Abstract. The semiring-based formalism to model soft constraint has been introduced in 1995 by Ugo Montanari and the authors of this paper. The idea was to make constraint programming more flexible and widely applicable. We also wanted to define the extension via a general formalism, so that all its instances could inherit its properties and be easily compared. Since then, much work has been done to study, extend, and apply this formalism. This papers gives a brief summary of some of these research activities. 1 Before soft constraints: a brief introduction to constraint programming

We present a formal model to represent and solve the unicast/multicast routing problem in networks with Quality of Service (QoS) requirements. To attain this, first we translate the network adapting it to a weighted graph (unicast) or and-or graph (multicast), where the weight on a connector corresponds to the multidimensional cost of sending a packet on the related network link: each component of the weights vector represents a different QoS metric value (e.g. bandwidth, delay, packet loss). The second step consists in writing this graph as a program in Soft Constraint Logic Programming: the engine of this framework is then able to find the best paths/trees by optimizing their costs and solving the constraints imposed on them (e.g. delay ≤ 40msec), thus finding a solution to QoS routing problem. Soft constraints, and related c-semiring structures,

Multitrust provides a flexible approach to encoding trust metrics whereby definitions for trust propagation and aggregation are specified in terms of a semiring. Determining the degree of trust between principals across a trust network is, in turn, programmed as a (semiring based) soft-constraint satisfaction problem. In this paper we consider the use of semiring-based metrics in reasoning about trust between coalition-forming principals. The configurable nature of multitrust makes it well-suited to modeling trust within coalitions: whether adding more principals to a coalition increases trust or decreases trust is captured by the definition of trust aggregation within the semiring. Copyright c ○ 0000 John Wiley & Sons, Ltd. KEY WORDS: