Algebraic Logic Research Articles

THE NEW BASIC REALIZATIONS OF OPERATIONS “EQUIVALENCE” OF NEURO-FUZZY AND BIOINSPIRED NEURO-LOGICS TO CREATE HARDWARE ACCELERATORS OF ADVANCED EQUIVALENTAL MODELS OF NEURAL STRUCTURES AND MACHINE VISION SYSTEMS

The perspective of neural networks equivalental models (EM) base on vector-matrix procedure with basic operations of continuous and neuro-fuzzy logic (equivalence, absolute difference) are shown. Capacity on base EMs exceeded the amount of neurons in 4-10 times. This is larger than others neural networks paradigms. Amount neurons of this neural networks on base EMs may be 10 – 100 thousand. The base operations in EMs are normalized equivalence operations. The family of new operations “equivalence” and “non-equivalence” of neuro-fuzzy logic’s, which we have elaborated on the based of such generalized operations of fuzzy-logic’s as fuzzy negation, t-norm and s-norm are shown. Generalized rules of construction of new functions (operations) “equivalence” which uses operations of t-norm and s-norm to fuzzy negation are proposed. Despite the wide variety of types of operations on fuzzy sets and fuzzy relations and the related variety of new synthesized equivalence operations based on them, it is possible and necessary to select basic operations, taking into account their functional completeness in the corresponding algebras of continuous logic, as well as their most effective circuitry implementations. Among these elements the following should be underlined: 1) the element which fulfills the operation of limited difference; 2) the element which algebraic product (intensifier with controlled coefficient of transmission or multiplier of analog signals); 3) the element which fulfills a sample summarizing (uniting) of signals (including the one during normalizing). The basic element of pixel cells for the construction of hardware accelerators EM NM is a node on the current-reflecting mirrors (CM), which implements the operation of a limited difference (LD) of continuous logic (CL). Synthesized structures which realize on the basic of these elements the whole spectrum of required operations: t-norm, s-norm and new operations – “equivalence” are shown. These realizations on the basic of CMOS transistors current mirror represent the circuit with analog and time-pulse optical input signals. Possibilities of “equivalence” circuits synthesis by such functions limited difference cells are shown. Such circuits consist of several dozen CMOS transistors, have low power supply voltage (1.8…3.3V), the range of an input photocurrent is 0.1…24 μA, the transformation time is less than 1 μs, low power consumption (microwatts). The circuits and the simulation results of their design with OrCAD are shown.

Fuzzy Logical Algebra and Study of the Effectiveness of Medications for COVID-19

A fuzzy logical algebra has diverse applications in various domains such as engineering, economics, environment, medicine, and so on. However, the existing techniques in algebra do not apply to delta-algebra. Therefore, the purpose of this paper was to investigate new types of cubic soft algebras and study their applications, the representation of cubic soft sets with δ-algebras, and new types of cubic soft algebras, such as cubic soft δ-subalgebra based on the parameter λ (λ-CSδ-SA) and cubic soft δ-subalgebra (CSδ-SA) over η. This study explains why the P-union is not really a soft cubic δ-subalgebra of two soft cubic δ-subalgebras. We also reveal that any R/P-cubic soft subsets of (CSδ-SA) is not necessarily (CSδ-SA). Furthermore, we present the required conditions to prove that the R-union of two members is (CSδ-SA) if each one of them is (CSδ-SA). To illustrate our assumptions, the proposed (CSδ-SA) is applied to study the effectiveness of medications for COVID-19 using the python program.

Cybernetic and economic mathematical models of energy autarky in the context of integral ecology tasks

