Propositional Argumentation Systems and Symbolic Evidence Theory

60 4 ASSUMPTION–BASED SYSTEMS
hint is equivalent to the given assumption–based system, i.e.
A ≡ H = ⊗H i = (N A , N P , ⊗Γ i ). (4.60)
4.7.2 From Hints to Assumption–Based Systems
Transforming a given hint into an assumption–based system is always possible,
but not always straightforward. Let n P and n A be the numbers of
different propositions and assumptions respectively used in an assumption–
based system, n = n P + n A . In this case 2 2n is the number of possible
hypotheses h ∈ H, and 2 2n A
is the number of different arguments a ∈ A.
A hint with n Ω different interpretations in Ω and n Θ different answers in Θ
has 2 n Θ
possible hypotheses and 2 n Ω
arguments. Thus, it is often impossible
to define the sets P of propositions and A of assumptions so that 2 2n = 2 n Θ
and 2 2n A
= 2 n Ω, with n ∈ IN and n A ∈ IN. In addition, no direct link exists
between the elements of sets generated by A and Ω or P and Θ respectively
like in the case described in the previous subsection. However, there is still
an intuitive way to transform hints into assumption–based systems.
Let H = (Ω, Θ, Γ) be the given symbolic hint. If we set A = Ω and P = Θ,
then we can construct the following clauses for each interpretation ω i ∈ Ω:
ω i → (∨{θ ∈ Γ(ω i )}), (4.61)
ω i → ∼θ, ∀θ /∈ Γ(ω i ). (4.62)
It also works if the ω i are conjunctions of interpretations obtained from
a previous combination of two or more hints. The fact that exactly one
interpretation in Ω is the correct interpretation can be described by ξ(Ω),
the exclusive–or of all interpretations in Ω (see Section 3.2):
ξ(ω) = ⊕{ω i : ω i ∈ Ω}. (4.63)
The knowledge base Σ consists of the clauses obtained from (4.61), (4.62),
and (4.63). The triple (P, A, Σ) forms an assumption–based system A that
represents the information from the given symbolic hint H. Thus, A and H
are equivalent and we can write
H ≡ A = (Ω, Θ, Σ). (4.64)
Note that interpretations ω i are not independent assumptions. This is an
important remark only if probabilities are assigned to the assumptions (see
Section 2.6) and if independence between the assumptions is assumed.