Constraint Solving over Semirings

We introduce a general framework for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction , and others can be easily cast. The framework is based on a semiring structure, where the set of the semiring specifies the values to be associated to each tuple of values of the variable domain, and the two semiring operations (+ and x) model constraint projection and combination respectively. Local consistency algorithms, as usually used for classical CSPs, can be exploited in this general framework as well, provided that some conditions on the semiring operations are satisfied. We then show how this framework can be used to model both old and new constraint solving schemes, thus allowing one both to formally justify many informally taken choices in existing schemes, and to prove that the local consistency techniques can be used also in newly defined schemes. 1 Introduction Classical constraint satisfaction problems (CSPs) [Mon-tanari, 1974; Mackworth, 1988] are a very expressive and natural formalism to specify many kinds of real-life problems. In fact, problems ranging from map coloring , vision, robotics, job-shop scheduling, VLSI design , etc., can easily be cast as CSPs and solved using one of the many techniques that have been developed for such problems or subclasses of them However, they also have evident limitations, mainly due to the fact that they are not very flexible when trying to represent real-life scenarios where the knowledge is not completely available nor crisp. In fact, in such situations, the ability of stating whether an instantiation of values to variables is allowed or not is not enough or sometimes not even possible. For these reasons, it is natural to try to extend the CSP formalism in this direction. CSPs have been extended with the ability to associate to each tuple, or to each constraint, a level of preference, and with the possibility of combining constraints using min-max operations. This extended formalism has been called Fuzzy CSPs (FCSPs). Other extensions concern the ability to model incomplete knowledge of the real problem [Fargier and Lang, 1993], to solve overconstrained problems [Freuder and Wallace, 1992], and to represent cost optimization problems. In this paper we define a constraint solving framework where all such extensions, as well as classical CSPs, can be cast. However, we do not relax the assumption of a finite domain for the variables of the constraint problems. The main idea is based …