Skip to main content
SpringerLink
Log in

Solving for Set Variables in Higher-Order Theorem Proving

  • Conference paper
  • First Online:
Automated Deduction—CADE-18 (CADE 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2392))

Included in the following conference series:

Abstract

In higher-order logic, we must consider literals with flexible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. Using the Knaster-Tarski Fixed Point Theorem [15], constraints whose solutions require recursive definitions can be solved as fixed points of monotone set functions. In this paper, we consider an approach to higher-order theorem proving which intertwines conventional theorem proving in the form of mating search with generating and solving set constraints.

This material is based upon work supported by the National Science Foundation under grants CCR-9732312 and CCR-0097179.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

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.

References

  1. Peter B. Andrews. On connections and higher-order logic. Journal of Automated Reasoning, 5:257–291, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  2. Peter B. Andrews. Classical type theory. In Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume 2, chapter 15, pages 965–1007. Elsevier Science, 2001.

    Google Scholar 

  3. Peter B. Andrews, Matthew Bishop, Sunil Issar, Dan Nesmith, Frank Pfenning, and Hongwei Xi. TPS: A theorem proving system for classical type theory. Journal of Automated Reasoning, 16:321–353, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  4. Sidney C. Bailin and Dave Barker-Plummer. Z-match: An inference rule for incrementally elaborating set instantiations. Journal of Automated Reasoning, 11:391–428, 1993. Errata: JAR 12 (1994), 411–412.

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Bartels, A. Dold, F. W. v. Henke, H. Pfeifer, and H. Rueß. Formalizing Fixed-Point Theory in PVS. Ulmer Informatik-Berichte 96-10, Universität Ulm, Fakultät für Informatik, 1996.

    Google Scholar 

  6. Christoph Benzmüller and Michael Kohlhase. System description: LEO — a higher-order theorem prover. In Claude Kirchner and Hélène Kirchner, editors, Proceedings of the 15th International Conference on Automated Deduction, volume 1421 of Lecture Notes in Artificial Intelligence, pages 139–143, Lindau, Germany, 1998. Springer-Verlag.

    Google Scholar 

  7. Wolfgang Bibel. Matings in matrices. Communications of the ACM, 26:844–852, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  8. W. W. Bledsoe. A maximal method for set variables in automatic theorem proving. In J. E. Hayes, Donald Michie, and L. I. Mikulich, editors, Machine Intelligence 9, pages 53–100. Ellis Harwood Ltd., Chichester, and John Wiley & Sons, 1979.

    Google Scholar 

  9. W. W. Bledsoe and Guohui Feng. Set-Var. Journal of Automated Reasoning, 11:293–314, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  10. Alonzo Church. A formulation of the simple theory of types. Journal of Symbolic Logic, 5:56–68, 1940.

    Article  MATH  MathSciNet  Google Scholar 

  11. Amy Felty. Proof search with set variable instantiation in the calculus of constructions. In M. A. McRobbie and J. K. Slaney, editors, Automated Deduction: CADE-13, volume 1104 of Lecture Notes in Artificial Intelligence, pages 658–672. Springer, 1996.

    Google Scholar 

  12. Amy Felty. The calculus of constructions as a framework for proof search with set variable instantiation. Theoretical Computer Science, 232:187–229, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. H. Geuvers. The calculus of constructions and higher order logic. In Ph. de Groote, editor, The Curry-Howard Isomorphism, pages 139–191. Academia, Louvain-la-Neuve (Belgium), 1995.

    Google Scholar 

  14. Sunil Issar. Operational Issues in Automated Theorem Proving Using Matings. PhD thesis, Carnegie Mellon University, 1991. 147 pp.

    Google Scholar 

  15. B. Knaster. Une théorème sur les fonctions d’ensembles. Annales Soc. Polonaise Math., 6:133–134, 1927.

    Google Scholar 

  16. Dale A. Miller. A compact representation of proofs. Studia Logica, 46(4):347–370, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  17. Neil V. Murray and Erik Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 40(3):504–535, July 1993.

    Google Scholar 

  18. H. J. Ohlbach. SCAN—elimination of predicate quantifiers. In M. A. McRobbie and J. K. Slaney, editors, Automated Deduction: CADE-13, volume 1104 of Lecture Notes in Artificial Intelligence, pages 161–165. Springer, 1996.

    Google Scholar 

  19. Lawrence C. Paulson. A fixedpoint approach to implementing (co)inductive definitions. In Alan Bundy, editor, Proceedings of the 12th International Conference on Automated Deduction, pages 148–161, Nancy, France, June 1994. Springer-Verlag LNAI 814.

    Google Scholar 

  20. Lawrence C. Paulson. Set theory for verification: II. Induction and recursion. Journal of Automated Reasoning, 15(2):167–215, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  21. Lawrence C. Paulson. Mechanizing coinduction and corecursion in higher-order logic. Journal of Logic and Computation, 7(2):175–204, March 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Chad E. Brown

Authors
  1. Chad E. Brown
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Manchester, Oxford Rd, Manchester, M13 9PL, UK

    Andrei Voronkov

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brown, C.E. (2002). Solving for Set Variables in Higher-Order Theorem Proving. In: Voronkov, A. (eds) Automated Deduction—CADE-18. CADE 2002. Lecture Notes in Computer Science(), vol 2392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45620-1_33

Download citation

  • DOI: https://doi.org/10.1007/3-540-45620-1_33

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43931-8

  • Online ISBN: 978-3-540-45620-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation