Logic Foundation of Artificial Intelligence - Advanced Artificial Intelligence - page 63

(1) ¾ A can not be deduced from ¾ (A∧B); and;

(2) What can be deduced from { ¾ P ã Q, ¬Q} is surprising.

In order to overcome these problems, McDermott and Doyle introduced
another modal operator Ʈ called necessity. The relationship between ¾ and Ʈ is as
follows:

Ʈ P ≡ ¬ ¾ ¬P
¾ P ≡ ¬ ¬P

Here the first definition states that P is necessary if and only if its negation is
incompatible; the second definition states that P is compatible if and only if its
negation is not necessary.

Ʈ

2.8 Autoepistemic Logic

2.8.1 Moore System ќ B

Autoepistemic logic was proposed by Moore as an approach to represent and
reason about the knowledge and beliefs of agents (Moore, 1985). It can be treated
as a modal logic with a modal operator B which is informally interpreted as
“believe” or “know”. Once the beliefs of agents are represented as logical
formulas, then a basic task of autoepistemic logic is to describe the conditions
which should be satisfied by these formulas. Intuitively, an agent should believe
these facts that can be deduced from its current beliefs. Furthermore, is an agent
believe or do not believe some fact, then the agent should believe that it believe
or do not believe this fact.
An autoepistemic theory T is sound with respect to an initial set of premises
A if and only if every autoepistemic interpretation of T in which all the formulas
of A are true is an autoepistemic model of T. The beliefs of an ideally rational
agent should satisfy the following conditions:

(1) if P 1 ··· P n ∈ T, and P 1 ··· P n ũ Q then Q∈T, (where ũ means ordinary
tautological consequence).