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Furthermore, according to these rules, Γ is inconsistent if Th(Γ) does not
exist. The definition of Th(Γ) presented in (8) can also be rewrited as
follows:
(9) Th(Γ) = {P | Γ |∼ P}
where Γ |∼P represent P∈Th(Γ). We also use FP(Γ) to denote the set {S |
NMΓ(S)=S } and call each element of this set as a fixed point of the theory Γ.
There are three major schools on nonmonotonic reasoning: the
circumscription theory proposed by McCarthy, the default logic proposed by
Reiter, and the autoepistemic logic proposed by Moore. In the circumscription
theory, a formula S is true with respect to a limited range if and only if S cannot
be proved to be true w.r.t. a bigger range. In the default logic, “a formula S is true
in default” means that “S is true if there is no evidence to prove the false of S”. In
the autoepistemic logic, S is true if S is not believed and there are no facts which
are inconsistent with S.
Various nonmonotonic logic systems have beed proposed by embracing the
nonmonotonic reasoning into formal logics. These nonmonotonic logics can be
roughly divided into two categories: nonmonotonic logics based on minimization,
and nonmonotonic logics based on fixed point. Nonmonotonic logics based on
minimization can again be devided into two groups: one is these based on the
minimization of model, such as the logic with the closed world assumption and
the circumscription proposed by McCarthy, and the other is these based on the
minimization of knowledge model, such as the ignorance proposed by Konolige.
Nonmonotonic logics based on fixed point can be devided into default logics and
autoepistemic logics. The nonmonotonic logic NML proposed by McDermott
and Doyle is a general default logic and was used for study the general
foundation of nonmonotonic logics, and the default logic proposed by Reiter is a
first-order formalization of default rules. Autoepistemic logic was firstly
proposed by Moore to solve the so-called Hanks-McDermott problem on
nonmonotinic logics.
2.4 Closed World Assumption
With respect to any base set KB of beliefs, the closed world assumption (CWA)
provides an approach to complete the theory T(KB) which is defined by KB.
