Propositional Argumentation Systems and Symbolic Evidence Theory

108 7 APPLICATIONS
produced, can be modeled as m 1 = ∼out 1 . Another possible failure mode,
for example, is the case in which the opposite of the correct output is always
produced. This can can be expressed by m 2 = out 1 ↔ ∼(in 1 ∧ in 2 ).
Thus, a component C with k failure modes leads to a set {m 0 , m 1 , . . . , m k }
of logical formulae. m 0 represents the normal behaviour, and m 1 to m k the
failure modes. At any given moment, exactly one failure mode is present, but
it is unknown which. This uncertainty can be modeled through assumptions.
For that purpose, there are two different approches:
(1) For each mode i a corresponding assumption a i is introduced. a i means
“component C is in mode i”. Then, the entire component can be
described as follows (⊕ denotes the logical exclusive–or):
a 0 → m 0 , a 1 → m 1 , . . . , a k → m k ,
a 0 ⊕ a 1 ⊕ · · · ⊕ a k . (7.15)
The last expression represents the fact that only one mode can be
present at any given moment. Note that the exclusive–or can also be
written as collection of k 2 + k + 1 implications or clauses.
(2) Only k − 1 assumptions a 0 to a k−1 are introduced. a 0 means “component
C is in mode 0”, a 1 means “if component C is not in mode 0,
then it is in mode 1”, a 2 means “if component C is not in mode 0 and
not in mode 1, then it is in mode 2”, etc. This leads to the following
implications:
a 0 → m 0 ,
∼a 0 ∧ a 1 → m 1 ,
.
∼a 0 ∧ ∼a 1 ∧ ∼a 2 ∧ · · · ∧ a k−1 → m k−1 ,
∼a 0 ∧ ∼a 1 ∧ ∼a 2 ∧ · · · ∧ ∼a k−1 → m k . (7.16)
The strength of the first approach is that the meaning of the assumptions
a 0 to a k is intuitive and coherent. However, there are also two major disadvantages:
first, the growing number of clauses produced by the exclusive–or
enlarges the set of necessary clauses; secondly, the assumptions a 0 to a k
are not independent, which is an important restriction if probabilities are
introduced.