Abstract
Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS).
In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method.
On the practical side, it can be easily automated using as ingredients standard interpretations and general-purpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several non-trivial examples and to obtain methods like dummy elimination or dependency pairs (without the dependency graph refinement) as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO.
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References
Arts, T., Giesl, J.: Automatically proving termination where simplification orderings fail. In: Bidoit, M., Dauchet, M. (eds.) CAAP 1997, FASE 1997, and TAPSOFT 1997. LNCS, vol. 1214, pp. 261–272. Springer, Heidelberg (1997)
Arts, T., Giesl, J.: Modularity of termination using dependency pairs. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 226–240. Springer, Heidelberg (1998)
Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 133–178 (2000)
Bachmair, L., Dershowitz, N.: Commutation, transformation, and termination. In: Siekmann, J.H. (ed.) CADE 1986. LNCS, vol. 230, pp. 5–20. Springer, Heidelberg (1986)
Bellegarde, F., Lescanne, P.: Termination by completion. Applicable Algebra in Engineering, Communication and Computing 1, 79–96 (1990)
Baader, F., Nipkow, T.: Term Riwriting and all that. Cambridge University Press, Cambridge (1998)
Comon, H.: Solving symbolic ordering constraints. International Journal of Foundations of Computer Science 1(4), 387–411 (1990)
Dershowitz, N.: Orderings for term-rewriting systems. Theoretical Computer Science 17(3), 279–301 (1982)
Dershowitz, N.: Termination of rewriting. Journal of Symbolic Computation 3, 69–116 (1987)
Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, ch. 6, pp. 244–320. Elsevier Science Publishers, Amsterdam (1990)
Ferreira, M., Zantema, H.: Dummy elimination: making termination easier. In: Reichel, H. (ed.) FCT 1995. LNCS, vol. 965, pp. 243–252. Springer, Heidelberg (1995)
Geser, A.: On a monotonic semantic path ordering. Technical Report 92-13, Ulmer Informatik-Berichte, Universität Ulm, Ulm, Germany (1992)
Giesl, J., Ohlebusch, E.: Pushing the frontiers of combining rewrite systems farther outwards. In: Proceedings of the Second International Workshop on Frontiers of Combining Systems (FroCoS 1998), Amsterdam, The Netherlands. Logic and Computation Series, pp. 141–160. Research Studies Press, John Wiley & Sons (1998)
Gramlich, B.: Generalized sufficient conditions for modular termination of rewriting. Applicable Algebra in Engineering, Communication and Computing 5, 131–158 (1994)
Jouannaud, J.-P., Rubio, A.: The higher-order recursive path ordering. In: 14th IEEE Symposium on logic in Computer Science (LICS), Trento, Italy, pp. 402–411 (1999)
Kamin, S., Levy, J.-J.: Two generalizations of the recursive path ordering. Unpublished note, Dept. of Computer Science, Univ. of Illinois, Urbana, IL (1980)
Klop, J.W.: Term rewriting systems. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of logic in Computer Science, vol. 2, pp. 1–116. Oxford University Press, Oxford (1992)
Kusakari, K., Nakamura, M., Toyama, Y.: Argument filtering transformation. In: Nadathur, G. (ed.) PPDP 1999. LNCS, vol. 1702, pp. 47–61. Springer, Heidelberg (1999)
Kruskal, J.B.: Well-quasi-ordering, the Tree Theorem, and Vazsonyi’s conjecture. Transactions of the American Mathematical Society 95, 210–225 (1960)
Middeldorp, A., Ohsaki, H., Zantema, H.: Transforming termination by self-labelling. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS (LNAI), vol. 1104, pp. 373–386. Springer, Heidelberg (1996)
Ohlebusch, E., Claves, C., Marché, C.: Talp: a tool for the termination analysis of logic programs. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833. Springer, Heidelberg (2000)
Ohlebusch, E.: On modularity of termination of term rewriting systems. Theoretical Computer Science 136(2), 333–360 (1994)
Rubio, A.: A fully syntactic AC-RPO. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 133–147. Springer, Heidelberg (1999)
Steinbach, J.: Automatic termination proofs with transformation orderings. In: Hsiang, J. (ed.) RTA 1995. LNCS, vol. 914, pp. 11–25. Springer, Heidelberg (1995)
Xi, H.: Towards automated termination proofs through freezing. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 271–285. Springer, Heidelberg (1998)
Zantema, H.: Termination of term rewriting: interpretation and type elimination. Journal of Symbolic Computation 17, 23–50 (1994)
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Borralleras, C., Ferreira, M., Rubio, A. (2000). Complete Monotonic Semantic Path Orderings. In: McAllester, D. (eds) Automated Deduction - CADE-17. CADE 2000. Lecture Notes in Computer Science(), vol 1831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10721959_27
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DOI: https://doi.org/10.1007/10721959_27
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