• Mani, A. Mereological Methods for Education Research, and School and College-Level Mathematics. In Proceedings of ICME 15, Sydney 2023-24

Mereology is the study of parts, wholes, related predicates and predications. While the subject finds wide application in specialized areas, it has not been explored in matching depth in education research, and specifically in mathematics education. A recent paper, due to the present author, explores potential areas of development for STEAM, in general. Applications to the signed number problem, and modeling students’ and teachers’ languages of discourse, are additionally explored by her in recent work. These are explored further towards the development of reasonably defined practices for adoption in teaching, evaluation and education research contexts.

• Mani, A. Algebraic Models for Qualified Aggregation in General Rough Sets, and Reasoning Bias Discovery. In: Campagner, A., Others (eds.) Rough Sets. IJCRS 2023, LNAI, vol. 14481, pp. 137–153. Springer Nature, DOI Preprint Link.

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.

• Mani, A. Algebraic, Topological, and Mereological Foundations of Existential Granules. In: Campagner, A., Others (eds.) Rough Sets. IJCRS 2023, LNAI 14481, 185–200. Springer Nature (2023), arXiv:2308.16157 DOI Preprint Version.

In this research, new concepts of existential granules that determine themselves are invented, and are characterized from algebraic, topological, and mereological perspectives. Existential granules are those that determine themselves initially, and interact with their environment subsequently. Examples of the concept, such as those of granular balls, though inadequately defined, algorithmically established, and insufficiently theorized in earlier works by others, are already used in applications of rough sets and soft computing. It is shown that they fit into multiple theoretical frameworks (axiomatic, adaptive, and others) of granular computing. The characterization is intended for algorithm development, application to classification problems and possible mathematical foundations of generalizations of the approach. Additionally, many open problems are posed and directions provided.

• Mani, A. and Mitra, S. Large Minded Reasoners for Soft and Hard Cluster Validation Annals of Computer and Information Sciences, Vol 36, PTI, 16pp, 2023. DOI Link

In recent research, validation methods for soft and hard clustering through general granular rough clusters are proposed by the first author. Large-minded reasoners are introduced and studied in the context of new concepts of non-stochastic rough randomness in a separate paper by her. In this research, the methodologies are reviewed and new low-cost scalable methodologies and algorithms are invented for computing granular rough approximations of soft clusters for many classes of partially ordered datasets. Specifically, these are applicable to datasets in which attribute values are numeric, vector-valued, lattice-ordered or partially ordered. Additionally, new research directions are indicated.

• Mani, A. Approximate Reasoning and the Ethnomathematics of Cooking Prometeica - Revista de Filosofía y Ciencias, 27, 295-305, 2023

Several methods of approximate reasoning are known and many remain to be discovered across domains. Some are tied to reasoning with uncertainty. The developments over the last fifty years or so in the development of approximate reasoning methods, and their application across multiple STEM domains suggest that it is necessary to introduce them early in school. While efforts towards building the infrastructure for the process have been limited, the bigger question is “What should be introduced?”. If concepts are always approximated in real life, then how should they be done mathematically? If the real numbers are not always necessary to approximate concepts or even quantities, then how should they be approximated, and what should be taught in schools? Specifically, functional representations of approximate reasoning are not introduced early. This additionally impedes diversity, ethnomathematical explorations, learning from experience, and through models. The objective of this research is to explain the problem with a focus on the ethnomathematics of cooking, and associated haptic methods. Further, it is argued through the context that of the many approaches to approximate reasoning, intrusive methods are best avoided, and that general rough sets is better suited for modeling such knowledge. Intrusive methods such as those based on fuzzy or probability theory are those that indulge in an excess of unjustified numeric assumptions. This builds on earlier work of the present author on modeling approximate reasoning. Her axiomatic approach to granularity are additionally applicable to approximate recipes that possess desired properties.

In the contexts of ethnomathematical discourse involving haptic reasoning (as in cooking), it is shown that such an approach can model the essence of context without oversimplifications or outright dismissal of the efforts involved. Further, it has the potential to help in inventing expressive structured languages for the ethnomathematics, and associated model eliciting activities. Distinct facets othree specific contexts are theoretically explored to illustrate aspects of the reasoning, and meta aspects.

• Mani, A. Granular Knowledge and Rational Approximation in General Rough Sets-1 Journal of Applied Non-Classical Logics, Dec’2023-24, Vol 34, 294-329 Link

Rough sets are used in numerous knowledge representation contexts, and are then empowered with varied ontologies. These may be intrinsically associated with ideas of rationality under certain conditions. In recent papers, specific granular generalizations of graded and variable precision rough sets are investigated by the present author from the perspective of rationality of approximations (and the associated semantics of rationality in approximate reasoning). The studies are extended to ideal-based approximations (sometimes referred to as subsethood-based approximations). It is additionally shown that co-granular or point-wise approximations defined by sigma-ideals/filters (for an arbitrary relation sigma) fit easily into the entire scheme. Concepts of the rationality of objects (vague or crisp) and their types are introduced, and are shown to be applicable to most general rough sets by the present author. Surprising results on these are proved on these by her in this part of the research paper. The present paper is the first of a three part study on the topic.

• Mani, A. Rough Randomness and its Application J. Cal. Math. Soc. 20(2), 143–154 (2024) 2023 Link

Abstract: A number of generalizations of stochastic and information-theoretic randomness are known in the literature. However, they are not compatible with handling meaning in vague and dynamic contexts of rough reasoning (and therefore explainable artificial intelligence and machine learning). In this research, new concepts of rough randomness that are neither stochastic nor based on properties of strings are introduced by the present author. Her concepts are intended to capture a wide variety of rough processes (applicable to both static and dynamic data), construct related models, and explore the validity of other machine learning algorithms. The last mentioned is restricted to soft/hard clustering algorithms in this paper. Two new computationally efficient algebraically-justified algorithms for soft and hard cluster validation that involve rough random functions are additionally proposed in this research. A class of rough random functions termed large-minded reasoners have a central role in these.

• Mani, A. Granular Directed Rough Sets, Concept Organization and Soft Clustering AMAI 2022, 32pp, Preprint Link

Abstract: Up-directed rough sets are introduced and studied by the present author in earlier papers. This is extended by her in two different granular directions in this research with a surprising algebraic semantics. The granules are based on ideas of generalized closure under up-directedness that may be read as a form of weak consequence. This yields approximation operators that satisfy cautious monotony, while pi-groupoidal approximations (that additionally involve strategic choice and algebraic operators) have nicer properties. The study is primarily motivated by possible structure of concepts in distributed cognition perspectives, real or virtual classroom learning contexts, and student-centric teaching. Rough clustering techniques for datasets that involve up-directed relations (as in the study of Sentinel project image data) are additionally proposed. This research is expected to see significant theoretical and practical applications in related domains.

• Mani, A. Granularity and Rational Approximation: Rethinking Graded Rough Sets Transactions on Rough Sets, XXIII, 2022, 36pp, Springer (Accepted) Preprint Link Paper Link

Abstract: The concept of rational discourse is typically determined by subjective, normative, and rule based constraints in the context under consideration. It is typically determined by related ontologies, and coherence between associated concepts employed in the discourse. Classical rough approximations, and variants of variable precision rough sets (VPRS) including graded rough sets embody at least some aspects of potentially useful concepts of rational approximation, but can be very lacking in application contexts, and rough set theoretical frameworks for cluster validation. While the literature on knowledge from general rough perspectives is rich and diverse, not much work has been done from the perspective of rationality in explicit terms. In this research, the gap is addressed by the present author in variants of high granular partial algebras. Specifically, the nature of optimal concepts of rational approximations is examined, and formalized by her in such frameworks. Graded rough sets are generalized from a granular perspective, and the compatibility of the introduced concepts are studied over it. Further aspects of algebraic semantics of granular graded rough sets are examined. Some incorrect results in graded rough sets in the literature are also corrected.

• Mani, A. General Rough Modeling of Cluster Analysis Forthcoming, 2021, 15pp

Abstract: In this research a general theoretical framework for abstract clustering is proposed over specific partial algebraic systems by the present author. Her theory helps in isolating minimal assumptions necessary for different concepts of clustering information in any form to be realized in a situation (and therefore in a semantics). It is well-known that of the limited number of proofs in the theory of hard and soft clustering that are known to exist, most involve statistical assumptions. Many methods seem to work because they seem to work in specific empirical practice. Two new general rough methods of analyzing clusterings are invented, and this opens the subject to clearer conceptions and contamination-free theoretical proofs. This is also illustrated with two abstract and real data sets. In addition, basic ideas of rough clustering in the literature are generalized from a theoretical and methodological perspective.

• Mani, A. Comparative Approaches to Granularity in General Rough Sets In IJCRS'2020, E. Bello Et Al (Editors) LNAI 12179, pages 500-518, 2020 Springer Link . Preprint Link

Abstract: A number of nonequivalent perspectives on granular computing are known in the literature, and many are in states of continuous development. Further related concepts of granules and granulations may be incompatible in many senses. This expository paper is intended to explain basic aspects of these from a critical perspective, their range of applications and provide directions relative to general rough sets and related formal approaches to vagueness. General granular principles related to knowledge are also mentioned.

• Mani, A. Towards Student Centric Rough Concept Inventories In IJCRS'2020, E. Bello Et Al (Editors) LNAI, Springer, 2020, LNAI 12179, pages 251-266, 2020 Springer Link . Preprint Link

Abstract: In the context of education research, a concept inventory is an instrument (that consists of a number of multiple-choice questions) designed to test the understanding of concepts (and possibly the reasons for failure to understand) by learners. Subject to a few caveats they are known to be somewhat effective in non student centric learning environments. In this research the issue of adapting the subject/concept-specific instruments to make room for diverse response patterns (including vague ones) is explored in some detail by the present author. It is shown that higher granular operator spaces (or partial algebras) with additional temporal and key operators are well suited for handling them. An improved version of concept inventory called rough concept inventory that can handle vague subjective responses is also proposed in this research.

• Mani, A. Functional Extensions of Knowledge Representation in General Rough Sets In IJCRS'2020, E. Bello Et Al (Editors) LNAI 12179, pages 19–34, 2020 Springer Link . Preprint Link

Abstract: A number of low and high-level models of general rough sets can be used to represent knowledge. Often binary relations between attributes or collections thereof have deeper properties related to decisions, inference or vision that can be expressed in ternary functional relationships (or groupoid operations)-- this is investigated from a minimalist perspective in this research by the present author. General approximation spaces and reflexive up-directed versions thereof are used by her as the basic frameworks. Related semantic models are invented and an interpretation is proposed in this research. Further granular operator spaces and variants are shown to be representable as partial algebras through the method. An analogous representation for all covering spaces does not necessarily hold. Applications to education research contexts that possibly presume a distributed cognition perspective are also outlined.

• Mani, A. High Granular Operator Spaces and Less-Contaminated Rough Mereologies Forthcoming, ArXiv Link, 2019

Abstract: Granular operator spaces and variants had been introduced and used in theoretical investigations on the foundations of general rough sets by the present author over the last few years. In this research, higher order versions of these are presented uniformly as partial algebraic systems. They are also adapted for practical applications when the data is representable by data table-like structures according to a minimalist schema for avoiding contamination. Issues relating to valuations used in information systems or tables are also addressed. The concept of contamination introduced and studied by the present author across a number of her papers, concerns mixing up of information across semantic domains (or domains of discourse). Rough inclusion functions (\textsf{RIF}s), variants, and numeric functions often have a direct or indirect role in contaminating algorithms. Some solutions that seek to replace or avoid them have been proposed and investigated by the present author in some of her earlier papers. Because multiple kinds of solution are of interest to the contamination problem, granular generalizations of RIFs are proposed, and investigated. Interesting representation results are proved and a core algebraic strategy for generalizing Skowron-Polkowski style of rough mereology (though for a very different purpose) is formulated. A number of examples have been added to illustrate key parts of the proposal in higher order variants of granular operator spaces. Further algorithms grounded in mereological nearness, suited for decision making in human-machine interaction contexts, are proposed by the present author. Applications of granular \textsf{RIF}s to partial/soft solutions of the inverse problem are also invented in this paper.

• Mani, A. Comparing Dependencies in Probability Theories and Rough Sets: Part-A Forthcoming, 2018-19, 1-69 Arxiv:1804.02322

Abstract: The problem of comparing concepts of dependence in general rough sets with those in probability theory had been initiated by the present author in some of her recent papers. This problem relates to the identification of the limitations of translating between the methodologies and possibilities in the identification of concepts. Comparison of ideas of dependence in the approaches had been attempted from a set-valuation based minimalist perspective by the present author. The deviant probability framework has been the result of such an approach. Other Bayesian reasoning perspectives (involving numeric valuations) and frequentist approaches are also known. In this research, duality results are adapted to demonstrate the possibility of improved comparisons across implications between ontologically distinct concepts in a common logic-based framework by the present author. Both positive and negative results are proved that delimit possible comparisons in a clearer way by her.

• Mani, A. Algebraic Methods in Granular Rough Sets In A. Mani, I. Düntsch, and G. Cattaneo, editors, Algebraic Methods in General Rough Sets, Trends in Mathematics, pages 123-303 Springer International, 2018

Abstract: At least three concepts of granular computing have been studied in the literature. The axiomatic approach in algebraic approaches to general rough sets had been introduced in an explicit formal way by the present author. Most of the results and techniques that are granular in this sense are considered critically in some detail in this research chapter by her. It is hoped that this work will serve as an important resource for all researchers in rough sets and allied fields.

• 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.

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