The Relative Complement Problem for Higher-Order Patterns

We address the problem of complementing higher-order patterns without repetitions of free variables. Differently from the first-order case, the complement of a pattern cannot, in general, be described by a pattern, or even by a finite set of patterns. We therefore generalize the simply-typed -calculus to include an internal notion of strict function so that we can directly express that a term must depend on a given variable. We show that, in this more expressive calculus, finite sets of patterns without repeated variables are closed under complement and unification. Our principal application is the transformational approach to negation in higher-order logic programs. 1 Introduction In most functional and logic programming languages the notion of a pattern, together with the requisite algorithms for matching or unification, play an important role in the operational semantics. Besides unification other problems such as generalization or complement also arise frequently. In this paper w..