Generalized Definite Set Constraints

• Authors: Jean-Marc Talbot;Philippe Devienne;Sophie Tison
• Venue: Constraints
• Year: 2000

Set constraints (SC) arelogical formul\ae in which atoms are inclusions between set expressions.Those set expressions are built over a signature \Sigma,variables and various set operators. On a semantical point ofview, the set constraints are interpreted over sets of treesbuilt from \Sigma and the inclusion symbol is interpretedas the subset relation over those sets. By restricting the syntaxof those formul\ae and/or the set of operators that can occurin set expressions, different classes of set constraints areobtained. Several classes have been proposed and studied forsome problems such as satisfiability and entailment. Among thoseclasses, we focus in this article on the class of definite SC‘sintroduced by Heintze and Jaffar, and the class of co-definiteSC‘s studied by Charatonik and Podelski. In spite oftheir name, those two classes are not dual from each other, neitherthrough inclusion inversion nor through complementation. In thisarticle, we propose an extension for each of those two classesby means of an intentional set construction, so called membershipexpression. A membership expression is an expression \{x\mid\Phi(x)\}. The formula \Phi(x) is a positivefirst-order formula built from membership atoms t \in Sin which S is a set expression. We name those twoclasses respectively generalized definite and generalized co-definiteset constraints. One of the main point concerning those so-extendedclasses is that the two generalized classes turn out to be dualthrough complementation. First, we prove in this article thatgeneralized definite set constraints is a proper extension ofthe definite class, as it is more expressive in terms of setsof solutions. But we show also that those extensions preservesome main properties of the definite and co-definite class. Hencefor instance, as definite set constraints, generalized definiteSC‘s have a least solution whereas the generalizedco-definite SC‘s have a greatest solution, just asco-definite ones. Furthermore, we devise an algorithm based ontree automata that solves the satisfiability problem for generalizeddefinite set constraints. Due to the dualization, the algorithmsolves the satisfiability problem for generalized co-definiteset constraints as well. This algorithm proves first that forthose generalized classes, the satisfiability problem remainsDEXPTIME-complete. It provides also a proof for regularity ofthe least solution of generalized definite constraints and so,by dualization for the greatest solution for the generalizedco-definite SC‘s.