Logic Programming Based on Higher-Order Hereditary Harrop Formulas
Hereditary Harrop formulas are an extension to Horn clauses in which the body of clauses can contain implications and universal quantifiers. These formulas can further be extended by embedding them in a higher-order logic; that is, by permitting quantification over function symbol occurrences and some predicate symbol occurrences, and by replacing first-order terms with simply typed λ-terms. Our justification for considering this rich extension of Horn clause theory as a satisfactory logic programming language is provided by a proof-theoretic notion we call "uniform proofs". This notion will be defined and motivated. This extended language can provide very natural and direct implementations of various kinds of abstraction mechanisms. For example, higher-order hereditary Harrop formulas (hohh) can be used to support aspects of modular programming, abstract data types, and higher-order programming. We have designed and built a logic programming system which implements hohh in much the same way Prolog implements first-order Horn clauses. This language and its interpreter, collectively called λProlog, will be described. We will present several example programs where λProlog provides a much more immediate and satisfactory implementation language than first-order Prologs. These examples are taken from theorem proving and program transformation. Finally, we will describe some aspects of our implementation of λProlog.