Propositional Argumentation Systems and Symbolic Evidence Theory

44 4 ASSUMPTION–BASED SYSTEMS
the same, which in general is not the case.
Example 4.4 (Sequence of Binary Inverters): Let us now study how symbolic
arguments can be helpful in the simple diagnostics problem of Example 4.2. First,
we may be interested in sets of components which, intact or faulty, explain the
observed behaviour of the system. Such explanations are called diagnoses. They
can be represented by conjunctions of literals of the assumptions ok 1 and ok 2 . Here,
the diagnoses are given by the support of the tautology,
sp[⊤] = [(ok 1 ∧ ∼ok 2 ) ∨ (∼ok 1 ∧ ok 2 )]. (4.18)
In this example, it is easy to see to that ok 1 ∧ ∼ok 2 and ok 1 ∧ ∼ok 2 are the only
possible diagnoses. If a system of two inverters connected in series produces 1 for
an input 0, it is clear that exactly one component is faulty and the other is intact.
It may also be interesting to know sets of possibly intact or faulty components,
which are in contradiction to the system description and the given observations.
Such sets are called conflicts, and they are given by the quasi–support of the
contradiction:
qs[⊥] = ∼sp[⊤] = [(ok 1 ∧ ok 2 ) ∨ (∼ok 1 ∧ ∼ok 2 )]. (4.19)
Obviously, ok 1 and ok 2 can neither be true nor false at the same time. Therefore,
it is clear that ok 1 ∧ ok 2 and ∼ok 1 ∧ ∼ok 2 are conflicts, and that in this example,
these are the only conflicts.
4.4 Representing and Computing Symbolic Arguments
In the previous section we defined an evidence theory based on propositional
logic. It assigns arguments (represented by formulae in L A ) to hypotheses
(represented by formulae in L N ) in the sense of the general evidence theory.
Although all results of Chapter 2 remain valid for assumption–based systems,
there are still some important questions to be answered, for example:
(1) How are arguments obtained from the knowledge base and the given
hypothesis?
(2) How can arguments be represented or stored efficiently?
(3) Is there a way to store efficiently all arguments for all possible hypotheses?