Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved, and fit naturally within constraint logic programming. However, in general, filtering that is sound for a global constraint can be unsound when the constraint is open. This paper provides a simple characterization, called contractibility, of the constraints where filtering remains sound when the constraint is open. With this characterization we can easily determine whether a constraint has this property or not. In the latter case, we can use it to derive a contractible approximation to the constraint. We demonstrate this work on both hard and soft constraints. In the process, we formulate two general classes of soft constraints.

The classical probabilistic entailment problem is to We apply the family of minimal violation measures from determine upper and lower bounds on the probability [Potyka, 2014] since they allow us to extend the classical notion of formulas, given a consistent set of probabilistic of models of a probabilistic knowledge base to inconsistent assertions. We generalize this problem ones. Intuitively, the generalized models are those probability by omitting the consistency assumption and, thus, functions that minimally violate the knowledge base provide a general framework for probabilistic reasoning [Potyka and Thimm, 2014]. We incorporate integrity constraints under inconsistency. To do so, we utilize and study a family of generalized entailment problems inconsistency measures to determine probability for probabilistic knowledge bases. More specifically, functions that are closest to satisfying the knowledge the contributions of this work are as follows: base. We illustrate our approach on several 1. We introduce the computational problem of generalized examples and show that it has both nice formal and entailment with integrity constraints in probabilistic logics computational properties.