Skip to main content

Advertisement

Springer Nature Link
Log in

Generalized sufficient conditions for modular termination of rewriting

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union ℛ1 ⊕ ℛ2 of two (finitely branching) terminating term rewriting systems ℛ1, ℛ2 is non-terminating, then one of the systems, say ℛ1, enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. ℛ1 ⊕ {itG(x, y)→ x, G(x, y) → y} is non-terminating, and the other system ℛ2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of potential counterexamples in disjoint unions may be arbitrarily high which shows that interaction of systems in such disjoint unions may be very subtle. Finally, extensions and generalizations of our main results in various directions are discussed. In particular, we show how to generalize the main results to some restricted form of non-disjoint combinations of term rewriting systems, namely for ‘combined systems with shared constructors’.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Dershowitz, N.: A note on simplification orderings. Information Processing Letters,9(5), 212–215 (1979)

    Google Scholar 

  2. Dershowitz, N.: Orderings for term-rewriting systems. Theoret Comput Sci 279–301 (1982)

  3. Dershowitz, N.: Termination of rewriting. J. Symbolic Comput.3(1), 69–116 (1987)

    Google Scholar 

  4. Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.). Formal models and semantics, Handbook of Theoretical Computer Science, vol. B, Chap. 6, pp. 243–320. Elsevier, The MIT Press 1990

  5. Drosten, K.: Termersetzungssysteme. Informatik-Fachberichte vol. 210. Berlin, Heidelberg, New York: Springer 1989

    Google Scholar 

  6. Ganzinger, H., Giegerich, R.: A note on termination in combinations of heterogeneous term rewriting systems. Bull. European Assoc. Theoret. Comput. Sci.31, 22–28 (1987)

    Google Scholar 

  7. Gramlich, B.: A structural analysis of modular termination of term rewriting systems. SEKI Report SR-91-15, Dept. of Comp. Science, Univ. of Kaiserslautern, 1991

  8. Gramlich, B.: Generalized sufficient conditions for modular termination of rewriting. In: Kirchner, H., Levi, G. (eds). Proc. of 3rd Int. Conf. on Algebraic and Logic Programming, Pisa, Italy, vol. 632. Lecture Notes in Computer Science, pp. 53–68. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  9. Gramlich, B.: Relating innermost, weak, uniform and modular termination of term rewriting systems. In: Voronkov, A. (ed.) International Conference on Logic Programming and Automated Reasoning, St. Petersburg, vol.624. Lecture Notes in Artificial Intelligence, pp. 285–296. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  10. Gramlich, B.: Sufficient conditions for modular termination of conditional term rewriting systems. In: Rusinowitch, M., Remy, J. L. (eds) Proc. of the 3rd International Workshop on Conditional Term Rewriting Systems, Pont-à-Mousson, vol.656. Lecture Notes in Computer Science, pp. 128–142. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  11. Huet, G., Lankford, D.: On the uniform halting problem for term rewriting systems. Technical Report 283, INRIA, 1978

  12. Huet, G., Oppen, D. C.: Equations and rewrite rules: A survey. In: Ronald V. Book (ed.) Formal Languages, Perspectives And Open Problems, pp. 349–405. New York: Academic Press 1980

    Google Scholar 

  13. Klop, J. W.: Term rewriting systems. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds). Handbook of Logic in Computer Science, volI. Oxford: University Press 1990

    Google Scholar 

  14. Kurihara, M., Kaji, I.: Modular term rewriting systems and the termination. Inf. Proc. Lett.34, 1–4 (1990)

    Google Scholar 

  15. Kurihara, M., Ohuchi, A.: Modularity of simple termination of term rewriting systems. J. IPS, Jpn34, 632–642 (1990)

    Google Scholar 

  16. Kurihara, M., Ohuchi, A.: Modularity of simple termination of term rewriting systems with shared constructors. Technical Report SF-36, Hokkaido University, Sapporo, 1990. Also in TCS103, 273–282 (1992)

