Partial Fuzzy Sets Research Articles

Solvability of systems of partial fuzzy relational equations revisited – a short note

Fuzzy inference systems have been widely investigated from different perspectives, including their logical correctness. It is not surprising that the logical correctness led mostly to the questions on the preservation of modus ponens. Indeed, whenever such a system processes an input equivalent to one of the rule antecedents, it is natural to expect the modus ponens to be preserved and the inferred output to be identical to the respective rule consequent. This leads to the related systems of fuzzy relational equations where the antecedent and consequent fuzzy sets are known values, the inference is represented either by the direct product related to the compositional rule of inference or by the Bandler-Kohout subproduct, and the fuzzy relation that represents the fuzzy rule base is the unknown element in the equations. The most important question is whether such systems are solvable, i.e., whether there even exists a fuzzy relation that models the given fuzzy rule base in such a way that the modus ponens is preserved.Partiality allows dealing with partially defined truth values which allows to deal with situations, where we cannot define the truth value for a given predicate. The partial logics have been recently extended to partial fuzzy logics and the partial fuzzy set theory has been developed. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe.This background leads naturally to the problem of solvability of partial fuzzy relational equations, which are equations with partial fuzzy sets in the role of antecedents and consequents and with partial fuzzy relation as the model of the given fuzzy rule base. Such a setting led to a recent publication that uncovers the solvability and even the shape of the solutions. In this contribution, we revisit the problem and consider a specific case that allows partiality only in the input. In other words, antecedents, as well as consequents expressing the knowledge, are fully defined. Thus, the model of the fuzzy rules is one of the standard and fully defined relations. We investigate what happens if the input differs from one of the antecedents only by a few undefined values. This mimics the situation when a description of an observed object misses a few values in its feature vector. We show that under specific conditions, we still may preserve the modified modus ponens, i.e., that the inferred output is identical with the fully defined consequent of the respective rule.

Preservation of properties of residuated algebraic structure by structures for the partial fuzzy set theory

This paper addresses the preservation of numerous essential properties of a residuated lattice structure in extended algebras for partial fuzzy set theory and partial fuzzy logics. The preservation includes the residuated lattice axioms, the identities narrowing the classes of the residuated lattices, and some well-known additional properties. In this paper, we consider nine algebras for partial fuzzy logics which incorporate handling undefined values in a bit different way. In particular, we consider the Bochvar, the Bochvar external, the Sobociński, the Kleene, the McCarthy, the Nelson, and the Łukasiewicz algebras, and two recently developed ones, namely the Lower estimation and the Dragonfly algebras. We summarize the obtained results in a comprehensible form which allows readers to easily check the information for the preserved and non-preserved properties in a certain partial algebraic structure. The resulting shape of the contribution is a sort of “atlas book” that aims at providing researchers with a comfortable and comprehensible form of an overview of the (non)preservation of fundamental properties of residuated structures.

Three-Way Fuzzy Sets and Their Applications (II)

Recently, the notion of a three-way fuzzy set is presented, inspired by the basic ideas of three-way decision and various generalized fuzzy sets, including lattice-valued fuzzy sets, partial fuzzy sets, intuitionistic fuzzy sets, etc. As the new theory of uncertainty, it has been used in attribute reduction and as a new control method for the water level. However, as an extension of a three-way decision, this new theory has not been used in multi-criteria decision making (MCDM for short). Based on the previous work, in this paper, we present rough set models based on three-way fuzzy sets, which extend the existing fuzzy rough set models in both complete and incomplete information systems. Furthermore, the new models are used to solve the issue of MCDM. Firstly, three-way fuzzy relation rough set and three-way fuzzy covering rough set models are presented for complete and incomplete information systems. Because almost all existing fuzzy rough set models are proposed under complete information, the new proposed models can be seen as a supplement to these existing models. Then, a relationship between the three-way fuzzy relation rough set and the three-way fuzzy covering rough set is presented. Finally, a novel method for the issue of MCDM is presented under the novel three-way fuzzy rough set models, which is used in paper defect diagnosis.

On solvability of systems of partial fuzzy relational equations

Systems of fuzzy relational equations belong to key mathematical problems related to the correct behavior of fuzzy systems, especially of fuzzy inference systems. Indeed, if we find a fuzzy relation that solves a given system of fuzzy relational equations, we found a model of a fuzzy rule base related to the given system that will behave correctly in the sense that it will preserve the fundamental modus ponens property. This problem with numerous results becomes absolutely unexplored as soon as we allow partiality in the system. Partial fuzzy sets have only partially defined membership degrees, i.e., for some elements, the membership degree is undefined, i.e., partial propositions have only partially defined truth-values. There are many origins from which we may obtain propositions with undefined truth-values, e.g., non-denoting or irrelevant terms, inconsistent (both true and false) propositions, or truth-values are simply unknown. Partial logics belong to classical topics in logic so, the extensions to partial fuzzy logic in recent years were not surprising. However, we should not dare to deal with partial fuzzy sets in fuzzy systems without any guarantee of the preservation of fundamental properties such as modus ponens. This article is focusing on filling this gap.

