Recent zbMATH articles in MSC 03https://zbmath.org/atom/cc/032022-09-13T20:28:31.338867ZUnknown authorWerkzeugTransition to advanced mathematicshttps://zbmath.org/1491.000012022-09-13T20:28:31.338867Z"Diedrichs, Danilo R."https://zbmath.org/authors/?q=ai:diedrichs.danilo-r"Lovett, Stephen"https://zbmath.org/authors/?q=ai:lovett.stephen-tPublisher's description: This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics.
The authors implement the practice recommended by the Committee on the Undergraduate Program in Mathematics (CUPM) curriculum guide, that a modern mathematics program should include cognitive goals and offer a broad perspective of the discipline.
Part I offers:
\begin{itemize}
\item An introduction to logic and set theory.
\item Proof methods as a vehicle leading to topics useful for analysis, topology, algebra, and probability.
\item Many illustrated examples, often drawing on what students already know, that minimize conversation about ``doing proofs''.
\item An appendix that provides an annotated rubric with feedback codes for assessing proof writing.
\end{itemize}
Part II presents the context and culture aspects of the transition experience, including:
\begin{itemize}
\item 21st century mathematics, including the current mathematical culture, vocations, and careers.
\item History and philosophical issues in mathematics.
\item Approaching, reading, and learning from journal articles and other primary sources.
\item Mathematical writing and typesetting in LaTeX.
\end{itemize}
Together, these Parts provide a complete introduction to modern mathematics, both in content and practice.A structure-oriented construction of the classical number domains. With a view to order structures, algebraic and topological structureshttps://zbmath.org/1491.000022022-09-13T20:28:31.338867Z"Maurer, Christian"https://zbmath.org/authors/?q=ai:maurer.christianPublisher's description: Dieses Buch entwickelt systematisch die Konstruktion der klassischen Zahlenbereiche mit Blick auf die wichtigsten mathematischen Strukturen: Ordnungsstrukturen, algebraische Strukturen und topologische Strukturen. Kurze Zusammenfassungen je Kapitel/Abschnitt erleichtern die Übersicht und das Verinnerlichen der Inhalte. Das Buch bietet einen weit vernetzten Überblick über die fachwissenschaftlichen Grundlagen und deren zentrale Zusammenhänge; damit dient es Studierenden im Fach- und insbesondere im Lehramtsstudium Mathematik für Grundschule und Sekundarstufe I als wertvolle Ergänzung und Begleitung während der ersten Semester. Lehrende finden hier eine Alternative zum klassischen Einstieg ins Studium. Darüber hinaus ist das Buch auch für Quereinsteiger -- etwa Lehrkräfte anderer Fächer -- zur berufsbegleitenden Weiterbildung geeignet.Book review of: C. J. Posy, Mathematical intuitionismhttps://zbmath.org/1491.000102022-09-13T20:28:31.338867Z"Cook, Roy T."https://zbmath.org/authors/?q=ai:cook.roy-tReview of [Zbl 1465.03001].Book review of: K. Hossack, Knowledge and the philosophy of number. What numbers are and how they are knownhttps://zbmath.org/1491.000112022-09-13T20:28:31.338867Z"Franklin, James"https://zbmath.org/authors/?q=ai:franklin.jamesReview of [Zbl 1451.00002].Book review of: P. Rusnock and J. Šebestík, Bernard Bolzano. His life and his workhttps://zbmath.org/1491.000142022-09-13T20:28:31.338867Z"Lapointe, Sandra"https://zbmath.org/authors/?q=ai:lapointe.sandraReview of [Zbl 1410.01026].Book review of: P. Weingartner (ed.) and H.-P. Leeb (ed.), Kreisel's interests. On the foundations of logic and mathematicshttps://zbmath.org/1491.000192022-09-13T20:28:31.338867Z"Prawitz, Dag"https://zbmath.org/authors/?q=ai:prawitz.dagReview of [Zbl 1456.03005].Book review of: V. Benci and M. Di Nasso, How to measure the infinite. Mathematics with infinite and infinitesimal numbershttps://zbmath.org/1491.000342022-09-13T20:28:31.338867Z"Wenmackers, Sylvia"https://zbmath.org/authors/?q=ai:wenmackers.sylviaReview of [Zbl 1429.26001].Some paradoxes of infinity revisitedhttps://zbmath.org/1491.000382022-09-13T20:28:31.338867Z"Sergeyev, Yaroslav D."https://zbmath.org/authors/?q=ai:sergeyev.yaroslav-dSummary: In this article, some classical paradoxes of infinity such as Galileo's paradox, Hilbert's paradox of the Grand Hotel, Thomson's lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirahã, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light.
Editorial remark: For more information on the notion of grossone, introduced by \textit{Y. D. Sergeyev}, see [Arithmetic of infinity. Cosenza: Edizioni Orizzonti Meridionali (2003; Zbl 1076.03048); EMS Surv. Math. Sci. 4, No. 2, 219--320 (2017; Zbl 1390.03048)]; see also [\textit{A. E. Gutman} and \textit{S. S. Kutateladze}, Sib. Mat. Zh. 49, No. 5, 1054--1076 (2008; Zbl 1224.03045); translation in Sib. Math. J. 49, No. 5, 835--841 (2008)].George Spencer Brown's \textit{Laws of form} between mathematics and philosophy. Substance -- genesis -- influencehttps://zbmath.org/1491.010042022-09-13T20:28:31.338867Z"Rathgeb, Martin"https://zbmath.org/authors/?q=ai:rathgeb.martinPublisher's description: With [Laws of form. London: George Allen and Unwin Ltd (1969; Zbl 0207.00701)], \textit{George Spencer-Brown} presented a calculus of indication that can partly be viewed as axiomatization of the structure of Boolean Algebras.
The first section of the present thesis deals with the CONTENT of this calculus of indications and thus can be regarded in the paradigm of ``mathematics as structure''. Here, the calculus will be examined as system of propositions in relation of mathematics as product. This approach is based on common proof theory and will in turn be contrasted with an alternative approach based upon equational logic that shows to grasp the text far better.
The calculus of indications starts out with a two-valued arithmetic and opens to considerations of extension by so-called Brownian values. In the meantime, various inner and extra-mathematical applications of this calculus and its extensions have been put forward. It is not my intention to further these extensions by another contribution.
Instead, the second section of my thesis deals with the GENESIS of the calculus and thus can be related to the paradigm ``mathematics as activity''. In the context of mathematics as process, the calculus will be examined for the potentialities of its language. Influenced by Josef Simon's Zeichenphilosophie I will firstly discuss the book reversely, starting in the middle, and will then in turn concretize the genesis from a linguistic perspective according to Ladislav Kvasz starting at the beginning.
The mathematical essay, Laws of Form, can especially be regarded as mathematics in a nutshell and as a proposal of mathematical foundations. The laws of forms mentioned in the title can then be interpreted as laws of sign usage, especially of indication and distinction. Both, indication and distinction, are discussed in terms of their methodical application.
The last section of this thesis deals then with the VALIDITY of the mathematics discussed by Brown and relates it to the discourse concerned with philosophy and question of meaning. Thus, I will analyze the conception of the text on the basis of Peter Reisinger's Rationalitätstypologie, its beginning by means of the philosophy of mathematics by Pirmin Stekeler-Weithofer and subsequently Spencer-Brown's mathematics in the light of Josef Simon's Zeichenphilosophie.The epistemology from within. Blends in honor of Hourya Benis-Sinaceurhttps://zbmath.org/1491.010072022-09-13T20:28:31.338867ZPublisher's description: This book, which brings together historians, philosophers and mathematicians, is a tribute to the works of Hourya Benis-Sinaceur, internationally recognized specialist of history and philosophy of mathematics.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Cortois, Paul}, Towards a less truncated interpretation of the truncated work of Caveillès, 27-48 [Zbl 07583715]
\textit{Szczeciniarz, Jean-Jacques}, Hourya Benis-Sinaceur, on Cavaillès and Spinoza, 49-72 [Zbl 07583716]
\textit{Grosholz, Emily R.}, Innovation and necessity. Hourya Benis-Sinaceur's perspective on Cavaillès' work on Dedekind and Cantor, and a response to this perspective, 73-99 [Zbl 07583717]
\textit{Fernández de Castro, Max}, Frege and Dedekind, varieties of logicism, 101-140 [Zbl 07583718]
\textit{Panza, Marco}, Anzahlen and Werthverläufe: some remarks on the \S\S I. 34--40 of the Grundgesetze of Frege, 141-189 [Zbl 07583719]
\textit{Roy, Marie-Françoise}, Quantifier elimination versus Hilbert's seventeenth problem, 191-211 [Zbl 07583720]
\textit{Marquis, Jean-Pierre}, Metamathematics and categorical logic, 213-247 [Zbl 07583721]
\textit{Rashed, Roshdi}, Fermat's critique of Descartes' geometry, 251-263 [Zbl 07583722]
\textit{Rabouin, David}, Logic as ars inveniendi. A Leibnizian legacy, 265-300 [Zbl 07583723]
\textit{Durand-Richard, Marie-José}, Symbolic foundations of algorithmic practices in England (1801--1860), 301-339 [Zbl 07583724]
\textit{Haffner, Emmylou; Schlimm, Dirk}, Dedekind and the creation of the arithmetic continuum, 341-378 [Zbl 07583725]
\textit{Scholz, Erhard}, The attempt by E. Cartan to build a bridge between Einstein and the Cosserats, or how translational curvature became torsion, 379-419 [Zbl 07583726]
\textit{Eckes, Christophe}, The mathematical sources of Albert Lautman, 421-450 [Zbl 07583727]
\textit{Schwartz, Elisabeth}, A meeting in three stages, 453-474 [Zbl 07583728]
\textit{Heinzmann, Gerhard}, Thinking as we collide: two versions of anti-foundationalism, 475-484 [Zbl 07583729]
\textit{Salanskis, Jean-Michael}, History, mathematics, philosophy, 485-500 [Zbl 07583730]
\textit{Chemla, Karine}, Hourya, a testimony and a review of bodies and models, 501-513 [Zbl 07583731]
\textit{Benis-Sinaceur, Hourya}, Axiomatics and philosophy. Kant, Hilbert Vuillemin., 517-556 [Zbl 07583732]Logic works. A rigorous introduction to formal logichttps://zbmath.org/1491.030012022-09-13T20:28:31.338867Z"Falkenstein, Lorne"https://zbmath.org/authors/?q=ai:falkenstein.lorne"Stapleford, Scott"https://zbmath.org/authors/?q=ai:stapleford.scott"Kao, Molly"https://zbmath.org/authors/?q=ai:kao.mollyPublisher's description: Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines.
The book covers classical first-order logic and alternatives, including intuitionistic, free, and many-valued logic. It also considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, independent study as a course text, it covers more material than is typically covered in an introductory course. It also covers this material at greater length and in more depth with the purpose of making it accessible to those with no prior training in logic or formal systems.
Online support material includes a detailed student solutions manual with a running commentary on all starred exercises, and a set of editable slide presentations for course lectures.
Key Features
\begin{itemize}
\item Introduces an unusually broad range of topics, allowing instructors to craft courses to meet a range of various objectives
\item Adopts a critical attitude to certain classical doctrines, exposing students to alternative ways to answer philosophical questions about logic
\item Carefully considers the ways natural language both resists and lends itself to formalization
\item Makes objectual semantics for quantified logic easy, with an incremental, rule-governed approach assisted by numerous simple exercises
\item Makes important metatheoretical results accessible to introductory students through a discursive presentation of those results and by using simple case studies
\end{itemize}The elements of advanced mathematicshttps://zbmath.org/1491.030022022-09-13T20:28:31.338867Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-georgePublisher's description: This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students having learned out of early editions, the author continues to find ways to modernize and maintain a unique presentation.
What sets the book apart is the excellent writing style, exposition, and unique and thorough sets of exercises. This edition offers a more instructive preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included.
As would be expected in a fifth edition, the overall content and structure of the book are sound.
This new edition offers a more organized treatment of axiomatics. Throughout the book, there is a more careful and detailed treatment of the axioms of set theory. The rules of inference are more carefully elucidated.
Additional new features include:
\begin{itemize}
\item An emphasis on the art of proof.
\item Enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin prime conjecture, and so forth.
\item The discussion of equivalence relations is revised to present reflexivity, symmetry, and transitivity before we define equivalence relations.
\item The discussion of the RSA cryptosystem in Chapter 8 is expanded.
\item The author introduces groups much earlier. Coverage of group theory, formerly in Chapter 11, has been moved up; this is an incisive example of an axiomatic theory.
\end{itemize}
Recognizing new ideas, the author has enhanced the overall presentation to create a fifth edition of this classic and widely-used textbook.
See the reviews of the first, second and third editions in [Zbl 0860.03001; Zbl 0988.03002; Zbl 1243.03001]. For the fourth edition see [Zbl 1375.03002].One true logic. A monist manifestohttps://zbmath.org/1491.030032022-09-13T20:28:31.338867Z"Griffiths, Owen"https://zbmath.org/authors/?q=ai:griffiths.owen"Paseau, A. C."https://zbmath.org/authors/?q=ai:paseau.alexander-christopherPublisher's description: Logical monism is the claim that there is a single correct logic, the `one true logic' of our title. The view has evident appeal, as it reflects assumptions made in ordinary reasoning as well as in mathematics, the sciences, and the law. In all these spheres, we tend to believe that there are determinate facts about the validity of arguments. Despite its evident appeal, however, logical monism must meet two challenges. The first is the challenge from logical pluralism, according to which there is more than one correct logic. The second challenge is to determine which form of logical monism is the correct one.
