ON THE USE OF SYNTACTIC POSSIBILISTIC FUSION FOR
COMPUTING POSSIBILISTIC QUALITATIVE DECISION
Salem Benferhat
CRIL, Universit´e d’Artois, Rue Jean Souvraz, S.P. 18 62307 LENS Cedex, France
Faiza Haned-Khellaf, Aicha Mokhtari, Ismahane Zeddigha
LRIA, Universit´e des Sciences et de la Technologie Houari Boumediene, BP 32 El Alia, Alger, Algeria
K
eywords:
Qualitative possibilistic decision, pessimistic criteria, data fusion.
Abstract:
This paper describes the use of syntactical data fusion to computing possibilistic qualitative decisions. More
precisely qualitative possibilistic decisions can be viewed as a data fusion problem of two particular possibility
distributions (or possibilistic knowledge bases): the first one representing the beliefs of an agent and the second
one representing the qualitative utility. The proposed algorithm computes a pessimistic optimal decisions
based on data fusion techniques. Weshow that the computation of optimal decisions is equivalent to computing
an inconsistency degree of possibilistic bases representing the fusion of agent’s beliefs and agent’s preferences.
1 INTRODUCTION
This paper presents a computation of pessimistic
decisions based on syntactic possibilistic fusion
operations. Qualitative possibilistic decisions can be
viewed as a data fusion problem of two particular
possibility distributions: the first one representing the
beliefs of an agent and the second one representing
the qualitative utility (or agent’s preferences). A
possibilistic decision model (Dubois and Prade,
1995) allows a gradual expression of both agent’s
preferences and knowledge. The preferences and the
available knowledge about the world are expressed
in ordinal way. In (Dubois and Prade, 1995), the
authors have proposed two qualitative criteria for
ordinal decision approaches under uncertainty: the
pessimistic and the optimistic decisions criteria. The
first one being more cautious than the second one for
computing optimal decisions.
A method for computing optimal decisions, based
on ATMS, has been proposed in (Dubois et al.,
1998). Using the pessimistic criteria, the procedure is
translated to a problem tractable by an ATMS (Kleer,
1986a)(Kleer, 1986b). In (Berre, 2000), Le Berre
has implemented the optimistic algorithm and the
pessimistic one. This implementation can not deal
with an important number of variables (Berre, 2000).
The rest of this paper is organized as follow. Sec-
tion 2 gives a brief backgrounds on possibilistic logic,
qualitative decision and data fusion in possibilistic
logic. Section 3 contains an efficient and unified way
of computing pessimistic qualitative decisions based
on syntactic counterpart of data fusion problem. Sec-
tion 4 concludes the paper.
2 BACKGROUNDS
2.1 Possibilistic Logic
This section gives a brief refresher on possibilistic
logic and qualitative decision theory. See (Dubois
et al., 1994b) for more details on possibilistic logic.
A possibility distribution (Dubois et al., 1994b) π is a
mapping from a set of interpretations Ω into the unit
interval [0,1]. π(ω) represents the degree of compat-
ibility of the interpretation ω with available pieces of
information.
Given a possibility distribution π, two dual mea-
sures are defined on the set of propositional formulas:
• The possibility measure of a formula φ, defined
by:
Π(φ) = max{π(ω) : ω |= φ and ω ∈ Ω} (1)
356
Benferhat S., Haned-Khellaf F., Mokhtari A. and Zeddigha I. (2008).
ON THE USE OF SYNTACTIC POSSIBILISTIC FUSION FOR COMPUTING POSSIBILISTIC QUALITATIVE DECISION.
In Proceedings of the Third International Conference on Software and Data Technologies - PL/DPS/KE, pages 356-359
DOI: 10.5220/0001880403560359
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c
 SciTePress

• The necessity measure of a formula φ, defined by:
N(φ) = 1 − Π(¬φ) (2)
A possibilistic knowledge base Σ is a set of
weighted formulas:
Σ = {(φ
i
,α
i
) : i = 1,...,n},
where φ
i
is a propositional formula and α
i
∈]0,1]
represents the certainty level of φ
i
.