An extended three-way decision and its application in member selection

The three-way decision method is a kind of new developed uncertain decision theory that combines fuzzy sets and rough sets together in recent years. It divides a set of objects into three regions, called the acceptance, rejection and uncertain regions, respectively. However, all the current researc hes deal with the three regions in the same way and do not provide further analysis on the most important uncertain region. In this paper, we propose an extended three-way decision method, called partial fuzzy sets, which regard acceptance region and rejection region as certain so as to focus on the uncertain region. Both certainty and uncertainty are simultaneously considered, that is more consistent with human thinking in decision making. Then, we establish two operational algorithms, called approximate approach and enhanced approximate approach, respectively. The approximate approach considers that the group accepts the certain information as far as possible. Specially speaking, based on approximate approach, if other members in a group are all uncertain to a judgment except one or two, who accept or reject it surely, then the group would agree with the certain judgment, even that they might be the minority. On the base of approximate approach, enhanced approximate approach reduces the possible bias through removing the maximal and the minimal membership grades in the group evaluation before aggregation. Obviously, the proposed methods provide more effective ways for three-way decisions than the current researches by avoiding some redundant and unnecessary computations, especially for the case with abundant of alternatives. Two numerical examples are provided to illustrate the practicality and validity of the proposed methods.

Fuzzy set approaches to classification of rock masses

Rock mass classification is analogous to multi-feature pattern recognition problem. The objective is to assign a rock mass to one of the pre-defined classes using a given set of criteria. This process involves a number of subjective uncertainties stemming from: (a) qualitative (linguistic) criteria; (b) sharp class boundaries; (c) fixed rating (or weight) scales; and (d) variable input reliability. Fuzzy set theory enables a soft approach to account for these uncertainties by allowing the expert to participate in this process in several ways. Hence, this study was designed to investigate the earlier fuzzy rock mass classification attempts and to devise improved methodologies to utilize the theory more accurately and efficiently. As in the earlier studies, the Rock Mass Rating (RMR) system was adopted as a reference conventional classification system because of its simple linear aggregation. The proposed classification approach is based on the concept of partial fuzzy sets representing the variable importance or recognition power of each criterion in the universal domain of rock mass quality. The method enables one to evaluate rock mass quality using any set of criteria, and it is easy to implement. To reduce uncertainties due to project- and lithology-dependent variations, partial membership functions were formulated considering shallow (<200 m) tunneling in granitic rock masses. This facilitated a detailed expression of the variations in the classification power of each criterion along the corresponding universal domains. The binary relationship tables generated using these functions were processed not to derive a single class but rather to plot criterion contribution trends (stacked area graphs) and belief surface contours, which proved to be very satisfactory in difficult decision situations. Four input scenarios were selected to demonstrate the efficiency of the proposed approach in different situations and with reference to the earlier approaches.

Fuzzy set approaches to classification of rock masses

Rock mass classification is analogous to multi-feature pattern recognition problem. The objective is to assign a rock mass to one of the pre-defined classes using a given set of criteria. This process involves a number of subjective uncertainties stemming from: (a) qualitative (linguistic) criteria; (b) sharp class boundaries; (c) fixed rating (or weight) scales; and (d) variable input reliability. Fuzzy set theory enables a soft approach to account for these uncertainties by allowing the expert to participate in this process in several ways. Hence, this study was designed to investigate the earlier fuzzy rock mass classification attempts and to devise improved methodologies to utilize the theory more accurately and efficiently. As in the earlier studies, the Rock Mass Rating (RMR) system was adopted as a reference conventional classification system because of its simple linear aggregation. The proposed classification approach is based on the concept of partial fuzzy sets representing the variable importance or recognition power of each criterion in the universal domain of rock mass quality. The method enables one to evaluate rock mass quality using any set of criteria, and it is easy to implement. To reduce uncertainties due to project- and lithology-dependent variations, partial membership functions were formulated considering shallow (<200 m) tunneling in granitic rock masses. This facilitated a detailed expression of the variations in the classification power of each criterion along the corresponding universal domains. The binary relationship tables generated using these functions were processed not to derive a single class but rather to plot criterion contribution trends (stacked area graphs) and belief surface contours, which proved to be very satisfactory in difficult decision situations. Four input scenarios were selected to demonstrate the efficiency of the proposed approach in different situations and with reference to the earlier approaches.