At the beginning of the creation of science, philosophers also tried to define life itself and the product of its evolution, i.e. man. The cybernetic definition of life is considered one of the best because it is above religions. Of all the religions of the world, physics is "the most religious," and again, on the other hand, empirical physics in tandem with experimental chemistry is the most anti-religious. Integral ecology is this good straight path of a methodological approach to solving problems and technological failures. Among all possible relations there is and the class of all possible sets consisting of objects. Unfortunately, there is no axiomatization for second-order logic. The calculation of the truths of this logic cannot be automated. Moving on to reasoning around multi-valued logic, based on Flaszen's statement that every absurdity has its own logic, we see how colorful, not black and white logic is! A third logical value often appears between true and false. Nature, its conditions cause the manifestation of evolution. It is still difficult for philosophers to say where it comes from, they even say that it comes from outer space, then a lot of mysteries where it comes from, even on nearby planets? This is why social life is under the constant scrutiny of not only philosophers, but also physicists, biophysicists, chemists, etc. They try to establish their views on changes in the state of the Earth, on the appearance of proteins, living organisms. At the same time, some physicists build thermodynamic models to characterize entropy and negentropy and their role in evolution, and then the role of negentropy in generating revolutions and wars. This is another utopia and just as communism or capitalism are anti-human. Communist theories and most other theories fail. In order to develop massifs of data taken from social life, the practical application of the principles of set theory is required. This is where algebraic logic comes into play. It is used to organize mathematical proofs. Polish scientists implement multi-valued logics. At the same time, most representatives of social sciences and even lawyers forget about the logic of simple sentences. There is a hypothetical factor in the prognosis of energy and financial sovereignty, etc. The manifestation of known phenomena can be described in simplification and roughly as an area of logical ignorance. But again: logic is the science of verifying and falsifying hypotheses. We will develop this thesis further along with the methods of proving. We will introduce the concept of propositional logic. Sentences can be of various types - statements, questions, orders, etc. Claims may be true or untrue. For commands and questions, determining their truthfulness often does not make sense, because they can define actions and objects. Only autarkic models of economic sectors, for example energy, can help to apply static modeling in financial and even economic bifurcation, in the conditions of a market economy. What can we say about modeling the state of macroeconomics.

DEFINITION OF THE CLASS OF INFORMATION OBJECTS OF CONSTRUCTION

Building the definition of the class of an information object in construction based on automated tools is a separate complex task. The work generalizes the concept of a task in terms of information environments for construction. A certain approach to the presentation of such concepts as the problem of construction, the problem of construction, ontology of construction, idea, concept, classification of problems, current directions, objects included in the formulation of the problem, taking into account the specifics of the problems solved within the framework of the developed information technologies in construction problems. In the work, the means and methods of information theory are determined and investigated, in fact, the tasks of construction, design, modeling, monitoring and their totality within the framework of the problems of the construction industry. The relevance of the task from a theoretical point of view is determined by the expansion of the ontology of construction as a science; from an applied point of view, it is determined by ensuring the possibility of accurate formalization of regulatory information and documentation in construction. The main difference between the study, the results of which are presented in this work, is cognitive-semantic analysis based on category theory, mathematical logic and universal algebra, algebra of sets, algebra of tuples and relational algebra, namely: the construction of knowledge representation in open languages in the construction of information relations sources – tasks – classes of information objects.

Seurat games on Stockmeyer graphs

AbstractWe define a family of vertex colouring games played over a pair of graphs or digraphs by players and . These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number such that always has a winning strategy in the game with colours whenever . This is related to the reconstruction conjecture for graphs and the degree‐associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with for graphs, and the same is true for the degree‐associated reconstruction conjecture and our conjecture for digraphs. We show (for any ) that the 2‐colour game can distinguish certain nonisomorphic pairs of graphs that cannot be distinguished by the ‐dimensional Weisfeiler–Leman algorithm. We also show that the 2‐colour game can distinguish the nonisomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.

Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted

We evaluate the usefulness of holographic stabilizer codes for practical purposes by studying their allowed sets of fault-tolerantly implementable gates. We treat them as subsystem codes and show that the set of transversally implementable logical operations is contained in the Clifford group for sufficiently localized logical subsystems. As well as proving this concretely for several specific codes, we argue that this restriction naturally arises in any stabilizer subsystem code that comes close to capturing certain properties of holography. We extend these results to approximate encodings, locality-preserving gates, certain codes whose logical algebras have non-trivial centers, and discuss cases where restrictions can be made to other levels of the Clifford hierarchy. A few auxiliary results may also be of interest, including a general definition of entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code.

Steps and traces

Abstract In the theory of coalgebras, trace semantics can be defined in various distinct ways, including through algebraic logics, the Kleisli category of a monad or its Eilenberg–Moore category. This paper elaborates two new unifying ideas: (i) coalgebraic,draftrules trace semantics is naturally presented in terms of corecursive algebras, and (ii) all three approaches arise as instances of the same abstract setting. Our perspective puts the different approaches under a common roof and allows to derive conditions under which some of them coincide.

Weakly Free Multialgebras

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \(\Sigma\), there exist algebras over \(\Sigma\) which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \(\Sigma\)-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of \(\Sigma\)-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \(\mathcal{U}\), from the category of \(\Sigma\)-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \(\mathcal{U}\) does not have a left adjoint.

General model of a transport-technological grain store route node for control systems programs

The paper concerns a problem to develop software for engineering control systems based upon standard industrial controllers and SCADA means at enterprises for grain storage and processing. The problem generalizing approaches to develop algorithms controlling equipment as a part of technological routes has been considered. A system to control transport-technological grain route at grain storage is the research object. Control of transport-technological equipment at a route level is the research subject. 
 Purpose is to develop general control model for nodes of a network of transport-technological grain storage routes to formalize tasks of software creation for industry specific engineering systems.
 Analysis, classification, and generalization of processing equipment have been carried out; approaches to implement functional structures of elevator routes have been studied. General model to control transport-technological grain storage route has been proposed in the form of a graph of states. Transitional conditions in the form of algebraic logic have been developed for this graph as well as algorithms to control certain mechanisms in terms of transitional states of the route.
 The control model developed in the form of graph involving transitional conditions and algorithms of transitional states has obtained its software implementation; moreover, it has been introduced in the context of industry specific engineering control systems at operating enterprises of grain storage and processing. The introduction results have demonstrated the model stability under the conditions of different enterprises engaged in grain storage and processing from the viewpoint of its implementation in the system software.
 The realized field experiments have demonstrated the system effectiveness, which allows to recommend it to solve the problems of software development for engineering control systems based upon standard industrial controllers and SCADA.

Omitting types algebraically and more about amalgamation for modal cylindric algebras

AbstractLet α be an arbitrary infinite ordinal, and . In [26] we studied—using algebraic logic—interpolation and amalgamation for an extension of first order logic, call it , with α many variables, using a modal operator of a unimodal logic that contributes to the semantics. Our algebraic apparatus was the class of modal cylindric algebras. Modal cylindric algebras, briefly , are cylindric algebras of dimension α, expanded with unary modalities inheriting their semantics from a unimodal logic such as , or . When modal cylindric algebras based on are just cylindric algebras, that is to say, . This paper is a sequel to [26], where we study algebraically other properties of . We study completeness and omitting types (s) for s by proving several representability results for so‐called dimension complemented and locally finite . Furthermore, we study the notion of atom‐canonicity for , the variety of n‐dimensional modal cylindric algebras. Atom canonicity, a well known persistence property in modal logic, is studied in connection to for , which is restricted to the first n variables. We further continue our study of interpolation in [26] for algebraizable extensions of by studying using both algebraic logic and category theory. Our main results on are Theorems 3.7, 4.4 & 4.6, while our main results on amalgamation are Theorems 5.7, 5.10, 5.13 & 5.16.

Residuation in finite posets

Abstract When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.

Boolean logic algebra driven similarity measure for text based applications.

In Information Retrieval (IR), Data Mining (DM), and Machine Learning (ML), similarity measures have been widely used for text clustering and classification. The similarity measure is the cornerstone upon which the performance of most DM and ML algorithms is completely dependent. Thus, till now, the endeavor in literature for an effective and efficient similarity measure is still immature. Some recently-proposed similarity measures were effective, but have a complex design and suffer from inefficiencies. This work, therefore, develops an effective and efficient similarity measure of a simplistic design for text-based applications. The measure developed in this work is driven by Boolean logic algebra basics (BLAB-SM), which aims at effectively reaching the desired accuracy at the fastest run time as compared to the recently developed state-of-the-art measures. Using the term frequency–inverse document frequency (TF-IDF) schema, the K-nearest neighbor (KNN), and the K-means clustering algorithm, a comprehensive evaluation is presented. The evaluation has been experimentally performed for BLAB-SM against seven similarity measures on two most-popular datasets, Reuters-21 and Web-KB. The experimental results illustrate that BLAB-SM is not only more efficient but also significantly more effective than state-of-the-art similarity measures on both classification and clustering tasks.

On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results

Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n<\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.