One True Logic is the first monograph to explicitly articulate a version of logical monism and defend it against the first challenge. It provides a critical overview of the monism vs pluralism debate and argues for the former. It also responds to the second challenge by defending a particular monism, based on a highly infinitary logic. It breaks new ground on a number of fronts and unifies disparate discussions in the philosophical and logical literature. In particular, it generalises the Tarski-Sher criterion of logicality, provides a novel defence of this generalisation, offers a clear new argument for the logicality of infinitary logic and replies to recent pluralist arguments.A unified approach to structural limits and limits of graphs with bounded tree-depthhttps://zbmath.org/1491.030042022-09-13T20:28:31.338867Z"Nešetřil, Jaroslav"https://zbmath.org/authors/?q=ai:nesetril.jaroslav"Ossona de Mendez, Patrice"https://zbmath.org/authors/?q=ai:ossona-de-mendez.patriceThis monograph sets up a framework for studying limits of infinite sequences of finite relational structures, such as graphs, by associating, when possible, a ``limit object'', called modeling, to such a sequence. This framework generalizes the frameworks considered by graph theorists when studying the notions of graphon and graphing. To say something about the main results we need to know something about three central concepts of this framework.
A sequence of finite structures \((A_n)_{n \in \mathbb{N}}\) (having the same finite relational signature) is FO-convergent if for every first-order formula \(\varphi(x_1, \ldots, x_k)\), the proportion of \(k\)-tuples in \(A_n\) which satisfy \(\varphi\) converges as \(n\) tends to infinity.
A modeling is a relational structure with the additional features that its domain is a Borel space equipped with a probability measure and every first-order definable relation is measurable in the corresponding product \(\sigma\)-algebra.
A modeling \(M\) is called a modeling FO-limit of an FO-convergent sequence of finite structures \((A_n)_{n \in \mathbb{N}}\) if for every first-order formula the proportion of tuples in \(A_n\) which satisfy it converges to the probability (according to \(M\)) of the relation defined by that formula in \(M\).
The main definitions and results are collected in the first section of the monograph. The first of these states that if \(\mathcal{C}\) is a monotone class of finite graphs such that every \(FO\)-convergent sequence of graphs from \(\mathcal{C}\) has a modeling FO-limit, then \(\mathcal{C}\) is nowhere dense. It follows that there are FO-convergent sequences of graphs without a modeling FO-limit. It is conjectured that the converse implication also holds and Section 5.2 in the appendix reports on recent progress towards verifying the conjecture.
The remaining main results are concerned with (colored) graphs: (a) Every FO-convergent sequence of finite graphs with a fixed maximum degree has a modeling FO-limit, and (b) every FO-convergent sequence of finite colored trees with a fixed maximum height has a modeling FO-limit. The statement (b) is generalized (see Theorem 4.3.6), via the use of interpretations, to say roughly that (c) every FO-convergent sequence of finite colored graphs with fixed maximum tree height has a modeling FO-limit. Conjecture 1.3 proposes conditions under which a modeling should be a modeling FO-limit of an FO-convergent sequence of finite graphs of with fixed maximum degree. In the case of the statements (b) and (c), this monograph proves converses of the implications. More precisely: If \(M\) is a modeling such that its underlying first-order structure is a rooted colored tree with finite height and if \(M\) satisfies the Finitary Mass Transport Principle (FMTP) (Definition 4.20), then \(M\) is a modeling FO-limit of a sequence of finite rooted colored trees. Via the use of interpretations this result, which is technically hard to prove, can be generalized (in Theorem 4.3.6) to roughly the following: If \(M\) is a modeling that can be interpreted in a modeling \(T\) with the FMTP such that the underlying first-order structure of \(T\) is a rooted colored tree, then \(M\) is the modeling FO-limit of a sequence of finite graphs with a fixed bound on the tree height.
Section 2 develops general methods (for possible future use) involving (Lindenbaum-Tarski) Boolean algebras, Stone spaces, Ehrenfeucht-Fraisse games, the Gaifmain locality theorem and the notion of an interpretation of one structure in another. Section 3 studies modelings and their relationship to the concepts of Vapnik-Chervonenkis dimension, nowhere denseness, and random freeness. Section 4 studies sequences and limits of (colored) graphs with bounded tree-height, in particular rooted colored trees with bounded height, and the results mentioned above about such structures are proved here. The final Section 5 discusses open problems related to the theory developed in the monograph.
Reviewer: Vera Koponen (Uppsala)Thinking and calculating. Essays in logic, its history and its philosophical applications in honour of Massimo Mugnaihttps://zbmath.org/1491.030052022-09-13T20:28:31.338867ZPublisher's description: This volume collects 22 essays on the history of logic written by outstanding specialists in the field. The book was originally prompted by the 2018--2019 celebrations in honor of Massimo Mugnai, a world-renowned historian of logic, whose contributions on Medieval and Modern logic, and to the understanding of the logical writings of Leibniz in particular, have shaped the field in the last four decades. Given the large number of recent contributions in the history of logic that have some connections or debts with Mugnai's work, the editors have attempted to produce a volume showing the vastness of the development of logic throughout the centuries. We hope that such a volume may help both the specialist and the student to realize the complexity of the history of logic, the large array of problems that were touched by the discipline, and the manifold relations that logic entertained with other subjects in the course of the centuries. The contributions of the volume, in fact, span from Antiquity to the Modern Age, from semantics to linguistics and proof theory, from the discussion of technical problems to deep metaphysical questions, and in it the history of logic is kept in dialogue with the history of mathematics, economics, and the moral sciences at large.
The articles of this volume will be reviewed individually.Categories for the working philosopherhttps://zbmath.org/1491.030062022-09-13T20:28:31.338867ZThe publication of \textit{S. Mac Lane} [Categories for the working mathematician. New York-Heidelberg-Berlin: Springer- Verlag (1971; Zbl 0232.18001)] was a monumental event in the history of category theory, convincing the reader that there had emerged a well-established body of material regarded as basic category theory. The book transmitted a meta-mathematical lingua franca to every mathematician.
The present book seeks to convey the fundamental concepts of category theory to a broad audience, especially philosophers. The book consists of 18 chapters.
\begin{itemize}
\item[1.] The role of set theories in mathematics by Colin McLarty
\item[2.] Revisiting the philosophy of geometry by David Corfield
\item[3.] Homotopy type theory: a synthetic approach to higher equalities by Michael Shulman
\item[4.] Structuralism, invariance, and univalence by Steve Awodey
\item[5.] Category theory and foundations by Michael Ernst
\item[6.] Canonical maps by Jean-Pierre Marquis
\item[7.] Categorical logic and model theory by John L. Bell
\item[8.] Unfolding FOLDs: a foundational framework for abstract mathematical concepts by Jean-Pierre Marquis
\item[9.] Categories and modalities by Kohei Kishida
\item[10.] Proof theory of the cut rule by J. R. B. Cockett and R. A. G. Seely
\item[11.] Contextuality: at the borders of paradox by Samson Abramsky
\item[12.] Categorical quantum mechanics. I :causal quantum processes by Bob Coecke and Aleks Kissinger
\item[13.] Category theory and the foundations of classical space-time theories by James Owen Weatherall
\item[14.] Six-dimensional Lorentz geometry by Joachim Lambek
\item[15.] Applications of categories to biology and cognition by Andrée Ehresmann
\item[16.] Categories as mathematical models by David I. Spivak
\item[17.] Categories of scientific theories by Hans Halvorson and Dimitris Tsementzis
\item[18.] Structural realism and category mistakes by Elaine Landry
\end{itemize}
The first six chapters (Chapters 1--6) are concerned with the meaning of category theory for the foundations of mathematics. Chapter 1 discusses how the notion of set differs in ZFC (Zermelo-Fraenkel set theory with the axiom of choice) and Lawvere's ETCS (elementary theory of the category of sets). Chapter 2 argues for category theory and homotopy type theory (HoTT) as direct successors of Klein's Erlangen Program for geometry. Chapter 3 elucidates the origins of HoTT, while Chapter 4 presents the univalence axiom of HoTT as the ultimate form of structuralism. Chapter 5 aims to provide a condensed summary of the foundational debate, considering the primary motivations for advocating categorical foundations. Chapter 6 argues that the notion of canonical maps illustrates how category theory provides deep insight into the structural character of mathematics.
The second five chapters (Chapters 7--11) are concerned with the relationships between category theory and logic. Chapter 7 summarizes the advances in categorical logic from its scratch in the 1960s up to roughly 1990, addressing Lawvere's functorial semantics and the topos-theoretic description of logic. Chapter 8 discusses ideas by Michael Makkai, who developed categorical model theory together with Gonzalo Reyes and others. Chapter 9 makes use of category theory to describe modal logic and its semantics, where the syntax is a functor and the models are certain natural transformations, while Lawvere views the syntax of logic as a category and its models as functors from this category to some target category. Chapter 10 explains how symmetric monoidal categories are to be used to develop a categorical semantics for linear logic. Chapter 11 analyzes the border-crossing notion of contextuality in terms of the local/global distinction.
The concluding seven chapters (Chapters 12--18) are concerned with the application of category theory to physics and the relevance of category theory for philosphy of science. Chapter 12 reviews the work of the so-called Oxford school, beginning with a process ontology and framing quantum mechanics as a category-theoretic theory of systems, processes and their interactions. The authors use the formalism to argue for comparibility between quantum and relativity theory, allowing of recovering a notion of causality which is to be used to establish the no-signaling theorem. Chapter 13 discusses how statements about differences between various fragments of physical theories are to be articulated in the language of forgetful functor. Chapter 14 presents a three-dimensional time to achieve a six-dimensional Lorentz geometry, wherein spacetime, including basic physical quantities, are represented by six vectors forming the objects of an additive category in which Lorentz transformations appear as arrows, allowing for a unifying account of the structure of both known and unknown particles. Chapter 15 uses category theory to present global dynamical models for living systems, framing a memory evolution system defined by a hierarchy of configuration categories with parital transition functors between them. Chapter 16 aims to give philosophers an intuitive idea about how category theory is to be thought of as a universal modeling language in which the relationships between objects are paramount. Chapter 17 describes how syntactic and semantics categories are to be associated with certain logical theories, using topos-theoretic technieques to address the relationship between such categories. Chapter 18 discusses uses and abuses of category theory within the metaphysics of science, in particular, with respect to radical ontic structural realism.
Reviewer: Hirokazu Nishimura (Tsukuba)2021 European summer meeting of the Association for Symbolic Logic (Logic Colloquium '21). Adam Mickiewicz University Poznań, Poland, July 19--24, 2021https://zbmath.org/1491.030072022-09-13T20:28:31.338867ZAbstracts of invited and contributed talks.A missed opportunity. The intuition of the dialectic in Brouwer's intuitionismhttps://zbmath.org/1491.030082022-09-13T20:28:31.338867Z"Keller, Olivier"https://zbmath.org/authors/?q=ai:keller.olivierSummary: Au cours des débats provoqués par la crise des fondements qui secoue le monde des mathématiciens depuis le début du XX\(^e\) siècle, le mathématicien hollandais Luitzen Egbertus Jan Brouwer (1881--1966) s'est rendu célébre comme chef de file du courant dit ``intuitionniste'', avec comme adversaire principal David Hilbert (1862--1943), che revenir sur ce débat déjà maintes fois décrit et commenté, mais d'attirer l'attention sur un aspect trop méconnu de la profonde originalité de Brouwer, à savoir son intuition de l'unité des contraires en mathématiques. \par La réaction de Brouwer à la crise des fondements est peut-être la plus intéressante parmi toutes les réactions de mathématiciens ou de logiciens. Il ne s'agit pas pour lui de fabriquer une axiomatique en posant des interdits, un lit de Procuste pour une théorie déjà développée (Zermelo, Fraenkel), mais de proposer un édifice complètement nouveau; non pas de se retrancher derrière des montages de signes vides de sens (Hilbert), mais de rechercher le fondement dans une activité humaine, et donner par conséquent sa place au sein même des mathématiques au ``sujet créateur'', donc d'une certaine façon à l'histoire; non pas de tenter de régresser jusqu'à des éléments premiers purement logiques (Frege, Russell), mais de critiquer la logique traditionnelle elle-même parce que, nous dit Brouwer, cette logique provoque l'illusion de l'omniscience. Ce qui vient en premier, selon lui, ce ne sont pas des éléments -- objets, axiomes ou postulats posés --, mais une intuition active, comme nous le verons. Il est nécessaire, dit-il, de reconnaître le primat de la construction, de l'activité réelle, par opposition à l'existence d'entités découlant du principe du tiers exclu et à la sèche déduction à partir de ces supposées entités; par opposition aussi à l'existence d'entités simplement posées a priori, comme l'ensemble des parties d'un ensemble. \par Développement d'une activité humaine qui ne consiste en rien d'autre qu'en construction intellectuellement ``palpables'', histoire: autant de façons d'introduire le mouvement, donc le changement, non seulement en tant que point de vue sur les mathématiques, et ceci par la considération d'entités que ne sont pas forcément déterminées d'avance, comme dans les mathématiques classiques, mais en cours de détermination dans l'esprit du ``sujet créateur''. Mais on changeant, une entité se change, elle devient autre qu'ellemême; c'est donc de dialectique, au sens d'unité des contraires, qu'il s'agit, même si ce n'est jamais explicite chez Brouwer. C'est cette tentative totalement inédite d'introduire de fait, sinon en droit. la dialectique, qui suggère de qualifier la réaction de Brouwer à la crise comme la plus intéressante de toutes.A new algorithmic decision for categorical syllogisms via Carroll's diagramshttps://zbmath.org/1491.030092022-09-13T20:28:31.338867Z"Kircali Gursoy, Necla"https://zbmath.org/authors/?q=ai:gursoy.necla-kircali"Senturk, Ibrahim"https://zbmath.org/authors/?q=ai:senturk.ibrahim"Oner, Tahsin"https://zbmath.org/authors/?q=ai:oner.tahsin"Gursoy, Arif"https://zbmath.org/authors/?q=ai:gursoy.arifSummary: In this paper, we propose a new effective algorithm for the categorical syllogisms by using a calculus system Syllogistic Logic with Carroll Diagrams, which determines a formal approach to logical reasoning with diagrams, for representations of the fundamental Aristotelian categorical syllogisms. We show that this logical reasoning is closed under the syllogistic criterion of inference. Therefore, the calculus system is implemented to let the formalism which comprises synchronically bilateral and trilateral diagrammatical appearance and naive algorithmic nature. And also, there is no need specific knowledge or exclusive ability to understand this decision procedure as well as to use it in an algorithmic system. Consequently, the empirical contributions of this paper are to design a polynomial-time algorithm at the first time in the literature to conduce to researchers getting into the act in different areas of science used categorical syllogisms such as artificial intelligence, engineering, computer science and etc.Recursion schemes and the WMSO+U logichttps://zbmath.org/1491.030102022-09-13T20:28:31.338867Z"Parys, Paweł"https://zbmath.org/authors/?q=ai:parys.pawelSummary: We study the weak MSO logic extended by the unbounding quantifier (WMSO+U), expressing the fact that there exist arbitrarily large finite sets satisfying a given property. We prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+U.
