Arguing over Goals for Negotiation: Adopting an Assumption-Based Argumentation Decision Support System - Efficient Decision Support Systems: Practice and Challenges from Current to Future - page 218

Definition 8 (Decision framework) . A decision framework is a tuple
DF = DL

,

P

sm ,

I

,

T

,

P

,

RV

,where:

•

DL = G∪D∪B

is a set of predicate symbols called the decision language ,wherewedistinguish

between goals (

G

), decisions (

D

) and beliefs (

B

);

•

P

sm is a set of atomic formulae built upon predicates in

DL

called presumptions ;

•

I

is the incompatibility relation , i.e. a binary relation over atomic formulae in

G

,

B

or

D

.

I

is not necessarily symmetric;

•

T

is a logic theory built upon

DL

; statements in

T

are clauses, each of which has a distinguished

name ;

•

P⊆G×G

is the priority relation, namely a transitive, irreflexive and asymmetric relation over
atomic formulae in G ;

RV G • is a set of literals built upon predicates in , called the reservation value .

Let us summarize the intuitive meaning of the various components of the framework. The
language DL is composed by:

•thesetof goal predicates, i.e. some predicate symbols which represent the features that a
decision must exhibit;
•theset D of decision predicates, i.e. some predicate symbols which represent the actions

which must be performed or not; different atoms built on D represent different alternatives ;

•theset of beliefs , i.e. some predicate symbols which represent epistemic statements;.

In this way, we can consider multiple objectives which may or not be fulfilled by a set of
decisions under certain circumstances.
We explicitly distinguish presumable (respectively non-presumable ) literals which can
(respectively cannot) be assumed to hold, as long as there is no evidence to the contrary.
Decisions as well as some beliefs can be assumed.

B

In this way, DF can model the

incompleteness of knowledge.
The most natural way to represent conflicts in our object language is by means of some
forms of logical negation. We consider two types of negation, as usual, e.g., in extended
logic programming, namely strong negation

¬

(also called explicit or classical negation ), and

weak negation , also called negation as failure . As a consequence we will distinguish between

strong literals, i.e. atomic formulae possibly preceded by strong negation, and weak literals,
i.e. literals of the form

∼

∼ L ,where L is a strong literal. The intuitive meaning of a strong literal

¬

L is “L is definitely not the case”, while

∼

L intuitively means “There is no evidence that L

is the case”.
The set I of incompatibilities contains some default incompatibilities related to negation on

the one hand, and to the nature of decision predicates on the other hand. Indeed, given an
atom A ,wehave A I¬ ¬ I I∼ A as well as A A .Moreover, L L , whatever L is, representing

∼

the intuition that L is evidence to the contrary of L . Notice, however, that we do not have

∼ I L , as in the spirit of weak negation. Other default incompatibilities are related to

decisions, since different alternatives for the same decision predicate are incompatible with
one another. Hence, D

L

( ) I ( ) ( ) I ( ) D ,and

a 1 and a 2 being different constants representing different 2 alternatives for D . Depending on
the particular decision problem being represented by the framework,

a 1 D a 2 and D a 2 D a 1 , D being a decision predicate in

I

may contain further

2 Notice that in general a decision can be addressed by more than two alternatives.