Higher-order equational logic programming

Abstract

Higher-order equational logic programming is a paradigm which combines first-order equational and higher-order logic programming, where higher-order logic programming is based on a subclass of simply typed l-terms, called higher-order patterns. Central to the notion of higher-order equational logic programming is the so-called higher-order equational unification. This paper extends several important classes of first-order equational unification algorithms to the higher-order setting: only problems of the extensions are discussed and first-order equational unifications are viewed as black boxes whenever possible.We first extend narrowing and show that the completeness of many higher-order narrowing strategies reduces to that of their underlying first-order counterparts. Then we propose an algorithm or higher-order equational unification of free higher-order patterns in an arbitrary equational theory. Finally a general approach to extend first-order unification combination algorithms is sketched informally. The termination property of the above higher-order extensions is considered in a uniform way.