Logic Foundation of Artificial Intelligence - Advanced Artificial Intelligence - page 69

(1) ɂ ŗ ϕ if and only if ◇ ɂ ⊆ ◇ ϕ ;

(2){ ɂ , ϕ } is satisfiable if and only if ◇ ɂ ◇ ϕ ∅ .

Now, from the point of set theory, we can redefine the semantics of
autoepistemic logic according to the following theorem.

Theorem 2.9 Let ɂ and ϕ be objective formulas, and let W,w be a model with

L L

W ⊆ 2 and w ∈ 2 . Then:

(1) W,w ɂ iff w ∈ ◇ ɂ ;

(2) W,w ɂ ∧ ϕ iff W,w ɂ and W,w ϕ ;

ɂ § ◇ ɂ

(3) W,w ¬ iff W ;

(4) W,w B ɂ iff W ⊆ ◇ ɂ .

Next we introduce the O -property.
Definition 2.24

ɂ

Let be a basic formula which is represented in the disjunctive

normal form i (1 ≤ i ≤ k) is of the form B ϕ i,1

∧ ··· ∧ B ϕ i,mi ∧ ¬ B ϕ i,1 ∧ ··· ∧ ¬ B ϕ i,ni ∧ ϕ ii with

ɂ ɂ 2 ∨ … ∨ ɂ k , where each ɂ

1 ∨

ɂ

ii an objective formula. Let

J be a subset of {1, ···, k}. We say that J has the O -property if and only if the

following conditions hold:

(1) ∪ j ∈ J ◇ jj _ B ϕ r ,1 ∧ ··· ∧ B ϕ r,mr ∧ ¬ B ϕ r ,1 ∧ ··· ∧ ¬ B ϕ rnr for each r ∈ J , and

(2) ∪ j ∈ J ◇ jj e B ϕ t,1 ∧ ··· ∧ B ϕ t,mt ∧ ¬ B ϕ t,1 ∧ ··· ∧ ¬ B ϕ t,nt for each t ∉ J.

The O -property of J can be decided according to the following two
approaches:
Lemma 2.4 -property if and only if the

following conditions hold (here ◇ J is the abbreviation of ∪ j ∈ J ◇

(The set theory approach) J has the O

ɂ jj ):