Negation and Constraint Logic Programming

Almost all constraint logic programming systems include negation, yet nowhere has a sound operational model for negation in CLP been discussed. The SLDNF approach of only allowing ground negative subgoals to execute is very restrictive in constraint logic programming where most variables appearing in a derivation never become ground. By describing a scheme for constructive negation in constraint logic programming we give a sound and complete operational model for negation in these languages. Constructive negation was rst formulated for logic programming in the Herbrand Universe and involves introducing disequality constraints. Constraint logic programming thus provides a much more natural framework for describing constructive negation. In this paper we describe a framework for constructive negation for constraint logic programming over arbitrary structures which is sound and complete with respect to the three-valued consequences of the completion of a program. Through this description, we arrive at a simpler, more e cient form of constructive negation for the Herbrand Universe. Finally we characterize what makes a structure particularly suited to the use of constructive negation and show this suitability condition for a number of structures and classes of structures. The constraint viewpoint of constraint logic programming is well matched with constructive negation. Not only is constructive negation easier to understand from this point of view, but more importantly it gives the rst clean approach to negation in constraint logic programming. 3 List of Symbols : negation ^ conjunction _ disjunction leftarrow ! rightarrow $ leftrightarrow 9 exists 8 forall W big disjunction V big conjunction , double leftrightarrow = equals 6= not equals > greater than < less than greater than or equal to less than or equal to + plus times minus t bold t u bold u f bold f vertical separator " uparrow # downarrow A calligraphic A L calligraphic L R calligraphic R HU calligraphic HU double left angle brackets double right angle brackets ! omega capital Phi < reals alpha beta gamma 4