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11.5.3 Rough Set Based Nonmonotonic Logic
Since the development of rough set theory, its researchers thought much of its
logic, and they attempted to construct rough logic with the help of rough sets. A
lot of papers in this field were published. For example, Z. Pawlak published a
paper titled by “Rough Logic” in 1987. In this paper, he gave the semantic
explanation of rough logic formulas: true, false, rough true, rough false and
rough negative. The five values can be viewed as different degrees of similarity,
but they are lack of mathematical description. Z. Pawlak and etc regarded rough
logic—rough set based imprecise inference logic—as the most important topic in
their review paper published by Communication of The ACM (Pawlak,1995).
T. Y. Lin, Q. Liu and etc defined rough lower approximate operator L and
rough upper approximate operator H based on topology. Semantic properties of
the two operators are very similar with the necessity operator Ʈ and possibility
operator in modal logic. Therefore, the logic formulas with L and H operators
are called as rough logic formulas. Moreover, they constructed the axiomatized
rough logic deductive system, which is similar with mode logic, and parallel
deductive rules. However, because for L and H, the rough logic defined is vague,
it cannot be explained by mathematic language. Although the work had some
shortages, it gave a research direction “ Approximation Proof ”, which is giving
the mathematical meanings of L and H. So that, the logic formulas constructed
by L and H can have mathematical meanings. Furthermore, Q. Liu and etc
defined similar degrees ȹ * and ȹ * based on rough set theory. The degrees were
composed with professional fields based imprecise numbers and experiential
numbers, and then rough numbers were generated. They also discussed the
properties of rough logic and the values of ȹ [ ȹ * , ȹ * ] in the explanation of logic
formulas. Besides, Q. Liu proposed an accuracy operators based rough logic
(AORL) in the fifth international conference of rough set held in Japan in 1996,
and presented the process of resolution reasoning.
A. Nakamura and etc proposed incomplete information system based rough
mode logic R 1 and
1 is based on some equivalence relation
of incomplete information systems. The work presented some properties and the
decision process of
R
2 . The main idea of
R
R 1 and an axiomatized deduction system. The completeness
and correctness of
is mainly based on algebraic structure of
interval sets. The work proposed the definitions of incomplete information
system's mode operations and the decision process of
R 1 were proved.
R 2
R
2 , which was different
with those of
R
1 . Moreover, the reduction of
R
2 axoimatized system was explored.
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