our third, recursive, algorithm, since this approach re-
duces the caching by deﬁnition. But many other as-
pects of this implementation have to be investigated,
in term of performance. Typically, the structure of the
set of combinations should be considered for optimiz-
ing the strategies of the computation ﬂow. Moreover,
an incremental computation of the combinations, may
be also investigated through computation ﬂows more
complex than map-reduce. From this viewpoint, the
reactivity of this parallel computation on possibly
complex single combinations is also a piece of per-
formance to be evaluated precisely or optimized in the
future, in regards to non-parallel approaches.
5 CONCLUSIONS
In this paper, we proposed a generic distributed
processing approach for computing belief combi-
nation rules. The approach is based on a map-
reduce paradigm, and has been implemented in
scala/SPARK. It is derived from the concept of referee
function, introduced in a previous work with the aim
of separating the deﬁnition of the combination rule
from its actual implementation. This work has been
completed by the proposal of a new recursive formal-
ism for the deﬁnition of the rules, and an improved
map-reduce generic implementation of the rules pro-
cessing. Some tests have been made for the rule of
Dubois & Prade, which illustrated this computation
improvement. More tests will be investigated in the
future. Moreover, our intention is to extend this work
to general data ﬂow paradigms for computation.
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APPENDIX
A Dubois & Prade Rule
Dubois and Prade reﬁned the conjunctive rule by re-
distributing disjunctively the conﬂict:
m
1
⊕
DP
m
2
(X) =
∑
Y
1
,Y
2
:
n
Y
1
∩Y
2
6=
/
0
Y
1
∩Y
2
=X
m
1
(Y
1
)m
2
(Y
2
)
+
∑
Y
1
,Y
2
:
n
Y
1
∩Y
2
=
/
0
Y
1
∪Y
2
=X
m
1
(Y
1
)m
2
(Y
2
) .
Map-reduce Implementation of Belief Combination Rules
149