Propositional Argumentation Systems and Symbolic Evidence Theory

4.4 Representing and Computing Symbolic Arguments 49
tion, i.e. a conjunction with zero literals, represents the tautology ⊤. C A
is a partially ordered set with the logical entailment |= as its ordering relation.
If A contains r elements, then C A contains 3 r different conjunctions.
Figure 4.2 shows the partially ordered set C A for A = {a 1 , a 2 }.
Figure 4.2: Example of a partially ordered set.
The implicants of an arbitrary formula f ∈ L A form a subset F ⊆ C A . If
f, for example, is the quasi–support of a hypothesis h ∈ L N , then F forms
the set of all supporting arguments of h. By µF we denote the subset of
conjunctions in F , which subsume no other conjunction of the set, i.e. which
are minimal within the set, and obviously we have µF = Ψ(f). In the
following µF is used to represent symbolic arguments.
Figure 4.3 shows the subset F ⊆ C A for the symbolic argument f = a 1 ∨ a 2 .
F is entirely determined by the minimal conjunctions a 1 and a 2 , i.e. µF =
Ψ(f) = {a 1 , a 2 }.
Each possible symbolic argument f ∈ L A (or [f] ∈ [L A ]) has a corresponding
set µF ⊆ C A of minimal conjunctions, and, inversely, each set µF of
minimal conjunctions represents a symbolic argument. Furthermore, a conjunction
of literals of assumptions can also be regarded as a set: the set of
its literals. Therefore, the entire theory of assumption–based systems can
also be developed in terms of sets of literals and set–operations, rather than
in terms of logical formulae and logical operations.
The minimal elements µF of a subset F ⊆ C A can also be interpreted in
the following sense. Let a ∈ C A be an arbitrary conjunction of literals of
different assumptions. L(a) denotes the corresponding set of literals. An
assumption a i ∈ A is called positive relative to a, if a i ∈ L(a); it is called
negative relative to a, if ∼a i ∈ L(a); and it is called irrelevant relative