For the entire collection see [Zbl 1381.68010].A proof of Bel'tyukov-Lipshitz theorem by quasi-quantifier elimination. I: Definitions and GCD-lemmahttps://zbmath.org/1491.030112022-09-13T20:28:31.338867Z"Starchak, M. R."https://zbmath.org/authors/?q=ai:starchak.m-rSummary: This paper is the first part of a new proof of decidability of the existential theory of the structure \(\langle \mathbb{Z} ; 0, 1, +, -, \leq , \vert \rangle \), where \(\vert\) corresponds to the binary divisibility relation. The decidability was proved independently in 1976 by \textit{A. P. Bel'tyukov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 60, 15--28 (1976; Zbl 0345.02035)] and \textit{L. Lipshitz} [Trans. Am. Math. Soc. 235, 271--283 (1978; Zbl 0374.02025)]. In 1977, \textit{V. I. Mart'yanov} [Algebra Logic 16, 395--405 (1977; Zbl 0394.03038); translation from Algebra Logika 16, 588--602 (1977)] proved an equivalent result by considering the ternary GCD predicate instead of divisibility (the predicates are interchangeable with respect to existential definability).
Generalizing in some sense the notion of quantifier elimination (QE) algorithm, we construct a quasi-QE algorithm to prove decidability of the positive existential theory of the structure \(\langle \mathbb{Z}_{ > 0} ; 1, \{ a \cdot \}_{a \in \mathbb{Z}_{ > 0}}, \operatorname{GCD}\rangle \). We reduce to the decision problem for this theory the decision problem for the existential theory of the structure \(\langle \mathbb{Z} ; 0, 1, +, -, \leq , \operatorname{GCD}\rangle \). A quasi-QE algorithm, which performs this reduction, will be constructed in the second part of the proof.
Transformations of formulas are based on a generalization of the Chinese remainder theorem to systems of the form \(\operatorname{GCD} (a_i, b_i + x) = d_i\) for every \(i \in [1..m]\), where \(a_i\), \(b_i\), \(d_i\) are some integers such that \(a_i \ne 0\), \(d_i > 0\).A proof of Bel'tyukov-Lipshitz theorem by quasi-quantifier elimination. II: The main reductionhttps://zbmath.org/1491.030122022-09-13T20:28:31.338867Z"Starchak, M. R."https://zbmath.org/authors/?q=ai:starchak.m-rSummary: This paper is the second part of a new proof of the Bel'tyukov-Lipshitz theorem, which states that the existential theory of the structure \(\left\langle \mathbb{Z};0,1, + , - , \leqslant ,| \right\rangle\) is decidable. We construct a quasi-quantifier elimination algorithm (the notion was introduced in the first part of the proof) to reduce the decision problem for the existential theory of \(\left\langle \mathbb{Z};0,1, + , - , \leqslant , \operatorname{GCD} \right\rangle\) to the decision problem for the positive existential theory of the structure \(\left\langle \mathbb{Z}_{ > 0};1,\{ a \cdot \}_{a \in\mathbb{Z}_{> 0}},\operatorname{GCD} \right\rangle \). Since the latter theory was proved decidable in the first part, this reduction completes the proof of the theorem. Analogues of two lemmas of Lipshitz's proof are used in the step of variable isolation for quasi-elimination. In the quasi-elimination step we apply GCD-lemma, which was proved in the first part.
For Part I see [the author, Vestn. St. Petersbg. Univ., Math. 54, No. 3, 264--272 (2021; Zbl 1491.03011); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 8(66), No. 3, 455--466 (2021)].Graded epistemic logic with public announcementhttps://zbmath.org/1491.030132022-09-13T20:28:31.338867Z"Benevides, Mário"https://zbmath.org/authors/?q=ai:benevides.mario-r-f"Madeira, Alexandre"https://zbmath.org/authors/?q=ai:madeira.alexandre"Martins, Manuel A."https://zbmath.org/authors/?q=ai:martins.manuel-aSummary: This work introduces a new fuzzy epistemic logic with public announcement with fuzzyness on both transitions and propositions. The interpretation of the connectives is done over the Gödel algebra and the interpretation of public announcements in this logic generalises the traditional update one. The core idea is that, the effect of a public announcement is reflected on the transitions degrees of the models. The update takes in account not only the truth degree of the announcement, at a target state, but also the degree of the transitions reaching that state. We prove the soundness of all axioms of the multi-agent epistemic logic with public announcements with respect to this graded semantics. Finally, we introduce the notion of bisimulation and prove the modal invariance property for our logic.Unification in pretabular extensions of S4https://zbmath.org/1491.030142022-09-13T20:28:31.338867Z"Bashmakov, Stepan I."https://zbmath.org/authors/?q=ai:bashmakov.stepan-igorevichThe paper is dedicated to the study of pretabular extensions of modal logic \(\mathrm{S4}\) described in [\textit{L. L. Maksimova}, Algebra Logic 14, 16--33 (1976; Zbl 0319.02019); translation from Algebra Logika 14, 28--55 (1975)]:
\[
\begin{array}{l} \mathrm{PM1 := S}4.3 + \mathcal{G}rz,\\
\mathrm{PM2} := \mathcal{G}rz + \sigma_2,\\
\mathrm{PM3} := \mathcal{G}rz + [\Box r \lor \Box(\Box r \to \sigma_2)] + (\Box\Diamond p \Leftrightarrow \Diamond\Box p),\\
\mathrm{PM4 := S}4 +[\Box p \lor \Box (\Box p \to \Box q \lor \Box \Diamond \neg q)] + (\Box \Diamond p \leftrightarrow \Diamond\Box p),\\
\mathrm{PM5 := S}5, \end{array}
\]
where \(\mathcal{G}rz := [\Box(\Box(p \to \Box p) \to p) \to p]\)\\
and \(\sigma_2 := [\Box p \lor \Box (\Box p \to \Box q \lor \Box \Diamond \neg q)]\).
It is proven that the logics \(\mathrm{PM2}\) and \(\mathrm{PM3}\) have a finitary unification type, while the logics \(\mathrm{PM1, PM4}\) and \(\mathrm{PM5}\) have a unitary unification type, and any unifiable in \(\mathrm{PM1, PM4}\) or \(\mathrm{PM5}\) formula is projective.
Reviewer: Alex Citkin (Warren)The modal logics of Kripke-Feferman truthhttps://zbmath.org/1491.030152022-09-13T20:28:31.338867Z"Nicolai, Carlo"https://zbmath.org/authors/?q=ai:nicolai.carlo"Stern, Johannes"https://zbmath.org/authors/?q=ai:stern.johannesIn response to the liar's paradox, Kripke developed the fixed-point semantics for languages with a truth predicate. The axiomatic theory ``Kripke-Feferman'' was subsequently developed by Feferman, to capture in proof-theoretic terms Kripke's model-theoretic framework. Building on previous work by the two authors [the first author, Stud. Log. 106, No. 1, 101--130 (2018; Zbl 1437.03162); the second author, Rev. Symb. Log. 7, No. 2, 299--318 (2014; Zbl 1345.03044)] the present paper discusses in depth the link between the Feferman-Kripke theory of truth and modal logic. The bridge is made through theorems establishing that a modal formula \(\phi\) is provable in a suitable system of modal logic if and only if its translation \(\phi^{\star}\) obtained by replacing \(\Box\) with the truth predicate is provable in the corresponding system of truth. This echoes Solovay's celebrated result making a similar connection between Peano arithmetic (PA) and the modal logic GL [\textit{R. M. Solovay}, Isr. J. Math. 25, 287--304 (1976; Zbl 0352.02019)]. Here the challenge is to devise a suitable semantics for the system of modal logic under consideration. The proposed semantics resembles so-called impossible world semantics, and allows for both classical and non-classical worlds. The present paper discusses at lenght and in great technical detail the above connection. The reader is invited to compare the discussion there with similar proposals, dealing with e.g. so-called revision theory [\textit{S. Standefer}, Rev. Symb. Log. 8, No. 3, 467--487 (2015; Zbl 1347.03049)].
Reviewer: Xavier Parent (Vienna)Axiomatization and polynomial solvability of strictly positive fragments of certain modal logicshttps://zbmath.org/1491.030162022-09-13T20:28:31.338867Z"Svyatlovskii, Mikhail V."https://zbmath.org/authors/?q=ai:svyatlovskii.mikhail-vSummary: The fragment of the language of modal logic that consists of all implications \(A\to B\), where \(A\) and \(B\) are built from variables, the constant \(\top\) (truth), and the connectives \(\wedge\) and \(\diamondsuit_1,\diamondsuit_2,\dots,\diamondsuit_m\). For the polymodal logic \(S5_m\) (the logic of \(m\) equivalence relations) and the logic \(K4.3\) (the logic of irreflexive linear orders), an axiomatization of such fragments is found and their algorithmic decidability in polynomial time is proved.Grounding rules for (relevant) implicationhttps://zbmath.org/1491.030172022-09-13T20:28:31.338867Z"Poggiolesi, Francesca"https://zbmath.org/authors/?q=ai:poggiolesi.francescaThe relation of \textit{grounding} -- holding between two facts or propositions when one can be explained in virtue of the other -- is one that naturally admits formal analysis (see \textit{e.g.} [\textit{F. Poggiolesi}, Boston Stud. Philos. Hist. Sci. 318, 291--309 (2016; Zbl 1436.03068)] for a survey). Some such explanatory relationships -- when a complex formula can be explained in virtue of its deduction from certain premises -- are \textit{logical} in nature, described as ``formal'' grounding in [\textit{F. Correia}, Rev. Symb. Log. 7, No. 1, 31--59 (2014; Zbl 1344.03005)]. In previous work [Synthese 193, No. 10, 3147--3167 (2016; Zbl 1380.03021)], the author has investigated a refinement of this species of the grounding grounding: ``complete and immediate'' grounding, relating a multiset of premises to a conclusion when it includes exactly the truths necessary to ground the conclusion in virtue of a single, nontrivial step in a proof. This condition reflects the author's understanding of grounding as following from joint conditions of derivability and the complexity of the derivation.
The author has recently suggested in [the author, J. Appl. Non-Class. Log. 31, No. 1, 26--55 (2021; Zbl 07368165)] that an account of the grounding of natural language conditionals---which presuppose a connection between antecedent and consequent -- can be illuminated by an account of the conditions on formal grounding in the context of \textit{relevant logics}. Relevant logics (see [\textit{A. R. Anderson} et al., Entailment. The logic of relevance and necessity. Vol. I. Princeton, NJ: Princeton University Press (1975; Zbl 0323.02030)] for a survey) characterize such a connection by insisting that validity of a conditional requires relevance between the antecedent and consequent.
The present work develops an application of the author's earlier work by giving a precise account of complete and immediate formal grounding of first-degree formulae in the relevant logic \(\mathsf{R}\) manifested by a natural deduction calculus \(\mathcal{RGD}\). This calculus is formulated in a language in which the language of propositional relevant logic is supplemented with a new binary connective \(\triangleright\) with an intended interpretation as ``\textit{completely and immediately because}.''
The author introduces a natural deduction calculus and argues that its deducibility corresponds to the intended understanding of formal explanation in \(\mathsf{R}\). This calculus and a standard natural deduction calculus for \(\mathsf{R}\) constitute the two components that, when synthesized, determines the system \(\mathcal{RGD}\). The author concludes the paper by demonstrating several formal features of \(\mathcal{RGD}\), including its conservativity over propositional \(\mathsf{R}\), its consistency, and an interesting ``grounding deduction theorem'' that \(\Gamma\triangleright\varphi\) is valid in \(\mathcal{RGD}\) (\textit{i.e.}, \(\Gamma\) is a complete and immediate grounds for \(\varphi\)) precisely when there exists an appropriate deduction in \(\mathsf{R}\) of \(\varphi\) from \(\Gamma\) (\textit{i.e.} a deduction acting as a formal explanation of \(\varphi\)).
Reviewer: Thomas Ferguson (Amsterdam/St. Andrews)Identity in Mares-Goldblatt models for quantified relevant logichttps://zbmath.org/1491.030182022-09-13T20:28:31.338867Z"Standefer, Shawn"https://zbmath.org/authors/?q=ai:standefer.shawnRelevant logics like Anderson and Belnap's \(\mathsf{R}\) described in [\textit{A. R. Anderson} et al., Entailment. The logic of relevance and necessity. Vol. I. Princeton, NJ: Princeton University Press (1975; Zbl 0323.02030)] are motivated by a thesis that valid entailments \(\varphi\rightarrow\psi\) should require some sort of relevance between the antecedent and consequent. In the propositional case, one necessary (but not necessarily sufficient) formal indicator of relevance is the \textit{variable sharing property} that an entailment's antecedent and consequent share a common propositional atom. Thus, a formula like \(p\rightarrow(q\rightarrow q)\) may be rejected on grounds of relevance despite the tautologous nature of its consequent.
When moving to more complicated languages, formal characterizations of relevance are less obvious. However, some cases of irrelevance are nevertheless recognizable. In the case of \textit{identity}, the same intuitions that cause a relevant logician to reject \(p\rightarrow(q\rightarrow q)\) in general suggest that \(p\rightarrow (t=t)\) may be rejected on similar grounds.
Having appropriate accounts of identity is particularly pressing for its importance in distinguishing \textit{relevant predication} from merely \textit{accidental} predication as discussed in [\textit{J. M. Dunn}, J. Philos. Log. 16, 347--381 (1987; Zbl 0638.03003)]. One can characterize the relevant predication of \(P(x)\) of \(t\) by the truth of the formula \(\forall x(x=t\rightarrow P(x))\). Thus, an account of \textit{e.g.} \(\mathsf{R}\) in which both quantification and identity receive appropriate treatment is important.
The author focuses two frameworks for quantified relevant logic: the \textit{Mares-Goldblatt} models described in [\textit{E. D. Mares} and \textit{R. Goldblatt}, J. Symb. Log. 71, No. 1, 163--187 (2006; Zbl 1100.03011)] and the \textit{Fine-Mares} models emerging from the works [\textit{K. Fine}, J. Philos. Log. 3, 347--372 (1974; Zbl 0296.02013); \textit{K. Fine}, J. Philos. Log. 17, No. 1, 27--59 (1988; Zbl 0646.03013); \textit{E. D. Mares}, Stud. Log. 51, No. 1, 1--20 (1992; Zbl 0788.03028)]. Although the Fine-Mares model theory includes an account of identity, the Mares-Goldblatt semantics lacks identity. The author thus investigates equipping the Mares-Goldblatt model theory with a satisfactory mechanism to capture extensions of \(\mathsf{R}\) with quantification and identity.
The author iteratively builds on the framework, first considering \(\mathsf{R}^{=}\), an extension of \(\mathsf{R}\) with an identity predicate that allows substitution of identicals in extensional contexts and proving soundness and completeness with respect to a modified model theory. Secondly, an extension \(\mathsf{R}^{=}_{sub}\) in which substitution of identicals is valid in all contexts (\textit{i.e.} in formulae in which relevant entailment \(\rightarrow\) appears) and shows soundness and completeness with respect to a modification of Mares-Goldblatt models with conditions placed on propositions. The development of the model theory concludes with an extension \(\mathsf{QR}^{=}_{sub}\) including both quantifiers and identity with substitution in general. The extension is shown to be sound and complete with respect to the author's modification of the Mares-Goldblatt model theory.
\(\mathsf{QR}^{=}_{sub}\) is important as it provides sufficient expressivity for an appropriate account of relevant predication. The author concludes by demonstrating additional value for the framework by showing that the inclusion of identity in the Mares-Goldblatt semantics exposes subtle distinctions between the approach and the Fine-Mares semantics. In particular, the author uses this point of comparison to discuss a more nuanced view of the constant domains of the Mares-Goldblatt model theory.
Reviewer: Thomas Ferguson (Amsterdam/St. Andrews)A reinterpretation of the semilattice semantics with applicationshttps://zbmath.org/1491.030192022-09-13T20:28:31.338867Z"Weiss, Yale"https://zbmath.org/authors/?q=ai:weiss.yaleThe goal of the paper is, using an idea of semilattice semantics introduced by \textit{A. Urquhart} [J. Symb. Log. 37, 159--169 (1972; Zbl 0245.02028)], for relevant logics, to construct semilattice semantics for intuitionistic propositional logic \(\mathbf{J}\) and Jankov's logic \(\mathbf{KC}\). In addition, semilattice semantics for (semi)-relevant logics \(\mathbf{SJ}\) and \(\mathbf{SKC}\) are presented.
A semilattice frame is a structure \(\mathfrak{F} = \langle S, 0, \sqcup \rangle\) where \(\langle S, \sqcup \rangle\) is a join-semilattice and \(0 \in S\) is lattice bottom element.
Let \(\mathcal{P}(S)\) be a power set of \(S\) and \(\Pi\) be a set of all propositional variables.
An intuitionistic semilattice model is a structure \(\mathfrak{M} = \langle \mathfrak{F},V \rangle\) where \(\mathfrak{F} = \langle S, 0, \sqcup \rangle\) is a semilattice frame and \(V :\Pi \cup \{\bot\} \longrightarrow \mathcal{P}(S)\) is subject to the following conditions:
\begin{itemize}
\item[1.] \(x \in V(p)\) implies \(x \sqcup y \in V(p)\)
\item[2.] \(x \in V(\bot)\) implies \(x \sqcup y \in V(\bot)\)
\item[3.] \(x \in V(\bot)\) implies \(x \in V(p)\) (for all \(p \in \Pi\))
\end{itemize}
The truth conditions for model \(\mathfrak{M}\) are defined as follows:
\begin{itemize}
\item[(a)] \(\models_x^\mathfrak{M} p\) if and only if \(x \in V(p)\)
\item[(b)] \(\models_x^\mathfrak{M} \bot\) if and only if \(x \in V(\bot)\)
\item[(c)] \(\models_x^\mathfrak{M} \varphi \land \psi\) if and only if \(\models_x^\mathfrak{M} \varphi\) and \(\models_x^\mathfrak{M} \psi\)
\item[(d)] \(\models_x^\mathfrak{M} \varphi \lor \psi\) if and only if \(\models_x^\mathfrak{M} \varphi\) or \(\models_x^\mathfrak{M} \psi\)
\item[(c)] \(\models_x^\mathfrak{M} \varphi \to \psi\) if and only if for all \(y \in S\), \(\not\models_y^\mathfrak{M} \varphi\) or \(\models_{x \sqcup y}^\mathfrak{M} \psi\)
\end{itemize}
As usual, formula \(\varphi\) is valid in model \(\mathfrak{M}\) if \(\models_0^\mathfrak{M} \varphi\). It is proven that semilattice models form a sound and complete semantics for \(\mathbf{J}\).
If we add to definition of model restriction \(x \in V(P)\) implies \(x \sqcup y \in V(p)\), we obtain a sound and complete semantics for Jankov's logic.
Reviewer: Alex Citkin (Warren)A first-order logic for reasoning about higher-order upper and lower probabilitieshttps://zbmath.org/1491.030202022-09-13T20:28:31.338867Z"Savić, Nenad"https://zbmath.org/authors/?q=ai:savic.nenad"Doder, Dragan"https://zbmath.org/authors/?q=ai:doder.dragan"Ognjanović, Zoran"https://zbmath.org/authors/?q=ai:ognjanovic.zoranSummary: We present a first-order probabilistic logic for reasoning about the uncertainty of events modeled by sets of probability measures. In our language, we have formulas that essentially say that ``according to agent Ag, for all \(x\), formula \(\alpha(x)\) holds with the lower probability at least \(\frac{1}{3}\)''. Also, the language is powerful enough to allow reasoning about higher order upper and lower probabilities. We provide corresponding Kripke-style semantics, axiomatize the logic and prove that the axiomatization is sound and strongly complete (every satisfiable set of formulas is consistent).
For the entire collection see [Zbl 1367.68004].Saturated models of first-order many-valued logicshttps://zbmath.org/1491.030212022-09-13T20:28:31.338867Z"Badia, Guillermo"https://zbmath.org/authors/?q=ai:badia.guillermo"Noguera, Carles"https://zbmath.org/authors/?q=ai:noguera.carlesSummary: This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a \(\kappa\)-saturated model, i.e. a model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski-Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of \(\kappa \)-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.\( \alpha \)-paramodulation method for a lattice-valued logic \(L_nF(X)\) with equalityhttps://zbmath.org/1491.030222022-09-13T20:28:31.338867Z"He, Xingxing"https://zbmath.org/authors/?q=ai:he.xingxing"Xu, Yang"https://zbmath.org/authors/?q=ai:xu.yang"Liu, Jun"https://zbmath.org/authors/?q=ai:liu.jun.2"Li, Yingfang"https://zbmath.org/authors/?q=ai:li.yingfangSummary: In this paper, \( \alpha \)-paramodulation and \(\alpha \)-GH paramodulation methods are proposed for handling logical formulae with equality in a lattice-valued logic \(L_nF(X)\), which has unique ability for representing and reasoning uncertain information from a logical point of view. As an extension of the authors' work [``Alpha-lock paramodulation for lattice-valued propositional logic'', in: Proceedings of the 10th international conference on intelligent systems and knowledge engineering, ISKE 2015, Taipei, Taiwan, 24--27 November 2015. Piscataway, NJ: IEEE Computer Society. 18--20 (2015; \url{doi:10.1109/ISKE.2015.90}); ``\(\alpha \)-Lock paramodulation for a lattice-valued first order logic \(L_nF(X)\)'', in: Uncertainty modelling in knowledge engineering and decision making. Proceedings of the 12th international FLINS conference, Roubaix, France, 24--26 August 2016. Singapore: World Scientific. 477--482 (2016; \url{doi:10.1142/9789813146976_0077})], a new form of \(\alpha \)-equality axioms set is proposed. The equivalence between \(\alpha \)-equality axioms set and \(E_{\alpha}\)-interpretation in \(L_nF(X)\) with an appropriate level is also established, which may provide a key foundation for equality reasoning in lattice-valued logic. Based on its equivalence, \(E_{\alpha}\)-unsatisfiability equivalent transformation is given. Furthermore, \( \alpha \)-paramodulation and its restricted method (i.e., \( \alpha \)-GH paramodulation) are given. The soundness and completeness of the proposed methods are also examined.The distributivity of extended operations on fuzzy truth valueshttps://zbmath.org/1491.030232022-09-13T20:28:31.338867Z"Liu, Zhi-Qiang"https://zbmath.org/authors/?q=ai:liu.zhiqiang"Wang, Xue-Ping"https://zbmath.org/authors/?q=ai:wang.xue-ping|wang.xueping|wang.xueping.1Summary: In this paper we give necessary and sufficient conditions for the distributivity of extended operations on fuzzy truth values. Applications are given to the distributivity problem of extended t-norms and t-conorms.Residual implications on lattice \(\mathcal{L}\) of intuitionistic truth values based on powers of continuous t-normshttps://zbmath.org/1491.030242022-09-13T20:28:31.338867Z"Singh, Vishnu"https://zbmath.org/authors/?q=ai:singh.vishnu-pratap"Mesiar, Radko"https://zbmath.org/authors/?q=ai:mesiar.radko"Dutta, Bapi"https://zbmath.org/authors/?q=ai:dutta.bapi"Goh, Mark"https://zbmath.org/authors/?q=ai:goh.markSummary: Residual implications are a special class of implications on the lattice \(\mathcal{L}\) of intuitionistic truth values which possess interesting theoretical and practical properties. Many studies have investigated the properties of the types of implications on \(\mathcal{L}\) and established the relationships among them. In this paper, the powers of the continuous t-norms \(\mathcal{T}\) on \(\mathcal{L}\) are introduced, and their properties studied. A new type of implication on \(\mathcal{L} \), termed the \(\mathcal{T}\)-power based implication, is derived from the powers of the continuous t-norms \(\mathcal{T} \), as denoted by \(\mathtt{I}_{\mathtt{I}\mathcal{T}}\) and satisfies certain properties of the residual implications defined on the interval \([0, 1]\) under certain conditions. Some important properties are analyzed. These results collectively reveal that they do not intersect the most well-known classes of fuzzy implications. Finally, we investigate the solutions of some Boolean-like laws for \(\mathtt{I}_{\mathtt{I}\mathcal{T}}\).Risk analysis via Łukasiewicz logichttps://zbmath.org/1491.030252022-09-13T20:28:31.338867Z"Vitale, Gaetano"https://zbmath.org/authors/?q=ai:vitale.gaetanoSummary: In this paper, we apply logical methods to risk analysis. We study \textit{generalized events}, i.e. not yes-no events but \textit{continuous} ones. We define on this class of events a risk function and a measure over it to analyse risk in this context. We use Riesz MV-algebras as algebraic structures and their associated logic in support of our research, thanks to their relations with other applications. Moreover, we investigate on decidability of consequence problem for our class of risk.A note on computable distinguishing coloringshttps://zbmath.org/1491.030262022-09-13T20:28:31.338867Z"Bazhenov, N."https://zbmath.org/authors/?q=ai:bazhenov.n-a"Greenberg, N."https://zbmath.org/authors/?q=ai:greenberg.noam"Melnikov, A."https://zbmath.org/authors/?q=ai:melnikov.alexander-g"Miller, R."https://zbmath.org/authors/?q=ai:miller.russell-g|miller.russel-g"Ng, K. M."https://zbmath.org/authors/?q=ai:ng.kengmengSummary: An \(\alpha \)-coloring \(\xi\) of a structure \(\mathcal{S}\) is \textit{distinguishing} if there are no nontrivial automorphisms of \(\mathcal{S}\) respecting \(\xi \). In this note we prove several results illustrating that computing the distinguishing number of a structure can be very hard in general. In contrast, we show that every computable Boolean algebra has a \(0^{\prime\prime}\)-computable distinguishing 2-coloring. We also define the notion of a computabile distinguishing 2-coloring of a separable space; we apply the new definition to separable Banach spaces.Punctual dimension of algebraic structures in certain classeshttps://zbmath.org/1491.030272022-09-13T20:28:31.338867Z"Dorzhieva, M. V."https://zbmath.org/authors/?q=ai:dorzhieva.marina-valerianovna"Issakhov, A. A."https://zbmath.org/authors/?q=ai:issakhov.a-a"Kalmurzayev, B. S."https://zbmath.org/authors/?q=ai:kalmurzaev.birzhan-s"Kornev, R. A."https://zbmath.org/authors/?q=ai:kornev.ruslan-aleksandrovich"Kotov, M. V."https://zbmath.org/authors/?q=ai:kotov.matveiSummary: Punctual presentations of algebraic structures in several familiar classes are studied. It is proved that the punctual dimension of punctual structures from the following classes is either 1 or \(\infty \): equivalence structures, linear orders, torsion-free abelian groups, abelian \(p\)-groups, Boolean algebras.Punctual categoricity relative to a computable oraclehttps://zbmath.org/1491.030282022-09-13T20:28:31.338867Z"Kalimullin, I. Sh."https://zbmath.org/authors/?q=ai:kalimullin.iskander-sh"Melnikov, A. G."https://zbmath.org/authors/?q=ai:melnikov.alexander-gSummary: We are studying the punctual structures, i.e., the primitive recursive structures on the whole set of integers. The punctual categoricity relative to a computable oracle \(f\) means that between any two punctual copies of a structure there is an isomorphism which togeteher with its inverse can be derived via primitive recursive schemes augmented with \(f\). We will prove that the punctual categoricity relative to a computable oracle can hold only for finitely generated or locally finite structures. We will show that the punctual categoricity of finitely generated structures is exhaused by the computable oracles with primitive recursive graph. We also present an example of locally finite structure where the punctual categoricity is provided by a primitive recursively bounded computable oracle.On first order rigidity for linear groupshttps://zbmath.org/1491.030292022-09-13T20:28:31.338867Z"Plotkin, Eugene"https://zbmath.org/authors/?q=ai:plotkin.eugene|plotkin.eugene-b|plotkin.eugene-iSummary: The paper is a short survey of recent developments in the area of first order descriptions of linear groups. It is aimed to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups.Fixed-point selection functionshttps://zbmath.org/1491.030302022-09-13T20:28:31.338867Z"Arslanov, M. M."https://zbmath.org/authors/?q=ai:arslanov.marat-mSummary: Let \(\simeq\) be a binary relation between sets of integers, and \(\leq_R\) be a Post reducibility, i.e. a reflexive and transitive relation between sets of integers such that if \(A\leq_RB\) then the computational complexity of recognition of elements of \(A\) is easier than (or equal to) the recognition of elements of \(B\). Suppose that for a class \({\mathcal{A}}\) of arithmetical sets, which have an effective enumeration \(\{\Omega_e\}_{e\in\omega} \), there are \(R\)-complete sets, i.e. such sets \(D\) that for any \(A\in{\mathcal{A}}, A\leq_RD\). Earlier we considered completeness criteria for such reducibilities roughly of the following type: For any \(A\in{\mathcal{A}}, A\) is \(R\)-complete if and only if there is a function \(f\), defined on \(\omega\) such that \(f\leq_RD\) and \(\Omega_{f(i)}\not\simeq\Omega_i\) for all \(i\in\omega \). This means that for any set \(A\in{\mathcal{A}} \), if it is non-complete, then any function \(f\leq_RA\) has a fixed-point \(e\): \( \Omega_{f(e)}\simeq\Omega_e\). In this paper we introduce a notion of fixed-point selection function for sequences of such sets and study their complexity characteristics.Turing degree spectra of minimal subshiftshttps://zbmath.org/1491.030312022-09-13T20:28:31.338867Z"Hochman, Michael"https://zbmath.org/authors/?q=ai:hochman.michael"Vanier, Pascal"https://zbmath.org/authors/?q=ai:vanier.pascalSummary: Subshifts are shift invariant closed subsets of \(\varSigma ^{\mathbb {Z}^d}\), with \(\varSigma\) a finite alphabet. Minimal subshifts are subshifts in which all points contain the same patterns. It has been proved by Jeandel and Vanier that the Turing degree spectra of non-periodic minimal subshifts always contain the cone of Turing degrees above any of its degrees. It was however not known whether each minimal subshift's spectrum was formed of exactly one cone or not. We construct inductively a minimal subshift whose spectrum consists of an uncountable number of cones with incomparable bases.
For the entire collection see [Zbl 1362.68016].Theories of Rogers semilattices of analytical numberingshttps://zbmath.org/1491.030322022-09-13T20:28:31.338867Z"Bazhenov, N. A."https://zbmath.org/authors/?q=ai:bazhenov.n-a"Mustafa, M."https://zbmath.org/authors/?q=ai:mustafa.manat"Tleuliyeva, Zh."https://zbmath.org/authors/?q=ai:tleuliyeva.zhSummary: The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings. Under the assumption of Projective Determinacy, we prove that for every non-zero natural number \(n\), there are infinitely many pairwise elementarily non-equivalent Rogers semilattices for \(\Sigma^1_n\)-computable families.Absolute and relative properties of negatively numbered familieshttps://zbmath.org/1491.030332022-09-13T20:28:31.338867Z"Faizrahmanov, M. Kh."https://zbmath.org/authors/?q=ai:faizrahmanov.marat-khaidarovich"Puzarenko, V. G."https://zbmath.org/authors/?q=ai:puzarenko.vadim-gSummary: We study in this paper negative \(\mathbb{A} \)-numberings where \(\mathbb{A}\) are admissible structures. We establish that a series of classical assumptions is remained to hold for negative \(\mathbb{A} \)-numberings in the case of the application of certain limits as for numberings as for admissible structures. We find examples of admissible structures \(\mathbb{A}\) whose families of all \(\Sigma \)-subsets have negative non-decidable minimal computable \(\mathbb{A} \)-numberings. The admissible structures from this series have negative computable \(\mathbb{A} \)-numberings whose numbered equivalence differs from corresponding ones of any computable \(\mathbb{A} \)-numbering of family of total functions.Unbalanced polarized relationshttps://zbmath.org/1491.030342022-09-13T20:28:31.338867Z"Garti, Shimon"https://zbmath.org/authors/?q=ai:garti.shimonRecall, that the polarized partition relation \(\binom{\alpha}{\beta}\rightarrow\binom{\gamma_0\;\;\gamma_1}{\delta_0\;\;\delta_1}\) means that for every coloring \(c:\alpha\times\beta\to 2\) there are \(A\subseteq\alpha\), \(B\subseteq\beta\) and \(i\in\{0,1\}\) such that \(\hbox{otp}(A)=\gamma_i\), \(\hbox{otp}(B)=\delta_i\) and \(c\upharpoonright (A\times B)\) is the constant \(i\). If \((\gamma_0,\delta_0)\neq (\gamma_1,\delta_1)\) then one speaks of an unbalanced relation. Otherwise, the relation is said to be balanced and one writes \(\binom{\alpha}{\beta}\rightarrow\binom{\gamma}{\delta}\),
Extending a long line of results and answering to the negative a question of \textit{P. Erdős} et al. [Acta Math. Acad. Sci. Hung. 16, 93--196 (1965; Zbl 0158.26603)], the author of the paper under review shows that consistently \(\binom{\mu^+}{\mu}\not\rightarrow\binom{\mu^+ \;\;\omega_1}{\mu\;\;\;\;\mu}\) where \(\mu\) is a strong limit cardinal of countable cofinality and \(2^\mu=\mu^+\). The theorem implies that the following earlier result of \textit{A. L. Jones} [Proc. Am. Math. Soc. 136, No. 4, 1445--1449 (2008; Zbl 1137.03028)] is optimal: If \(\mu\) is a strong limit cardinal of countable cofinality and \(2^\mu=\mu^+\), then \(\binom{\mu^+}{\mu}\rightarrow\binom{\mu^+ \;\;\tau}{\mu\;\;\;\;\mu}\) for every \(\tau\in\omega_1\). The special instance of the above stated theorem from the paper under review, in which \(\mu\) is an \(\omega\)-limit of measurable cardinals, gives a negative answer to a question of \textit{S. Garti} et al. [Electron. J. Comb. 27, No. 2, Research Paper P2.8, 11 p. (2020; Zbl 1439.05085)] and so implies that in a certain sense the following ZFC result of \textit{S. Shelah} [Fundam. Math. 155, No. 2, 153--160 (1998; Zbl 0897.03050)] is also optimal: If \(\mu\) is a singular limit of measurable cardinals and \(\tau\in \mu^+\), then \(\binom{\mu^+}{\mu}\rightarrow \binom{\tau}{\mu}\).
Reviewer: Vera Fischer (Wien)When \(P_\kappa(\lambda)\) (vaguely) resembles \(\kappa\)https://zbmath.org/1491.030352022-09-13T20:28:31.338867Z"Matet, Pierre"https://zbmath.org/authors/?q=ai:matet.pierreFor two uncountable cardinals \(\kappa,\lambda\), where \(\kappa\) is regular and \(\kappa<\lambda,\) let \(P_\kappa(\lambda)\) denotes the family of all subsets of \(\kappa\) of size less than \(\lambda\). The paper under review is devoted to finding a way of transferring certain properties of \(\kappa\) to \(P_\kappa(\lambda)\) by finding an appropriate stationary coding set.
In particular, following the ideas from the paper of \textit{W. S. Zwicker} [Contemp. Math. 31, 243--259 (1984; Zbl 0536.03030)], the author constructs (assuming the existence of a cofinal subset of \(P_\kappa(\lambda)\) of size \(\lambda\)) a stationary subset \(B\subseteq P_\kappa(\lambda)\) such that the restriction of the nonstationary ideal \(NS_{\kappa,\lambda}\) to \(B\) resembles the restriction of the nonstationary ideal \(NS_{\kappa}\) to the set \(E^\kappa_\mu=\left\{\alpha<\kappa:\alpha\text{ is limit }\&\ \text{cf}(\alpha)=\mu\right\}\) (for a regular cardinal \(\mu<\kappa\)). His construction uses an improved version of Zwicker's coding function, i.e. a function \(h:\lambda\to P_\kappa(\lambda)\) such that the range of \(h\) is cofinal in \(P_\kappa(\lambda)\). In the remaining part of the paper, different configurations of \(\kappa,\lambda\) and \(\mu\) are investigated.
Reviewer: Jan Kraszewski (Wrocław)Strongly compact cardinals and the continuum functionhttps://zbmath.org/1491.030362022-09-13T20:28:31.338867Z"Apter, Arthur W."https://zbmath.org/authors/?q=ai:apter.arthur-w"Dimopoulos, Stamatis"https://zbmath.org/authors/?q=ai:dimopoulos.stamatis"Usuba, Toshimichi"https://zbmath.org/authors/?q=ai:usuba.toshimichiIn this article, the authors investigate the behavior of the continuum function in the presence of non-supercompact strongly compact cardinal. First, they show that if \(\kappa\) is strongly compact and \(2^\kappa=\kappa^+\), then we can make \(2^\kappa\) arbitrarily large while preserving the strong compactness of \(\kappa\). Before this result, it was not known if such a model can be obtained without a stronger large cardinal assumption. Then, using the method in the proof, they prove various consistency results about strongly compact cardinals and the continuum functions.
Reviewer: Tetsuya Ishiu (Oxford, Ohio)Computational procedure for solving fuzzy equationshttps://zbmath.org/1491.030372022-09-13T20:28:31.338867Z"Abbasi, F."https://zbmath.org/authors/?q=ai:abbasi.fazlollah"Allahviranloo, T."https://zbmath.org/authors/?q=ai:allahviranloo.tofighSummary: The classical methods for solving fuzzy equations are very limited because, often, there are no solutions or very strong conditions for the equations it is placed to have a solution. In addition, the solution's support obtained in these methods is large. All of this is due to the consideration of operations related to equations based on the principle of extension, which is due to the absence of ineffective members. These high points are our motive for achieving a new approach to solving fuzzy equations. We will solve the fuzzy equations, taking into account the fuzzy operations involved in the equation based on the transmission average by \textit{F. Abbasi} et al. [J. Intell. Fuzzy Syst. 29, No. 2, 851--861 (2015; Zbl 1354.16048)]. In this paper, a computational procedure is proposed to solve the fuzzy equations that meets the defects of previous techniques, specially reluctant to question whether the answer is valid in the equation. The proposed approach is implemented on the fuzzy equations as \(AX+B=C, AX^2+BX+C=D, AX^3+BX^2=CX\). At the end, it is shown that the solution of the proposed method in comparison with other methods of solving fuzzy equations are more realistic, that is, they have smaller support.Representing uncertainty about fuzzy membership gradehttps://zbmath.org/1491.030382022-09-13T20:28:31.338867Z"Aggarwal, Manish"https://zbmath.org/authors/?q=ai:aggarwal.manishSummary: A novel uncertainty representation framework is introduced based on the inter-linkage between the inherent fuzziness and the agent's confusion in its representation. The measure of fuzziness and this confusion is considered to be directly related to the lack of distinction between membership and non-membership grades. We term the proposed structure as confidence fuzzy set (CFS). It is further generalized as generalized CFS, quasi CFS and interval-valued CFS to take into consideration the DM's individualistic bias in the representation of the underlying fuzziness. The operations on CFSs are investigated. The usefulness of CFS in multi-criteria decision making is discussed, and a real application in supplier selection is included.A fuzzy function of C-level subsetshttps://zbmath.org/1491.030392022-09-13T20:28:31.338867Z"AL-Hur Kadum, Intissar Abd"https://zbmath.org/authors/?q=ai:al-hur-kadum.intissar-abd"Abdulwahab, Azal Taha"https://zbmath.org/authors/?q=ai:abdulwahab.azal-taha"Alkfari, Batool Hatem Akar"https://zbmath.org/authors/?q=ai:alkfari.batool-hatem-akarSummary: The definition of fuzzy function concept as a function of the fuzzy C-level of the elements of fuzzy sets is considered the main goal of this work. The research is composited as follows : In section 1, we define the fuzzy concept, in section 2, we introduce basic information about extension principle and fuzzy sets. However, fuzzy function of the fuzzy C-level and a related fuzzy function to this concept are investigated and some significant results are proved in section 3.Investigation and corrigendum to some results related to \(g\)-soft equality and \(gf\)-soft equality relationshttps://zbmath.org/1491.030402022-09-13T20:28:31.338867Z"Al-Shami, Tareq M."https://zbmath.org/authors/?q=ai:al-shami.tareq-mohammedSummary: Since Molodtsov defined the concept of soft sets, many types of soft equality relations between two soft sets were discussed. Among these types are \(g\)-soft equality and \(gf\)-soft equality relations introduced in [\textit{M. Abbas} et al., ibid. 28, No. 6, 1191--1203 (2014; Zbl 1459.03086)] and [\textit{M. Abbas} et al., ibid. 31, No. 19, 5955--5964 (2017; Zbl 07459988)], respectively. In this paper, we first aim to show that some results obtained in [Zbl 1459.03086, loc. cit.; Zbl 07459988, loc. cit.] need not be true, by giving two counterexamples. Second, we investigate under what conditions these results are correct. Finally, we define and study the concepts of \(gf\)-soft union and \(gf\)-soft intersection for arbitrary family of soft sets.Temporal intuitionistic fuzzy pairshttps://zbmath.org/1491.030412022-09-13T20:28:31.338867Z"Atanassov, Krassimir"https://zbmath.org/authors/?q=ai:atanassov.krassimir-todorov"Atanassova, Vassia"https://zbmath.org/authors/?q=ai:atanassova.vassiaSummary: The concept of a Temporal Intuitionistic Fuzzy Pair (TIFP) is introduced as an extension of the concept of an intuitionistic fuzzy pair. Some geometrical interpretations of the TIFPs are given. The basic relations, operations and operators are defined over TIFPs.Generating nullnorms on some special classes of bounded lattices via closure and interior operatorshttps://zbmath.org/1491.030422022-09-13T20:28:31.338867Z"Çaylı, Gül Deniz"https://zbmath.org/authors/?q=ai:cayli.gul-denizSummary: In this article, we introduce different methods for constructing nullnorms on some special classes of bounded lattices by using closure and interior operators. As a by-product, we obtain new classes of idempotent nullnorms. Furthermore, we give some interesting examples for a better understanding of these new classes of nullnorms. In particular, the results presented here provide different approaches to the suggestion put forward by \textit{Y. Ouyang} and \textit{H.-P. Zhang} [Fuzzy Sets Syst. 395, 93--106 (2020; Zbl 1452.03120)].A study of similarity measures through the paradigm of measurement theory: the fuzzy casehttps://zbmath.org/1491.030432022-09-13T20:28:31.338867Z"Coletti, Giulianella"https://zbmath.org/authors/?q=ai:coletti.giulianella"Bouchon-Meunier, Bernadette"https://zbmath.org/authors/?q=ai:bouchon-meunier.bernadetteSummary: We extend to fuzzy similarity measures the study made for classical ones in a companion paper [ibid. 23, No. 16, 6827--6845 (2019; Zbl 1418.03156)]. Using a classic method of measurement theory introduced by Tversky, we establish necessary and sufficient conditions for the existence of a particular class of fuzzy similarity measures, representing a binary relation among pairs of objects which expresses the idea of ``no more similar than''. In this fuzzy context, the axioms are strictly dependent on the combination operators chosen to compute the union and the intersection.Fuzzy weighted attribute combinations based similarity measureshttps://zbmath.org/1491.030442022-09-13T20:28:31.338867Z"Coletti, Giulianella"https://zbmath.org/authors/?q=ai:coletti.giulianella"Petturiti, Davide"https://zbmath.org/authors/?q=ai:petturiti.davide"Vantaggi, Barbara"https://zbmath.org/authors/?q=ai:vantaggi.barbaraSummary: Some similarity measures for fuzzy subsets are introduced: they are based on fuzzy set-theoretic operations and on a weight capacity expressing the degree of contribution of each group of attributes. For such measures, the properties of dominance and \(T\)-transitivity are investigated.
For the entire collection see [Zbl 1367.68004].Generalized trapezoidal hesitant fuzzy numbers and their applications to multi criteria decision-making problemshttps://zbmath.org/1491.030452022-09-13T20:28:31.338867Z"Deli, Irfan"https://zbmath.org/authors/?q=ai:deli.irfan"Karaaslan, Faruk"https://zbmath.org/authors/?q=ai:karaaslan.farukSummary: Generalized hesitant trapezoidal fuzzy number whose membership degrees are expressed by several possible trapezoidal fuzzy numbers, is more adequate or sufficient to solve real-life decision problem than real numbers. Therefore, in this paper, to model the some multi-criteria decision-making (MCDM) problems, we define concept of generalized trapezoidal hesitant fuzzy (GTHF) number, whose membership degrees of an element to a given set are expressed by several different generalized trapezoidal fuzzy numbers in the set of real numbers \(R\). Then, we introduce some basic operational laws of GTHF-numbers and some properties of them. Also, we propose a decision-making method to solve the MCDM problems in which criteria values take the form of GTHF information. To use in proposed decision-making method, we first give definitions of some concepts such as score, standard deviation degree, deviation degree of GTHF-numbers. We second develop some GTHF aggregation operators called the GTHF-number weighted geometric operator, GTHF-number weighted arithmetic operator, GTHF-number weighted geometric operator, GTHF-number weighted arithmetic operator. Finally, we give a numerical example for proposed MCDM to validate the reasonable and applicable of the proposed method.A novel dissimilarity measure on picture fuzzy sets and its application in multi-criteria decision makinghttps://zbmath.org/1491.030462022-09-13T20:28:31.338867Z"Duong, Truong Thi Thuy"https://zbmath.org/authors/?q=ai:duong.truong-thi-thuy"Thao, Nguyen Xuan"https://zbmath.org/authors/?q=ai:nguyen-xuan-thao.Summary: With the increase in complexity of real problems, more and more, the decision makers are involved remarkably in the decision-making processes. It is required more efficient technique or tool to give reliable outcomes. This paper proposes a new dissimilarity measure on picture fuzzy sets. A multi-criteria decision-making problem is utilized to apply this new measure to select the optimal alternative. Finally, an example uses the proposed measure to evaluate and select the optimal market segment.Łukasiewicz logic and the divisible extension of probability theoryhttps://zbmath.org/1491.030472022-09-13T20:28:31.338867Z"Frič, Roman"https://zbmath.org/authors/?q=ai:fric.romanSummary: We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the ``classical core'' of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.A complete ranking method for interval-valued intuitionistic fuzzy numbers and its applications to multicriteria decision makinghttps://zbmath.org/1491.030482022-09-13T20:28:31.338867Z"Huang, Weiwei"https://zbmath.org/authors/?q=ai:huang.weiwei"Zhang, Fangwei"https://zbmath.org/authors/?q=ai:zhang.fangwei"Xu, Shihe"https://zbmath.org/authors/?q=ai:xu.shiheSummary: In this study, a complete ranking method for interval-valued intuitionistic fuzzy numbers (IVIFNs) is introduced by using a score function and three types of entropy functions. This work is motivated by the work of \textit{V. L. G. Nayagam} et al. [Soft Comput. 21, No. 23, 7077--7082 (2017; Zbl 1382.91035)] in which a novel non-hesitant score function for the theory of interval-valued intuitionistic fuzzy sets was introduced. The authors claimed that the proposed non-hesitant score function could overcome the shortcomings of some familiar methods. By using some examples, they pointed out that the non-hesitant score function is better compared with Sahin's and Zhang et al.'s approaches. It is pointed out that although in some specific cases, the cited method overcomes the shortcomings of several of the existing methods mentioned, it also created new defects that can be solved by other methods. The main aim of this study is to give a complete ranking method for IVIFNs which can rank any two arbitrary IVIFNs. At last, two examples to demonstrate the effectiveness of the proposed method are provided.Shadowed sets with higher approximation regionshttps://zbmath.org/1491.030492022-09-13T20:28:31.338867Z"Ibrahim, M. A."https://zbmath.org/authors/?q=ai:ibrahim.mohamed-a|ibrahim.mahmoud-a|ibrahim.muhammed-a|ibrahim.m-a-k|ibrahim.mohammed-ali-faya|ibrahim.mohd-asrul-hery"William-West, T. O."https://zbmath.org/authors/?q=ai:william-west.tamunokuro-opubo"Kana, A. F. D."https://zbmath.org/authors/?q=ai:kana.a-f-d"Singh, D."https://zbmath.org/authors/?q=ai:singh.d-r|singh.deep|singh.deen-dayal|singh.deshanand-p|singh.dasharath|singh.dalip|singh.david-j|singh.dilbag|singh.dharmvir|singh.deepak-kumar|singh.devender|singh.dr-d-p|singh.d-b|singh.daljeet|singh.deobrat|singh.d-v|singh.deo-karan|singh.dinesh-chandra|singh.d-m|singh.divya|singh.dhirendra-kumar|singh.dharam|singh.d-kingsly-jeba|singh.daljit|singh.dhiraj-k|singh.darshan|singh.david-e|singh.dilbaj|singh.dhiraj-kumar|singh.dhaneshwar|singh.d-n|singh.daya-s|singh.deepti|singh.devinder|singh.dhananjay|singh.dilip|singh.dharm-veer|singh.devraj|singh.dharmendra|singh.derek|singh.dipti|singh.devendra-pratap|singh.devi|singh.daroga|singh.digvijay|singh.dimple|singh.dhan-pal|singh.deeksha|singh.deepika|singh.diwakar|singh.davinder|singh.dharamender|singh.dharmveer|singh.debabrata|singh.deepa|singh.didar|singh.dwesh-k|singh.dharSummary: This paper mainly discusses three points involving shadowed set approximation of a given fuzzy set. Firstly, a principle of uncertainty balance, which guarantees that preservation of uncertainty in the induced shadowed set is studied. Secondly, an alternative formulation for determining the optimum partition thresholds of shadowed sets is suggested. This formulation helps us study principle of uncertainty balance in shadowed sets with higher approximation regions. Thirdly, five-region shadowed set, which effectively deals with the issue of uncertainty balance, is introduced. We provide a closed-form formula for determining its optimum partition thresholds and generalize it to \(n (\ge 5)\)-region shadowed sets. Finally, some examples from synthetic and real dataset are provided to demonstrate the feasibility of the suggested methods.Fuzzy \(\alpha \)-cut and related mathematical structureshttps://zbmath.org/1491.030502022-09-13T20:28:31.338867Z"Jana, Purbita"https://zbmath.org/authors/?q=ai:jana.purbita"Chakraborty, Mihir K."https://zbmath.org/authors/?q=ai:chakraborty.mihir-kumarSummary: This paper deals with the notions called fuzzy \(\alpha \)-cut, fuzzy strict \(\alpha \)-cut and their properties. Algebraic structures arising out of the family of fuzzy \(\alpha \)-cuts and fuzzy strict \(\alpha \)-cuts have been investigated. Some significance and usefulness of fuzzy \(\alpha \)-cuts are discussed.Multi-valued picture fuzzy soft sets and their applications in group decision-making problemshttps://zbmath.org/1491.030512022-09-13T20:28:31.338867Z"Jan, Naeem"https://zbmath.org/authors/?q=ai:jan.naeem"Mahmood, Tahir"https://zbmath.org/authors/?q=ai:mahmood.tahir"Zedam, Lemnaouar"https://zbmath.org/authors/?q=ai:zedam.lemnaouar"Ali, Zeeshan"https://zbmath.org/authors/?q=ai:ali.zeeshanSummary: Soft set theory initiated by Molodtsov in 1999 has been emerging as a generic mathematical tool for dealing with uncertainty. A noticeable progress is found concerning the practical use of soft set in decision-making problems. The purpose of this manuscript is to