The aim of this paper is to show the fruitfulness and fecundity of the authors' proof-theoretic analysis of logic programming (both for definite and normal programs). It is based on a simple logical framework that goes under the name of regular search spaces. The challenge faced here is to give a treatment in proof-theoretic terms of the issue of negation, which has been one of the toughest problems that has plagued logic programming from its very beginning. While negation-as-failure (NF) has been overwhelmingly the more widespread answer, its intrinsic limitations have made it a rather unsatisfactory solution. In the present paper it is first contended that the notion of regularity offers a better understanding of the traditional theory of NF, and second a firm yet very simple and natural basis for a form of constructive negation, in the sense of Chan, Stuckey and Harland. A version of constructive negation is presented, based on the notion of regular splitting, a transformation technique where the failure axiom(s) of a predicate occurring negatively in a program are split into new clauses according to a covering of the underlying signature.

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