The possibility distribution associated with a
weighted formula (φ
i
,α
i
) is (Dubois et al., 1994b):
∀ω ∈ Ω,
π
(φ
i
,α
i
)
(ω) =

1− α
i
if ω 6|= φ
i
1 otherwise
(3)
More generally, the possibility distribution associated
with a possibilistic knowledge base Σ is the result
of combining possibility distributions associated with
each weighted formula (φ
i
,α
i
) of Σ, namely ∀ω ∈ Ω:
π
Σ
(ω) = ⊕{π
(φ
i
,α
i
)
(ω) : (φ
i
,α
i
) ∈ Σ}. (4)
where ⊕ is in general either equal to the minimum op-
erator (in standard possibilistic logic), or to the prod-
uct operator (*).
2.2 Qualitative Decision
Let D = {l
i
} be a set of decision variables, where l
i
are distinguished variables of the language L. Let d ⊆
D, then the decision d
∧
is the logical conjunction of
literals in the chosen subset. Each set of decision d
induces a possibility distribution π
K
d
in the following
way (Dubois et al., 1994a):
π
K
d
(ω) =



1 if ∀(φ
i
,α
i
) ∈ K,ω |= φ
i
and ω |= d
∧
min
(φ
i
,α
i
)∈K/ω|=¬φ
i
(1− α
i
) if ω |= d
∧
0 if ω 6|= d
∧
Where α
i
represents the degrees of necessity of the
formulas in the corresponding layers of K ∪ {(d,1)}.
The utility function µ is built over Ω in a similar way:
µ(ω) =

1 if ∀(ψ
j
,β
j
) ∈ P,ω |= ψ
j
min
(ψ
j
,β
j
)∈P/ω|=¬ψ
j
(1− β
j
)
where β
j
represents a degree of priority of a formulas
in P.
Making a decision amounts to choosing a subset d of
the decision set D. The objective is to rank-order de-
cisions on the basis of K and P.
The pessimistic utility function is expressed in terms
of the possibility distribution π
K
d
and the utility func-
tion µ (Dubois et al., 1999):
u
∗
(d) = min
ω∈Ω
max(1− π
K
d
(ω),µ(ω)) (5)
In the pessimistic case, the decision d must satisfy
(Dubois et al., 1997):
K
∧
α
∧ d
∧
⊢ P
∧
>(1−α)
(6)
The decision d associated with the most certain part of
K entails the satisfaction of the goals, even those with
low priority. The pessimistic utility u
∗
of decision d,
defined at the syntactic level, takes the form (Dubois
et al., 1997):
u
∗
(d) =
(
max
K
∧
α
∧d
∧
⊢P
∧
>(1−α)
,K
∧
α
∧d
∧
6=⊥
α
0 if {K
∧
α
∧ d
∧
⊢ P
∧
>(1−α)
,K
∧
α
∧ d
∧
6=⊥} =
/
0
2.3 Fusion in Possibilistic Logic
Let Σ
1
, Σ
2
be two possibilistic bases and π
1
, π
2
be
their associated possibility distributions. Let ⊕ be a
two-place function whose domain is [0,1]×[0,1], to
be used for aggregating π
1
and π
2
. The only require-
ments for ⊕ are the following properties (Benferhat
et al., 1997):
• 1 ⊕ 1 = 1,
• if ∀ω,ω
′
if π
1
(ω) ≥ π
1
(ω
′
) and π
2
(ω) ≥ π
2
(ω
′
),
then π
1
(ω) ⊕ π
2
(ω) ≥ π
1
(ω
′
) ⊕ π
2
(ω
′
).
