Negation in Combining Constraint Systems

Stephan Kepser

To appear in Frontiers of Combining Systems, Proceedings 98. Maarten de Rijke and Dov Gabbay (Eds.). Kluwer Academic Publishers, 1998.

In a recent paper, Baader and Schulz presented a general method for the combination of constraint systems for purely positive constraints. But negation plays an important role in constraint solving. E.g., it is vital for constraint entailment. Therefore it is of interest to extend their results to the combination of constraint problems containing negative constraints. We show that the combined solution domain introduced by Baader and Schulz is a domain in which one can solve positive and negative ``mixed'' constraints by presenting an algorithm that reduces solvability of positive and negative ``mixed'' constraints to solvability of pure constraints in the components. The existential theory in the combined solution domain is decidable if solvability of literals with so-called linear constant restrictions is decidable in the components. We also give a criterion for ground solvability of mixed constraints in the combined solution domain. The handling of negative constraints can be significantly simplified if one can show that the solution domain owns the independence of negative constraints property. We provide a modularity result giving sufficient conditions for the component systems in order for the combined solution domain to have the independence property.