Abstract
Up to now, all existing completeness results for ordered paramodulation and Knuth–Bendix completion have required term ordering ≻ to be well founded, monotonic, and total(izable) on ground terms. For several applications, these requirements are too strong, and hence weakening them has been a well-known research challenge.
Here we introduce a new completeness proof technique for ordered paramodulation where the only properties required on ≻ are well-foundedness and the subterm property. The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS).
By a careful further analysis of our technique, we obtain the first Knuth–Bendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly non-totalizable) reduction ordering ≻ whenever it exists. Note that being a reduction ordering is the minimal possible requirement on ≻, since a TRS terminates if, and only if, it is contained in a reduction ordering.
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Bofill, M., Godoy, G., Nieuwenhuis, R. et al. Paramodulation and Knuth–Bendix Completion with Nontotal and Nonmonotonic Orderings. Journal of Automated Reasoning 30, 99–120 (2003). https://doi.org/10.1023/A:1022515030222
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DOI: https://doi.org/10.1023/A:1022515030222