Realizing the Dependently Typed Λ-calculus

Dependently typed λ-calculi such as the Edinburgh Logical Framework (LF) can encode relationships between terms in types and can naturally capture correspondences between formulas and their proofs. Such calculi can also be given a logic programming interpretation: the system is based on such an interpretation of LF. We have considered whether a conventional logic programming language can also provide the benefits of a Twelf-like system for encoding type and term dependencies through dependent typing, and whether it can do so in an efficient manner. In particular, we have developed a simple mapping from LF specifications to a set of formulas in the higher-order hereditary Harrop (hohh) language, that relates derivations and proof-search between the two frameworks. We have shown that this encoding can be improved by exploiting knowledge of the well-formedness of the original LF specifications to elide much redundant type-checking information. The resulting logic program has a structure that closely follows the original specification, thereby allowing LF specifications to be viewed as meta-programs that generate hohh programs. We have proven that this mapping is correct, and, using the Teyjus implementation of λProlog, we have shown that our translation provides an efficient means for executing LF specifications, complementing the ability the Twelf system provides for reasoning about them. In addition, the translation offers new avenues for reasoning about such specifications, via reasoning over the generated hohh programs.