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Abstract

We define a strong and versatile termination order for term rewriting systems, called theImproved General Path Order, which simplifies and strengthens Dershowitz/Hoot's General Path Order. We demonstrate the power of the Improved General Path Order by proofs of termination of non-trivial examples, among them a medium-scale term rewriting system that models a lift control.

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References

  1. Arts, T., Zantema, H.: Termination of logic programs via labelled term rewriting systems. Technical Report UU-CS-1994-20, Universiteit Utrecht, The Netherlands, 1994

    Google Scholar 

  2. Avenhaus, J., Madlener, K.: Term rewriting and equational reasoning. In: Banerji, R. B. (ed) Formal Techniques in Artificial Intelligence: A Source-Book. Elsevier Science Publishers, North-Holland, 1990

    Google Scholar 

  3. Bachmair, L., Dershowitz, N., Plaisted, D.A.: Completion without failure. In: Ait-Kaci, H., Nivat, M. (eds) Resolution of Equations in Algebraic Structures 2: Rewriting Techniques, pages 1–30. Academic Press, 1989

  4. Bellegarde, F., Lescanne, P.: Termination by completion. AAECC 1, 79–96 (1990)

    Google Scholar 

  5. Broy, M.: An example for the design of distributed systems in a formal setting: The lift problem. Technical Report MIP-8804, Universität Passau, Germany, 1988

    Google Scholar 

  6. Dauchet, M.: Simulation of Turing machines by a regular rule. Theoret. Comput. Sci.,103, 409–420 (1992)

    Google Scholar 

  7. Dershowitz, N.: A note on simplification orderings. Inform. Process. Lett.9, 212–215, 1979

    Google Scholar 

  8. Dershowitz, N.: Orderings for term rewriting systems. Theoret. Comput. Sci.17(3), 279–301 (March 1982)

    Google Scholar 

  9. Dershowitz, N.: Termination of rewriting. J. Symb. Comput.3(1–2), 69–115, Feb./April 1987. Corrigendum: 4 (3), 409–410, Dec. 1987

    Google Scholar 

  10. Dershowitz, N., Hoot, C.: Topics in termination. In: Kirchner, C. (ed), 5th Int. Conf. Rewriting Techniques and Applications, pp. 198–212. LNCS vol. 690, Berlin, Heidelberg, New York: Springer 1993

    Google Scholar 

  11. Dershowitz, N., Hoot, C.: Natural termination. Theoret. Comput. Sci.142 (2), 179–207, 1995

    Google Scholar 

  12. Dershowitz, N., Jouannaud, J.-P.: Notations for rewriting. Bulletin of the EATCS,43, 162–172, (1991)

    Google Scholar 

  13. Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, volume B (Formal Models and Semantics), pp. 243–320. Elsevier — The MIT Press, 1991

  14. Dershowitz, N., Jouannaud, J.-P., Klop, J. W.: Problems in rewriting III. In: Hsiang, J. (ed), 6th Int. Conf. Rewriting Techniques and Applications, pp. 457–471. LNCS vol. 914, Berlin, Heidelberg, New York: Springer 1995

    Google Scholar 

  15. Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Communications of the ACM,22(8), 465–476 (1979)

    Google Scholar 

  16. Dershowitz, N., Okada, M., Sivakumar, G.: Canonical conditional rewrite systems. In 9th Int. Conf. Automated Deduction, pp 538–549. LNCS vol. 310, Berlin, Heidelberg, New York: Springer 1988

    Google Scholar 

  17. Fraus, U.: Verifying the specification of a technical software system by induction. Research report FORWISS, Universität Passau, Germany, 1992

    Google Scholar 

  18. Fraus, U., Inductive theorem proving for algebraic specifications-TIP system user's manual. Technical Report MIP-9401, Universität Passau, Germany, Feb. 1994

    Google Scholar 

  19. Geser, A.: On a monotonie semantic path ordering. Technical Report 92-13, Ulmer Informatik-Berichte, Universität Ulm, Germany, 1992

    Google Scholar 

  20. Geser, A.: An improved general path order. Technical Report MIP-9407, Universität Passau, Germany, June 1994

    Google Scholar 

  21. Hofstadter, D. R.: Gödel, Escher, Bach: An eternal golden braid. Basic Books, New York, 1979

    Google Scholar 

  22. Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J Assoc. Comput. Mach.27, 797–821 (1980)

    Google Scholar 

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

    Google Scholar 

  24. Huet, G., Oppen, D. C: Equations and rewrite rules — a survey. In: Book, R. (ed) Formal Languages: Perspectives and Open Problems, pp 349–405. Academic Press, 1980

  25. Jouannaud, J.-P., Lescanne, P., Reinig, F.: Recursive decomposition ordering. In: Bjørner, D. (ed), Formal description of programming concepts 2, pp 331–348. North-Holland, 1982

  26. Kamin, S., Lévy, J.-J.: Attempts for generalizing the recursive path orderings. Manuscript; copy available at Dept. of Computer Science, Univ. of Illinois at Urbana-Champaign, Feb. 1980

    Google Scholar 

  27. Kapur, D., Narendran, P., Sivakumar, G.: A path ordering for proving termination of term rewriting system. In: 10th Colloquium on Trees in Algebra and Programming, pp 173–185. LNCS vol. 185, Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  28. Klop, J. W.: Term rewrite systems. In: Abramsky, S., Gabbay, D. M., Maibaum, T. (eds), Handbook of Logic in Computer Science, volume II, pp 1–116. Clarendon Press, Oxford, UK, 1992

    Google Scholar 

  29. Knuth, D. E., Bendix, P. B.: Simple word problems in universal algebras. In: Leech, J. (ed), Computational Problems in Abstract Algebra, pp 263–297. Pergamon Press, 1970

  30. Lankford, D. S.: On proving term rewriting systems are noetherian. Technical Report MTP-3, Louisiana Technical University, Math. Dept., Ruston, LA, 1979

    Google Scholar 

  31. Lescanne, P.: Uniform termination of term rewriting systems: Recursive decomposition ordering with status. In: Courcelle, B. (ed), 6th Colloquium on Trees in Algebra and Programming, pp 181–194, Bordeaux, France, Mar. 1984

  32. Lescanne, P.: On the recursive decomposition ordering with lexicographic status and other related orderings. J. Automated Reasoning,6, 39–49 (1990)

    Google Scholar 

  33. Plaisted, D. A.: Term rewriting systems: In: Gabbay, D. M., Hogger, C. J., Robinson, J. A.(eds), Handbook of Logic in Artificial Intelligence and Logic Programming, volume 4, Chap. 2. Clarendon Press, Oxford, UK, 1993

    Google Scholar 

  34. Rusinowitch, M.: Path of subterms ordering and recursive decomposition ordering revisited. J. Symb. Comput,3(1–2), 117–131 (1987)

    Google Scholar 

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

    Google Scholar 

  36. Wirth, C.-P., Grämlich, B.: A constructor-based inductive validity in positive/negative-conditional equational specifications. J. Symb. Comput.11, 1994

  37. Zantema, H.: Termination of term rewriting by semantic labelling. Fundamenta Informaticae24, 89–105 (1995)

    Google Scholar 

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Geser, A. An Improved General Path Order. AAECC 7, 469–511 (1996). https://doi.org/10.1007/BF01293264

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