Skip to main content
SpringerLink
Log in

Uniform Normalisation beyond Orthogonality

  • Conference paper
  • First Online:
Rewriting Techniques and Applications (RTA 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2051))

Included in the following conference series:

Abstract

A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A well-known fact is uniform normalisation of orthogonal non-erasing term rewrite systems, e.g. the λI-calculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and non-erasingness to the non-linear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first- and second-order term rewrite systems as well as to a λ-calculus with explicit substitutions.

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. F. Baader and T. Nipkow. Term Rewriting and All That. CUP, 1998.

    Google Scholar 

  2. H. Barendregt. The Lambda Calculus, Its Syntax and Semantics. NH, 1984.

    Google Scholar 

  3. Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. λv, a calculus of explicit substitutions which preserves strong normalisation. JFP, 6(5):699–722, 1996.

    MATH  MathSciNet  Google Scholar 

  4. R. Bloo. Preservation of Termination for Explicit Substitution. PhD thesis, Technische Universiteit Eindhoven, 1997.

    Google Scholar 

  5. C. Böhm and B. Intrigila. The ant-lion paradigm for strong normalization. I&C, 114(1):30–49, 1994.

    MATH  Google Scholar 

  6. E. Bonelli. Perpetuality in a named lambda calculus with explicit substitutions. MSCS, to appear.

    Google Scholar 

  7. Alonzo Church. The Calculi of Lambda-Conversion. PUP, 1941.

    Google Scholar 

  8. Georges Gonthier, Jean-Jacques Lévy, and Paul-André Melliès. An abstract standardisation theorem. In LICS’92, pages 72–81, 1992.

    Google Scholar 

  9. Bernhard Gramlich. Termination and Confluence Properties of Structured Rewrite Systems. PhD thesis, Universität Kaiserslautern, 1996.

    Google Scholar 

  10. Gérard Huet. Confluent reductions: Abstract properties and applications to term rewriting systems. JACM, 27(4):797–821, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  11. Gérard Huet and Jean-Jacques Lévy. Computations in orthogonal rewriting systems, I. In Computational Logic: Essays in Honor of Alan Robinson, pages 395–414. MIT Press, 1991.

    Google Scholar 

  12. Z. Khasidashvili. On the longest perpetual reductions in orthogonal expression reduction systems. TCS, To appear.

    Google Scholar 

  13. Z. Khasidashvili, M. Ogawa, and V. van Oostrom. Perpetuality and uniform normalization in orthogonal rewrite systems. I&C, To appear. http://www.phil.uu.nl/~oostrom/publication/ps/pun-icv2.ps.

  14. Jan Willem Klop. Combinatory Reduction Systems. PhD thesis, Rijksuniversiteit Utrecht, 1980. Mathematical Centre Tracts 127.

    Google Scholar 

  15. J.W. Klop. Term rewriting systems. In Handbook of Logic in Computer Science, volume 2, pages 1–116. OUP, 1992.

    MathSciNet  Google Scholar 

  16. Yves Lafont. From proof-nets to interaction nets. In Advances in Linear Logic, pages 225–247. CUP, 1995.

    Google Scholar 

  17. Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192:3–29, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  18. Paul-André Melliès. Description Abstraite des Systèmes de Réécriture. Thèse de doctorat, Université Paris VII, 1996.

    Google Scholar 

  19. Paul-André Melliès. Personal communication, 1999.

    Google Scholar 

  20. Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. Academisch proefschrift, Vrije Universiteit, Amsterdam, 1994.

    Google Scholar 

  21. Vincent van Oostrom. Development closed critical pairs. In HOA’95, volume 1074 of LNCS, pages 185–200. Springer, 1996.

    Google Scholar 

  22. Vincent van Oostrom. Normalisation in weakly orthogonal rewriting. In RTA’99, volume 1631 of LNCS, pages 60–74. Springer, 1999.

    Google Scholar 

  23. F. van Raamsdonk. Confluence and Normalisation for Higher-Order Rewriting. Academisch proefschrift, Vrije Universiteit, Amsterdam, 1996.

    Google Scholar 

  24. M.H. Sörensen. Effective longest and infinite reduction paths in untyped lambda-calculi. In CAAP’96, volume 1059 of LNCS, pages 287–301. Springer, 1996.

    Google Scholar 

  25. Yoshihito Toyama. Strong sequentiality of left-linear overlapping term rewriting systems. In LICS’92, pages 274–284, 1992.

    Google Scholar 

  26. J.B. Wells and Robert Muller. Standardization and evaluation in combinatory reduction systems, 2000. Working paper.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Zurab Khasidashvili

  2. PRESTO and NTT Communication Science Laboratories, Japan Science and Technology Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa, 243-0198, Japan

    Mizuhito Ogawa

  3. Department of Philosophy, Utrecht University, P.O. Box 80089, 3508 TB, Utrecht, The Netherlands

    Vincent van Oostrom

Authors
  1. Zurab Khasidashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mizuhito Ogawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vincent van Oostrom
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Information Sciences and Electronics, University of Tsukuba, Tsukuba, 305-8573, Japan

    Aart Middeldorp

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Khasidashvili, Z., Ogawa, M., van Oostrom, V. (2001). Uniform Normalisation beyond Orthogonality. In: Middeldorp, A. (eds) Rewriting Techniques and Applications. RTA 2001. Lecture Notes in Computer Science, vol 2051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45127-7_11

Download citation

  • DOI: https://doi.org/10.1007/3-540-45127-7_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42117-7

  • Online ISBN: 978-3-540-45127-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation