Algebraic Methods for Granular Rough Sets

Abstract

In the context of general rough sets, the act of combining two things to form another is not straightforward. The situation is similar for other theories that concern uncertainty and vagueness. Such acts can be endowed with additional meaning that go beyond structural conjunction and disjunction as in the theory of * -norms and associated implications over L-fuzzy sets. In the present research, algebraic models of acts of combining things in generalized rough sets over lattices with approximation operators (called rough convenience lattices) is invented. The investigation is strongly motivated by the desire to model skeptical or pessimistic, and optimistic or possibilistic aggregation in human reasoning, and the choice of operations is constrained by the perspective. Fundamental results on the weak negations and implications afforded by the minimal models are proved. In addition, the model is suitable for the study of discriminatory/toxic behavior in human reasoning, and of ML algorithms learning such behavior.

Dialectical Rough Sets, Parthood and Figures of Opposition-I

“…• The view that dialectical negation cannot be reduced to classical negation (see Ioan (1998)). Indeed, in rough sets many kinds of negations and partial negations have been used in the literature (see for example, Banerjee and Chakraborty (1996); Cattaneo and Ciucci (2004); Mani (2005Mani ( , 2018a; Pagliani (1998Pagliani ( , 2000, Cattaneo and Ciucci (2018); Cattaneo, Ciucci, and Dubois (2011); Mani (2008Mani ( , 2009aMani ( , 2011; Pagliani (2016); Ślȩzak and Wasilewski (2007) and these lead to many contradictions as in (1) contradictions Pagliani (1998) which are not false but that represent topological boundaries; (2) contradictions Pagliani (2000) which are not false but lie between an absolute and local false; (3) contradictions Pagliani (2016) which lead to at least a paraconsistent and a paracomplete logic. • The view that dialectical negation is glutty negation (example Brandom (2008);…”

“…Many generalizations of classical rough sets from granular perspectives have been studied in the literature. For an overview the reader is referred to Mani (2018a). These were part of the motivations for the invention of the concept of granular operator spaces by the present author in Mani (2015) and developed further in Mani (2016bMani ( , 2016cMani ( , 2017b.…”

“…In simple terms, granules are the subsets that generate approximations and granulations are the collections of all such granules in the context. In the present author's view, there are at least three main classes of granular computing (five classes have been considered by her in Mani (2018a)). The three main classes are • Primitive Granular Computing Processes (PGCP): in which the problem requirements are not rigid, effort on formalization is limited, scope of abstraction become limited and concept of granules are often vague, though they maybe concrete or abstract (relative to all materialist viewpoints).…”

“…In general, a given set of approximations may be compatible with multiple concepts of definite and rough objects -the eventual choice depends on one's choice of semantic domain. A detailed treatment, due to the present author, can be found in Mani (2012bMani ( , 2018a. In Chakraborty (2016), some of these definitions of rough objects are regarded as imperfect for the purpose of expressing both multiple extensions of concepts and rough objects -this problem relates to representation (given a restricted view of granularity) within the classical semantic domain of Mani (2012b).…”

“…For this reason higher order variants of general granular operator spaces have been defined in Mani (2017a) by the present author. A detailed account can be found in Mani (2018a).…”