Algebraic Logic Research Articles

On Subtractive Derivations of Rl‐Monoids

This paper is intended to introduce the subtractive derivations and study some of their algebraic properties on R‐monoids. Also, we give some characterizations of subtractive derivations on the Gödel center. Moreover, Gödel algebras are characterized by a fixed set of subtractive derivations. Finally, we discuss the relationship between subtractive derivations and other derivations for R‐monoids. These results of the paper can provide the common properties of subtractive derivations in the t‐norm‐based fuzzy logical algebras.

Amalgamation and Robinson property in universal algebraic logic

Abstract There is a well-established correspondence between interpolation and amalgamation for algebraizable logics that satisfy certain additional assumptions. In this paper, we introduce the Robinson property of a logic and show that a conditionally algebraizable logic without any additional assumptions has the Robinson property if and only if the corresponding class of Lindenbaum–Tarski algebras has the amalgamation property. Moreover, we give the logical characterization of the strong amalgamation property, solving an open problem of Andréka–Németi–Sain. It is also shown that given the mentioned extra assumptions the Robinson property implies the interpolation property. As conditionally algebraizable logics cover algebraizable logics as well as various quantifier logics such as classical first order logic, our results yield a generalization of some of the results concerning interpolation and amalgamation.

ON THE THEORY OF SYNTHESIS OF MINIMAL SCHEMES OF SYSTEMS CONTROL OF HYDRAULIC AND PNEUMATIC DRIVES

Showed the strict compliance of the scientific direction "Synthesis of minimum control schemes of hydraulic and pneumatic drive systems" developed by the author with the point of view of general algebra, algebra of logic, graph theory and automata theory. The synthesis of the minimum graph of operations, which is a mathematical model of the control system, has been proved. The legitimacy of the methods of undivided decomposition of equations describing the scheme of the control system has been proved. The control system is considered as a cyclic Moore finite automaton. By a cyclic automaton (CA) we will understand the mathematical model of a device designed to control cyclic processes, which are a set of technological operations performed in a certain sequence. In this regard, the automaton at each clock necessarily passes into some new state, and for a finite number of cycles the target reaches any state, and its graph contains a contour, covering all states. In general, the CA may contain several circuits, so that each circuit is interpreted either as one of the possible sequences of technological operations due to the corresponding mode of operation, or as an independent and simultaneous execution of a number of sets of technological operations. A sequential decomposition of the CA is presented in order to represent it by the sequential operation of automata with one internal state. Such a consideration of the function of transitions will naturally lead to a decrease in the number of elements in the implementation of the CA. The study will be subjected to the CA, the graph of which consists of a single circuit, since the results obtained are easily generalized to multi-circuit CA. Obtaining a breakdown of the states of a cyclic automaton in the manner indicated above is performed directly according to any automaton description without any additional calculations, tables and other constructions.

Lesson on the theme "Elements of circuitry. Logic diagrams"

The article presents one of the lessons of the "Elements of algebra of logic" section of the informatics course for 10th grade. The content of the lesson corresponds to the teaching materials on informatics by L. L. Bosova, A. Yu. Bosova. The lesson studies such concepts as a logical element (gate), a logical device, a logical circuit, an inverter, a conjunctor, a disjunctor. A feature of the lesson is the use of thematic videos (with and without sound) for viewing in order to study new material and for self-voicing. Organized frontal, independent and group work is aimed at studying the ways of compiling logical formulas for logical circuits and compiling logical circuits for given logical formulas. During the lesson, cadets draw up a summary on the topic and carry out self-assessment using a checklist.

AN ABSTRACT ALGEBRAIC LOGIC STUDY OF DA COSTA’S LOGIC AND SOME OF ITS PARACONSISTENT EXTENSIONS

AbstractTwo famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$ . On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as , , or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$ , with , , and .

