Missing values and dragonfly operations in fuzzy relational compositions

Abstract

Three-valued logics were found by logicians as an important topic focusing on dealing with truth-values different from the standard True and False values. The variety of such values, including “Irrelevant”, “Non-applicable”, “Indeterminable”, “Incosistent”, “Graded truth” or “Unknown”, generated a wide variety of distinct three-valued logics, each focusing on a distinct type of the third value and the consequent aspects of the related logic. Indeed, there is no single approach that would correctly model all the motivating situations and serve perfectly to all practical problems. Furthermore, one has to keep in mind that these logical or even only purely algebraic approaches serve as a sort of approximation of the modeled real situation. Indeed, some of them might deserve very complex approaches using several other techniques and scientific fields related to the uncertainty theories.However, the logical/algebraic approaches may serve as very appropriate, comprehensible, elegant and efficient way to treat such truth values that are neither True, nor False. Following some of the previous works, we will call such values by the word “undefined” and make a short revision of the three-valued logics dealing with such undefined values. Secondly, we will review some extensions of these three-valued logics to many-valued logics, i.e., in particular partial fuzzy logics, which extend typical, usually [0,1]-valued fuzzy logics by a dummy value ⋆ in order to represent the undefined truth value. Furthermore, we recall that none of them is primarily proposed in order to deal with the missing values in fuzzy relational compositions and thus, the first attempts to deal with such values in fuzzy relational compositions was built on a combination of two algebras for partial fuzzy logics, namely Bochvar and Sobociński. However, it is clear that this combination of two algebras in the definition of fuzzy relational compositions is a sort of higher-level construction of a rather heuristic origin. Therefore, in this paper, we go back one level lower and design a new set of operations for the purpose of dealing with missing values. This algebra employs the lower estimation approach and it is designed in order to preserve as many properties from the residuated lattices as possible. Further properties of the proposed operations are provided and formally proved. Finally, the application potential is demonstrated on a real example of the taxonomical classification of dragonflies. Based on the primary application, we call the proposed algebra of operations as Dragonfly algebra or simply Dragonfly operations.