| The Free Site | vBuddy - social networking for webmasters | Cheap Web Hosting - starting at $5 |
We develop the relation between generalised cover based \textsf{AUAI} rough sets (\cite{IM}) and different enhanced approximation spaces including dynamic spaces for possible algebraic semantics. \textsf{AUAI}-rough set theory (RST) is also extended to accommodate local determination of universes. The results obtained are also significant in the representation theory of general granular RST, for the problems of multi source RST and Ramsey-type combinatorics.
Mani, A. "Integrated Dialectical Semantics for Relativised Rough Set Theory"In this research paper we introduce two new semantics of rough set theory (RST) relative a fused object and meta level of understanding. The motivations can be traced to application contexts (where dual interpretations may be seen to be in action) as well as philosophical considerations on the nature of conjunction and disjunction in rough logic. The results of this paper are extended to general RST in the longer version of this paper (\cite{AM699}). More importantly this is also a semantics for relativised or multi RST in which discernibility is ordered.
Mani, A. "Algebraic Semantics of Bitten Rough Sets"We develop different algebraic semantics for bitten rough set theory (\cite{SW}) over similarity spaces and their abstract granular versions. Connections with the choice based generalized rough semantics developed in \cite{AM99} by the present author are also considered.
Mani, A. "Meaning, Choice and Algebraic Semantics of Similarity Based Rough Set Theory"Both algebraic and computational approaches for dealing with similarity spaces are well known in generalized rough set theory. However, these studies may be said to have been confined to particular perspectives of distinguishability in the context. In this research, the essence of an algebraic semantics that can deal with all possible concepts of distinguishability over similarity spaces is progressed. Key to this is the addition of choice-related operations to the semantics that have connections to modal logics as well. In this presentation, I will focus on a semantics based on local clear distinguishability over similarity spaces.
Mani, A. "Consistency in Knowledge Frameworks and Euclidean Granular Rough Semantics"A rough semantics over Euclidean domains and a theory of mutual and relative consistency of knowledge is developed in this research paper. This is a continuation of the granular action based rough semantics developed earlier by the present author. In particular we consider the case of application contexts in which the domain has granular entities with graded existence (or meaning) corresponding to the points. We also develop a theory of mutual consistency of knowledge creating operators (and so of generalized knowledge). The research is about knowledge consistency and the euclidean granular rough set theory developed helps in illustrating certain features.
Mani, A. "Esoteric Rough Set Theory: Algebraic Semantics of a Generalized VPRS and VPRFS"In different theories involving indiscernibility, it is assumed that at some level the objects involved are actually assignable distinct names. This can prove difficult in different application contexts if the main semantic level is distinct from the semantic-naming level. Set-theoretically too this aspect is of much significance. In the present research paper we develop a framework for a generalized form of rough set theory involving partial equivalences on two different types of approximation spaces. The theory is also used to develop an algebraic semantics for variable precision rough set and variable precision fuzzy rough set theory. A quasi-inductive concept of relativised rough approximation is also introduced in the last section. Its relation to esoteric rough sets is considered.
In [22] the problem of the logics corresponding to topological quasi-boolean algebras [27, 1] has been recently solved by the present author. The semantics provided involved \emph{convex amalgams} of boolean algebras with additional total and partial operations. Canonical extensions of the structure was also investigated. In the present research, this semantics is generalized to a wide class of logics including distributive logics. It is also shown that the semantics is a proper generalization of the general theory of algebraizable logics due to Blok-Pigozzi [5] and Czelakowski [7].
In this research paper different concepts of dialectically presentable logics are introduced and progressed. The methodological content of different dialectical philosophies especially Marxist dialectics are abstracted in the process. We also identify fundamentally distinct methods in the formalization of dialectical logics. This is a contribution to the thesis that every logic is essentially dialectical and beautifully so.
In this research a generalized theory of algebraic semantics of a logic is developed. This is sometimes a proper generalization of the classical Lindenbaum-Tarski algebraisation procedure. The theory is largely influenced by the recently developed \emph{super rough semantics} and it's extension to generalized rough sets, recently developed by the present author. The semantics is in a sense getting to exact semantics by properly presenting the dialectics of some approximate parts. The eventual algebraic semantics is developed via many deep results in convexity in ordered structures. The relation with other general algebraization theories is also estabilished.
In this research a new algebraic semantics of rough set theory including additional meta aspects is proposed. The semantics is based on enhancing the standard rough set theory with notions of 'relative ability of subsets of approximation spaces to approximate'. The eventual algebraic semantics is developed via many deep results in convexity in ordered structures. A new variation of rough set theory, namely 'ill-posed rough set theory' in which it may suffice to know some of the approximations of sets, is eventually introduced.
In the initial section of this research paper rough equalities from partially ordered approximation spaces are investigated. Special types of rough equalities are characterized via convex and other types of sets. Extension of these to all types of rough equalities is also indicated. Two new theories of `Rough Difference Orders' which are often more general and distinct from that of `Rough Orders are also developed in the last section by the present author.
In this research we develop different concepts of rough theoretical versions of the natural and the different real number system. The intent is at applications in formal semantics of rough sets and direct real-life applications. We develop the necessary philosophical basis for the semantics and then the different possible semantics too.
Mani, A. "A Partial-Algebraic Logic of TQBAs"In the present research, we develop an axiomatic logical system corresponding to the topological quasi-boolean algebras (TQBA) in a sense. In the process we extend the concept of algebraic semantics of a logic to partial algebraic semantics in yet another way. Here we have a single consequence operation associated as opposed to the two consequences in the dialgebraic semantics developed by the present author. The logic developed has interesting connections with the different algebraic semantics of rough set theory and generalized versions thereof.
Mani, A. "Constrained Abstract Representation Problems in Semigroups and Partial Groupoids" Glasnik Math. 39 (59) 2004, 245--255In this research paper different constrained abstract representation theorems for partial groupoids and semigroups are proved by the present author. Methods for improving the retract properties of the structures are also developed in the process. These have strong class-theoretical implications for many types of generalized periodic semigroups,and related partial semigroups in particular.The results are significant in a model-theoretical setting and without too.
In this research, we generalise the notion of partial well-orderability and consider its relation to partial difference operations possibly definable. Results on these and systems of invariants for V-PWO posets are also formulated. These are relevant in partial algebras with differences and pseudo-natural number systems for very generalised abstract model theory.
A theory of \emph{rough difference orders} was recently introduced by the present author in [AM1]. In the present paper an algebraic semantics is developed for the same in particular. This in particular paves the way for a possible sequent calculus. A concept of \emph{representational completeness} is also introduced. A form of algebraically representable difference orders with interesting possibilities in universal algebra is also developed in the paper.