Propositional Argumentation Systems and Symbolic Evidence Theory

40 4 ASSUMPTION–BASED SYSTEMS
Formulae R 1 to R 6 can easily be transformed into a set Σ = {ξ 1 , . . . , ξ 10 } of clauses.
This forms the entire knowledge base that can be used to solve the diagnostics
problem (see Example 4.4 of Section 4.3).
ξ 1 : ∼ok 1 ∨ ∼in ∨ ∼v ξ 6 : ∼ok 2 ∨ ∼v ∨ ∼out
ξ 2 : ∼ok 1 ∨ in ∨ v ξ 7 : ∼ok 2 ∨ v ∨ out
ξ 3 : ok 1 ∨ ∼in ∨ v ξ 8 : ok 2 ∨ ∼v ∨ out
ξ 4 : ok 1 ∨ in ∨ ∼v ξ 9 : ok 2 ∨ v ∨ ∼out
ξ 5 : ∼in ξ 10 : out
4.3 Hypotheses and Arguments in Assumption–Based Systems
Given an assumption–based system A = (P, A, Σ), we may be interested in
certain hypotheses h about the knowledge contained in A. Such hypotheses
can be expressed as logical formulae in L N . What can we learn from Σ about
the possible truth of h? If some assumptions are considered to be either true
or false, then h can possibly be deduced from Σ. Such combinations of true
and false assumptions or, more generally, logical formulae which are only
composed by assumptions can be regarded as arguments in favour of h.
This is the link to the general evidence theory of Chapter 2: the possible
hypotheses are logical formulae in L N , and the possible arguments are logical
formulae in L A . More precisely, the finite Boolean algebras H of hypotheses
and A of arguments correspond to the sets [L N ] and [L A ] of equivalence
classes of formulae in L N and L A respectively.
In view of these remarks let us consider logical formulae a ∈ L A , such that
a ∧ ξ |= h, where ξ is the conjunction of the clauses contained in Σ, and
h ∈ L N represents a hypothesis. Such formulae a allow to deduce h from
Σ. They are called supporting arguments for h relative to Σ. If a is the
coarsest (least precise) supporting argument of h, i.e. if there is no other
supporting argument a ′ of h with a ′ |= a and a ′ ≠ a, then the equivalence
class [a] of a is called quasi–support of h relative to Σ, denoted by qs[h, Σ]
(instead of [qs(h, Σ)]). It can be shown (Kohlas & Monney, 1995) that
quasi–support satisfies the conditions (Q1) to (Q4) from Chapter 2,
(Q1) qs[⊤, Σ] = [⊤],
(Q2) qs[h 1 ∧ h 2 , Σ] = qs[h 1 , Σ] ∧ qs[h 2 , Σ],
(Q3) if h 1 |= h 2 , then qs[h 1 , Σ] |= qs[h 2 , Σ],