Skip to main content
SpringerLink
Log in

The longest perpetual reductions in Orthogonal Expression Reduction Systems

  • Conference paper
  • First Online:
Logical Foundations of Computer Science (LFCS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 813))

Included in the following conference series:

Abstract

We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the λ-calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs.

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. Barendregt H. P. The Lambda Calculus, its Syntax and Semantics. North-Holland, 1984.

    Google Scholar 

  2. Barendregt H. P., Bergstra J., Klop J. W., Volken H. Some notes on lambda-reduction, In: Degrees, reductions, and representability in the lambda calculus. Preprint no. 22, University of Utrecht, Department of mathematics, p. 13–53, 1976.

    Google Scholar 

  3. Bergstra J. A., Klop J. W. Strong normalization and perpetual reductions in the Lambda Calculus. J. of Information Processing and Cybernetics 18, 1982, p. 403–417.

    Google Scholar 

  4. De Groote P. The Conservation Theorem revisited. In: M. Bezem, J. F. Groote, eds. Proc. of the International Conference on Typed Lambda Calculi and Applications. Springer LNCS, vol. 664, Utrecht, 1993, p. 163–178.

    Google Scholar 

  5. Huet G., Lévy J.-J. Computations in Orthogonal Rewriting Systems. In: J.-L. Lassez and G. Plotkin, eds. Computational Logic, Essays in Honor of Alan Robinson. MIT Press, 1991, p. 395–443.

    Google Scholar 

  6. hasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

    Google Scholar 

  7. hasidashvili Z. Optimal normalization in orthogonal term rewriting systems. In: C. Kirchner, ed. Proc. of 5th International Conference on Rewriting Techniques and Applications. Springer LNCS, vol. 690, Montreal, 1993, p. 243–258.

    Google Scholar 

  8. Khasidashvili Z. Perpetuality and strong normalization in orthogonal term rewriting systems. In: P. Enjalbert, E. W. Mayr, and K. W. Wagner eds. Proc. of 11th Symposium on Theoretical Aspects of Computer Science. Springer LNCS, vol. 775, Caen, 1994, p. 163–174.

    Google Scholar 

  9. Khasidashvili Z. Perpetual reductions in orthogonal combinatory reduction systems. CWI report CS-R9349, 1993.

    Google Scholar 

  10. Klop J. W. Combinatory Reduction Systems. Mathematical Centre Tracts n. 127, CWI, Amsterdam, 1980.

    Google Scholar 

  11. Klop J. W. Term Rewriting Systems. In: S. Abramsky, D. Gabbay, and T. Maibaum eds. Handbook of Logic in Computer Science, vol. II, Oxford University Press, 1992, p. 1–116.

    Google Scholar 

  12. Klop J. W., van Oostrom V., van Raamsdonk F. Combinatory reduction Systems: introduction and survey. In: To Corrado Böhm, J. of Theoretical Computer Science 121, 1993, p. 279–308.

    Google Scholar 

  13. Maranget L. La stratégie paresseuse, Thèse de l'Université de Paris VII, 1992.

    Google Scholar 

  14. Nederpelt R. P. Strong Normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

    Google Scholar 

  15. O'Donnell M. J. Computing in systems described by equations. Springer LNCS 58, 1977.

    Google Scholar 

  16. Van der Pol J. Termination proofs for Higher-order Rewrite Systems. To appear in: Proc. of International Workshop on Higher-order Algebra, Logic and Term Rewriting, Amsterdam, 1993. Available as logic group preprint series No. 93, Utrecht, 1993.

    Google Scholar 

  17. De Vrijer R. C. Surjective Pairing and Strong Normalization: Two Themes in Lambda Calculus. Ph.D. Thesis, Amsterdam, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Systems, UEA, Norwich NR4 7TJ, England

    Zurab Khasidashvili

Authors
  1. Zurab Khasidashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Anil Nerode Yu. V. Matiyasevich

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Khasidashvili, Z. (1994). The longest perpetual reductions in Orthogonal Expression Reduction Systems. In: Nerode, A., Matiyasevich, Y.V. (eds) Logical Foundations of Computer Science. LFCS 1994. Lecture Notes in Computer Science, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58140-5_20

Download citation

  • DOI: https://doi.org/10.1007/3-540-58140-5_20

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58140-6

  • Online ISBN: 978-3-540-48442-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation