Alexei Muravitsky - Academia.edu (original) (raw)

Papers by Alexei Muravitsky

Conference on Mathematical Foundations of Informatics, 2017

The younger generation from Indian Sundarban delta is mostly opting for migrant labour jobs outsi... more The younger generation from Indian Sundarban delta is mostly opting for migrant labour jobs outside the region. • Individual educational attainment is found to have an inverse relationship with activities like fish and crab collection. Education enables the local youth to gather more information and to move outward. • A policy of incentivizing and facilitating basic education among youth in these remote rural locations may increase their livelihood resilience and help in reducing ecological stress. Disclaimer: The presentation of material and details in maps used in this book does not imply the expression of any opinion whatsoever on the part of the Publisher or Author concerning the legal status of any country, area or territory or of its authorities, or concerning the delimitation of its borders. The depiction and use of boundaries, geographic names and related data shown on maps

Larisa Maksimova on Implication, Interpolation, and Definability, 2018

We follow along the line of research begun with the pioneering work Maksimova and Rybakov (1974) ... more We follow along the line of research begun with the pioneering work Maksimova and Rybakov (1974) which has initiated comparative analysis of the lattices of normal extensions of S4 and of intermediate logics. Grounding on the Godel–McKinsey–Tarski embedding, for any S4-logic, we define the notion of \(\tau \)-decomposition and that of a modal component of the logic. Then, we investigate conditions, when a modal logic has its least modal component.

We show that the modalized Heyting calculus [2] admits a normal axiomatization. Then, we prove th... more We show that the modalized Heyting calculus [2] admits a normal axiomatization. Then, we prove that the inference rules ◻α α and ◻α → α α are admissible in this calculus. Finally, we show that this calculus and intuitionistic propositional calculus are assertorically equipollent, which leads to a variant of limited separation property for the modalized Heyting calculus.

The paper is devoted to an algebraic interpretation of Kuznetsov's theorem which establishes the ... more The paper is devoted to an algebraic interpretation of Kuznetsov's theorem which establishes the assertoric equipollence of intuitionistic and proof-intuitionistic propositional calculi. Given a Heyting algebra, we define an enrichable Heyting algebra, in which the former algebra is embedded. Moreover, we show that both algebras generate one and the same variety of Heyting algebras. This algebraic result is equivalent to the Kuznetsov theorem. The proposed construction of the enrichable "counterpart" of a given Heyting algebra allows one to observe that some properties of the original algebra are preserved by this embedding in the counterpart algebra.

arXiv: Logic, 2019

During his brief life, the Polish mathematician and logician Adolf Lindenbaum (1904--1941) contri... more During his brief life, the Polish mathematician and logician Adolf Lindenbaum (1904--1941) contributed to mathematical logic, among other things, by several significant achievements. Some results of Lindenbaum's, which bear his name, were published without proofs by other people from the Lvov--Warsaw School and the proofs later were provided by some others, though the authorship of Lindenbaum has never been challenged. Many may have heard about Lindenbaum's lemma, asserting the existence of Lindenbaum's extension, and Lindenbaum-Tarski algebra; less known is Lindenbaum's logical matrix. This tutorial is devoted to the two last concepts rather than the first one. However, the latter can be understood in a purely algebraic fashion, if one employs the notion of Lindenbaum-Tarski algebra. In general, the notions of Lindenbaum matrix and Lindenbaum-Tarski algebra have paved a way to further algebraization of logic, which had been begun by George Boole in the 19th century,...

These results are a contribution to the model theory of matrix consequence. We give a semantic ch... more These results are a contribution to the model theory of matrix consequence. We give a semantic characterization of uniform and couniform consequence relations. These properties have never been treated individually, at least in a semantic manner. We consider these notions from a purely semantic point of view and separately, introducing the notion of a uniform bundle/atlas and that of a couniform class of logical matrices. Then, we show that any uniform bundle defines a uniform consequence; and if a structural consequence is uniform, then its Lindenbaum atlas is uniform. Thus, any structural consequence is uniform if, and only if, it is determined by a uniform bundle/atlas. On the other hand, any couniform set of matrices defines a couniform structural consequence. Also, the Lindenbaum atlas of a couniform structural consequence is couniform. Thus, any structural consequence is couniform if, and only if, it is determined by a couniform bundle/atlas. We then apply these observations to...

We consider the representation of each extension of the modal logic S4 as sum of two components. ... more We consider the representation of each extension of the modal logic S4 as sum of two components. The first component in such a representation is always included in Grzegorczyk logic and hence contains "modal resources" of the logic in question, while the second one uses essentially the resources of a corresponding intermediate logic. We prove some results towards the conjecture that every S4-logic has a representation with the least component of the first kind.

Innovations and Advances in Computer Sciences and Engineering, 2009

A security solution (SS) is a mechanism, process or procedure to handle security problems for an ... more A security solution (SS) is a mechanism, process or procedure to handle security problems for an organization. The decision makers are responsible to choose and implement the SS for their organizations. For the selection of a decision, handling of information plays a very important role. The decision makers collect information both in explicit and implicit form, then take their decision based on trusting or distrusting that collected information. The way of collecting information and the way of using it are not well structured for them. Sometimes they do know how to collect information, but do not collect and analyze information in a structural way while making their security solution decisions (SSDs). Very often they collect information as knowledge, experience, and recommendation in both forms (explicit and implicit). This paper focuses on SSDs and in particular, how information is gathered and used in such decision processes. This paper also works on trust, how trust can reflect the status of a certain piece of information based on knowledge, experience, and recommendation. This paper conducts a survey to investigate how the decision makers (experienced and inexperienced participants in the survey) use empirical data (explicit information) and their knowledge and experience (implicit information) to deal with SSDs. The survey further studies the effect of implicit information in the answers provided by the experienced participates and observes that the variation in the answers provided by the experienced participants is larger than the answers provided by the inexperienced participants.

