Three-way decision with incomplete information based on similarity and satisfiability
Junfang Luo, Mengjun Hu, Keyun Qin
TL;DR
This work extends three-way decision in rough set theory from complete to incomplete information by parallel generalizations of two core formulations: a computational one based on similarity degrees and a conceptual one grounded in satisfiability degrees. It introduces two computational pathways—$\alpha$-similarity classes and object approximability—and two conceptual pathways—$\alpha$-meaning sets and formula confidence—each yielding interpretable, rule-based three-way decisions under incomplete data. The paper formalizes set-valued representations of incomplete tables, defines quantifiable similarity and satisfiability measures using $T$-norms and fuzzy implications, and demonstrates how to derive acceptance, rejection, or non-commitment rules from description regions. The proposed framework offers new directions for learning and interpreting three-way decision rules in practical settings with missing or partial information and suggests future work on thresholds, redundancy reduction, and applicability to varied incomplete-information semantics.
Abstract
Three-way decision is widely applied with rough set theory to learn classification or decision rules. The approaches dealing with complete information are well established in the literature, including the two complementary computational and conceptual formulations. The computational formulation uses equivalence relations, and the conceptual formulation uses satisfiability of logic formulas. In this paper, based on a briefly review of these two formulations, we generalize both formulations into three-way decision with incomplete information that is more practical in real-world applications. For the computational formulation, we propose a new measure of similarity degree of objects as a generalization of equivalence relations. Based on it, we discuss two approaches to three-way decision using alpha-similarity classes and approximability of objects, respectively. For the conceptual formulation, we propose a measure of satisfiability degree of formulas as a quantitative generalization of satisfiability with complete information. Based on it, we study two approaches to three-way decision using alpha-meaning sets of formulas and confidence of formulas, respectively. While using similarity classes is a common method of analyzing incomplete information in the literature, the proposed concept of approximability and the two approaches in conceptual formulation point out new promising directions.