The syntactic counterpart of the fusion of π
1
and
π
2
is the following possibilistic base, denoted by
Σ
⊕
= Σ
1
⊕ Σ
2
, which is made of the union of :
• the initial bases, however with new necessity de-
grees defined by :
{(φ
i
,1− (1− α
i
) ⊕ 1) : (φ
i
,α
i
) ∈ Σ
1
}∪
{(ψ
j
,1− (1− β
j
) ⊕ 1) : (ψ
j
,β
j
) ∈ Σ
2
}
• and the knowledge common to Σ
1
and Σ
2
defined
by :
{(φ
i
∨ ψ
j
,1 − (1 − α
i
) ⊕ (1 − β
j
)) : (φ
i
,α
i
) ∈ Σ
1
and (ψ
j
,β
j
) ∈ Σ
2
}
The conjunctive operators exploit the symbolic
complementarities between sources.
⊕ is said to be a conjunctive operator if
∀a ∈ [0,1],⊕(a,1) = ⊕(1,a) = a.
The operator minimum (min) is an idempo-
tent conjunctive one. At the syntactic level,
the base associated to π
min
, such that π
min
(ω) =
min(π
1
(ω),π
2
(ω)), is Σ
min
= Σ
1
∪ Σ
2
(Benferhat
et al., 1997);
ON THE USE OF SYNTACTIC POSSIBILISTIC FUSION FOR COMPUTING POSSIBILISTIC QUALITATIVE
DECISION
357

3 COMPUTATION OF
QUALITATIVE PESSIMISTIC
OPTIMAL DECISIONS BASED
ON DATA FUSION TECHNICAL
A good pessimistic decision d maximizing u
∗
(d) is
such that:
u
∗
(d) = min
ω∈Ω
max(1− π
K
d
(ω),µ(ω)) (7)
which is equivalent to:
u
∗
(d) = 1 − max
ω∈Ω
min(π
K
d
(ω),1− µ(ω)) (8)
Besides, the syntactic counterpart of
min(π
1
(ω),π
2
(ω)) is the possibilistic base
Σ
min
= Σ
1
∪ Σ
2
. Thus, combining these re-
sults, the corresponding base Σ
min
associated to
min(π
K
d
(ω),1 − µ(ω)) is the possibilistic base
K ∪n
P
∪{(d,1)}, such that n
P
is the possibilistic base
corresponding to the utility function 1− µ(ω).
3.1 Transformations Steps
In this subsection, we define the possibilistic base n
P
corresponding to the utility function 1 − µ(ω), from
the preferences base P.
Let P = {(φ
i
,α
i
) : i = 1,..n} be a preferences
base. We assume that: α
0
= 0 < α
1
< ... < α
n
. The
following definition gives the possibilistic knowledge
base associated with the negation of P.
Definition 1. The negated base of P = {(φ
i
,α
i
) :
i = 1,..n} is a possibilistic base, denoted by n
p
, and
defined by:
n
P
= {(d
i
,1− α
i
) : i = 1,...,n} ∪ {(⊥,1− α
n
)}
where d
i
= ¬φ
i
∨ ¬φ
i+1
∨ ... ∨ ¬φ
n
.
The following proposition shows that n
P
is indeed
encodes the negation of P:
Proposition 1. Let P = {(φ
i
,α
i
) : i = 1,..n} be a pref-
erence base, and n
p
its negated base obtained using
definition 7. Let µ
P
and µ
n
P
be the utility distributions
associated with P and n
P
respectively. Then:
∀ω ∈ Ω, µ
P
(ω) = 1 − µ
n
P
(ω)
The obtained base n
P
must be put in clausal form.
So, we get C
n
P
.
If α
n
is different to 1, then the utility function
1 − µ(ω) is not normalized. In this case, it will be
necessary to add contradiction to the possibilistic
base C
n
P
with priority degree 1− α
n
. Let C
′
n
P
be this
base. Then C
n
P
and C
′
n
P
are equivalents.
