Skip to main content

Quotient Types: A Modular Approach

  • Conference paper
  • First Online:
Theorem Proving in Higher Order Logics (TPHOLs 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2410))

Included in the following conference series:

  • 486 Accesses

Abstract

In this paper we introduce a new approach to axiomatizing quotient types in type theory. We suggest replacing the existing monolithic rule set by a modular set of rules for a specially chosen set of primitive operations. This modular formalization of quotient types turns out to be much easier to use and free of many limitations of the traditional monolithic formalization. To illustrate the advantages of the new approach, we show how the type of collections (that is known to be very hard to formalize using traditional quotient types) can be naturally formalized using the new primitives. We also show how modularity allows us to reuse one of the new primitives to simplify and enhance the rules for the set types.

This work was partially supported by AFRL grant F49620-00-1-0209 and ONR grant N00014-01-1-0765.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Stuart F. Allen. A Non-Type-Theoretic Semantics for Type-Theoretic Language. PhD thesis, Cornell University, 1987.

    Google Scholar 

  2. Stuart F. Allen. A Non-type-theoretic Definition of Martin-Löf’ s Types. In Proceedings of the Second Symposium on Logic in Computer Science, pages 215–224. IEEE, June 1987.

    Google Scholar 

  3. Roland Backhouse. A note of subtypes in Martin-Löf’ s theory of types. Technical Report CSM-90, University of Essex, November 1984.

    Google Scholar 

  4. Roland Backhouse. On the meaning and construction of the rules in Martin-Löf’ s theory of types. In A. Avron, editor, Workshop on General Logic, Edinburgh, February 1987, number ECS-LFCS-88-52. Department of Computer Science, University of Edinburgh, May 1988.

    Google Scholar 

  5. Roland C. Backhouse, Paul Chisholm, Grant Malcolm, and Erik Saaman. Do-it-yourself type theory. Formal Aspects of Computing, 1:19–84, 1989.

    Article  Google Scholar 

  6. Ken Birman, Robert Constable, Mark Hayden, Jason J. Hickey, Christoph Kreitz, Robbert van Renesse, Ohad Rodeh, and Werner Vogels. The Horus and Ensemble projects: Accomplishments and limitations. In DARPA Information Survivability Conference and Exposition (DISCEX 2000), pages 149–161. IEEE, 2000.

    Google Scholar 

  7. Robert L. Constable. Mathematics as programming. In Proceedings of the Workshop on Programming and Logics, Lectures Notes in Computer Science 164, pages 116–128. Springer-Verlag, 1983.

    Google Scholar 

  8. Robert L. Constable. Types in logic, mathematics, and programming. In S. R. Buss, editor, Handbook of Proof Theory, chapter X, pages 683–786. Elsevier Science B.V., 1998.

    Google Scholar 

  9. Robert L. Constable, Stuart F. Allen, H.M. Bromley, W.R. Cleaveland, J.F. Cremer, R.W. Harper, Douglas J. Howe, T.B. Knoblock, N.P. Mendler, P. Panangaden, James T. Sasaki, and Scott F. Smith. Implementing Mathematics with the NuPRL Development System. Prentice-Hall, NJ, 1986.

    Google Scholar 

  10. Pierre Courtieu. Normalized types. In L. Fribourg, editor, Computer Science Logic, Proceedings of the 10th Annual Conference of the EACSL, volume 2142 of Lecture Notes in Computer Science, pages 554–569. Springer-Verlag, 2001. http://link.springer-ny.com/link/service/series/0558/tocs/t2142.htm.

    Google Scholar 

  11. Jason J. Hickey. NuPRL-Light: An implementation framework for higer-order logics. In William McCune, editor, Proceedings of the 14th International Conference on Automated Deduction, volume 1249 of Lecture Notes on Artificial Intelligence, pages 395–399, Berlin, July 13–17 1997. Springer. CADE’97. An extended version of the paper can be found at http://www.cs.caltech.edu/~jyh/papers/cade14_nl/default.html.

