*Chapter 3 in "Types in Logic Programming" (F. Pfenning, ed.), *MIT Press, pp. 91-140, 1992

In this chapter, we propose a framework for logic programming with different type systems. In this framework a typed logic program consists of a type specification and a Horn clause program which is well-typed with respect to the type specification. The type specification defines all types which can be used in the logic program. Relations between types are expressed by equations on the level of types. This permits the specification of many-sorted, order-sorted, polymorphic and polymorphically order-sorted type systems. We present the declarative semantics of our framework and two proof procedures (deduction and resolution) for typed logic programs. An interesting application is a type system that combines parametric polymorphism with order-sorted typing and permits higher-order logic programming. Moreover, our framework sheds some new light on the role of types in logic programming.

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