Lower Approximations by Fuzzy Consequence Operators

Peer ReviewedPostprint (author's final draft

Stone-type representations and dualities for varieties of bisemilattices

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent

LearnFCA: A Fuzzy FCA and Probability Based Approach for Learning and Classification

Formal concept analysis(FCA) is a mathematical theory based on lattice and order theory used for data analysis and knowledge representation. Over the past several years, many of its extensions have been proposed and applied in several domains including data mining, machine learning, knowledge management, semantic web, software development, chemistry ,biology, medicine, data analytics, biology and ontology engineering. This thesis reviews the state-of-the-art of theory of Formal Concept Analysis(FCA) and its various extensions that have been developed and well-studied in the past several years. We discuss their historical roots, reproduce the original definitions and derivations with illustrative examples. Further, we provide a literature review of it’s applications and various approaches adopted by researchers in the areas of dataanalysis, knowledge management with emphasis to data-learning and classification problems. We propose LearnFCA, a novel approach based on FuzzyFCA and probability theory for learning and classification problems. LearnFCA uses an enhanced version of FuzzyLattice which has been developed to store class labels and probability vectors and has the capability to be used for classifying instances with encoded and unlabelled features. We evaluate LearnFCA on encodings from three datasets - mnist, omniglot and cancer images with interesting results and varying degrees of success. Adviser: Dr Jitender Deogu

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page

LEARNFCA: A FUZZY FCA AND PROBABILITY BASED APPROACH FOR LEARNING AND CLASSIFICATION

Formal concept analysis(FCA) is a mathematical theory based on lattice and order theory used for data analysis and knowledge representation. Over the past several years, many of its extensions have been proposed and applied in several domains including data mining, machine learning, knowledge management, semantic web, software development, chemistry ,biology, medicine, data analytics, biology and ontology engineering. This thesis reviews the state-of-the-art of theory of Formal Concept Analysis(FCA) and its various extensions that have been developed and well-studied in the past several years. We discuss their historical roots, reproduce the original definitions and derivations with illustrative examples. Further, we provide a literature review of it’s applications and various approaches adopted by researchers in the areas of dataanalysis, knowledge management with emphasis to data-learning and classification problems. We propose LearnFCA, a novel approach based on FuzzyFCA and probability theory for learning and classification problems. LearnFCA uses an enhanced version of FuzzyLattice which has been developed to store class labels and probability vectors and has the capability to be used for classifying instances with encoded and unlabelled features. We evaluate LearnFCA on encodings from three datasets - mnist, omniglot and cancer images with interesting results and varying degrees of success. Adviser: Jitender Deogu