Weak Inflationary BL-Algebras and Filters of Inflationary (Pseudo) General Residuated Lattices

After the research on naBL-algebras gained by the non-associative t-norms and overlap functions, inflationary BL-algebras were also studied as a recent kind of non-associative generalization of BL-algebras, which can be obtained by general overlap functions. In this paper, we show that not every inflationary general overlap function can induce an inflationary BL-algebra by a counterexample and thus propose the new concept of weak inflationary BL-algebras. We prove that each inflationary general overlap function corresponds to a weak inflationary BL-algebra; therefore, two mistaken results in the previous paper are revised. In addition, some properties satisfied by weak inflationary BL-algebras are discussed, and the relationships among some non-classical logic algebras are analyzed. Finally, we establish the theory of filters and quotient algebras of inflationary general residuated lattice (IGRL) and inflationary pseudo-general residuated lattice (IPGRL), and characterize the properties of some kinds of IGRLs and IPGRLs by naBL-filters, (weak) inflationary BL-filters, and weak inflationary pseudo-BL-filters.

Anti-hesitant Fuzzy Subalgebras, Ideals and Deductive Systems of Hilbert Algebras

The Hilbert algebra, one of several algebraic structures, was first described by Diego in 1966 [7] and has since been extensively studied by other mathematicians. Torra [18] was the first to suggest the idea of hesitant fuzzy sets (HFSs) in 2010, which is a generalization of the fuzzy sets defined by Zadeh [20] in 1965 as a function from a reference set to a power set of the unit interval. The significance of the ideas of hesitant fuzzy subalgebras, ideals, and filters in the study of the different logical algebras aroused our interest in applying these concepts to Hilbert algebras. In this paper, the concepts of HFSs to subalgebras (SAs), ideals (IDs), and deductive systems (DSs) of Hilbert algebras are introduced in terms of anti-types. We call them anti-hesitant fuzzy subalgebras (AHFSAs), anti-hesitant fuzzy ideals (AHFIDs), and anti-hesitant fuzzy deductive systems (AHFDSs). The relationships between AHFSAs, AHFIDs, and AHFDSs and their lower and strong level subsets are provided. As a result of the study, we found their generalization as follows: every AHFID of a Hilbert algebra Ω is an AHFSA and an AHFDS of Ω. We also study and find the conditions for the complement of an HFS to be an AHFSA, an AHFID, and an AHFDS. In addition, the relationships between the complements of AHFSAs, AHFIDs, and AHFDSs and their upper and strong level subsets are also provided.

Pseudo General Overlap Functions and Weak Inflationary Pseudo BL-Algebras

General overlap functions are generalized on the basis of overlap functions, which have better application effects in classification problems, and the (weak) inflationary BL-algebras as the related algebraic structure were also studied. However, general overlap functions are a class of aggregation operators, and their commutativity puts certain restrictions on them. In this article, we first propose the notion of pseudo general overlap functions as a non-commutative generalization of general overlap functions, so as to extend their application range, then illustrate their relationship with several other commonly used aggregation functions, and characterize some construction methods. Secondly, the residuated implications induced by inflationary pseudo general overlap functions are discussed, and some examples are given. Then, on this basis, we show the definitions of inflationary pseudo general residuated lattices (IPGRLs) and weak inflationary pseudo BL-algebras, and explain that the weak inflationary pseudo BL-algebras can be gained by the inflationary pseudo general overlap functions. Moreover, they are more extensive algebraic structures, thus enriching the content of existing non-classical logical algebra. Finally, their related properties and their relations with some algebraic structures such as non-commutative residuated lattice-ordered groupoids are investigated. The legend reveals IPGRLs include all non-commutative algebraic structures involved in the article.

Complexity of arbitrary functions of the logic algebra of a small number of variables

Complexity of arbitrary functions of the logic algebra of a small number of variables

Equivalence between Varieties of Łukasiewicz–Moisil Algebras and Rings

Abstract The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order $p$ and $p$-rings, $p$ prime and lift this result to the axled Łukasiewicz–Moisil algebra $L \cong B_s \times P$ and the ring $\prod ^s F_2 \times \prod ^l F_p$, where $B_s$ is a Boolean algebra of order $2^s$, $P$ a $p$-valued Post algebra of order $p^l$ and $F_p$ is the prime field of order $p$.

A combinatorial property of logical algebras

Logic gives a technique for the articial intelligence to make the computers simulate human being in dealing with certainty and uncertainty in information. Various logical algebras have been proposed and researched as the semantical systems of non-classical logical systems. In this paper using the concept of commutator, we introduce the Engel algebra and then study a condition on infinite subsets of infinite algebras. We also show that some logical algebras satisfy to this condition but do not have the properties associated with that condition.

