Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

proposed this logic and called it non-monotonic logic I (NML I). The logic,
which is an extension of Predicate Logic, includes a new operator, called the
consistency operator, symbolized by . To illustrate the importance of the
operator, let us consider the following statement.
∀ X Bird (X) ∧ Fly (X) → Fly (X)
which means that if X is a bird and if it is consistent that X can fly then infer
that X will fly.
It is to be noted that NML I has the capability to handle default
assumptions. Thus the Default Logic of Reiter [11], to be presented shortly, is
a special case of NML I.
Let us now attempt to represent the notion of the non-monotonic inferencing
mechanism by the consistency operator. We will consider that
T |~/ ¬ A
⇒ T |~ A,
which means that if ¬ A is not non-monotonically derivable from the
axiomatic theory T, then infer that A is consistent with any of the theorems
provable from T.
7.4 Fixed Points in Non-Monotonic Reasoning
To understand the concept on fixed points (also called fixpoints), let us
consider an example, following McDermott and Doyle [8].
Example 7.2: Let T be an axiomatic theory, which includes
P → ¬ Q and Q → ¬ P.
Formally, T = { P → ¬ Q , Q → ¬ P },
which means that if P is consistent with the statements of the axiomatic
theory T then¬Q and if Q is consistent with the statements in T then ¬ P.
McDermott and Doyle called the system having two fixed points ( P, ¬ Q)
and (¬P, Q). On the other hand, if T = { P→ ¬ P}, then there is no
fixed points in the system.