Compiling All Possible Conflicts of a CSP

In interactive decision-making settings, such as product configuration, users are stating preferences, or foreground constraints, over a set of possible solutions, as defined by background constraints. When the foreground constraints introduce inconsistencies with the background constraints, we wish to find explanations that help the user converge to a solution. In order to provide satisfactory explanations, it can be useful to know one or several subsets of conflicting constraints; such a subset is called a conflict. When computing such conflicts is intractable in an interactive context, we can choose to compile the problem so as to allow faster response times. In this paper we propose a new representation, which implicitly encompasses all conflicts possibly introduced by a user's choices. We claim that it can help in situations where extra information about conflicts is needed, such as when explanations of inconsistency are required.