Abstract
In this article we investigate the properties of unification in sort theories. A sort is represented by a set of monadic predicates, called sort symbols. Sorts are attached to variables restricting their respective domain to the intersection of the denotations of the sort symbols. A sort theory consists of a set of declarations that are atoms starting with a sort symbol. Two terms are unifiable with respect to some sort theory, if they are unifiable in the standard sense and the assignments of the unifier do respect the declarations in the sort theory. Therefore, the new sorted unification algorithm is formed by standard unification augmented by extra rules that consider the information in the sort theory. We prove the new sorted unification algorithm to be correct and complete and establish complexity results for several different, syntactically characterized sort theories. The notions of a sort and a sort theory are developed in such a way that sort symbols are used like ordinary monadic predicate symbols. To this end, sorts may denote empty sets, and the sort theory is not a static part of the signature. It may dynamically change during a deduction process. The applicability of the approach is demonstrated for the resolution and the tableau calculus.
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