RESEARCH OF DATA TYPE CLASSIFICATION METHODS WHEN DEVELOPING COMPUTER ENGINEERING SOFTWARE

The paper deals with the problems of increasing the efficiency of software development, in particular, the issue of reducing the time for developing programs and using automated synthesis of programs, which will avoid the revision of the original product. The software should be tested along with other system components in all combinations that may occur. Testing is time-consuming because hidden bugs are revealed through unexpected interactions between software components. With structural analysis, data flow diagrams are not the end result, they are a developer tool. First, diagrams are built, and then mechanisms are developed to ensure the required system behavior. A graphical approach to solving the problem of automation of software development is being developed, based on the involvement of visual forms of program presentation. For any program object, you can select a finite number of states in which it is at each moment of time. The program progress is associated with the transition of an object from one state to another. The graph replaces the textual form of the description of the program algorithm, while the visual representation of the algorithm is realized. The specification of data structures, as well as the setting of intermodular interfaces according to data, is separated from the description of the structure of the algorithm and controls. Basic modules and data types are used. Basic modules are local calculable functions, on the basis of which all other technology objects are generated. Data types describe the syntactic and semantic aspects of constructing data used in base functions. Algorithms for finding routes on directed graphs are considered. When defining routes from the root vertex to the final ones, the properties of the algebra of three-valued logic were used. Based on the considered approach, as well as taking into account its shortcomings, a method for classifying data types was proposed, based on the implementation of a partial enumeration of the routes of the graph of program links and a method for designing software based on it, taking into account minimizing the time and cost of the project.
 Keywords: software, computer engineering, information systems, components, partial enumeration of graph routes, development costs.

Fragments of Quasi-Nelson: The Algebraizable Core

Abstract This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.

On ultrafilter extensions of first-order models and ultrafilter interpretations

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jonsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Muller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Cech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Muller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456 ). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017).

EVENT-DRIVEN MANAGEMENT OF ECONOMICS AND THE EXIT OF ECONOMICS FROM STAGNATION

The work describes the developed cortege and mathematical model for managing the quality of development and exit of the system from stagnation. The scientific problem of economics development is formulated on the basis of artificial intelligence, algebra of logic and logical-probabilistic calculus.

A MLP-Hedge-Algebras Admittance Controller for Physical Human–Robot Interaction

Recently, the identification of inertia and damping matrices (IIDM) and safety issues, as well as natural cooperation, are interestingly considered to enhance the quality of the physical human–robot interaction (pHRI). To cover all of these issues, advanced admittance controllers, such as those based on fuzzy logic or hedge algebras, have been formulated and successfully applied in several industrial problems. However, the inference mechanism of those kinds of controllers causes the discreteness of the super surface describing the input–output relationship in the Cartesian coordinates. As a consequence, the quality of the safe-natural cooperation between humans and robots is negatively affected. This paper presents an alternative admittance controller for pHRI by using a combination of hedge algebras and multilayer perceptron neural network (MLP), whose purpose is to create a more accurate inference mechanism for the admittance controller. To our best knowledge, this is the first time that such a neural network is considered for the inference mechanism of hedge algebras and also the first time that such an admittance controller is used for pHRI. The proposed admittance controller is verified on a teaching task using a 6-DOF manipulator. Experimental results have shown that the proposed method provides better cooperation compared with previous methods.

FIBRED ALGEBRAIC SEMANTICS FOR A VARIETY OF NON-CLASSICAL FIRST-ORDER LOGICS AND TOPOLOGICAL LOGICAL TRANSLATION

AbstractLawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural (or Lambekian) algebraic logic and hyperdoctrinal (or Lawverian) categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics.

Development of Resilient Control Systems for Technological Plants

The aims of the research are increasing resiliency of industrial process control systems, the de-velopment of methods of creating resilient industrial process control systems and creating a resil-ient control system of the distillation column for the production of food alcohol. The set of ob-jectives were achieved without the use of hardware backup, but using the developed algorithms which were based on the use of the model of dynamics of technological plants in the software of the control system and modern theory of multivariable automatic control systems. The opti-mal linear-quadratic controller with disturbances model and a state observer was used as a full controller. The most important result was improvement of the survivability of industrial process control systems using the developed algorithm. Also, the resilient distillation column control sys-tem was developed. For the further researches significant results are formulas which allow de-fining new mode values of controls in a distillation column at failure event of corresponding control output channels and formulas of logic algebra for a choice of optimum configuration of control system of a distillation column of food spirit industry. The basis of the developed tech-nique was reconfiguration of the control system with inclusion, instead of the failed, additional devices, which were not being used in a current configuration, but were carried out the similar function taking into account dynamics coupling of a technological plant. The significance of these results is that they can be applied to the development of cyber-production systems that do not require permanent maintenance by personnel.