Acta Informatica

, Volume 44, Issue 2, pp 91–121 | Cite as

Infinitary rewriting: meta-theory and convergence

Original Article

Abstract

When infinitary rewriting was introduced by Kaplan et al. (Principles of Programming Languages, ACM, New York, pp. 250–259, 1989) at the beginning of the 1990s, its term universe was explained as the metric completion of a metric on finite terms. The motivation for this connection to topology was that it allowed to import other well-studied notions from metric spaces, in particular the notion of convergence as a replacement for normalisation. This paper generalises the approach by parameterising it with a term metric, and applying the process of metric completion not only to terms but also to operations on and relations between terms. The resulting meta-theory is studied, leading to a revised notion of infinitary rewrite system. For these systems a method is devised to prove their convergence.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold A. and Nivat M. (1980). Metric interpretations of infinite trees and semantics of nondeterministic recursive programs. Theor. Comput. Sci. 11(2): 181–205 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baader F. and Nipkow T. (1998). Term Rewriting and All That. Cambridge University Press, Cambridge Google Scholar
  3. 3.
    Brainerd W.S. (1969). Tree generating regular systems. Inform. Control 14: 217–231 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Copson E. (1968). Metric Spaces. Cambridge University Press, Cambridge MATHGoogle Scholar
  5. 5.
    Corazza P. (1999). Introduction to metric-preserving functions. Am. Math. Mon. 106(4): 309–323 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Day J.M. and Franklin S.P. (1967). Spaces of continuous relations. Mathematische Annalen 169: 289–293 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dershowitz, N., Kaplan, S.: Rewrite, Rewrite, Rewrite, Rewrite, Rewrite, ... In: Principles of Programming Languages, pp. 250–259. ACM, New York (1989)Google Scholar
  8. 8.
    Dershowitz N., Kaplan S. and Plaisted D. (1991). Rewrite, Rewrite, Rewrite, .... Theor. Comput. Sci. 83(1): 71–96 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goguen, J.: Fuzzy sets. In: Mamdani, E., Gaines, B. (eds.) Fuzzy reasoning and its applications, pp. 67–115. Academic, New York (1981)Google Scholar
  10. 10.
    Grimeisen G. (1972). Continuous relations. Mathematische Zeitschrift 127: 35–44 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hahn H. (1948). Reelle Funktionen. Chelsea Publishing Company, New York Google Scholar
  12. 12.
    Jänich K. (1984). Topology. Springer, Heidelberg MATHGoogle Scholar
  13. 13.
    (2003). Haskell 98, Languages and libraries, The revised report. Cambridge University Press, Cambridge Google Scholar
  14. 14.
    Kahrs, S.: The variable containment problem. In: Higher-Order Algebra, Logic and Term Rewriting, Second International Workshop, Selected Papers. LNCS, vol. 1074, pp. 109–123 (1995)Google Scholar
  15. 15.
    Kamperman, J.: Compilation of term rewriting systems. Ph.D. thesis, University of Amsterdam (1996)Google Scholar
  16. 16.
    Kamperman, J., Walters, H.: Lazy rewriting and eager machinery. In: Rewriting Techniques and Applications. LNCS, vol. 914, pp. 147–162 (1995)Google Scholar
  17. 17.
    Kelley J. (1950). Convergence in topology. Duke Math. J. 17(3): 277–283 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kennaway, R.: On transfinite abstract reduction systems. Tech. Rep. CS-R9205, Centrum voor Wiskunde en Informatica, Amsterdam (1992)Google Scholar
  19. 19.
    Kennaway, R., de Vries, F.J.: Term rewriting systems, chap. Infinitary Rewriting, pp. 668–711. Cambridge University Press, Cambridge (2003)Google Scholar
  20. 20.
    Ketema J.(2006). Böhm-like trees for rewriting. Ph.D. thesis, Vrije Universiteit AmsterdamGoogle Scholar
  21. 21.
    Klop, J.W.: Combinatory reduction systems. Ph.D. thesis, Centrum voor Wiskunde en Informatica (1980)Google Scholar
  22. 22.
    Kuratowski K. (1966). Topology. Academic, New York Google Scholar
  23. 23.
    Meinke K. (1992). Universal algebra in higher types. Theor. Comput. Sci. 100(2): 385–417 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Smyth M. (1992). Topology. In: Abramsky, S., Gabbay, D. and Maibaum, T. (eds) Handbook of Logic in Computer Science, vol 1., pp 641–762. Oxford University Press, Oxford Google Scholar
  25. 25.
    Stoltenberg-Hansen V. and Tucker J. (1995). Effective algebras. In: Abramsky, S., Gabbay, D.M. and Maibaum, T. (eds) Handbook of Logic in Computer Science, vol. 4., pp 357–526. Oxford University Press, Oxford Google Scholar
  26. 26.
    (2003). Term Rewriting Systems. Cambridge University Press, Cambridge Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Kent at CanterburyCanterburyUK

Personalised recommendations