Mani, A. *"Types of Probabilities Associated with Rough Membership Functions"* Forthcomin'2015, 12pp.

In this research paper, connections between various meta theories of probability and rough membership functions are critically reviewed and variants are proposed by the present author. These are relevant for rethinking the various *probabilistic rough theories* and related methodologies. The problem of contamination reduction was proposed in \cite{AM240} and related papers by the present author. In this study the scope of the problem within probabilistic rough sets (PSTs) is clarified by her. A new definition of artificial intelligence applicable in rough perspectives is also proposed on the basis of recent advances in algebraic semantics related to rough membership functions.

Mani, A. *"Antichain Based Semantics for Rough Sets" *
in RSKT 2015, D. Ciucci, G. Wang, S. Mitra, and W. Wu, Eds. Springer-Verlag, 2015, 319--330.

The idea of using antichains of rough objects was suggested by the present author in her earlier papers. In this research basic aspects of such semantics are considered over general rough sets and general approximation spaces over quasi-equivalence relations. Most of the considerations are restricted to semantics associated with maximal antichains and their meaning. It is shown that even when the approximation operators are poorly behaved, some semantics with good structure and computational potential can be salvaged.

Mani, A. *"Ontology, Rough Y-Systems and Dependence" *
International J of Computer Science and Appl. (Technomath Foundation),
"Special Issue of IJCSA on Computational Intelligence", 11, 2, 2014, 114--136 (was part of keynote talk at ICCI'2014).

In this research paper, we explore the philosophical connections between Rough Y-Systems(RYS), mereology and concepts in applied ontology, introduce the concept of contamination- free rough dependence and compare this to possible concepts of probabilistic dependence. The nature of granular rough dependence is also characterized and the reason for breakdown of comparison of rough set models with probabilistic models are made clearer. From this we can test the validity of related comparisons in a semantic way.

Mani, A. *"Approximation Dialectics of Proto-Transitive Rough Sets" *
In *Facets of Uncertainties and Applications'2013*}, M. K. Chakraborty et. al. (Eds)
Springer Proceedings in Mathematics and Statistics 125, Springer Verlag, 1--11.

Rough Sets over generalized transitive relations like proto-transitive ones have been initiated by the present author in \cite{AM270} and detailed semantics have been developed in forthcoming papers \cite{AM2400}. In this research paper, approximation of proto-transitive relations by other relations is investigated and the relation with rough approximations is developed towards constructing semantics that can handle \emph{fragments of structure}. It is also proved that difference of approximations induced by some approximate relations need not induce rough structures.

Mani, A. *" L-Computing over Rough Y-Systems" *
Submitted' 2013 20pp

We introduce a new kind of nature inspired computing based on interaction of safety critical systems and independently on people communicating under specific kinds of constraints, emotional structure and conflicts using relatively vague expression and involved semiotics. We realize the computing process in a abstract way as new kinds of correspondences with evolution between rough Y-systems (\textsf{RYS}). Temporal aspects associated with process permit us to compare key kinds of correspondences that carry natural meaning. New methods and results on comparison of correspondences are also proved in the process. We also provide more detailed explanations of various ontological aspects of \textsf{RYS} and their realization in practice. The developed method may also be expected to be applicable for studying cognitive development of the evolution of specific languages for specialized domains and a wide variety of situations.

Mani, A. *<" Contamination-Free Measures and Algebraic Operations" *
Proceedings of FUZZIEEE'2013,Hyderabad Edited by N.Pal et. al. 16pp

An open concept of rough evolution and an axiomatic approach to granules was also developed in \cite{AM240} by the present author. Subsequently the concepts were used in the formal framework of rough Y-systems (\textsf{RYS}) for developing on granular correspondences in \cite{AM1800}. These have since been used for a new approach towards comparison of rough semantics across different semantic domains by way of correspondences that preserve rough evolution and try to avoid contamination. In this research paper, we propose methods and semantics for handling possibly contaminated operations and structured bigness. These would also be of natural interest for relative consistency of one collection of knowledge relative other.

Mani, A. *"Axiomatic Approach to Granular Correspondences" *
In Proceedings of RSKT'2012, edited by Li, T et. al, LNAI 7414, 2012, 482--487, Springer-Verlag.

An axiomatic approach towards granulation in general rough set theory (\textsf{RST}) was introduced by the present author in \cite{AM99} and extended in \cite{AM240} over general rough Y-systems (\textsf{RYS}). In the present brief paper a restricted first order version is formulated and granular correspondences between simpler \textsf{RYS} are considered. These correspondences are also relevant from the perspective of knowledge interpretation of rough sets, where we may find admissible concepts of a knowledge being a sub-object of another. Proofs will appear separately.

Mani, A. *"Dialectics of Counting and Mathematics of Vagueness"*

Transactions on Rough Sets Vol XV, LNCS 7255,'2012, 122--180

New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (\textsf{RST}) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general \textsf{RST} recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical \textsf{RST} can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of \emph{rough naturals} are also studied and variants are characterised in the penultimate section.