Matrix expression of finite BZMVdM-algebra

In this paper, we provide a systematic characterization of finite BZMVdM-algebra by using semi-tensor product of matrices. The abstract operation law about logic of the finite algebra is transformed into the simple operation of concrete logical matrices. In addition, we study some properties of BZMVdM-algebra, such as homomorphism, isomorphism, and the product of the BZMVdM-algebra. Through logical matrix operation, the direct verifiable conditions for detecting the above properties are given.

On Implicative Derivations of MTL-Algebras

This paper introduces the implicative derivations and gives some of their characterizations on MTL-algebras. Furthermore, we provide some representation of MTL-algebras by implicative derivations and obtain some representation of Boolean algebra via the algebra of all implicative derivations. Finally, we explore the relationship between implicative derivation and other operators on MTL-algebras and show that there exists a bijection between the sets of multiplier and implicative derivations on IMTL-algebras. The results of this paper can provide the common properties of implicative derivations in the t-norm-based fuzzy logical algebras.

A Brief Note On QS/BP/BOI-Algebras

In this paper, the three types of the logical algebra of interest are QS-algebras, BOI-algebras, and BP-algebras, which are continuously studied by algebraists. Since the terms for each of the above logical algebras appear to be independent of each other, it may be understood that they are different. Therefore, the main purpose of this brief note is to show that QS-algebras, BOI-algebras, and BP-algebras satisfying (BM) as one.

Non-Deterministic Matrices: Theory and Applications to Algebraic Semantics

AbstractWe call multioperation any operation that return for even argument a set of values instead of a single value. Through multioperations we can define an algebraic structure equipped with at least one multioperation. This kind of structure is called multialgebra. The study of them began in 1934 with the publication of a paper of Marty. In the realm of Logic, multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) and used as semantics tool for characterizing some logics which cannot be characterized by a single finite matrix. Carnielli and Coniglio introduced the semantics of swap structures for LFIs (Logics of Formal Inconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron’s semantics. In this thesis, we will introduce a new method of algebraization of logics based on multialgebras and swap structures that is similar to classical algebraization method of Lindenbaum-Tarski, but more extensive because it can be applied to systems such that some operators are non-congruential. In particular, this method will be applied to a family of non-normal modal logics and to some LFIs that are not algebraizable by the very general techniques introduced by Blok and Pigozzi. We also will obtain representation theorems for some LFIs and we will prove that, within out approach, the classes of swap structures for some axiomatic extensions of mbC are a subclass of the class of swap structures for the logic mbC.Abstract prepared by Ana Claudia de Jesus Golzio.E-mail: anaclaudiagolzio@yahoo.com.brURL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322436

(<img src=image/13426382_01.gif>)-Anti-Intuitionistic Fuzzy Soft b-Ideals in BCK/BCI-Algebras

Among many algebraic structures, algebras of logic form an essential class of algebras. BCK and BCI-algebras are two classes of logical algebras. They were introduced by Imai and Iséki [6, 7] in 1966 and have been extensively investigated by many researchers. The concept of fuzzy soft sets is introduced in [17] to generalize standard soft sets [21]. The concept of intuitionistic fuzzy soft sets is introduced by Maji et al. [18], which is based on a combination of the intuitionistic fuzzy set [2] and soft set models. The first section will discuss the origins and importance of studies in this article. Section 2 will review the definitions of a BCK/BCI-algebra, a soft set, a fuzzy soft set, and an intuitionistic fuzzy soft set and show the essential properties of BCK/BCI-algebras to be applied in the next section. In Section 3, the concept of an anti-intuitionistic fuzzy soft b-ideal (AIFSBI) is discussed in BCK/BCI-algebras, and essential properties are provided. A set of conditions is provided for an AIFSBI to be an anti-intuitionistic fuzzy soft ideal (AIFSI). The definition of quasi-coincidence of an intuitionistic fuzzy soft point with an intuitionistic fuzzy soft set (IFSS) is considered in a more general form. In Section 4, the concepts of an (<img src=image/13426382_01.gif>)-AFSBI and an (<img src=image/13426382_01.gif>)-AIFSBI of <img src=image/13426382_02.gif> are introduced, and some characterizations of (<img src=image/13426382_01.gif>)-AIFSBI are discussed using the concept of an AIFSBI with thresholds. Finally, conditions are given for a (<img src=image/13426382_01.gif>)-AIFSBI to be a (∈,∈)-AIFSBI.

