The completeness of logic programming with sort predicates

An order-sorted logic can be regarded as a generalized first-order predicate logic that includes many and ordered sorts (i.e. a sort-hierarchy). In the fields of knowledge representation and AI, this logic with sort-hierarchy has been used to design a logic-based language appropriate for representing taxonomic knowledge. By incorporating the sort-hierarchy, order-sorted resolution and sorted logic programming have been formalized that provide efficient reasoning mechanisms with structural representation. In this work, Beierle et al. developed an order-sorted logic to couple separated taxonomic knowledge and assertional knowledge. Namely, its language allows us to make use of sorts to denote not only the types of terms but also unary predicates (called sort predicates). In this paper, we propose a sorted logic programming language with sort predicates in order to improve the practicability of the logic proposed by Beierle et al. The linear resolution is obtained by adding inference relative to sort predicates and subsort relations. In the semantics, the terms and formulas that follow the sorted signature extended with sort predicates are interpreted over its corresponding Σ-structures. Finally, we build the Herbrand models of programs containing sort predicates, and thus prove the soundness and completeness of this logic programming. order-sorted logic, sort predicate, logic programming, knowledge base system