skip to main content
research-article

Higher-order representation of substructural logics

Published:27 September 2010Publication History
Skip Abstract Section

Abstract

We present a technique for higher-order representation of substructural logics such as linear or modal logic. We show that such logics can be encoded in the (ordinary) Logical Framework, without any linear or modal extensions. Using this encoding, metatheoretic proofs about such logics can easily be developed in the Twelf proof assistant.

Skip Supplemental Material Section

Supplemental Material

icfp-tues-1030-crary.mov

References

  1. }}Arnon Avron, Furio Honsell, and Ian A. Mason. Using typed lambda calculus to implement formal systems on a machine. Technical Report ECS-LFCS-87-31, Department of Computer Science. University of Edinburgh, July 1987.Google ScholarGoogle Scholar
  2. }}Arnon Avron, Furio Honsell, and Ian A. Mason. An overview of the Edinburgh Logical Framework. In Graham Birtwistle and P. A. Subrahmanyam, editors, Current Trends in Hardware Verification and Automated Theorem Proving. Springer. 1989. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. }}Arnon Avron, Furio Honsell, Marino Miculan, and Cristian Paravano. Encoding modal logics in logical frameworks. Studia Logica, 60(1), January 1998.Google ScholarGoogle Scholar
  4. }}Brian Aydemir, Arthur Charguéraud, Benjamin C. Pierce, Randy Pollack, and Stephanie Weirich. Engineering formal metatheory. In Thirty-Fifth ACM Symposium on Principles of Programming Languages, San Francisco, California, January 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. }}Iliano Cervesato and Frank Pfenning. A linear logical framework. In Eleventh IEEE Symposium on Logic in Computer Science, pages 264--275, New Brunswick, New Jersey, July 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. }}Karl Crary. Explicit contexts in LF. In Workshop on Logical Frameworks and Meta-Languages: Theory and Practice, Pittsburgh, Pennsylvania, 2008. Revised version at www.cs.cmu.edu/~crary/papers/2009/excon-rev.pdf.Google ScholarGoogle Scholar
  7. }}Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. }}Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. Journal of the ACM, 40(1):143--184, January 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. }}Robert Harper and Frank Pfenning. On equivalence and canonical forms in the LF type theory. ACM Transactions on Computational Logic, 6(1), 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. }}Tom Murphy, VII. Modal Types for Mobile Code. PhD thesis, Carnegie Mellon University, School of Computer Science. Pittsburgh, Pennsylvania, May 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. }}Aleksandar Nanevski, Frank Pfenning, and Brigitte Pientka. A contextual modal type theory. ACM Transactions on Computational Logic, 9(3), 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. }}Peter W. O'Hearn and David J. Pym. The logic of bunched implications. Bulletin of Symbolic Logic, 5(2), 1999.Google ScholarGoogle Scholar
  13. }}Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Carnegie Mellon University, School of Computer Science, December 1994.Google ScholarGoogle Scholar
  14. }}Frank Pfenning and Rowan Davies. A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science, 11(4):511--540. 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. }}Frank Pfenning and Conal Elliott. Higher-order abstract syntax. In 1988 SIGPLAN Conference on Programming Language Design and Implementation, pages 199--208, Atlanta, Georgia, June 1988. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. }}Frank Pfenning and Carsten Schürmann. Twelf User's Guide, Version 1.4, 2002. Available electronically at http://www.cs.cmu.edu/~twelf.Google ScholarGoogle Scholar
  17. }}Jeff Polakow. Ordered Linear Logic and Applications. PhD thesis, Carnegie Mellon University, School of Computer Science. Pittsburgh, Pennsylvania, August 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. }}Jeff Polakow and Frank Pfenning. Natural deduction for intuitionistic non-commutative linear logic. In 1999 International Conference on Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science. L'Aquila, Italy, April 1999. Springer. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. }}Alex Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.Google ScholarGoogle Scholar
  20. }}Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Carnegie Mellon University, School of Computer Science. Pittsburgh, Pennsylvania, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Higher-order representation of substructural logics

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM SIGPLAN Notices
        ACM SIGPLAN Notices  Volume 45, Issue 9
        ICFP '10
        September 2010
        382 pages
        ISSN:0362-1340
        EISSN:1558-1160
        DOI:10.1145/1932681
        Issue’s Table of Contents
        • cover image ACM Conferences
          ICFP '10: Proceedings of the 15th ACM SIGPLAN international conference on Functional programming
          September 2010
          398 pages
          ISBN:9781605587943
          DOI:10.1145/1863543

        Copyright © 2010 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 27 September 2010

        Check for updates

        Qualifiers

        • research-article

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      ePub

      View this article in ePub.

      View ePub
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!