Skip to main content
SpringerLink
Log in

Orthogonal higher-order rewrite systems are confluent

  • Conference paper
  • First Online:
Typed Lambda Calculi and Applications (TLCA 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 664))

Included in the following conference series:

Abstract

The results about higher-order critical pairs and the confluence of OHRSs provide a firm foundation for the further study of higher-order rewrite systems. It should now be interesting to lift more results and techniques both from term-rewriting and λ-calculus to the level of HRSs. For example termination proof techniques are much studied for TRSs and are urgently needed for HRSs; similarly the extension of our result to weakly orthogonal HRSs or even to Huet's “parallel closed” systems is highly desirable. Conversely, a large body of λ-calculus reduction theory has been lifted to CRSs [10] already and should be easy to carry over to HRSs.

Finally there is the need to extend the notion of an HRS to more general left-hand sides. For example the eta-rule for the case-construct on disjoint unions [15] case(U,λx.F(inl(x)),λy.G(inr(y))) → F(U) is outside our framework, whichever way it is oriented.

Research supported by ESPRIT BRA 3245, Logical Frameworks, and WG 6028, CCL.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, 1978.

    Google Scholar 

  2. H. P. Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, 2nd edition, 1984.

    Google Scholar 

  3. V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. 3rd IEEE Symp. Logic in Computer Science, pages 82–90, 1988.

    Google Scholar 

  4. N. Dershowitz and J.-P. Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Formal Models and Semantics, Handbook of Theoretical Computer Science, Vol. B, pages 243–320. Elsevier — The MIT Press, 1990.

    Google Scholar 

  5. D. J. Dougherty. Adding algebraic rewriting to the untyped lambda-calculus. In R. V. Book, editor, Proc. 4th Int. Conf. Rewriting Techniques and Applications, pages 37–48. LNCS 488, 1991.

    Google Scholar 

  6. A. Felty. A logic-programming approach to implementing higher-order term rewriting. In L.-H. Eriksson, L. Hallnäs, and P. Schroeder-Heister, editors, Extensions of Logic Programming, Proc. 2nd Int. Workshop, pages 135–158. LNCS 596, 1992.

    Google Scholar 

  7. J. Gallier. Constructive Logics. Part I. Technical Report 8, DEC Paris Research Laboratory, 1991.

    Google Scholar 

  8. J. Hindley and J. P. Seldin. Introduction to Combinators and λ-Calculus. Cambridge University Press, 1986.

    Google Scholar 

  9. G. Huet. Confluent reductions: Abstract properties and applications to term rewritingsystems. J. ACM, 27:797–821, 1980.

    Google Scholar 

  10. J. W. Klop. Combinatory Reduction Systems. Mathematical Centre Tracts 127. Mathematisch Centrum, Amsterdam, 1980.

    Google Scholar 

  11. J. W. Klop. Term rewriting systems. Technical Report CS-R9073, CWI, Amsterdam, 1990.

    Google Scholar 

  12. D. Miller. A logic programming language with lambda-abstraction, function variables, and simple unification. In P. Schroeder-Heister, editor, Extensions of Logic Programming, pages 253–281. LNCS 475, 1991.

    Google Scholar 

  13. F. Müller. Confluence of the lambdacalculus with left-linear algebraic rewriting. Information Processing Letters, 41:293–299, 1992.

    Google Scholar 

  14. G. Nadathur and D. Miller. An overview of λProlog. In R. A. Kowalski and K. A. Bowen, editors, Proc. 5th Int. Logic Programming Conference, pages 810–827. MIT Press, 1988.

    Google Scholar 

  15. T. Nipkow. Higher-order critical pairs. In Proc. 6th IEEE Symp. Logic in Computer Science, pages 342–349, 1991.

    Google Scholar 

  16. T. Nipkow. Functional unification of higher-order patterns. Technical report, TU München, Institut für Informatik, 1992.

    Google Scholar 

  17. L. C. Paulson. Isabelle: The next 700 theorem provers. In P. Odifreddi, editor, Logic and Computer Science, pages 361–385. Academic Press, 1990.

    Google Scholar 

  18. S. Stenlund. Combinators, λ-Terms, and Proof Theory. D. Reidel, 1972.

    Google Scholar 

  19. F. van Raamsdonk. A simple proof of confluence for weakly orthogonal combinatory reduction systems. Technical Report CS-R9234, CWI, Amsterdam, 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, Postfach 20 24 20, W-8000, München 2, Germany

    Tobias Nipkow

Authors
  1. Tobias Nipkow
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Marc Bezem Jan Friso Groote

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nipkow, T. (1993). Orthogonal higher-order rewrite systems are confluent. In: Bezem, M., Groote, J.F. (eds) Typed Lambda Calculi and Applications. TLCA 1993. Lecture Notes in Computer Science, vol 664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0037114

Download citation

  • DOI: https://doi.org/10.1007/BFb0037114

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56517-8

  • Online ISBN: 978-3-540-47586-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation