Skip to main content

An analysis of divisibility orderings and recursive path orderings

  • Session 7
  • Conference paper
  • First Online:
Advances in Computing Science — ASIAN'97 (ASIAN 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1345))

Included in the following conference series:

Abstract

We show that normal and analytic functors provide a foundation to the theory of divisibility orderings and recursive path orderings. These functors are used to give intrinsic definitions independent from particular syntactic presentations.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Becker and V. Weispfenning, Grôbner Bases, Graduate Texts in Mathematics 141, (Springer, 1993).

    Google Scholar 

  2. D. H. J. de Jongh and R. Parikh, Well-partial orderings and hierarchies, Indagationes Math. 39 (1977) 195–207.

    Article  Google Scholar 

  3. N. Dershowitz, Orderings for term-rewriting systems, Theoretical Computer Sci. 17 (1982) 279–301.

    Article  Google Scholar 

  4. N. Dershowitz and Z. Manna, Proving termination with multiset orderings, Communications ACM 22 (1979) 465–476.

    Article  Google Scholar 

  5. N. Dershowitz and M. Okada, Proof theoretic techniques for term rewriting theory, in: Third Annual Symposium on Logic in Computer Science, 1988, Edinburgh, Scotland, (IEEE, 1988) pp. 104–111.

    Google Scholar 

  6. J.-Y. Girard, II 21 -logic, part I; dilators, Ann. Math. Logic 21 (1981) 75–219.

    Article  Google Scholar 

  7. J.-Y. Girard, Normal functors, power series and λ-calculus, Ann. Pure Applied Logic 37 (1988) 129–177.

    Article  Google Scholar 

  8. J.-Y. Girard and J. Vauzeilles, Functors and ordinal notations II: A functorial construction of the Bachmann-hierarchy, J. Symbolic Logic 49 (1984) 713–729.

    Google Scholar 

  9. R. Hasegawa, Well-ordering of algebras and Kruskal's theorem, in: Logic, Language and Computation, Festschrift in Honor of Satoru Takasu, N. D. Jones, M. Hagiya, M. Sato, eds., Lecture Notes in Computer Science 792, (Springer, 1994) pp. 133–172.

    Google Scholar 

  10. G. Higman, Ordering by divisibility in abstract algebras, Proc. London Math. Soc., Third Series 2 (1952) 326–336.

    Google Scholar 

  11. A. Joyal, Une théorie combinatoire des séries formelles, Advances Math. 42 (1981) 1–82.

    Article  Google Scholar 

  12. A. Joyal, Foncteurs analytiques et espèces de structures, in: Combinatoire Enumérative, Proceedings, Montreal, Québec, Canada, 1985, G. Labelle, P. Leroux, eds., Lecture Notes in Mathematics 1234, (Springer, 1986) pp. 126–159.

    Google Scholar 

  13. J. B. Kruskal, Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture, Transactions American Math. Soc. 95 (1960) 210–225.

    Google Scholar 

  14. P. Lescanne, Uniform termination of term rewriting systems, recursive decomposition ordering with status, in: Ninth Colloquium on Trees in Algebra and Programming, 1984, Bordeaux, France, B. Courcelle ed., (Cambridge University Press, 1984) pp. 181–194.

    Google Scholar 

  15. S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic, A First Introduction to Topos Theory, (Springer, 1992).

    Google Scholar 

  16. U. Martin and E. Scott, The order types of termination orderings on monadic terms, strings and multisets, in: Proc. Eighth Annual IEEE Symposium on Logic in Computer Science, 1992, Montreal, Canada, (IEEE, 1993) pp. 356–363.

    Google Scholar 

  17. A. Middeldorp and H. Zantema, Simple termination of rewrite systems, Theoretical Computer Sci., 175 (1997) 127–158.

    Article  Google Scholar 

  18. C. St. J. A. Nash-Williams, On well-quasi-ordering finite trees, Proc. Cambridge Philosophical Soc. 59 (1963) 833–835.

    Google Scholar 

  19. K. Sakai, Knuth-Bendix algorithm for Thue system based on kachinuki ordering, ICOT Technical Memorandum: TM-0087, ICOT, Institute for New Generation Computer Technology, Dec. 1984.

    Google Scholar 

  20. S. G. Simpson, Nonprovability of certain combinatorial properties of finite trees, in: Harvey Friedman's research on the foundations of mathematics, L. A. Harrington, M. D. Morley, A. Scedrov, S. G. Simpson, eds., (North-Holland, 1985) pp. 87–117.

    Google Scholar 

  21. G. Takeuti, Proof Theory, Studies in Logic and the Foundations of Mathematics, Vol. 81, (North-Holland, 1975).

    Google Scholar 

  22. A. Weiermann, A functorial property of the Aczel-Buchholz-Feferman function, J. Symbolic Logic, 59 (1994) 945–955.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

R. K. Shyamasundar K. Ueda

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hasegawa, R. (1997). An analysis of divisibility orderings and recursive path orderings. In: Shyamasundar, R.K., Ueda, K. (eds) Advances in Computing Science — ASIAN'97. ASIAN 1997. Lecture Notes in Computer Science, vol 1345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63875-X_59

Download citation

  • DOI: https://doi.org/10.1007/3-540-63875-X_59

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63875-9

  • Online ISBN: 978-3-540-69658-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics