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has distinct extensions E1 and E2 such that E'1⊆ E1 and E'2⊆ E2.
2.6 Circumscription Logic
Circumscription logic (CIRC) is proposed by McCarthy for nonmonotonic
reasoning. The basic idea of circumscription logic is that “the objects that can be
shown to have a certain property P by reasoning from certain facts A are all the
objects that satisfy P” (McCarthy, 1980). During the process of human informal
reasoning, the objects that have been shown to have a certain property P are often
treated as all the objects that satisfy P; such a treatment will be used in the further
reasoning and will not be revised until other objects are discoveried to have the
property P. For example, it is ever guessed by the famous mathematician Erdos
that the mathematical equation xxyy = zz has only two trivial solution: x=1, y=z
and y=1, x=z. But later it was proved by Chinese mathematical Zhao He that this
mathematical equation has infinite number of trivial solutions and therefore
overthrew Erdos's guess.
Circumscription logic is based on minimization. In the following, starting
with a propositional circumscription which is based on minimal model, we first
introduce basic definitions of circumscription. Then we will introduce some basic
results on predicate cirsumscription.
Definition 2.12
Let p 1 p 2 be two satisfying truth assignments for a propositional
language L 0 . Then p 1 is called smaller than p 2 , written as p 1 X p 2 , if and only if
p 2 (x)=l for any proposition x which holds p 1 (x)=l.
Definition 2.13
Let p be a satisfying truth assignment of a formula A. We say
that p is a minimal satisfying assignment of A if and only if there is no other
satisfying truth assignment p' of A such that p' X p.
Definition 2.14
A formula B is called a minimal entailment of a foumula A,
written as A
ż
M B, if and only if B is true with respect to any minimal model of
A.
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