Background

Nowadays, several numerical tools are usually adopted in AI to represent and manage uncertainty. All of them originate from amendments of the well-known Probability measure, aimed at generalizing it to better fit different peculiarities of specific application fields [29,45]. The measures that achieved wider diffusion can be classified as:

-

Capacities;

-

Possibility and Necessity measures;

-

Probabilities;

-

Belief and Plausibility functions;

-

Lower and Upper probabilities.

Among all these measures (which are real-valued functions), Capacities [7] characterize the weakest notion. Indeed, Capacities are measures whose unique property is monotonicity with respect to the implication of events.

The class of Capacities includes all other classes.

Possibility measures (Necessity measures are their dual) come from Fuzzy theory [18,49] and originate from the need to express ``vagueness'' about the descriptions of situations instead of uncertainty about their truth.

Probabilities are characterized by the ``additivity'' property: Having judged the uncertainties $P(A)$ and $P(B)$ on any pair of disjoint situations $A$ and $B$, the uncertainty on their combination $A\vee B$ is defined as $P(A)+P(B)$.

Belief functions (whose dual are Plausibilities) are the base of Evidence theory [38].

Lower probabilities (whose dual are Upper probabilities) are instead adopted whenever one needs to consider as valid an entire family of probabilistic models in place of a single one. Such measures have been developed within the field of Imprecise probabilities [44].

As a matter of fact, each one of the framework described so far, can manage uncertainty and retains all of the expressive power of mathematical quantitative models. Though, inevitably, they suffer from the drawbacks often faced whenever numerical models are applied to practical problems: a) the difficulty of expressing a complete evaluation, and b) the hardness to elicit precise numerical values. The former problem can be circumvented by introducing the so called partial models, i.e. numerical evaluations defined only on some of the situations at hand, and intended to be a restriction of some of the complete models mentioned above. To obviate the second drawback of numerical models, qualitative approaches have been proposed in the last decades. The central idea of such methodologies is to grade uncertainty about the truth of propositions, through comparisons expressing the judgement of ``less or more believed to be true''. This operationally translates into the use of (partial) order relations in place of numerical grades.

Qualitative approaches are receiving wider and wider attention, either as theoretical tools to deal directly with belief management [2,10,14], or inside the more articulated framework of decision-making theory (see, for example, [16,15,19,25,30]). This is because, they better fit the nature of human judgments.

We show that representability of orders, defined on arbitrary finite sets of propositions, can be characterized by the specific properties (axioms)

Most of such axioms are of direct declarative reading, as they involve only logical and preference relations. As we will see, such a declarative character supports a straightforward translation of the axioms within the logical framework of Answer Set Programming. As a consequence, we immediately obtain an executable specification able to discriminate between the different uncertainty orders.