  12. Jason J. Hickey. The MetaPRL Logical Programming Environment. PhD thesis, Cornell University, Ithaca, NY, January 2001.

    Google Scholar 

  13. Jason J. Hickey, Brian Aydemir, Yegor Bryukhov, Alexei Kopylov, Aleksey Nogin, and Xin Yu. A listing of MetaPRL theories. http://metaprl.org/theories.pdf.

  14. Jason J. Hickey, Aleksey Nogin, Alexei Kopylov, et al. MetaPRL home page. http://metaprl.org/.

  15. Martin Hofmann. Extensional concepts in intensional Type theory. PhD thesis, University of Edinburgh, Laboratory for Foundations of Computer Science, July 1995.

    Google Scholar 

  16. Martin Hofmann. A simple model for quotient types. In Typed Lambda Calculus and Applications, volume 902 of Lecture Notes in Computer Science, pages 216–234, 1995.

    Chapter  Google Scholar 

  17. Douglas J. Howe. Semantic foundations for embedding HOL in NuPRL. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85–101. Springer-Verlag, Berlin, 1996.

    Chapter  Google Scholar 

  18. Alexei Kopylov and Aleksey Nogin. Markov’s principle for propositional type theory. In L. Fribourg, editor, Computer Science Logic, Proceedings of the 10th Annual Conference of the EACSL, volume 2142 of Lecture Notes in Computer Science, pages 570–584. Springer-Verlag, 2001. http://link.springer-ny.com/link/service/series/0558/tocs/t2142.htm.

    Google Scholar 

  19. Xiaoming Liu, Christoph Kreitz, Robbert van Renesse, Jason J. Hickey, Mark Hayden, Kenneth Birman, and Robert Constable. Building reliable, high-performance communication systems from components. In 17 th ACM Symposium on Operating Systems Principles, December 1999.

    Google Scholar 

  20. Per Martin-Löf. Constructive mathematics and computer programming. In Proceedings of the Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153–175, Amsterdam, 1982. North Holland.

    Google Scholar 

  21. J. McCarthy. A basis for a mathematical theory of computation. In P. Braffort and D. Hirschberg, editors, Computer Programming and Formal Systems, pages 33–70. Amsterdam: North-Holland, 1963.

    Google Scholar 

  22. Aleksey Nogin. Quotient types — A modular approach. Department of Computer Science http://cs-tr.cs.cornell.edu/Dienst/UI/1.0/Display/ncstrl.cornell/ TR2002-1869 TR2002-1869, Cornell University, April 2002. See also http://nogin.org/papers/quotients.html.

  23. Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. Accepted to TPHOLs 2002, 2002.

    Google Scholar 

  24. Bengt Nordström and Kent Petersson. Types and specifications. In IFIP’93. Elsvier, 1983.

    Google Scholar 

  25. Bengt Nordström, Kent Petersson, and Jan M. Smith. Programming in Martin-Löf’s Type Theory. Oxford Sciences Publication, Oxford, 1990.

    MATH  Google Scholar 

  26. Frank Pfenning and Rowan Davies. Judgmental reconstruction of modal logic. Mathematical Structures in Computer Science, 11(4), August 2001.

    Google Scholar 

  27. Simon Thompson. Type Theory and Functional Programming. Addison-Wesley, 1991.

    Google Scholar 

  28. Anne Sjerp Troelstra. Metamathematical Investigation of Intuitionistic Mathematics, volume 344 of Lecture Notes in Mathematics. Springer-Verlag, 1973.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nogin, A. (2002). Quotient Types: A Modular Approach. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2002. Lecture Notes in Computer Science, vol 2410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45685-6_18

Download citation

  • DOI: https://doi.org/10.1007/3-540-45685-6_18

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44039-0

  • Online ISBN: 978-3-540-45685-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics