Equality and Constrained Resolution

This paper looks at the addition of equality reasoning to systems that perform constrained resolution as defined within the substitutional framework. These systems reason with a constraint logic in which the constraints are interpreted relative to a constraint theory. First, a special case is considered when equality can be treated as a constraint. Then the general case is dealt with by developing and proving correct the rule of constrained paramodulation, which along with the rule of constrained resolution (and factoring) yields a refutationally complete set of inference rules for languages with equality. It is shown that if certain conditions are met, paramodulation into variables is not necessary and the functionally reflexive axioms need not be present. The modal case satisfies these conditions. If other weaker conditions are met, paramodulation into variables is necessary, but the functionally reflexive axioms are not needed. Some sorted logics satisfy these conditions. The analysis provides a means of extending restrictions on resolution and paramodulation (e.g. ordering restrictions) to constrained deduction, a relatively clean and simple mechanism for adding paramodulation to sorted logics and a proof of the conjectures of Walther and Schmidt-Schauss on the need for paramodulation into variables and the functionally reflexive axioms in the case of sorted logics.