Reduct and attribute order

Based on the principle of discernibility matrix, a kind of reduction algorithm with attribute order has been developed and its solution has been proved to be complete forreduct and unique for a given attribute order. Being called thereduct problem, this algorithm can be regarded as a mappingR=Reduct(S) from the attribute order space⊝ to thereduct spaceR for an information system (U, C∪ D), whereU is the universe andC andD are two sets of condition and decision attributes respectively. This paper focuses on the reverse problem ofreduct problemS=Order(R), i.e., for a givenreduct R of an information system, we determine the solution ofS=Order(R) in the space⊝. First, we need to prove that there is at least one attribute orderS such thatS=Order(R). Then, some decision rules are proposed, which can be used directly to decide whether the pair of attribute orders has the samereduct. The main method is based on the fact that an attribute order can be transformed into another one by moving the attribute for limited times. Thus, the decision of the pair of attribute orders can be altered to the decision of the sequence of neighboring pairs of attribute orders. Therefore, the basic theorem of neighboring pair of attribute orders is first proved, then, the decision theorem of attribute order is proved accordingly by the second attribute.