Dialectics of Counting and the Mathematics of Vagueness

2017

Rough Sets

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Abstract. Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the inverse problem. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical perspective, a few (as opposed to one) theoretical frameworks may be suitable for formulating the problem itself. Granular operator spaces have been recently introduced and investigated by the present author in her recent work in the context of antichain based and dialectical semantics for general rough sets. The nature of the inverse problem is examined from number-theoretic and combinatorial perspectives in a higher order variant of granular operator spaces and some necessary conditions are proved. The results and the novel approach would be useful in a number of unsupervised and semi supervised learning contexts and algorithms.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximations from Anywhere and General Rough Sets

2017

Rough Sets

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“…The concept of admissible granulation was defined for rough Y-systems RYS (a more general framework due to the present author in [1]) using parthoods instead of set inclusion and relative to RYS, P =⊆, P =⊂ in granular operator spaces [7]. The concept of generalized granular operator spaces has been introduced in [8] by the present author as a proper generalization of that of granular operator spaces.…”

Section: Definitionmentioning

confidence: 99%

“…This idea might work in some contexts -the developed/invented formalisms suggest some restrictions on possible contexts. Ideals and filters have been used by the present author in algebraic semantics of general rough sets in some of her earlier papers like [9,1,10,11]. Concepts of rough ideals have also been studied by different authors in specific algebras (see for example [12,13])-these studies involve the use of rough concepts within algebras.…”

Section: Introductionmentioning

confidence: 99%

Generalized Ideals and Co-granular Rough Sets

2017

Rough Sets

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Abstract. Lattice-theoretic ideals have been used to define and generate non granular rough approximations over general approximation spaces over the last few years by few authors. The goal of these studies, in relation based rough sets, have been to obtain nice properties comparable to those of classical rough approximations. In this research paper, these ideas are generalized in a severe way by the present author and associated semantic features are investigated by her. Granules are used in the construction of approximations in implicit ways and so a concept of co-granularity is introduced. Knowledge interpretation associable with the approaches is also investigated. This research will be of relevance for a number of logico-algebraic approaches to rough sets that proceed from point-wise definitions of approximations and also for using alternative approximations in spatial mereological contexts involving actual contact relations. The antichain based semantics invented in earlier papers by the present author also applies to the contexts considered.

“…To connect these with other rough objects, various kinds of negation-like operations (or generalizations thereof) are of interest. Such operations and predicates are of interest when rough parthoods are not partial or quasi orders Mani (2012bMani ( , 2016c; Polkowski (2011).…”

Section: Introductionmentioning

confidence: 99%

Dialectical Rough Sets, Parthood and Figures of Opposition-I

Transactions on Rough Sets XXI

2019

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In one perspective, the main theme of this research revolves around the inverse problem in the context of general rough sets that concerns the existence of rough basis for given approximations in a context. Granular operator spaces and variants were recently introduced by the present author as an optimal framework for antichain based algebraic semantics of general rough sets and the inverse problem. In the framework, various sub-types of crisp and non-crisp objects are identifiable that may be missed in more restrictive formalism. This is also because in the latter cases concepts of complementation and negation are taken for granted -while in reality they have a complicated dialectical basis. This motivates a general approach to dialectical rough sets building on previous work of the present author and figures of opposition. In this paper dialectical rough logics are invented from a semantic perspective, a concept of dialectical predicates is formalised, connection with dialetheias and glutty negation are established, parthood analyzed and studied from the viewpoint of classical and dialectical figures of opposition by the present author. Her methods become more geometrical and encompass parthood as a primary relation (as opposed to roughly equivalent objects) for algebraic semantics.