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Subtyping and intersection types revisited

Published:01 October 2007Publication History
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Abstract

Church's system of simple types has proven to be remarkably robust: call-by-name, call-by-need, and call-by-value languages, with or without effects, and even logical frameworks can be based on the same typing rules. When type systems become more expressive, this unity fractures. An early example is the value restriction for parametric polymorphism which is necessary for ML but not Haskell; a later manifestation is the lack of distributivity of function types over intersections in call-by-value languages with effects.

In this talk we reexamine the logical justification for systems of subtyping and intersection types and then explore the consequences in two different settings: logical frameworks and functional programming.

In logical frameworks functions are pure and their definitions observable, but complications could arise from the presence of dependent types. We show that this is not the case, and that we can obtain soundness and completeness theorems for a certain axiomatization of subtyping. We also sketch a connection to the type-theoretic notion of proof irrelevance.

In functional programming we investigate how the encapsulation of effects in monads interacts with subtyping and intersection types, providing an updated analysis of the value restriction and related phenomena. While at present this study is far from complete, we believe that its origin in purely logical notions will give rise to a uniform theory that can easily be adapted to specific languages and their operational interpretations.

References

  1. Rowan Davies and Frank Pfenning. Intersection types and computational effects. In P. Wadler, editor, Proceedings of the Fifth International Conference on Functional Programming (ICFP'00), pages 198--208, Montreal, Canada, September 2000. ACM Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. William Lovas and Frank Pfenning. A bidirectional refinement type system for LF. In B. Pientka and C. Schürmann, editors, Proceedings of the Second International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice, pages 11--25, Bremen, Germany, July 2007.Google ScholarGoogle Scholar
  3. Frank Pfenning and Rowan Davies. A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science, 11:511--540, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Noam Zeilberger. On the unity of duality. Annals of Pure and Applied Logic, 2007. To appear in a special issue on "Classical Logic and Computation".Google ScholarGoogle Scholar

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            • Published in

              cover image ACM SIGPLAN Notices
              ACM SIGPLAN Notices  Volume 42, Issue 9
              Proceedings of the ICFP '07 conference
              September 2007
              331 pages
              ISSN:0362-1340
              EISSN:1558-1160
              DOI:10.1145/1291220
              Issue’s Table of Contents
              • cover image ACM Conferences
                ICFP '07: Proceedings of the 12th ACM SIGPLAN international conference on Functional programming
                October 2007
                346 pages
                ISBN:9781595938152
                DOI:10.1145/1291151

              Copyright © 2007 ACM

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              Association for Computing Machinery

              New York, NY, United States

              Publication History

              • Published: 1 October 2007

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