Encoding a dependent-type λ-calculus in a logic programming language

Abstract

Various forms of typed λ-calculi have been proposed as specification languages for representing wide varieties of object logics. The logical framework, LF, is an example of such a dependent-type λ-calculus. A small subset of intuitionistic logic with quantification over simply typed λ-calculus has also been proposed as a framework for specifying general logics. The logic of hereditary Harrop formulas with quantification at all non-predicate types, denoted here as hh ^ω, is such a meta-logic that has been implemented in both the Isabelle theorem prover and the λProlog logic programming language. Both frameworks provide for specifications of logics in which details involved with free and bound variable occurrences, substitutions, eigenvariables, and the scope of assumptions within object logics are handled correctly and elegantly at the “meta” level. In this paper, we show how LF can be encoded into hh ^ω in a direct and natural way by mapping the typing judgments in LF into propositions in the logic of hh ^ω. This translation establishes a very strong connection between these two languages: the order of quantification in an LF signature is exactly the order of a set of hh ^ω clauses, and the proofs in one system correspond directly to proofs in the other system. Relating these two languages makes it possible to provide implementations of proof checkers and theorem provers for logics specified in LF by using standard logic programming techniques which can be used to implement hh ^ω.