A trisecting-and-acting model explains three-way decisions (3WD) in terms of two basic tasks. One task is to divide a universal set into three pair-wise disjoint regions called a trisection or a tri-partition of the universal set. The other task is to act upon objects in one or more regions by developing appropriate strategies. 3WD are a class of effective ways and heuristics commonly used in human problem solving and information processing. We argue that 3WD are built on solid cognitive foundations and offer cognitive advantages and benefits. We demonstrate the flexibility and general applicability of 3WD by using examples from across many fields and disciplines.

Three-Way Decisions and Cognitive Computing

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Fuzzy sets and systems

In situations with incomplete information, we may have only partial knowledge about a concept. This motivates the introduction of the notion of a partially-known concept represented by a set of known instances, a set of known non-instances, and a set of objects with unknown states. We present a common conceptual framework of the notions of interval sets and incomplete formal contexts for representing partially-known concepts. An interval set is interpreted as a family of sets bounded by a pair of sets, and any one in the family may possibly be the actual set of instances of the concept when the information or knowledge becomes complete. An incomplete formal context is interpreted as a family of complete formal contexts, and any one in the family may possibly be the actual formal context when the information or knowledge becomes complete. While a complete formal context is induced by a binary relation, an incomplete formal context is induced by an interval binary relation that is interpreted as a family of binary relations. Within the proposed framework, we identify four possible forms for representing partially-known concepts. We examine, interpret, and extend existing studies on concept analysis in complete formal contexts.

Interval sets and three-way concept analysis in incomplete contexts

Three-way decisions approach to multiple attribute group decision making with linguistic information-based decision-theoretic rough fuzzy set

Recently, three-way concept lattice is studied to handle the uncertainty and incompleteness in the given attribute set based on acceptation, rejection, and uncertain regions. This paper aimed at analyzing the uncertainty and incompleteness in the given fuzzy attribute set characterized by truth-membership, indeterminacy-membership, and falsity membership functions of a defined single-valued neutrosophic set. For this purpose a method is proposed to generate the component wise three-way formal fuzzy concept and their hierarchical order visualization in the fuzzy concept lattice using the properties of neutrosophic graph, neutrosophic lattice, and Godel residuated lattice with an illustrative example.

Three-way fuzzy concept lattice representation using neutrosophic set

Incomplete contexts are a kind of formal contexts in which the relationship between some objects and some attributes is unavailable or lost. Knowledge discovery in incomplete contexts is of interest because such databases are frequently encountered in the real world. This paper mainly focuses on two issues: approximate concept construction with three-way decisions and attribute reduction in incomplete contexts. The theory of three-way decisions is formulated based on the notions of acceptance, rejection and non-commitment. It is an extension of the commonly used binary-decision model with an added third option. Based on three-way decisions, we propose two models to construct approximate concepts in incomplete contexts, and the equivalence of the two is revealed. To simplify the representation of the approximate concept lattices, we further present the attribute reduction approaches.

Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts

The classical concept lattices express the precise relation between object sets and attribute sets, but fuzzy concept lattices express the uncertain relation between object sets and attribute sets. Therefore, it is important to study hierarchy fuzzy knowledge from a fuzzy formal context. In this paper, a kind of fuzzy decision formal context is proposed and (α, β) reduct based on this fuzzy decision formal context is defined. Furthermore, we propose a method to judge attribute consistent sets and reducts in fuzzy decision formal contexts. Finally, a Boolean method is also formulated to attribute reduction in fuzzy decision formal context from the view of the discernibility matrix.

Attribute reduction in fuzzy decision formal contexts

Concept lattice is an effective tool for knowledge discovery and has been applied to many fields successfully. The objects and attributes have clear relations in the classical formal context, but there are a lot of fuzzy information in real-life applications. Therefore, it is important to study the fuzzy formal context. As we know, the fuzzy set and the classical set can be transformed into each other based on the cut set. Thus, it is believed that the cut set can be used in the research of fuzzy concept lattice. In this paper, a preliminary exploration to use the cut set is carried out. In particular, the cut set is employed to construct the discernibility matrix of L-fuzzy formal context and then all the reducts can be obtained via the use of discernibility function. After that, all the attributes are classified into three types by their significances in constructing the L-fuzzy concept lattice. The characteristics of these types of attributes are also analyzed. The obtained results in this paper are beneficial for the research of fuzzy formal context based on the cut set in further.

On the use of cut set for attribute reduction in L-fuzzy concept lattice

A simple discernibility matrix for attribute reduction in formal concept analysis based on granular concepts

Knowledge reduction is a basic issue in knowledge representation and data mining. Al- though various methods have been developed to reduce the size of classical formal contexts, the reduction of formal fuzzy contexts based on fuzzy lattices remains a difficult problem owing to its complicated derivation operators. To address this problem, this paper proposes a method of knowl- edge reduction by reducing attributes in a formal fuzzy context based on the crisply generated fuzzy concept lattice. Employing the proposed approach, attributes which are non-essential to the struc- ture of the crisply generated fuzzy concept lattice are removed. Discernibility matrix and Boolean function are employed to compute the attribute reducts of the formal fuzzy contexts, by which all the attribute reducts of the formal fuzzy contexts are deter mined without changing the structure of the lattice. Further, all the attributes are classified into three types by their significance in construct- ing the crisply generated fuzzy concept lattice. The characteristics of these types of attributes are also analyzed. Finally, the proposed method is used to conduct knowledge reduction in the variable threshold concept lattices, which is a complement to the existing knowledge reduction methods.

Mei-Zheng Li

Papers

Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts

Attribute reduction in fuzzy decision formal contexts

On the use of cut set for attribute reduction in L-fuzzy concept lattice

A simple discernibility matrix for attribute reduction in formal concept analysis based on granular concepts

Knowledge Reduction in Crisply Generated Fuzzy Concept Lattices