Mints type deductive calculi for logic programming

AbstractMints (1986) has given a deductive calculus, a set of proof rules, for pure Prolog such that the goal X = A1,..., An succeeds in Prolog iff X is derivable in this calculus and X fails in Prolog iff (∼)X is derivable in this calculus. We summarise Mints' results and give appropriate modifications of his calculus to deal with (a) use of negated goals in Prolog, (b) SLD- resolution, (c) SLDNF-resolution, (d) extensions of SLDNF-resolution allowing negation as failure to be applied to nonground negative literals