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The order types of termination orderings on monadic terms, strings and multisets

Published online by Cambridge University Press:  12 March 2014

Ursula Martin  and
Elizabeth Scott
Ursula Martin
Affiliation:
Division of Computational Science, University of St Andrews, St Andrews, Fife KY16 6SX, Scotland, U.K., E-mail: um@dcs.st-and.ac.uk
Elizabeth Scott
Affiliation:
Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, England, U.K., E-mail: eas@dcs.rhbnc.ac.uk
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Abstract

We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order type ω, ω2 or ωω. We show that a familiar construction gives rise to continuum many such orderings of order type ω. We construct a new family of such orderings of order type ω2, and show that there are continuum many of these. We show that there are only four such orderings of order type ωω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on Nn which are preserved under vector addition. We show that any such ordering must have order type ωk for some 1 ≤ k ≤ n. We show that if k < n there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 624 - 635
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Cherifa, A. BEN and Lescanne, P., Termination of rewriting systems by polynomial interpretations and its implementation, Science of Computer Programming, vol. 9 (1987), pp. 137–160.CrossRefGoogle Scholar
[2]Cichon, E. A., Bounds on derivation lengths from termination proofs, International Journal of Foundations of Computer Science, to appear.Google Scholar
[3]Cichon, E. A. and Lescanne, P., Polynomial interpretations and the complexity of algorithms, Proceedings of the 11th international conference on computer aided deduction, Lecture Notes in Computer Science, no. 607, Springer-Verlag, 1992.Google Scholar
[4]Dauchet, M., Simulation of Turing machines by a left linear rewrite rule, Proceedings of the 3rd international conference on term rewriting systems, Lecture Notes in Computer Science, no. 355, Springer-Verlag, 1989, pp. 109–120.Google Scholar
[5]Dershowitz, N., Orderings for term-rewriting systems, Theoretical Computer Science, vol. 17 (1982), pp. 279–301.CrossRefGoogle Scholar
[6]Dershowitz, N., Termination of rewriting, Journal of Symbolic Computation, vol. 3 (1987), pp. 69–116.CrossRefGoogle Scholar
[7]Dershowitz, N. and Manna, Z., Proving termination with multiset orderings, Comm ACM, vol. 22 (1979), pp. 465–476.CrossRefGoogle Scholar
[8]Dershowitz, N. and Okada, M., Proof theoretic techniques for term rewriting theory, Proceedings of the third annual symposium on logic in computer science (Edinburgh), 07 1988.Google Scholar
[9]Dick, A. J. J., Kalmus, J. R., and Martin, U., Automating the Knuth Bendix ordering, Acta Informatica, vol. 28 (1990), pp. 95–119.CrossRefGoogle Scholar
[10]Ferreira, M. C. F. and Zantema, H., Total termination of term rewriting, Applicable Algebra in Engineering, Communication and Computing, vol. 7 (1996), pp. 133–162.Google Scholar
[11]Gallier, J. H., What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results on proof theory, Annals of Pure and Applied Logic, vol. 53 (1991).CrossRefGoogle Scholar
[12]Higman, G., Ordering by divisibility in abstract algebras, Proceedings of the London Mathematical Society, vol. 2 (1952), pp. 326–336.CrossRefGoogle Scholar
[13]Hofbauer, D. and Lautemann, C., Termination proofs and the lengths of derivations (preliminary version), Proceedings of the 3rd international conference on term rewriting systems, Lecture Notes in Computer Science, vol. 355, Springer-Verlag, 1989, pp. 167–77.Google Scholar
[14]Jouannaud, J.-P., Lescanne, P., and Reinig, F., Recursive decomposition ordering, Formal description of programming concepts (Bjorner, D., editor), vol. 2, Elsevier, 1982, pp. 331–348.Google Scholar
[15]Kamin, S. and Levy, J.-J., Two generalisations of the recursive path ordering, Department of Computer Science, University of Illinois, Urbana, IL, 1980.Google Scholar
[16]Knuth, D. and Bendix, P., Simple word problems in universal algebras, Computational problems in abstract algebra (Leech, J., editor), Pergamon Press, 1970.Google Scholar
[17]Lankford, D. S., On proving term rewriting systems are Noetherian, Tech report MTP-3, Louisiana Technical University, Ruston, 1979.Google Scholar
[18]Lautemann, C., A note on polynomial interpretation, Bulletin of the European Association for Theoretical Computer Science, vol. 4 (1988), pp. 129–131.Google Scholar
[19]Lescanne, P., Two implementations of the recursive path ordering on monadic terms, Proceedings of the 19th Allerton House conference on communication, control and computing, University of Illinois Press, 1981, pp. 634–643.Google Scholar
[20]Lescanne, P., Termination of rewrite systems by elementary interpretations, Proceedings of the 3rd international conference on algebraic and logic programming, Lecture Notes in Computer Science, vol. 463, Springer-Verlag, 1992, pp. 21–36.CrossRefGoogle Scholar
[21]Lothaire, M., Combinatorics on words, Encyclopaedia of mathematics and its applications, vol. 17, Addison Wesley, 1983.Google Scholar
[22]Martin, U., A geometrical approach to multiset orderings, Theoretical Computer Science, vol. 67 (1989), pp. 37–54.CrossRefGoogle Scholar
[23]Lescanne, P., On the diversity of orderings on strings, Fundamenta Informaticae, vol. 24 (1995), pp. 25–46.Google Scholar
[24]Morris, F. L. and Jones, C. B., An early proof by Alan Turing, Annals of the History of Computing, vol. 6 (1984), pp. 139–143.CrossRefGoogle Scholar
[25]Narendran, P. and Rusinowitch, M., Any ground associative-commutative theory has a finite canonical system, Proceedings of the 4th international conference on term rewriting systems, Springer Lecture Notes in Computer Science, vol. 488, pp. 423–434.Google Scholar
[26]Sakai, Ko, Knuth-Bendix algorithm for Thue system based on Kachinuki ordering, ICOT Technical Memorandum: TM-0087, 12 1984.Google Scholar
[27]Sims, C. C., Verifying nilpotence, Journal of Symbolic Computation, vol. 3 (1987), pp. 231–247.CrossRefGoogle Scholar
[28]Turing, A. M., Checking a large routine, Report of a conference on high speed automatic calculating machines, Univ. Math. Lab, Cambridge, 1949, pp. 67–69.Google Scholar
[29]Wilder, R. L., Introduction to the foundations of mathematics, John Wiley & Sons, New York, 1965.Google Scholar
[30]Zantema, H., Termination of term rewriting by interpretation: Interpretation and type elimination, Journal of Symbolic Commutation, vol. 17 (1994), pp. 23–50.CrossRefGoogle Scholar