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In formal concept analysis, the collection of protoconcepts of any context forms a double Boolean algebra (dBa) which is fully contextual. Semiconcepts of a context form a pure dBa. The present article is a study on topological representation results for dBas, and in particular, those for fully contextual and pure dBas. In this work, a context on topological spaces (CTS) is considered, and the focus is on a special kind of CTS in which the relation defining the context as well as the converse of the relation are continuous with respect to the topologies. Such CTS are denoted as CTSCR. It is observed that clopen object oriented protoconcepts of a CTSCR form a fully contextual dBa, while clopen object oriented semiconcepts form a pure dBa. Every dBa is shown to be quasi-embeddable into the dBa of clopen object oriented protoconcepts of a particular CTSCR. The quasi-embedding turns into an embedding in case of a contextual dBa, and into an isomorphism, when the dBa is fully contextual. For pure dBas, one obtains an isomorphism with the algebra of clopen object oriented semiconcepts of the CTSCR. Representation of finite dBas and Boolean algebras are also addressed in the process. Abstraction of properties of this CTSCR leads to the definition of Stone contexts. Stone contexts and CTSCR- homeomorphisms are seen to form a category, denoted Scxt. Furthermore, correspondences are observed between dBa isomorphisms and CTSCR-homeomorphisms. This motivates a study of categorical duality of dBas, constituting the second part of the article. Pure dBas and fully contextual dBas along with dBa isomorphisms form categories, denoted PDBA and FCDBA respectively. It is established that PDBA is equivalent to FCDBA, while FCDBA and PDBA are dually equivalent to Scxt., 40 pages, 2 figures

The article is a study of two algebraic structures, the `contrapositionally complemented pseudo-Boolean algebra' (ccpBa) and `contrapositionally $\vee$ complemented pseudo-Boolean algebra' (c$\vee$cpBa). The algebras have recently been obtained from a topos-theoretic study of categories of rough sets. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. We study properties of these algebras, give examples, and compare them with relevant existing algebras. `Intuitionistic Logic with Minimal Negation (ILM)' corresponding to ccpBas and its extension ILM-${\vee}$ for c$\vee$cpBas, are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce's logic. With a focus on properties of the two negations, two kinds of relational semantics for ILM and ILM-${\vee}$ are obtained, and an inter-translation between the two semantics is provided. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn's logical framework for the purpose, two logics $K_{im}$ and $K_{im-{\vee}}$ are presented, where the language does not include implication. $K_{im}$-algebras are reducts of ccpBas. The negations in the algebras are shown to occupy distinct positions in an enhanced form of Dunn's Kite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee}}$ are given, based on Dunn's compatibility frames. Finally, relationships are established between the different algebraic and relational semantics for the logics defined in the work., Comment: 36 pages

Possibility theory and modal logic are two knowledge representation frameworks that share some common features, such as the duality between possibility and necessity, as well as some obvious di?erences since possibility theory is graded but is not primarily a logical setting. In the last thirty years there have been a series of attempts, reviewed in this paper, for bridging the two frameworks in one way or another. Possibility theory relies on possibility distributions and modal logic on accessibility relations, at the semantic level. Beyond the observation that many properties of possibility theory have qualitative counterparts in terms of axioms of well-known modal logic systems, the first works have looked for (graded) accessibility relations that can account for the behavior of possibility and necessity measures. More recently, another view has emerged from the study of logics of incomplete information, which is no longer based on Kripke-like models. On the one hand, possibilistic logic, closely related to possibility theory, mainly handles beliefs having various strength. On the other hand, in the so-called meta-epistemic logic (MEL) an agent can express both beliefs and explicitly ignored facts (both without strength), by only using modal formulas of depth 1, and no objective ones; its semantics is based on epistemic states. The system MEL+ is an extension of MEL having the syntax of S5. Generalized possibilistic logic (GPL) extends both possibilistic logic and MEL, and has a semantics in terms of sets of possibility distributions. After a survey of these di?erent attempts, the paper presents GPL+, a graded counterpart of MEL+ that extends MEL by allowing objective (sub)formulas. The axioms of GPL+ are graded counterparts of those of S5 modal system, the semantics being based on pairs made of an interpretation (representing the real state of facts) and a possibility distribution (representing an epistemic state). Soundness and completeness are established. The paper also discusses the diference with S5 used as a logic for rough sets that accounts for indiscernibility rather than incomplete information, using also the square of opposition as a common structure underlying modal logic, possibility theory, and rough set theory.

A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets. Further, we show that negation with the Kleene property (called the `Kleene negation') always arises from the set theoretic complement. The corresponding propositional logic is then studied through a 3-valued and rough set semantics. It is also established that Kleene negation can be considered as a modal operator, and enables giving a perp semantics to the logic. One concludes with the observation that all the semantics for this logic are equivalent.

Fuzzy rough sets have been successfully applied in classification tasks, in particular in combination with OWA operators. There has been a lot of research into adapting algorithms for use with Big Data through parallelisation, but no concrete strategy exists to design a Big Data fuzzy rough sets based classifier. Existing Big Data approaches use fuzzy rough sets for feature and prototype selection, and have often not involved very large datasets. We fill this gap by presenting the first Big Data extension of an algorithm that uses fuzzy rough sets directly to classify test instances, a distributed implementation of FRNN-OWA in Apache Spark. Through a series of systematic tests involving generated datasets, we demonstrate that it can achieve a speedup effectively equal to the number of computing cores used, meaning that it can scale to arbitrarily large datasets.