In order to solve the errors of computing the core and attribute reduction which caused by the incompatibility
of decision table, the inconsistent decision tables are converted into consistent decision tables through adding a condition
attribute. Then a definition of the discernibility matrix is provided, and the properties of the core and attribute reduction are
proposed. It is proved that the core and attribute reduction acquired from the properties are the core and attribute reduction
based on positive region. Finally, the general algorithm and its improved algorithm for computing attribute reduction are
designed, and the example is used to explain the correctness of the method.