A Proof-Theoretic Analysis of Goal-Directed Provability

Uniform proofs have been presented as a basis for logic programming, and it is known that by restricting the class of formulae it is possible to guarantee that uniform proofs are complete with respect to provability in intuitionistic logic. In this paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform proofs arise naturally as a normal form for proofs in rst-order intuitionistic sequent calculus. Next we show that the class of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae from uniform proofs in two diierent ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proofs are guaranteed to be complete. Finally we brieey discuss some possibilities for a slightly more general approach using intermediate and innnitary logics.