### Paramodulation with Well-founded Orderings

M. Bofill, A. Rubio
2008 Journal of Logic and Computation
For many years, all existing completeness results for Knuth-Bendix completion and ordered paramodulation required the term ordering ≻ to be well-founded, monotonic and total(izable) on ground terms. Then, it was shown that well-foundedness and the subterm property were enough for ensuring completeness of ordered paramodulation. Here we show that the subterm property is not necessary either. By using a new restricted form of rewriting, we obtain a completeness proof of ordered paramodulation for
more » ... paramodulation for Horn clauses with equality, where wellfoundedness of the ordering suffices. Apart from the theoretical significance of this result, some potential applications motivating the interest of dropping the subterm property are given. The proof of the results included in this paper, being still technical in some parts, is pretty much shorter and easier to read than the one we have in the preliminary version of this work presented at the CADE 2002 conference [8] . Another typical situation is in deduction modulo built-in equational theories E. There, apart from replacing unification by E-unification, the ordering needs to be total up to E-equal ground terms and E-compatible as well as wellfounded and monotonic (see e.g. [17] ). Unfortunately having all these properties can be a too strong requirement. For example, the existence of such an ordering for the case where E consists of associativity and commutativity (AC) properties for some symbols remained open for a long time. 4 And, even worse, for many theories E such orderings cannot exist. For example, when E contains an idempotency axiom f (x, x) ≃ x, then, if s ≻ t, by monotonicity one should have f (s, s) ≻ f (s, t), which by E-compatibility implies s ≻ f (s, t) and, hence, well-foundedness is lost. The first results on ordered paramodulation and Knuth-Bendix completion with weaker orderings were given in [6, 7] . There, the monotonicity requirement was dropped and well-foundedness and the subterm property were shown to be sufficient for completeness. Note that any such ordering can be totalized without losing these two properties. After this, the fundamental question arises whether more requirements can be dropped. In this paper we prove that, for the case of Horn clauses with equality, the subterm property is not necessary either, that is, equations may be ordered from subterms to superterms such as in a → f (a), and ordered paramodulation remains refutation complete. The only requirement left for the ordering is being well-founded (note that a → f (a) can be oriented in the well-founded ordering induced by f m (a) ≻ f n (a) ≻ a ≻ f (a) with m > n > 1). This is a new important step in the theory of paramodulation, which shows the power of ordered paramodulation regardless of the properties of the ordering that is used, and leaves as the last question whether even well-foundedness is necessary. Apart from its theoretical value, this result has its most significant potential application in deduction modulo built-in equational theories E, since the requirement of a (only) well-founded E-compatible ordering does not exclude any theory. Note that, like any denumerable set, the set of E-congruence classes admits a well-founded ordering and, hence, there is a well-founded Ecompatible ordering for any theory E. Therefore, if our results are extended to deduction modulo E, the only remaining condition to obtain refutation complete E-paramodulation calculi relates to the need of computing complete sets of Eunifiers in the paramodulation steps. Moreover, we believe that our results can be shown to be compatible with basic strategies [5, 15] . Then, in fact, as shown in [19, 16] , if basic strategies are applied, only decidability of the E-unification problems is necessary. Another (less obvious) potential application of our results may be in goaloriented deduction since, in some cases, a goal-oriented (ordered) paramodulation proof can only be obtained if the ordering contradicts the subterm property. Let us illustrate this with a simple example: