One of the distinguishing features of logic programming seems to be the notion of goal-directed provability, i.e. that the structure of the goal is used to determine the next step in the proof search process. It is known that by restricting the class of formulae it is possible to guarantee that a certain class of proofs, known as 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 first-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 different ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proo...

It is known that larger classes of formulae than Horn clauses may be used as logic programming languages. One such class of formulae is hereditary Harrop formulae, for which an operational notion of provability has been studied, and it is known that operational provability corresponds to provability in intuitionistic logic. In this paper we discuss the notion of a normal form for this class of formulae, and show how this may be given by removing disjunctions and existential quantifications from programs. Whilst the normal form of the program preserves operational provability, there are operationally equivalent programs which are not intuitionistically equivalent. As it is known that classical logic is too strong to precisely capture operational provability for larger classes of programs than Horn clauses, the appropriate logic in which to study questions of equivalence is an intermediate logic. We explore the nature of the required logic, and show that this may be obtained by the addit...

Miller has shown that disjunctions are not necessary in a large fragment of hereditary Harrop formulae, a class of formulae which properly includes Horn clauses. In this paper we extend this result to include existential quantifications, so that for each program D, there is a program D 0 which is operationally equivalent, but contains no disjunctions or existential quantifications. We may think of this process as deriving a normal form for the program. This process is carried out by pushing the connectives outwards from the body of a clause, and this process leads to a normal form for goals as well. The properties of the search process used to find uniform proofs of goals (which generalises SLD-resolution) together with the normal form allow successful goals to be converted into program clauses, and so we may add successful goals to the program. The stored form of the goal requires a larger class of formulae, i.e. full first-order hereditary Harrop formulae, and so this lea...