    Google Scholar 

  17. Middeldorp, A.: Modular aspects of properties of term rewriting systems related to normal forms. In: Dershowitz, N. (ed.) Proceedings of the 3rd International Conference on Rewriting Techniques and Applications, vol.335. Lecture Notes in Computer Science, pp. 263–277. Berlin, Heidelberg, New York: Springer 1989

    Google Scholar 

  18. Middeldorp, A.: A sufficient condition for the termination of the direct sum of term rewriting systems. In: Proceedings of the 4th IEEE Symposium on Logic in Computer Science, pp. 396–401. Pacific Grove 1989

  19. Middeldorp, A.: Termination of disjoint unions of conditional term rewriting systems. Technical Report CS-R8959, Centre for Mathematics and Computer Science, Amsterdam 1989

    Google Scholar 

  20. Middeldorp, A.: Confluence of the disjoint union of conditional term rewriting systems. In: Kaplan, S., Okada, M. (eds). Proc. of the 2nd Int. Workshop on Conditional and Typed Rewriting Systems, vol.516. Lecture Notes in Computer Science, pp. 295–306. Berlin, Heidelberg, New York: Springer 1990

    Google Scholar 

  21. Middeldorp, A.: Modular Properties of Term Rewriting Systems. PhD thesis, Free University, Amsterdam 1990

    Google Scholar 

  22. Middeldorp, A.: Completeness of combinations of conditional constructor systems. In: Rusinowitch, M., Remy, J. L. (eds). Proc. of the 3rd International Workshop on Conditional Term Rewriting Systems, Pont-à-Mousson, vol.656 Lecture Notes in Computer Science, pp. 82–96. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  23. Middeldorp, A., Toyama, Y.: Completeness of combinations of constructor systems. In: Book, R. V. (ed). Proc. of the 4th Int. Conf. on Rewriting Techniques and Applications, vol.488. Lecture Notes in Computer Science, pp. 174–187. Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

  24. Ohlebusch, E.: Combinations of simplifying conditional term rewriting systems. In: Rusinowitch, M., Remy, J. L. (eds). Proc. of the 3rd International Workshop on Conditional Term Rewriting Systems, Pont-à-Mousson, vol.656. Lecture Notes in Computer Science, pp. 113–127. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  25. Ohlebusch, E.: A note on simple termination of infinite term rewriting systems. Technical Report 7, Univ. of Bielefeld 1992

  26. Rusinowitch, M.: On termination of the direct sum of term rewriting systems. Inf. Proc. Lett.26, 65–70 (1987)

    Google Scholar 

  27. Seifert, R.: Fachbereich Informatik, Univ. Bremen, personal communication, July 1992

  28. Toyama, Y.: Counterexamples to termination for the direct sum of term rewriting systems. Inf. Proc. Lett.25, 141–143 (1987)

    Google Scholar 

  29. Toyama, Y.: On the Church-Rosser property for the direct sum of term rewriting systems. J. ACM34(1), 128–143 (1987)

    Google Scholar 

  30. Toyama, Y., Klop, J. W., Barendregt, H. P.: Termination for the direct sum of left-linear term rewriting systems. In: Dershowitz, N. (ed.) Proc. of the 3rd Int. Conf. on Rewriting Techniques and Applications, vol.355. Lecture Notes in Computer Science, pp. 477–491. Berlin, Heidelberg, New York: Springer 1989

    Google Scholar 

  31. Zantema, H.: Termination of term rewriting by interpretation. In: Rusinowitch, M., Remy, J. L. (eds) Proc. of the 3rd International Workshop on Conditional Term Rewriting Systems, Pont-à-Mousson, vol.656. Lecture Notes in Computer Science, pp. 155–167. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Informatik, Universität Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Bernhard Gramlich

Authors
  1. Bernhard Gramlich
    View author publications

    You can also search for this author inPubMed Google Scholar

Additional information

This research was supported by the ‘Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt)’.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gramlich, B. Generalized sufficient conditions for modular termination of rewriting. AAECC 5, 131–158 (1994). https://doi.org/10.1007/BF01190827

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01190827

Keywords