Uniication in an Extensional Lambda Calculus with Ordered Function Sorts and Constant Overloading

We develop an order-sorted higher-order calculus suitable for automatic theorem proving applications by extending the extensional simply typed lambda calculus with a higher-order ordered sort concept and constant overloading. Huet's well-known techniques for unifying simply typed lambda terms are generalized to arrive at a complete transformation-based uniication algorithm for this sorted calculus. Consideration of an order-sorted logic with functional base sorts and arbitrary term declarations was originally proposed by the second author in a 1991 paper; we give here a corrected calculus which supports constant rather than arbitrary term declarations, as well as a corrected uniication algorithm, and prove in this setting results corresponding to those claimed there. 1 Introduction In the quest for calculi best suited for automating logic, the introduction of sorts has been one of the most promising developments. Sorts, which are intended to capture for automated deduction purposes the kinds of meta-level taxonomic distinctions that humans naturally assume structure the universe, can be employed to syntactically distinguish objects of diierent classes. The essential idea behind sorted logics is to assign sorts to objects and to restrict the ranges of variables to particular sorts, so that unintended inferences, which then violate the constraints imposed by this sort information, are disallowed. These techniques have been seen to dramatically reduce the search space associated with deduction in rst-order systems ((Wal88], Coh89], Sch89]). On the other hand, the inherently higher-order nature of many problems whose solutions one would like to deduce automatically has sparked an increasing interest in higher-order deduction. The behavior of sorted higher-order calculi, which boast both the expressiveness of higher-order logics and the eeciency of sorted calculi, is thus a natural topic of investigation. In this paper, we develop precisely such a calculus | an order-sorted lambda calculus supporting functional base sorts and constant