属性定向概念格的协调近似表示空间

Consistent approximate representation space for property oriented concept lattices

构造形式背景、协调决策形式背景和不协调决策形式背景的统一模型,并提出不同形式背景的概念格的属性约简的定义和方法.首先,定义对象幂集上的一种等价关系,并利用该等价关系给出属性定向概念格的协调近似表示空间的概念.其次,针对不同形式背景构造相应地协调近似表示空间.特别地,构造不协调决策形式背景的四类协调近似表示空间,分别为分布协调近似表示空间、最大分布协调近似表示空间、下协调近似表示空间和上协调近似表示空间.最后,给出协调近似表示空间的属性约简的定义,并利用可辨识矩阵给出属性约简集的判定定理.分析表明,形式背景和决策形式背景(包括协调的和不协调的)都可看作是协调近似表示空间的特殊形式,且形式背景的属性约简恰好对应协调近似表示空间的属性约简.因此,协调近似表示空间的属性约简方法可以在一定程度上降低由形式背景的复杂性带来的概念格的属性约简的复杂性.

Formal concept analysis is an effective tool for data analysis,knowledge discovery and information management.This paper mainly studies a unified structure for a formal context,a consistent formal decision context as well as an inconsistent formal decision context based on a property oriented concept lattice,and proposes a new approach to attribute reduction for property oriented concept lattices of different kinds of formal contexts.Using equivalence relations defined on the object power set,an approximate representation space is first introduced for a formal context,which is a quadruple including an object set,an attribute set,a family of equivalence relations on the object power set and an equivalence relation on the object power set.And then the notion of consistency of the approximate rep-resentation space is defined.Moreover,consistent approximate representation spaces are constructed corresponding to a formal context,a consistent formal decision context and an inconsistent formal decision context,respectively.In particular,four types of approximate representation spaces are defined for an inconsistent formal decision context,which are distribution approximate representation space,maximum approximate representation space,lower approximate representation space and upper approximate representation space.It is shown that a formal context,a consistent formal decision context and an inconsistent formal decision context are all special cases of consistent approximate representation spaces.Finally,a notion of attribute reduction is defined for consistent approximate representation spaces,and an approach to attribute reduction to calculate all attribute reducts is presented by means of discernibility matrices.It is proved that attribute reduction of six types of consistent approximate representation spaces are just corresponding to attribute reduction of a formal context,a consistent formal decision context and an inconsistent formal decision context.The result indicates that attribute reduction of consistent approximate representation space can be regarded as the unified approach to attribute reduction of different kinds of formal contexts,which can reduce the complexities of attribute reduction produced by differences of formal contexts.