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.
Supplemental Material
- }}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 Scholar
- }}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 Scholar
Digital Library
- }}Arnon Avron, Furio Honsell, Marino Miculan, and Cristian Paravano. Encoding modal logics in logical frameworks. Studia Logica, 60(1), January 1998.Google Scholar
- }}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 Scholar
Digital Library
- }}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 Scholar
Digital Library
- }}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 Scholar
- }}Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987. Google Scholar
Digital Library
- }}Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. Journal of the ACM, 40(1):143--184, January 1993. Google Scholar
Digital Library
- }}Robert Harper and Frank Pfenning. On equivalence and canonical forms in the LF type theory. ACM Transactions on Computational Logic, 6(1), 2005. Google Scholar
Digital Library
- }}Tom Murphy, VII. Modal Types for Mobile Code. PhD thesis, Carnegie Mellon University, School of Computer Science. Pittsburgh, Pennsylvania, May 2008. Google Scholar
Digital Library
- }}Aleksandar Nanevski, Frank Pfenning, and Brigitte Pientka. A contextual modal type theory. ACM Transactions on Computational Logic, 9(3), 2008. Google Scholar
Digital Library
- }}Peter W. O'Hearn and David J. Pym. The logic of bunched implications. Bulletin of Symbolic Logic, 5(2), 1999.Google Scholar
- }}Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Carnegie Mellon University, School of Computer Science, December 1994.Google Scholar
- }}Frank Pfenning and Rowan Davies. A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science, 11(4):511--540. 2001. Google Scholar
Digital Library
- }}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 Scholar
Digital Library
- }}Frank Pfenning and Carsten Schürmann. Twelf User's Guide, Version 1.4, 2002. Available electronically at http://www.cs.cmu.edu/~twelf.Google Scholar
- }}Jeff Polakow. Ordered Linear Logic and Applications. PhD thesis, Carnegie Mellon University, School of Computer Science. Pittsburgh, Pennsylvania, August 2001. Google Scholar
Digital Library
- }}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 Scholar
Digital Library
- }}Alex Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD thesis, University of Edinburgh, 1994.Google Scholar
- }}Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Carnegie Mellon University, School of Computer Science. Pittsburgh, Pennsylvania, 1999. Google Scholar
Digital Library
Index Terms
Higher-order representation of substructural logics
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