Abstract
In [12], Toyama proved that the union of two confluent term-rewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [5,11].
In this paper we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies in two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions; a modularity property of ordered completion, that allows to pairwise match the caps and alien substitutions of two equivalent terms.
We then show that Toyama’s modularity result scales up to rewriting modulo equations in all considered cases.
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References
Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 243–309. North-Holland, Amsterdam (1990)
Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. Journal of the ACM 27(4), 797–821 (1980)
Jouannaud, J.-P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM Journal on Computing 15(4), 1155–1194 (1986)
Jouannaud, J.-P., van Raasdon, F., Rubio, A.: Rewriting with types and arities (2005), available from the web
Klop, J.W., Middeldorp, A., Toyama, Y., de Vrijer, R.: Modularity of confluence: A simplified proof. Information Processing Letters 49(2), 101–109 (1994)
Lankford, D.S., Ballantyne, A.M.: Decision procedures for simple equational theories with permutative axioms: Complete sets of permutative reductions. Research Report Memo ATP-37, Department of Mathematics and Computer Science, University of Texas, Austin, Texas, USA (August 1977)
Marché, C.: Normalised rewriting and normalised completion. In: Proc. 9th IEEE Symp. Logic in Computer Science, pp. 394–403 (1994)
Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. Theoretical Computer Science 192(1), 3–29 (1998)
Middeldorp, A.: Modular aspects of properties of term rewriting systems related to normal forms. In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355, pp. 263–277. Springer, Heidelberg (1989)
Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. Journal of the ACM 28(2), 233–264 (1981)
Bezem, M., Kop, J.W., de Vrijer, R. (eds.): Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science 55. Cambridge University Press, Cambridge (2003)
Toyama, Y.: On the Church-Rosser property for the direct sum of term rewriting systems. Journal of the ACM 34(1), 128–143 (1987)
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Jouannaud, JP. (2006). Modular Church-Rosser Modulo. In: Pfenning, F. (eds) Term Rewriting and Applications. RTA 2006. Lecture Notes in Computer Science, vol 4098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11805618_8
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DOI: https://doi.org/10.1007/11805618_8
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