A Combinator-based Order-sorted Higher-order Uniication Algorithm

This paper develops a sound and complete transformation-based algorithm for uniication in an extensional order-sorted combinatory logic supporting constant overloading and a higher-order sort concept. Appropriate notions of order-sorted weak equality and extensionality | reeecting order-sorted-equality in the corresponding lambda calculus given by Johann and Kohlhase | are deened, and the typed combinator-based higher-order uniication techniques of Dougherty are modiied to accommodate uniication with respect to the theory they generate. The algorithm presented here can thus be viewed as a combinatory logic counterpart to that of Johann and Kohlhase, as well as a reenement of that of Dougherty, and provides evidence that combinatory logic is well-suited to serve as a framework for incorporating order-sorted higher-order reasoning into deduction systems aiming to capitalize on both the expressiveness of extensional higher-order logic and the eeciency of order-sorted calculi.