Constructing shadowed sets and three-way approximations of fuzzy sets

Shadowed sets, proposed by Pedrycz, are an example of three-way approximations of fuzzy sets. A fuzzy set is approximated by elevating membership grades at or above one threshold to 1, reducing membership grades at or below another threshold to 0, and mapping membership grades between the two thresholds to the unit interval [0, 1]. A fundamental issue in such a construction process of three-way approximations is the interpretation and determination of a pair of thresholds on the unit interval [0, 1]. In this paper, we adopt a generalized definition of three-valued sets by using a set of three values {n, m, p} to replace {0, [0, 1], 1}. We introduce an optimization-based framework for constructing three-way approximations. Within the framework, we critically review existing studies and results and present new formulations according to three principles, i.e., a principle of uncertainty invariance, a principle of minimum distance, and a principle of least cost. Finally, we propose a least-cost model based on a semantic distance function between membership grades in [0, 1] and values in {n, m, p}.