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On the unification problem for Cartesian closed categories

Published online by Cambridge University Press:  12 March 2014

Paliath Narendran ,
Frank Pfenning  and
Richard Statman
Paliath Narendran
Affiliation:
Institute of Programming and Logics, Department of Computer Science, State University of NY at Albany, Albany, NY 12222, USA, E-mail: dran@cs.albany.edu
Frank Pfenning
Affiliation:
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA, E-mail: fp@cs.cmu.edu
Richard Statman
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pa 15213, USA, E-mail: rstatman@cs.cmu.edu
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Abstract

Cartesian closed categories (CCCs) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce, Di Cosmo and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an open question. We also show that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible.

CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by Rittri. It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 636 - 647
Copyright
Copyright © Association for Symbolic Logic 1997

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References

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