Extending Soft Arc Consistency Algorithms to Non-invertible Semirings

We extend arc consistency algorithms proposed in the literature in order to deal with not necessarily invertible semirings. As a result, consistency algorithms can now be used as a preprocessing procedure in soft CSPs defined over a larger class of semirings: either partially ordered, or with non idempotent ×, or not closed ÷ operator, or constructed as cartesian product or Hoare power sets of any semiring (which can be used for multicriteria CSPs). To reach this objective, we first show that each semiring can be transformed into a new one where the + is instantiated with the Least Common Divisor (LCD) between the elements of the semiring. The LCD value corresponds to the amount we can “safely move” from the binary constraint to the unary one in the consistency algorithm (when × is not idempotent). We then propose an arc consistency algorithm which takes advantage of this operator.