• Mani, A. *Approximations from Anywhere and General Rough Sets,
* In IJCRS'2017, Ed L. Polkowski et. al Part II, LNAI 10314,
pp. 23–42, 2017. DOI: 10.1007/978-3-319-60840-2\_2, Springer International
Preprint: Arxiv:1704.05443

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.

• Mani, A. *Generalized Ideals and Co-Granular Rough Sets, *
In IJCRS'2017, Ed L. Polkowski et. al, Part II, LNAI 10314, pp. 3–22, 2017,
DOI: 10.1007/978-3-319-60840-2\_1, Springer International Preprint:
Arxiv:1704.05477

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.

• Mani, A. *Dialectical Rough Sets, Parthood
and Figures of Opposition-I, II* To Appear,
2017: 69pp

Abstract: In one perspective, the central problem pursued in this research is that of the inverse problem in the context of general rough sets. The problem is about the existence of rough basis for given approximations in a context. Granular operator spaces were recently introduced by the present author as an optimal framework for anti-chain based algebraic semantics of general rough sets and the inverse problem. In the framework, various subtypes of crisp and non crisp objects are identifiable that may be missed in more restrictive formalism. This is also because in the latter cases the concept of complementation and negation are taken for granted. This opens the door for 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 developed from a semantic perspective, concept of dialectical predicates is formalized, connection with dialethias and glutty negation established, parthood analyzed and studied from the point of view of classical and dialectical figures of opposition. Potential semantics through dialectical counting based on these figures are proposed building on earlier work 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. Dialectical counting strategies over antichains (a specific form of dialectical structure) for semantics are also proposed.

• Mani, A. *Knowledge and Consequence in AC Semantics for General Rough Set, * In Thriving Rough Sets--10th Anniversary--Honoring Professor Zdzislaw Pawlak's Life and Legacy and 35 years of Rough
Sets, edited by G. Wang and A. Skowron and Y. Yao and D. Slezak and Lech Polkowski
"Studies in Computational Intelligence" Series, Vol.708, Springer'2017. 237--268, doi = "10.1007/978-3-319-54966-8.

Abstract: Antichain based semantics for general rough sets was introduced recently by the present author. In her paper two different semantics, one for general rough sets and another for general approximation spaces over quasi-equivalence relations, were developed. These semantics are improved and studied further from a lateral algebraic logic and an algebraic logic perspective in this research. The framework of granular operator spaces is also generalized. The main results concern the structure of the algebras, deductive systems and the algebraic logic approach. The epistemological aspects of the semantics is also studied in this paper in some depth and revolve around nature of knowledge representation, Peircean triadic semiotics and temporal aspects of parthood. Examples have been constructed to illustrate various aspects of the theory and applications to human reasoning contexts that fall beyond information systems.

• Mani, A. *On Deductive Systems of AC
Semantics for Rough Sets* Arxiv:1610.02634 2016:
12pp Download Link

Abstract. Antichain based semantics for general rough sets were introduced recently by the present author. In her paper two different semantics, one for general rough sets and another for general approximation spaces over quasi-equivalence relations, were developed. These semantics are improved and studied further from a lateral algebraic logic perspective in this research. The main results concern the structure of the algebras and deductive systems in the context.

• Mani, A. *Probabilities, Dependence and Rough
Membership Functions, * Special Issue on
Computational Intelligence and Communications,International Journal of Computers and Applications,
Vol. 39, No.1, 2016, 17--39 Link .

The goal of this research is to study the reflection of probability theories through rough membership functions (RMF) in rough sets. Towards this, philosophy and variants of probability theories and their theorized connections with RMFs are critically analyzed. The concept of RMFs functions and rough dependence are also generalized to granular operator spaces, characterized and used for the same purpose in more general contexts. A new theory of \emph{dependence based deviant probability} is developed as a severe extension of the axiomatic approach to a dependence based probability introduced recently by the present author. It is shown that the theories of rough and deviant probabilist dependence are very distinct semantically and similarities are poorly justified. The problem of contamination reduction was proposed recently across many papers by the present author. In this study the scope of the problem within RMFs, probabilistic rough sets (PSTs) and three way decision making is also clarified and extended by her. Using the less intrusive deviant probability a nontrivial application to ovarian cancer diagnosis is developed in the last section. A new definition of AI applicable in rough perspectives is also proposed on the basis of recent advances in algebras of RMFs.

• Mani, A. *Combinatorial Aspects of
Distribution of Rough Objects *Forthcoming
2016: 23pp

The inverse problem of general rough sets, considered by the present author in some of her earlier papers, in one of its manifestations is essentially the question of when an agent's view about crisp and non crisp objects over a set of objects has a rough evolution. In this research the nature of the problem is examined from number-theoretic and combinatorial perspectives under very few assumptions about the nature of data and some necessary conditions are proved.

• Mani, A. *Algebraic Semantics of
Proto-Transitive Rough Sets, * Transactions on
Rough Sets XX, LNCS 10020, Springer'2016 51--108

Rough Sets over generalized transitive relations like proto-transitive ones have been initiated recently by the present author. In a recent paper, approximation of proto-transitive relations by other relations was investigated and the relation with rough approximations was developed towards constructing semantics that can handle \emph{fragments of structure}. It was also proved that difference of approximations induced by some approximate relations need not induce rough structures. In this research, the structure of rough objects is characterized and a theory of dependence for general rough sets is developed and used to internalize the Nelson-algebra based approximate semantics developed earlier by the present author. This is part of the different semantics of \textsf{PRAX} developed in this paper by her. The theory of rough dependence initiated in earlier papers is extended in the process. This paper is reasonably self-contained and includes proofs and extensions of representation of objects that have not been published earlier.

• Mani, A. *Types of Probabilities Associated
with Rough Membership Functions, * IEEEXplore,
ICRICN'2015, 175--180. Doi. 10.1109/ICRCICN.2015.7434231

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.