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g ,where g , g are different goals
(as cheap I expensive in the procurement example). To summarize, the incompatibility
relation captures the conflicts, either default or domain dependent, amongst decisions, beliefs
and goals.
The incompatibility relation can be easily lifted to set of sentences. We say that two sets of
sentences
non-default incompatibilities. For instance, we may have g
I
Φ 1 and
Φ 2 are incompatible (still denoted by
Φ 1 I Φ 2 ) iff there is a sentence
φ 1 in
Φ 1
and a sentence
φ 1 I φ 2 .
A theory gathers the statements about the decision problem.
φ 2 in
Φ 2 such that
Definition 9 (Theory) . A theory
T
is an extended logic program, i.e a finite set of rules R: L 0
L
L 1 ,..., L j ,
L j + 1 ,...,
L n with n
0, each L i (with i
0 ) being a strong literal in
. R, called
the unique name of the rule, is an atomic formula of
. All variables occurring in a rule are implicitly
universally quantified over the whole rule. A rule with variables is a scheme standing for all its ground
instances.
L
Considering a decision problem, we distinguish:
goal rules of the form R : G 0
>
G 1 ,..., G n with n
0, where each G i ( i
0) is a goal
DL
(or its strong negation). According to this rule, the goal G 0 is promoted (or
demoted) by the combination of the goal literals in the body;
epistemic rules of the form R : B 0
literal in
B 1 ,..., B n with n
0, where each B i ( i
0) is a belief
. According to this rule, B 0 is true if the conditions B 1 ,..., B n are satisfied;
decision rules of the form R : G
literal of
DL
0. The
head of the rule is a goal (or its strong negation). The body includes a set of decision
literals ( D i
D 1
(
a 1
)
,..., D m
(
a m
)
, B 1 ,..., B n with m
1, n
) and a (possibly empty) set of belief literals. According to this rule, the
goal is promoted (or demoted) by the decisions
(
a i
) ∈L
{
D 1
(
a 1
)
,..., D m
(
a m
) }
, provided that the
conditions B 1 ,..., B n are satisfied.
For simplicity, we will assume that the names of rules are neither in the bodies nor in the head
of the rules thus avoiding self-reference problems. Moreover, we assume that the elements
in the body of rules are independent (the literals cannot be deduced from each other), the
decisions do not influence the beliefs, and the decisions have no side effects.
Considering statements in the theory is not sufficient to make a decision. In order to evaluate
the previous statements, other relevant pieces of information should be taken into account,
such as the priority amongst goals. For this purpose, we consider the priority relation
P
over
the goals in
G 2 can be read “ G 1 has
priority over G 2 ”. There is no priority between G 1 and G 2 ,eitherbecause G 1 and G 2 are ex
æquo (denoted G 1
G
, which is transitive, irreflexive and asymmetric. G 1 P
G 2 ), or because G 1 and G 2 are not comparable. The priority corresponds
to the relative importance of the goals as far as solving the decision problem is concerned. For
instance, we can prefer a fast service rather than a cheap one. This preference can be captured
by the priority. The reservation is the minimal set of goals which needs to be reached. The
reservation value is the least favourable point at which one will accept a negotiated agreement.
It would mean the bottom line that one would be prepared to concede.
In order to illustrate the previous notions, we provide here the decision framework related to
the problem described in Section 3.1.
Example 3 (Decision framework) . We consider the procurement example which is described in
Section 3.1. The buyer's decision problem is captured by a decision framework
DF = DL
,
P
sm ,
I
,
T
,
P
,
RV
where:
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