Many “semiring-like” structures are used in Soft Constraint Satisfaction Problems (SCSPs). We review a few properties of semirings that are useful for dealing with soft constraints, highlighting the differences between alternative proposals in the literature.
We then extend the semiring structure by adding the notion of division as a weak inverse operation of product. In particular, division is needed to apply constraint relaxation when the product operation of the semiring is not idempotent. The division operator is introduced via residuation and it is also able to deal with partial orders, generalizing the approach given for Valued CSPs.
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