 22(3):1387–1400, 1983)
K.Bruce, R.DiCosmo and G.Longo. Provable isomorphism of types. Preprint LIENS-90-14, Ecole Normale Superieure, Paris, 1990.
R.Di Cosmo. Invertibility of Terms and valid isomorphisms. A prooftheoretic study on second order lambda calculus with surjective pairing and terminal object. Technical Report LIENS-91-10, Ecole Normale Superieure, Paris, 1991.
J.Lambek. Deductive Systems and Categories. II. Lect.Notes in Math.. 86:76–122,1969.
J.Lambek. Multicategories Revisited. In J.W.Gray and A.Scedrov, editors, Categories in Computer Science and Logic. AMS, Providence, 217–239.
M.E.Szabo. Algebra of Proofs. Studies in Logic and the Foundations of Mathematics, 88, North-Holland P.C., 1978.
G.E.Mints. Closed categories and Proof Theory. J.Soviet Math., 15:45–62, 1981.
A.A.Babaev. Equality of Morphisms in Closed categories. I. Izvestia AN Azerbaijanskoi SSR, 1:3–9, 1981.
A.A.Babaev. Equality of Morphisms in Closed categories.II. Izvestia AN Azerbaijanskoi SSR, 2:3–9, 1981.
C.B. Jay. The structure of free closed categories. Journal of Pure and Applied Algebra, 66(3):271–287, 1990.
G.M.Kelly and S. Mac Lane. Coherence in Closed Categories. Journal of Pure and Applied Algebra, 1(1):97–140, 1971.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Soloviev, S. (1993). A complete axiom system for isomorphism of types in closed categories. In: Voronkov, A. (eds) Logic Programming and Automated Reasoning. LPAR 1993. Lecture Notes in Computer Science, vol 698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56944-8_71
Download citation
DOI: https://doi.org/10.1007/3-540-56944-8_71
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56944-2
Online ISBN: 978-3-540-47830-0
eBook Packages: Springer Book Archive