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This paper describes a new approach to computation in a semiring-based system, which includes semiring-based CSPs (in particular weighted CSPs, fuzzy CSPs and standard CSPs) as well as Bayesian networks. The approach to computation is based on what we call semiring-labelled decision diagrams (SLDDs). These can be generated in a similar way to a standard search tree (decision tree) for solving a CSP, but some nodes are merged, creating a more compact representation; for certain classes of CSPs, the number of nodes in the resulting network will be a tiny fraction of the number of nodes in the corresponding search tree. A method is given for generating an SLDD that represents e.g., a particular instance of a semiring-based CSP; it is shown how this can be used to perform various computations of interest, such as solving a semiring-based CSP, finding optimal solutions, determining the possible values of each variable and counting solutions of a CSP.