Reasoning with sorted-pareto dominance and other qualitative and partially ordered preferences in soft constraints

In decision making problems where we need to choose a particular decision or alternative from a set of possible choices, we often have some preferences which determine if we prefer one decision over another. When these preferences give us an ordering on the decisions that is complete, then it is easy to choose the best or one of the best decisions. However it often occurs that the preferences relation is partially ordered, and we have no best decision. In this thesis, we look at what happens when we have such a partial order over a set of decisions, in particular when we have multiple orderings on a set of decisions, and we present a framework for qualitative decision making. We look at the different natural notions of optimal decision that occur in this framework, which gives us different optimality classes, and we examine the relationships between these classes. We then look in particular at a qualitative preference relation called Sorted-Pareto Dominance, which is an extension of Pareto Dominance, and we give a semantics for this relation as one that is compatible with any order-preserving mapping of an ordinal preference scale to a numerical one. We apply Sorted-Pareto dominance to a Soft Constraints setting, where we solve problems in which the soft constraints associate qualitative preferences to decisions in a decision problem. We also examine the Sorted-Pareto dominance relation in the context of our qualitative decision making framework, looking at the relevant optimality classes for the Sorted-Pareto case, which gives us classes of decisions that are necessarily optimal, and optimal for some choice of mapping of an ordinal scale to a quantitative one. We provide some empirical analysis of Sorted-Pareto constraints problems and examine the optimality classes that result.