Mani, A. *"Towards Logics of Some Rough Perspectives of Knowledge"*

In Series: Intelligent Systems Reference Library dedicated to the memory of Prof. Pawlak, (ed. Suraj, Z and Skowron, A.) '2011-12, 342--367

Pawlak had introduced a concept of knowledge as a state of relative exactness in classical rough set theory (\textsf{RST}) \cite{ZPB}. From a theory of knowledge and application perspective, it is of much interest to study concepts of relative consistency of knowledge, correspondences between evolvents of knowledges and problems of conflict representation and resolution. Semantic frameworks for dealing with these are introduced and developed in this research paper by the present author. New measures that deal with different levels of contamination are also proposed. Further, it is shown that the algebraic semantics are computationally very amenable. The proposed semantics would also be of interest for multi-agent systems, dynamic spaces and collections of general approximation spaces. Part of the literature on related areas is also critically surveyed.

Mani, A. *"Choice Inclusive General Rough Semantics"*

Information Sciences 181(6), Vol 181, 1097--1115, '2011

Similarity based rough set theory (\textsf{RST}) involving choice in the formation of approximations was recently introduced by the present author. Though the theory can be used to develop improved semantics and models of knowledge and belief with ontology, application requires \emph{a priori} concepts of granules and granulation as opposed to the more common \emph{a posteriori} or \emph{not a priori} concepts of the same prevalent in the literature. In this research, we clarify the desirable semantic features of a context for seamless application of the theory to more general situations, formalise them and refine the semantics. A new axiomatic theory of granules in general \textsf{RST} (including hybrid versions involving fuzzy set theories) is also developed in the process. Interesting new applications to human learning are also illustrated in this paper.

Mani, A. "Dialectics of Counting and Measures of Rough Theories"

Proceedings of NCETSC' 2011, 16 pp

New concepts of rough natural number systems, recently introduced by the present author, are used to improve most rough set-theoretical measures in general Rough Set theory (\textsf{RST}) and measures of mutual consistency of models of knowledge. In this research paper, the explicit dependence on the axiomatic theory of granules of \cite{AM99} is reduced and more results on the measures and representation of the numbers are proved.

Mani, A. "A Program in Dialectical Rough Set Theory"

Preprint' 2009, http://arxiv.org/abs/0909.4876

A dialectical rough set theory focussed on the relation between roughly equivalent objects and classical objects was introduced in \cite{AM699} by the present author. The focus of our investigation is on elucidating the minimal conditions on the nature of granularity, underlying semantic domain and nature of the general rough set theories (RST) involved for possible extension of the semantics to more general RST on a paradigm. On this basis we also formulate a program in dialectical rough set theory. The dialectical approach provides better semantics in many difficult cases and helps in formalizing a wide variety of concepts and notions that remain untamed at meta levels in the usual approaches. This is a brief version of a more detailed forthcoming paper by the present author.

Mani, A. "Towards an Algebraic Approach for Cover Based Rough Semantics and Combinations of Approximation Spaces"

In Sakai, H. et. al (Eds), RSFDGrC'09 New Delhi, LNAI 5908, 77--84, 2009

We introduce the concept of a synchronal approximation space (\textsf{SA}) and a \textsf{AUAI}-multiple approximation space and show that they are essentially equivalent to an \textsf{AUAI} rough system. Through this we have estabilished connections between general cover based systems, dynamic spaces and generalized approximation spaces (APS) for easier 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"

* Internat. Conference on Rough Sets, Tripura University, Agartala' * 2009

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"

* Fundamenta Informaticae * **97** (1-2) 2009, 177--197

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"

* International Conference in Logic and Applications, Chennai * 2009 (Refereed),
http://ali.cmi.ac.in/icla2009/

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"

* Preprint*'2009

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"

*Transactions in Rough Sets,* Vol-VIII,LNCS 5084, 2008, 175--223

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.

Mani, A. "Di-Algebraic Semantics of Logics"

*Fundamenta Informaticae* **70**, (4) 2006, 333--350

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].

Mani, A. "Dialectically Presentable Logics - Condensed Version"

*Preprint* 2006

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.

Mani, A. *"Super Algebraic Semantics"*

*Preprint*

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 algebraisation theories is also established.

Mani, A. "Super Rough Semantics"

*Fundamenta Informaticae* **65**, 2005, 1--13

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.

Mani, A. "Rough Equalities on Posets and Rough Difference Orders"

*Fundamenta Informaticae* **53** (3,4) 2002, 321--333

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.

Mani, A. "Definable and Applicable Rough Reals"

*Preprint,* 2006

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"

To be Submitted, 2007

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--255

In 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.

Mani, A. "V-Perspectives, Pseudo-Natural Number Systems and Partial Orders"

*Glasnik Math* Vol.**37** (57) 2002, 245-257

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.

Mani, A. *"Algebraic Semantics of Rough Difference Orders"*

*Internat. Symposium on Mathematics at the Cal.Math.Soc. Dec`20-22, 2002*

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.