Lemma 1. Let Σ = {(φ
i
,α
i
),i = 1,n} be a preferences
base and let α
1
,...,α
n
be the distinct valuations ap-
pearing in Σ, ranked increasingly : 0 ≤ α
1
,...,α
n
≤ 1
and let µ(ω) be the utility function associated to the
preferences base Σ. Let n
Σ
= {(ψ
i
,β
i
),i = 1,n} be
the preferences base associated to the utility function
µ
n
Σ
(ω) = 1− µ(ω). The base n
′
Σ
= n
Σ
∪{(⊥,1−α
n
)}
is equivalent to n
Σ
. We have:
∀ω ∈ Ω, µ
n
Σ
(ω) = µ
n
′
Σ
(ω).
3.2 Computation of Pessimistic
Decisions
We recall that:
u
∗
(d) = 1 − max
ω∈Ω
{π
min
(ω)} (9)
where π
min
(ω) = min(π
K
d
(ω)
,1 − µ(ω)) and Σ
min
=
K ∪C
n
P
∪ {(d,1)}.
On the other hand, the inconsistency degree of a pos-
sibilistic base K, Inc(K) is defined as follow (Dubois
et al., 1994b):
Inc(K) = 1− max{π
K
d
(ω)} (10)
Proposition 2. Then clearly, the pessimistic utility
function associated to decision d is :
u
∗
(d) = Inc(Σ
min
)
Where Inc(Σ
min
) represents the inconsistency degree
of the base K ∪C
n
P
∪ {(d,1)}.
Then, the computation of optimal pessimistic deci-
sions is obtained using the following algorithm.
Algorithm: Computation of Optimal Pessimistic
Decisions
Input : K:knowledge base,
n
P
:revers preferences base,
N:number of decision variables,
D:set of decisions,
Output : Decision: optimal decisions,
Begin
i := 1;
max := 0;
Inc := 1;
For i=1 to N do
Begin
Inconsi(K ∪ C
n
P
∪ {(d
i∈[1,n]
,1)},Inc,bool);
/* d
i
∈ D*/
if (bool=true) then
if(Inc > max)then
max := Inc;
Decision := {d
i
};
else
ICSOFT 2008 - International Conference on Software and Data Technologies
358

if(Inc=max)then
Decision := Decision∪ {d
i
};
endif;
endif
endif
end
return < Decision >;
end
The computation of inconsistency degree is per-
formed by a call to the function Inconsi(B ∪
{(¬φ,1)},Inc,bool). This function has three parame-
ters: a stratified knowledge base, an integer represent-
ing current inconsistency degree and a boolean vari-
able. More precisely, the function Inconsi is defined
as follows:
Function Inconsi(B ∪ {(¬φ,1)},Inc,bool) Input :
B:stratified base,
φ:weighted formula,
n: number of strate in base B,
Output :Inc: inconsistency degree,
bool:boolean,
Begin
l := 0; /*initially pointed on the last strate of the
base*/
u := n; /*initially pointed on first strate of the
base*/
bool := true;
while (l < u) do
Begin
r := [(l + u)/2];
/*pointer uses for dichotomy*/
if(B
∗
≥α
r
∧ ¬φ consistent)
/*B
∗
≥α
r
= {φ
i
/α
i
≥ α
r
}*/
then
u := r − 1;
/*check inconsistency in most big base*/
else
l := r;
/*check the inconsistency delimited by u,l*/
endif
end
if(α
r
< inc) then bool := false;
else Inc := α
r
; /*Inc = N(φ)*/
endif
return < Inc,bool >
end
4 CONCLUSIONS
The main contribution of this paper is a proposition of
a new approach to compute a qualitative pessimistic
decision problem. This problem is viewed as the
one of computing inconsistency degrees of particu-
lar bases in the framework of possibilistic logic. The
application exploits the syntactic counterparts of data
fusion techniques. Our approach avoids the use of the
ATMS to compute the pessimistic optimal qualitative
decision developed in (Dubois et al., 1999) which is
known to be a hard problem.
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DECISION
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