A Logic Program for Transforming Sequent Proofs to Natural Deduction Proofs

In this paper, we show that an intuitionistic logic with second-order function quantification, called hh2 here, can serve as a meta-language to directly and naturally specify both sequent calculi and natural deduction inference systems for first-order logic. For the intuitionistic subset of first-order logic, we present a set of hh2 formulas which simultaneously specifies both kinds of inference systems and provides a direct and concise account of the correspondence between cut-free sequential proofs and normal natural deduction proofs. The logic of hh2 can be implemented using such logic programming techniques as providing operational interpretations to the connectives and implementing unification on λ-terms. With respect to such an interpreter, our specification provides a description of how to convert a proof in one system to a proof in the other. The operation of converting a sequent proof to a natural deduction proof is functional in the sense that there is always one natural deduction proof corresponding to each sequent proof. Our specification, in fact, provides a direct implementation of the transformation in this direction.