Note on the signed occurrences of propositional variables

This note concerns the positive and negative occurrences of propositional variables. Just like the theory of infectious truth-values provides an algebraic understanding of the position according to which identity of subject-matter between two formulas can approximated syntactically by the identity of propositional variables occurring in these formulas, we develop an algebraic understanding of the similar position which considers signed occurrence (i.e., positive or negative) instead of mere occurrence. We apply our framework to classical logic, yielding this first (to our knowledge) semantic characterisation of the logic called SCL by Hornischer (2020). Moreover, we settle two conjectures by Humberstone (2014) which use signed occurrences to study the equational logic of the power algebra of the two-valued Boolean algebra.

Monteiro's algebraic notion of maximal consistent theory for Tarskian logics

Antonio Monteiro gave a characterization of maximal congruences in certain semisimple varieties in order to present a representation theorem for them. Under specific conditions, this theorem can be presented with the same proof for every semisimple variety considered by him. In this note, we show that this notion of maximal congruence is closely linked to Henkin's notion of maximal theories for certain families of logics from the literature of algebraic logic. To see this relation, we study the class of n-valued Hilbert algebras with supremum enriched with Moisil operators. For this class of algebras, we present a sound and complete propositional and first-order calculus. Moreover, we show how this relation works for logics from semisimple varieties studied in the Monteiro's school. Extending the scope of applications, we present soundness and completeness results for some first-order paraconsistent logics through non-deterministic matrix semantics. Despite the fact that these logics are not algebraizable with the Blok-Pigozzi's method, they display an algebraic behaviour that allows us to give a simplified presentation.

Обзор логических базисов и микросхем при построении комбинационного устройства с учётом надёжности

The fundamental laws of the positive algebra of logic are considered, including the rules relating to the elements of equivalence and non-equivalence. Logic gates used in practice are presented. The whole set of standard schemes is implemented, in particular, the frontal version, minimized, in the "OR–NOT" basis, in the "AND–NOT" basis based on the initial logical dependence: ¬(¬(A¬B) × ¬(¬CD) + ¬(¬AB) × ¬(C¬D)) + ¬(¬(AB) × ¬(CD) + ¬(¬A¬B) × ¬(¬C¬D)) . A combination device based on K176LE5 chips has been designed. It is noted that the implementation of this function requires at least 3 chips in the Pier basis, which is not optimal in terms of reliability and weight and size indicators. The transition from the minimized variant to the basis of "Non-Equivalence" + "Conjunction", as well as the basis of "Equivalence" + "Pierce function" is carried out. The results of the search for existing chips that implement the functions "Exclusive OR", "AND", "Exclusive OR–NOT" and "OR-NOT" are presented. 4 groups of chips were found, such as: chips based on logic with emitter coupling (ECL), based on transistor-transistor logic (TTL), based on a series of CMOS chips: with a supply voltage of 5-15 V, and also with a supply voltage of 3-5 V. The corresponding descriptions of each of the MC10H1xx chips (group 1 series); K155xx, xx74xx (group 2 series); K561xx, CD40xxxx (group 3 series); SN74LVCxxx (group 4 series) are presented. Implementation options for these groups of chips in both bases are presented. The cost analysis of the applied microcircuits is made. The calculation of the final cost of products is presented. The conclusion is made about the expediency of using one or another group of microcircuits, depending on the initial requirements set.

Universal Algebraic Geometry: Syntax and Semantics

In this paper, we give a general insight into the ideas that make ground for the developing of universal algebraic geometry and logical geometry. We specify the role of algebraic logic as one of the major instruments of the whole theory. The problem of the sameness of geometries of algebraic and definable sets for different algebras is considered as ans illuminating example how algebra, geometry, model theory, and algebraic logic work together.