Generalized Definite Set Constraints

Set constraints (SC) are logical formulae in which atoms are inclusions between set expressions. Those set expressions are built over a signature , variables and various set operators. On a semantical point of view, the set constraints are interpreted over sets of trees built from and the inclusion symbol is interpreted as the subset relation over those sets. By restricting the syntax of those formulae and/or the set of operators that can occur in set expressions, different classes of set constraints are obtained. Several classes have been proposed and studied for some problems such as satisfiability and entailment. Among those classes, we focus in this article on the class of definite SC's introduced by Heintze and Jaffar, and the class of co-definite SC's studied by Charatonik and Podelski. In spite of their name, those two classes are not dual from each other, neither through inclusion inversion nor through complementation. In this article, we propose an extension for each of those two classes by means of an intentional set construction, so called membership expression. A membership expression is an expression {x| (x)}. The formula (x) is a positive first-order formula built from membership atomst in S in which S is a set expression. We name those two classes respectively generalized definite and generalized co-definite set constraints. One of the main point concerning those so-extended classes is that the two generalized classes turn out to be dual through complementation. First, we prove in this article that generalized definite set constraints is a proper extension of the definite class, as it is more expressive in terms of sets of solutions. But we show also that those extensions preserve some main properties of the definite and co-definite class. Hence for instance, as definite set constraints, generalized definite SC's have a least solution whereas the generalized co-definite SC's have a greatest solution, just as co-definite ones. Furthermore, we devise an algorithm based on tree automata that solves the satisfiability problem for generalized definite set constraints. Due to the dualization, the algorithm solves the satisfiability problem for generalized co-definite set constraints as well. This algorithm proves first that for those generalized classes, the satisfiability problem remains DEXPTIME-complete. It provides also a proof for regularity of the least solution of generalized definite constraints and so, by dualization for the greatest solution for the generalized co-definite SC's.