Propositional Argumentation Systems and Symbolic Evidence Theory

66 5 KNOWLEDGE PROPAGATION IN VALUATION NETWORKS
(3) n is contained in every node along the path from E i to E j in the case
of E i , E j being two distinct elements of ℵ and n being within E i and
E j .
If (ℵ, Λ) is a Markov tree, then it is also called Markov tree representative
for the hypertree for ℵ. Every hypertree has at least one Markov tree
representative, but most of them have more than one. Figure 5.4 shows a
Markov tree representative derived from the right hypertree of Figure 5.3.
The symbols between the nodes of the Markov tree indicate the common
variables of the corresponding nodes. It is easy to prove that the tree from
Figure 5.4 satisfies the Markov tree conditions.
Figure 5.4: A Markov tree representative.
5.2 Valuations and Local Computation
In valuation networks, knowledge is represented by entities called valuations.
The term valuation is very general and abstract. A valuation can
be regarded as a piece of information or knowledge about certain questions.
Let X = {X 1 , . . . , X r } be a set of variables representing questions to be considered.
We denote by V x a valuation representing some information about
a subset x ⊆ X of variables. x is called domain of V x . Symbolic hints (see
Chapter 3) and assumption–based systems (see Chapter 4) are examples of
valuations. Other applications of valuations are Bayesian probability distributions,
Dempster–Shafer’s and Spohn’s epistemic belief functions, Zadeh’s
possibility functions, database relations, fuzzy relations, propositional or
algebraic formulae, etc.