Mathematical Notes of the Academy of Sciences of the USSR, 1984

Results on the completeness of superintuitionistic logics (see, for example, [I, 2]) have apparen... more Results on the completeness of superintuitionistic logics (see, for example, [I, 2]) have apparently never been connected (as in the case of modal logics) with the hope of extending the property of completeness to all superintuiti0nistic logics. The development of these results was delayed further by the appearance in 1977 of an incomplete superintuition~ istic logic (see [3]). A partially ordered Fine set with weak signification was used in [3] as a sufficient distinguishing medium. It should be said that the logic of the model used in [3], containing an infinite antichain, is not modellable. Below we show that in a certain sense such a model cannot be simpler. As a corollary, we obtain the results on completeness from [i, 2]. Problems on completeness, connected with the above (inparticular, countable modellability), are discussed in [4]. 0. Basic Definitions~ We fix a propositional language with connectives & (conjunction), \/ (disjunction),-+ (material implication) and J_ (falsity symbol), in which the formulae (A, B, C, ...) are constructed in the usual way, starting with a set v (not necessarily countable) of propositional variables. As usual, an (intuitionistie) model is a pair (~, n), where ~= (w,~)is some nonempty partially ordered set, and ~ is a mapping from V to ~*, where ~* in its turn is the set of all subsets of the set w which are closed with respect to increase in the sense of the ordering-~, and are sometimes called cones. Thus The mapping ~, called in future a signification on ~, satisfies the usual semantic rules of Gzhegorehik and Kripke, which assign a truth value to formulas at points of w (symbolically:

Following A. Kuznetsov's outline, we restore Kuznetsov's syntactic proof of the assertoric equipo... more Following A. Kuznetsov's outline, we restore Kuznetsov's syntactic proof of the assertoric equipollence of the intuitionistic propositional calculus and the proof-intuitionistic calculus KM (Kuznetsov's Theorem). Then, we show that this property is true for a broad class of modal logics on an intuitionistic basis, which includes, e.g., the modalized Heyting calculus mHC. The last fact is one of two key properties necessary for the commutativity of a diagram involving the lattices of normal extensions of four well-known logics. Also, we give an algebraic interpretation of the assertoric equipollence for subsystems of KM.

The book is devoted to the study of the field of application of the method, which arose from the ... more The book is devoted to the study of the field of application of the method, which arose from the concept of the Lindenbaum matrix by A. Lindenbaum and the Lindenbaum theorem, within the framework of the concept of a consequence relation by A. Tarski and in the context of the conception of separating tools by A. Kuznetsov. The unifying term Tarski-Lindenbaum method is intended to refer to the first two headings as the key topics of this study. Our implementation of the Tarski-Lindenbaum method aims to emphasize the role of the conception of separating tools.

arXiv.org, 2019

The paper is devoted to an algebraic interpretation of Kuznetsov's theorem which established ... more The paper is devoted to an algebraic interpretation of Kuznetsov's theorem which established the assertoric equipollence of intuitionistic and proof-intuitionistic propositional calculi. Given a Heyting algebra, we define an enrichable Heyting algebra, in which the former algebra is embedded; moreover, we show that both algebras generate one and the same variety of Heyting algebras. This algebraic result is equivalent to the Kuznetsov theorem. The proposed construction of an enrichable extension of a given Heyting algebra allows one to observe some properties which can be preserved in the passage from the given algebra to the proposed enrichable extension of it. Keywords: intuitionistic propositional logic, proof-intuitionistic logic (KM), Heyting algebra

This is my chapter contributed to the the Springer series Outstanding Contributions to Logic, vol... more This is my chapter contributed to the the Springer series Outstanding Contributions to Logic, vol. 15, dedicated to Larisa Maksimova.

Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without... more Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to Abstract algebraic logic.

We develop a multi-valued logic approach to data base design, that are tolerant to inconsistent i... more We develop a multi-valued logic approach to data base design, that are tolerant to inconsistent information. Also, we discuss possible knowledge transformers in one type of such data bases.

Fundamenta Informaticae

We propose a framework in terms of domain theory for semantic information models. We show how an ... more We propose a framework in terms of domain theory for semantic information models. We show how an artificial agent (the computer) can operate within such a model in a multiple attitude environment (fuzziness) where information is conveyed. We illustrate our approach by two examples — taking as the set of the degrees of reliability Kleene’s 3-valued strong logic and Belnap-Dunn’s 4-valued logic.

Mathematics of the USSR-Sbornik

Logica Universalis, 2017

We show that the lattices of the normal extensions of four well-known logics (propositional intui... more We show that the lattices of the normal extensions of four well-known logics (propositional intuitionistic logic Int, Grzegorczyk logic Grz, modalized Heyting calculus mHC and K4.Grz) can be joined in a commutative diagram. One connection of this diagram is an isomorphism between the lattices of the normal extensions of mHC and K4.Grz; we show some preservation properties of this isomorphism. Two other connections are join semilattice epimorphims of the lattice of the normal extensions of mHC onto that of Int and of the lattice of the normal extensions of K4.Grz onto that of Grz.