Order-sorted logic programming with predicate hierarchy

Order-sorted logic has been formalized as first-order logic with sorted terms where sorts are ordered to build a hierarchy (called a sort-hierarchy). These sorted logics lead to useful expressions and inference methods for structural knowledge that ordinary first-order logic lacks. Nitta et al. pointed out that for legal reasoning a sort-hierarchy (or a sorted term) is not sufficient to describe structural knowledge for event assertions, which express facts caused at some particular time and place. The event assertions are represented by predicates with n arguments (i.e., n-ary predicates), and then a particular kind of hierarchy (called a predicate hierarchy) is built by a relationship among the predicates. To deal with such a predicate hierarchy, which is more intricate than a sort-hierarchy, Nitta et al. implemented a typed (sorted) logic programming language extended to include a hierarchy of verbal concepts (corresponding to predicates). However, the inference system lacks a theoretical foundation because its hierarchical expressions exceed the formalization of order-sorted logic. In this paper, we formalize a logic programming language with not only a sort-hierarchy but also a predicate hierarchy. This language can derive general and concrete expressions in the two kinds of hierarchies. For the hierarchical reasoning of predicates, we propose a manipulation of arguments in which surplus and missing arguments in derived predicates are eliminated and supplemented. As discussed by Allen, McDermott and Shoham in research on temporal logic and as applied by Nitta et al. to legal reasoning, if each predicate is interpreted as an event or action (not as a static property), then missing arguments should be supplemented by existential terms in the argument manipulation. Based on this, we develop a Horn clause resolution system extended to add inference rules of predicate hierarchies. With a semantic model restricted by interpreting a predicate hierarchy, the soundness and completeness of the Horn-clause resolution is proven.