A Logic of Partially Satisfied Constraints

Soft constraints are recognised as being important for many constraints applications. These include (a) over-constrained problems, where we cannot satisfy all the constraints, (b) situations where a constraint can be partially satisfied, so that there are degrees of satisfaction, and (c) where the identity of a constraint is uncertain, so that it can be uncertain whether a constraint is satisfied or not by a tuple. This paper takes a logic-based approach to partially satisfied constraints, for situations where a tuple can be assigned partially ordered degrees of satisfaction for a constraint. A semantics is defined, which determines which constraints can be inferred from a set of constraints, and a sound and complete proof theory is given. This formalism is closely related to idempotent semiring-based CSPs and lattice-valued possibilistic logic.