A logic of soft constraints based on partially ordered preferences

Abstract

Representing and reasoning with an agent's preferences is important in many applications of constraints formalisms. Such preferences are often only partially ordered. One class of soft constraints formalisms, semiring-based CSPs, allows a partially ordered set of preference degrees, but this set must form a distributive lattice; whilst this is convenient computationally, it considerably restricts the representational power. This paper constructs a logic of soft constraints where it is only assumed that the set of preference degrees is a partially ordered set, with a maximum element 1 and a minimum element 0. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. A generalised possibilistic logic, based on partially ordered values of possibility, is also constructed, and shown to be formally very strongly related to the logic of soft constraints. It is shown how the machinery that exists for the distributive lattice case can be used to perform sound and complete deduction, using a compact embedding of the partially ordered set in a distributive lattice

Ordering based decision making – A survey

“…Because preference in most of case of reality appears as partially ordered structure due to incompleteness of knowledge, ambiguity of opinions, and so on [47,92], which makes the aggregation process much more complex and challenging. For decision making under this situation, a preference aggregation or inference procedure need to be applied to combine these partial orders to produce an overall preference ordering, and this again can be a partial order.…”

“…For example, R(a, b) is interpreted as the alternative a is preferred to b. Wilson [92] developed a logic of soft constraints where the set of preferences is only assumed to be a partially ordered set, with a minimum element and a maximum element. This means that there are no additional restrictions and operations, which will restrict the representational power, needed for the set of preferences to form a lattice.…”

Multi-Objective Constraint Optimization with Tradeoffs

“…Its corresponding primal graph is depicted in Figure 1(b). The Pareto set of the problem contains 8 solutions with undominated utility vectors: (3,24), (8,21), (9,19), (10,16), (11,14), (12,12), (13,8) and (14,6), respectively.…”

Sorted Pareto Dominance: An Extension to Pareto Dominance and Its Application in Soft Constraints

“…Firstly, a Preference Degree Structure (PDS) (based on [8]) is a tuple P = I, ⊗, , where I is a set of preference degrees, is a partial order on I, and ⊗ is a commutative and associative operator, monotonic with respect to (i.e., a b ⇒ a⊗c b⊗c) which is used to combine the preference degrees. We define a general Soft Constraints problem (based on [25]) to be a tuple F = V, D, C, P , where V is a set of problem variables, D is a set of variable domains, C is a multiset of soft constraints, and P is a PDS, and we describe this Soft Constraints problem briefly as follows.…”

“…Some work that looks at branch and bound algorithms for partially ordered Soft Constraints includes [10] and [25]. Both [23] and [17] look at algorithms for multi-criteria optimisation in Soft Constraints for approximating Pareto optimal solution sets, and the work in [5] looks at depth first branch and bound algorithms for the computation of leximin optimal solutions in Constraint Networks.…”