explore the novel of multi-valued picture fuzzy set (MPFS) and multi-valued picture fuzzy soft set (MPFSS) which are the generalizations of the notions of picture fuzzy soft set (PFSS) and multi-fuzzy soft set (MFSS). This notion can be used to express fuzzy information in more general and effective way. In particular, some basic operations such as union, intersection, complement and product of the proposed MPFSS are developed, and their properties are investigated. Furthermore, some aggregation operators corresponding to the proposed MPFSSs are called multi-picture fuzzy soft weighted averaging, multi-picture fuzzy soft ordered weighted averaging and multi-picture soft hybrid weighted averaging operators for a collections of MPFSSs are also developed. Moreover, based on these operators, we presented a new method to deal with the multi-attribute group decision-making problems under the multi-valued picture fuzzy soft environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods. The graphical interpretation of the explored approaches is also utilized with future directions.Fuzzy sets and presheaveshttps://zbmath.org/1491.030522022-09-13T20:28:31.338867Z"Jardine, John F."https://zbmath.org/authors/?q=ai:jardine.john-frederickSummary: This paper presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval. \par The system \(V_*(X)\) of Vietoris-Rips complexes for a data set \(X\) is both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.Quintuple implication principle on interval-valued intuitionistic fuzzy setshttps://zbmath.org/1491.030532022-09-13T20:28:31.338867Z"Jin, Jianhua"https://zbmath.org/authors/?q=ai:jin.jianhua"Ye, Mingfei"https://zbmath.org/authors/?q=ai:ye.mingfei"Pedrycz, Witold"https://zbmath.org/authors/?q=ai:pedrycz.witoldSummary: This paper mainly aims to introduce Quintuple Implication Principle (QIP) on interval-valued intuitionistic fuzzy sets (IVIFSs). Firstly, some algebraic properties of a class of interval-valued intuitionistic triangular norms are discussed in detail. In particular, a unified expression of residual interval-valued intuitionistic fuzzy implications generated by left-continuous triangular norms is presented. Secondly, Triple Implication Principles (TIPs) of both interval-valued intuitionistic fuzzy modus ponens (IVIFMP) and fuzzy modus tollens (IVIFMT) based on residual interval-valued intuitionistic fuzzy implications are analyzed. It is shown that the TIP solution of IVIFMP is recoverable, and the TIP solution of IVIFMT is only weakly local recoverable. Moreover, it sees by an illustrated example that the TIP method sometimes makes the computed solutions for IVIFMP and IVIFMT meaningless or misleading. To avoid the above shortcoming and enhance the recovery property of TIP solution of IVIFMT, QIP and \(\alpha \)-QIP for IVIFMP and IVIFMT are investigated and the corresponding expressions of solutions of them are also given, respectively. In addition, the QIP methods for IVIFMP and IVIFMT are recoverable and sound. Finally, QIP solutions of IVIFMP for multiple fuzzy rules are provided. An application example for medical diagnosis is given to illustrate the feasibility and effectiveness of the QIP of IVIFMP.Bipolar \(N\)-soft set theory with applicationshttps://zbmath.org/1491.030542022-09-13T20:28:31.338867Z"Kamacı, Hüseyin"https://zbmath.org/authors/?q=ai:kamaci.huseyin"Petchimuthu, Subramanian"https://zbmath.org/authors/?q=ai:petchimuthu.subramanianSummary: In this paper, the notion of bipolar \(N\)-soft set, which is the bipolar extension of \(N\)-soft set, and its fundamental properties are introduced. This new idea is illustrated with real-life examples. Moreover, some useful operations and products on the bipolar \(N\)-soft sets are derived. We thoroughly discuss the idempotent, commutative, associative, and distributive laws for these emerging operations and products. Also, we set forth two outstanding algorithms to handle the decision-making problems under bipolar \(N\)-soft set environments. We give potential applications and comparison analysis to demonstrate the efficiency and advantages of algorithms.A Zadeh's max-min composition operator for two 2 dimensional quadratic fuzzy numbershttps://zbmath.org/1491.030552022-09-13T20:28:31.338867Z"Kang, Chul"https://zbmath.org/authors/?q=ai:kang.chul-joong|kang.chul-goo|kang.chul-hee"Yun, Yong Sik"https://zbmath.org/authors/?q=ai:yun.yong-sikSummary: Generating triangular fuzzy numbers on \(\mathbb R\) to \(\mathbb R^2\) we define parametric operations between two regions valued a-cuts and obtain the parametric operations for two triangular fuzzy numbers defined on \(\mathbb R^2\). The results for the parametric operations are the generalization of Zadeh's extended algebraic operations. Also, we generate the quadratic fuzzy numbers on \(\mathbb R\) to \(\mathbb R^2\) and calculate the Zadeh's max-min composition operator for two 2-dimensional quadratic fuzzy numbers.Comments to \(\mathcal{N}\)-cubic sets with an NC-decision making problemhttps://zbmath.org/1491.030562022-09-13T20:28:31.338867Z"Karazma, F."https://zbmath.org/authors/?q=ai:karazma.f"Kologani, M. Aaly"https://zbmath.org/authors/?q=ai:kologani.mona-aaly"Borzooei, R. A."https://zbmath.org/authors/?q=ai:borzooei.rajab-ali"Jun, Y. B."https://zbmath.org/authors/?q=ai:jun.young-baeMeans of fuzzy numbers in the fuzzy information evaluation problemhttps://zbmath.org/1491.030572022-09-13T20:28:31.338867Z"Khatskevich, V. L."https://zbmath.org/authors/?q=ai:khatskevich.vladimir-lSummary: Based on means of systems of fuzzy numbers, we introduce and study a class of averaging functionals for the implementation of the fuzzy information evaluation problem. It is shown that these functionals have a number of special properties: idempotency, monotonicity, continuity, etc., typical of scalar aggregating functions.On the cross-migrativity of uninorms revisitedhttps://zbmath.org/1491.030582022-09-13T20:28:31.338867Z"Li, Wen-Huang"https://zbmath.org/authors/?q=ai:li.wen-huang"Qin, Feng"https://zbmath.org/authors/?q=ai:qin.feng.0Summary: Cross-migrative equation between aggregation operators (for example, t-norms) is a weaker form of the classical commuting equation. The work is dedicated to the study of cross-migrativity involving uninorms with continuous underlying operators. The investigation is presented in two separate parts: the first part focuses on the case where one of the uninorms belongs either to the set \(U_{\min}\) or \(U_{\max}\). The second one deals with the situation where both uninorms have continuous underlying operators. Full characterizations are provided.Entropy measure and TOPSIS method based on correlation coefficient using complex q-rung orthopair fuzzy information and its application to multi-attribute decision makinghttps://zbmath.org/1491.030592022-09-13T20:28:31.338867Z"Mahmood, Tahir"https://zbmath.org/authors/?q=ai:mahmood.tahir"Ali, Zeeshan"https://zbmath.org/authors/?q=ai:ali.zeeshanSummary: Entropy measure (EM) and similarity measure (SM) are important techniques in the environment of fuzzy set (FS) theory to resolve the similarity between two objects. The q-rung orthopair FS (q-ROFS) and complex FS are new extensions of FS theory and have been widely used in various fields. In this article, the EM, Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on the correlation coefficient is investigated. It is very important to study the SM of Cq-ROFS. Then, the established approaches and the existing drawbacks are compared by an example, and it is verified that the explored work can distinguish highly similar but inconsistent Cq-ROFS. Finally, to examine the reliability and feasibility of the new approaches, we illustrate an example using the TOPSIS method based on Cq-ROFS to manage a case related to the selection of firewall productions, and then, a situation concerning the security evaluation of computer systems is given to conduct the comparative analysis between the established TOPSIS method based on Cq-ROFS and previous decision-making methods for validating the advantages of the established work by comparing them with the other existing drawbacks.A novel entropy and divergence measures with multi-criteria service quality assessment using interval-valued intuitionistic fuzzy TODIM methodhttps://zbmath.org/1491.030602022-09-13T20:28:31.338867Z"Mishra, Arunodaya Raj"https://zbmath.org/authors/?q=ai:mishra.arunodaya-raj"Rani, Pratibha"https://zbmath.org/authors/?q=ai:rani.pratibha"Pardasani, Kamal Raj"https://zbmath.org/authors/?q=ai:pardasani.kamal-raj"Mardani, Abbas"https://zbmath.org/authors/?q=ai:mardani.abbas"Stević, Željko"https://zbmath.org/authors/?q=ai:stevic.zeljko"Pamučar, Dragan"https://zbmath.org/authors/?q=ai:pamucar.dragan-sSummary: Interval-valued intuitionistic fuzzy sets (IVIFSs) are proven to be the fastest growing research area and are more flexible way to handle the uncertainty. Information measures play vital role in the study of uncertain information; therefore, number of new interval-valued intuitionistic fuzzy divergence and entropy measures have been proposed in the literature and applied for different purposes. Recently, multi-criteria decision-making (MCDM) methods with IVIFSs have broadly studied by researchers and practitioners in various fields. In this paper, firstly surveys of IVIF-divergence and entropy measures are conducted and then demonstrated some counter-intuitive cases. Then, novel divergence and entropy measures are originated for IVIFSs to avoid the shortcomings of previous measures. Later on, systematic reviews of Portuguese for Interactive Multi-criteria Decision Making (TODIM) method are presented with recent fuzzy developments. Based on classical TODIM method, a new approach for MCDM is introduced under IVIF environment which considers the bounded rationality of decision makers. In the present method, the proposed entropy measure is utilized to compute the weight vector of the criteria, and the proposed divergence measure is applied in the calculation of dominance degrees. To illustrate the effectiveness of the present approach, a decision-making problem of vehicle insurance companies is presented where the evaluation values of the alternatives are given in terms of IVIF numbers. Comparison with some existing methods shows the applicability and consistency of the present method.\(L\)-valued quasi-overlap functions, \(L\)-valued overlap index, and Alexandroff's topologyhttps://zbmath.org/1491.030612022-09-13T20:28:31.338867Z"Paiva, Rui"https://zbmath.org/authors/?q=ai:paiva.rui-c|paiva.rui-pedro"Bedregal, Benjamín"https://zbmath.org/authors/?q=ai:bedregal.benjamin-rene-callejasSummary: In one recent work, the first author et al. generalized the notion of overlap functions to the context of lattices and introduced a weaker definition, called a quasi-overlap, that arises from the removal of the continuity condition [``Lattice-valued overlap and quasi-overlap functions'', Inf. Sci. 562, 180--199 (2021; \url{doi:10.1016/j.ins.2021.02.010})]. In this article, quasi-overlap functions on lattices are equipped with a topological space structure, namely, Alexandroff's spaces. Some examples are presented and theorems related to the migrativity and neutral element properties are provided. It is shown that, in these spaces, the concepts of overlap and quasi-overlap functions coincide. Also, the notion of overlap index is extended to the context of \(L\)-fuzzy sets. \(L\)-valued overlap indices are obtained by adding degrees of quasi-overlap functions on bounded lattice \(L\), as well as quasi-overlap functions are obtained via \(L\)-valued overlap indices and some examples are presented. Finally, the concepts of migrativity and convex sum are extended to the context of \(L\)-valued overlap index.A conceptual framework of convex and concave sets under refined intuitionistic fuzzy set environmenthttps://zbmath.org/1491.030622022-09-13T20:28:31.338867Z"Rahman, Atiqe Ur"https://zbmath.org/authors/?q=ai:rahman.atiqe-ur"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammad-sarmad|arshad.muhammad-junaid"Saeed, Muhammad"https://zbmath.org/authors/?q=ai:saeed.muhammad-tariq|saeed.muhammad-omer-bin|saeed.muhammad-sarwar(no abstract)An algebraic approach to modular inequalities based on interval-valued fuzzy hypersoft sets via hypersoft set-inclusionshttps://zbmath.org/1491.030632022-09-13T20:28:31.338867Z"Rahman, Atiqe Ur"https://zbmath.org/authors/?q=ai:rahman.atiqe-ur"Saeed, Muhammad"https://zbmath.org/authors/?q=ai:saeed.muhammad-omer-bin|saeed.muhammad-sarwar|saeed.muhammad-tariq"Khan, Khuram Ali"https://zbmath.org/authors/?q=ai:khan.khuram-ali"Nosheen, Ammara"https://zbmath.org/authors/?q=ai:nosheen.ammara"Mabela, Rostin Matendo"https://zbmath.org/authors/?q=ai:mabela.rostin-matendo(no abstract)Cubic bipolar fuzzy set with application to multi-criteria group decision making using geometric aggregation operatorshttps://zbmath.org/1491.030642022-09-13T20:28:31.338867Z"Riaz, Muhammad"https://zbmath.org/authors/?q=ai:riaz.muhammad-tanveer|riaz.muhammad-bilal|riaz.muhammad-mohsin"Tehrim, Syeda Tayyba"https://zbmath.org/authors/?q=ai:tehrim.syeda-tayybaSummary: Bipolar irrational emotions are implicated in a broad variety of individual actions. For example, two specific elements of decision making are the benefits and adverse effects. The harmony and respectful coexistence between these two elements is viewed as a cornerstone to a healthy social setting. For bipolar fuzzy characteristics of the universe of choices that rely on a small range of degrees, a bipolar fuzzy decision making method utilizing different techniques is accessible. The idea of a simplistic bipolar fuzzy set is ineffective in supplying consistency to the details about the frequency of the rating due to minimal knowledge. In this respect, we present cubic bipolar fuzzy sets (CBFSs) as a generalization of bipolar fuzzy sets. The plan of this research is to establish an innovative multi-criteria group decision making (MCGDM) based on cubic bipolar fuzzy set (CBFS) by unifying aggregation operators under geometric mean operations. The geometric mean operators are regarded to be a helpful technique, particularly in circumstances where an expert is unable to fuse huge complex unwanted information properly at the outset of the design of the scheme. We present some basic operations for CBFSs under dual order, i.e., \( \text{P} \)-Order and \(\text{R} \)-Order. We introduce some algebraic operations on CBFSs and some of their fundamental properties for both orders. We propose \(\text{P} \)-cubic bipolar fuzzy weighted geometric \(( \text{P} \)-CBFWG) operator and \(\text{R} \)-cubic bipolar fuzzy weighted geometric \(( \text{R} \)-CBFWG) operator to aggregate cubic bipolar fuzzy data. We also discuss the useability and efficiency of these operators in MCGDM problem. In human decisions, the second important part is ranking of alternatives obtained after evaluation. In this regard, we present an improved score and accuracy function to compare the cubic bipolar fuzzy elements (CBFEs). We also discuss a set theoretic comparison of proposed set with other theories as well as method base comparison of the proposed method with some existing techniques of bipolar fuzzy domain.Possibility distribution calculus and the arithmetic of fuzzy numbershttps://zbmath.org/1491.030652022-09-13T20:28:31.338867Z"Sgarro, Andrea"https://zbmath.org/authors/?q=ai:sgarro.andrea"Franzoi, Laura"https://zbmath.org/authors/?q=ai:franzoi.lauraSummary: Based on possibility theory and multi-valued logic and taking inspiration from the seminal work in probability theory by A. N. Kolmogorov, we aim at laying a hopefully equally sound foundation for fuzzy arithmetic. A possibilistic interpretation of fuzzy arithmetic has long been known even without taking it to its full consequences: to achieve this aim, in this paper we stress the basic role of the two limit-cases of possibilistic interactivity, namely deterministic equality versus non-interactivity, thus getting rid of weak points which have ridden more traditional approaches to fuzzy arithmetic. Both complete and incomplete arithmetic are covered.Generalized hesitant fuzzy rough sets (GHFRS) and their application in risk analysishttps://zbmath.org/1491.030662022-09-13T20:28:31.338867Z"Shaheen, Tanzeela"https://zbmath.org/authors/?q=ai:shaheen.tanzeela"Ali, Muhammad Irfan"https://zbmath.org/authors/?q=ai:ali.muhammad-irfan"Shabir, Muhammad"https://zbmath.org/authors/?q=ai:shabir.muhammadSummary: As a generalization of fuzzy rough sets, the concept of generalized hesitant fuzzy rough sets (GHFRS) is presented in this paper. It is an endeavor to define rough approximations of a collection of hesitant fuzzy sets over a given universe. To this end, elements of the universe are initially clustered using a set-valued map, and then, hesitant fuzzy sets are aggregated by using lower and upper approximation operators. These operators produce hesitant fuzzy sets which aggregate hesitant fuzzy elements. Structural and topological properties associated with GHFRS have been examined. The model is further employed to design a three-way decision analysis technique which preserves many properties of classical techniques but needs less effort and computation. Unlike the existing approaches, the alternatives can be clustered and selected jointly by using a set-valued mapping. This feature makes its application area broader. Moreover, this method is applied to an example, where risk analysis issue is discussed for the selection of energy projects.On the relationship between possibilistic and standard moments of fuzzy numbershttps://zbmath.org/1491.030672022-09-13T20:28:31.338867Z"Stoklasa, Jan"https://zbmath.org/authors/?q=ai:stoklasa.jan"Luukka, Pasi"https://zbmath.org/authors/?q=ai:luukka.pasi"Collan, Mikael"https://zbmath.org/authors/?q=ai:collan.mikaelSummary: In this paper we introduce a transformation of the center of gravity, variance and higher moments of fuzzy numbers into their possibilistic counterparts. We show that this transformation applied to the standard formulae for the computation of the center of gravity, variance, and higher moments of fuzzy numbers gives the same formulae for the computation of possibilistic moments of fuzzy numbers that were introduced by Carlsson and Fullér (2001) for the possibilistic mean and variance, and also the formulae for the calculation of higher possibilistic moments as presented by \textit{A. Saeidifar} and \textit{E. Pasha} [J. Comput. Appl. Math. 223, No. 2, 1028--1042 (2009; Zbl 1159.65013)]. We also present an inverse transformation to derive the formulae for standard measures of central tendency, dispersion, and higher moments of fuzzy numbers, from their possibilistic counterparts. This way a two-way transition between the standard and the possibilistic moments of fuzzy numbers is enabled. The transformation theorems are proven for a wide family of fuzzy numbers with continuous, piecewise monotonic membership functions. Fast computation formulae for the first four possibilistic moments of fuzzy numbers are also presented for linear fuzzy numbers, their concentrations and dilations.On algebraic properties and linearity of OWA operators for fuzzy setshttps://zbmath.org/1491.030682022-09-13T20:28:31.338867Z"Takáč, Zdenko"https://zbmath.org/authors/?q=ai:takac.zdenkoSummary: We deal with an ordered weighted averaging operator (OWA operator) on the set of all fuzzy sets. Our starting point is OWA operator on any lattice introduced in [\textit{I. Lizasoain} and \textit{C. Moreno}, Fuzzy Sets Syst. 224, 36--52 (2013; Zbl 1284.03246); \textit{G. Ochoa} et al., ``Some properties of lattice OWA operators and their importance in image processing'', in: Proceedings of the 16th world congress of the International Fuzzy Systems Association (IFSA) and the
9th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT). Amsterdam: Atlantis Press. 1261--1265 (2015; \url{doi:10.2991/ifsa-eusflat-15.2015.178})].
We focus on a particular case of lattice, namely that of all normal convex fuzzy sets in \([0,1]\), and study algebraic properties and linearity of the proposed OWA operator. It is shown that the operator is an extension of standard OWA operator for real numbers and it possesses similar algebraic properties as standard one, however, it is neither homogeneous nor shift-invariant, i.e., it is not linear in contrast to the standard OWA operator.Constructions of overlap functions on bounded latticeshttps://zbmath.org/1491.030692022-09-13T20:28:31.338867Z"Wang, Haiwei"https://zbmath.org/authors/?q=ai:wang.haiweiSummary: In this paper, we present two methods for constructing new overlap functions on bounded lattices from given ones. At first, we introduce the notion of overlap functions on bounded lattices, which is a generalization of overlap functions on the real unit interval. Then we provide the \(\wedge \)-extension of an overlap function on a subinterval and give the necessary and sufficient conditions for the \(\wedge \)-extension to be an overlap function. Finally, we propose a definition of ordinal sum of finitely many overlap functions on subintervals of a bounded lattice, where the endpoints of the subintervals constitute a chain. Necessary and sufficient conditions for the ordinal sum yielding again an overlap function are provided.An information-based score function of interval-valued intuitionistic fuzzy sets and its application in multiattribute decision makinghttps://zbmath.org/1491.030702022-09-13T20:28:31.338867Z"Wei, An-Peng"https://zbmath.org/authors/?q=ai:wei.an-peng"Li, Deng-Feng"https://zbmath.org/authors/?q=ai:li.dengfeng.1"Lin, Ping-Ping"https://zbmath.org/authors/?q=ai:lin.pingping"Jiang, Bin-Qian"https://zbmath.org/authors/?q=ai:jiang.binqianSummary: The score functions are often used to rank the interval-valued intuitionistic fuzzy sets (IVIFSs) in multiattribute decision making (MADM). The purpose of this paper is to develop an information-based score function of the IVIFS and apply it to MADM. Considering the information amount, the reliability, the certainty information, and the relative closeness degree, we propose an information-based score function of the IVIFS. Comparing the information-based score function with existing ranking methods, we find that the information-based score function can overcome the drawbacks of the existing ranking methods and can rank the IVIFSs well. Three illustrative examples of MADM with linear programming are examined to demonstrate the applicability and feasibility of the information-based score function. It is shown that the information-based score function is well defined and can be applied to MADM.Axiomatic framework of fuzzy entropy and hesitancy entropy in fuzzy environmenthttps://zbmath.org/1491.030712022-09-13T20:28:31.338867Z"Xu, Ting-Ting"https://zbmath.org/authors/?q=ai:xu.tingting"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.8|zhang.hui.10|zhang.hui|zhang.hui.7|zhang.hui.4|zhang.hui.11|zhang.hui.6|zhang.hui.3|zhang.hui.5|zhang.hui.1|zhang.hui.9|zhang.hui.2"Li, Bo-Quan"https://zbmath.org/authors/?q=ai:li.boquanSummary: Entropy is a vital concept to measure uncertainties, in order to measure the uncertainties of fuzzy sets (FSs), intuitionist fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs) more fully, in this paper, the axiomatic definition of fuzzy entropy of FSs is modified, the entropy measures of IFSs and PFSs are categorized as fuzzy entropy and hesitancy entropy, and the axiomatic definitions of these two entropy measures are also revised. Further, the axiomatic definitions of two overall entropies are given based on fuzzy entropy and hesitancy entropy, and the expressions of overall entropy of IFSs and PFSs are constructed by special functions. Then, it is shown that three existing overall entropy formulas can be constructed by three particular functions, and their rationality is proved. Finally, the effectiveness and feasibility of the proposed method and overall entropy are illustrated by an example and two comparative analyses.Some new basic operations of probabilistic linguistic term sets and their application in multi-criteria decision makinghttps://zbmath.org/1491.030722022-09-13T20:28:31.338867Z"Yue, Na"https://zbmath.org/authors/?q=ai:yue.na"Xie, Jialiang"https://zbmath.org/authors/?q=ai:xie.jialiang"Chen, Shuili"https://zbmath.org/authors/?q=ai:chen.shuiliSummary: This paper is concerned with the operations and methods to tackle the probabilistic linguistic multi-criteria decision making (PL-MCDM) problems where criteria are interactive. To avoid the defects of the existing operations of the probabilistic linguistic term sets (PLTSs) and make the operations easier, we redefine a family of operations for PLTSs and investigate their properties in-depth. Then, based on the probabilistic linguistic group utility measure, the probabilistic linguistic individual regret measure and the probabilistic linguistic compromise measure proposed in this paper, the probabilistic linguistic E-VIKOR method is developed. To make up for the deficiency of the above method, the improved probabilistic linguistic VIKOR method which can not only consider the distances between the alternatives and the positive ideal solution but also consider the distances between the alternatives and the negative ideal solution is developed to solve the correlative PL-MCDM problems. And then a case about the video recommender system is conducted to demonstrate the applicability and effectiveness of the proposed methods. Finally, the improved probabilistic linguistic VIKOR method is compared with the probabilistic linguistic E-VIKOR method, the general VIKOR method and the extended TOPSIS method to show its merits.Parametric operations for two 2-dimensional trapezoidal fuzzy setshttps://zbmath.org/1491.030732022-09-13T20:28:31.338867Z"Yun, Y. S."https://zbmath.org/authors/?q=ai:yun.yinshan|yun.young-sang|yun.young-sun|yun.youngsu|yun.yon-sik|yun.yong-sikSummary: In our earlier work, we calculated parametric operations for two 2-dimensional generalized triangular fuzzy sets [\textit{C. Kim} and \textit{Y. S. Yun}, ``Parametric operations for generalized 2-dimensional triangular fuzzy sets'', Int. J. Math. Anal. 11, No. 4, 189--197 (2017)] and for two 2-dimensional quadratic fuzzy numbers [\textit{C. Kang} and \textit{Y. S. Yun}, Far East J. Math. Sci. (FJMS) 101, No. 10, 2185--2193 (2017; Zbl 1491.03055)]. We also calculated parametric operations between 2-dimensional triangular fuzzy numbers and 2-dimensional trapezoidal fuzzy sets [\textit{H. S. Ko} and \textit{Y. S. Yun}, ``Parametric operations between 2-dimensional triangular fuzzy number and trapezoidal fuzzy set'', Far East J. Math. Sci. 102, 2459--2471 (2017]. \par The results for the parametric operations are the generalization of Zadeh's extended algebraic operations \textit{L. A. Zadeh} [Inf. Sci. 8, 199--249 (1975; Zbl 0397.68071)]. In this paper, we calculate the parametric operations for two 2-dimensional trapezoidal fuzzy sets. The results of this paper have been illustrated with the help of an example.Some further results about uninorms on bounded latticeshttps://zbmath.org/1491.030742022-09-13T20:28:31.338867Z"Zhao, Bin"https://zbmath.org/authors/?q=ai:zhao.bin|zhao.bin.1"Wu, Tao"https://zbmath.org/authors/?q=ai:wu.taoSummary: The main purpose of this paper is to solve the problem proposed by Çaylı about uninorms on bounded lattices and build close relationships among uninorms constructed in this paper. Based on the known construction methods and researchers' work, we obtain new uninorms on \(L\) with the given \(t\)-norm and \(t\)-conorm by using closure (interior) operators. The new construction methods provide answers to the problem presented by Çaylı. All classes of uninorms constructed via closure (interior) operators in this paper can be closely connected in a quadruple from the views of the logics of finite observations.Type-2 fuzzy numbers made simple in decision makinghttps://zbmath.org/1491.030752022-09-13T20:28:31.338867Z"Zhu, Bin"https://zbmath.org/authors/?q=ai:zhu.bin.7|zhu.bin.4|zhu.bin.1|zhu.bin.5|zhu.bin|zhu.bin.6"Ren, Peijia"https://zbmath.org/authors/?q=ai:ren.peijiaSummary: For the decision-making problems based on decision makers' judgments in terms of linguistic terms, we propose type-2 fuzzy numbers (T2FNs) that allow decision makers better formalize their judgments. A T2FN has two components: a primary membership and a secondary membership. Compared with T1FSs and interval type-2 fuzzy sets, T2FNs consider an additional dimension by introducing the secondary membership. The primary membership indicates the truth degree of judgment, and the secondary membership further indicates the reliability degree of the truth. We define simple operation rules on T2FNs such that they can be easily used to deal with decision-making problems, such as multi-criteria decision making and multi-stages decision making. Compared with existing related approaches, we verify our approach with several numerical examples.The conditional distributivity condition for T-uninorms revisitedhttps://zbmath.org/1491.030762022-09-13T20:28:31.338867Z"Zong, Wenwen"https://zbmath.org/authors/?q=ai:zong.wenwen"Su, Yong"https://zbmath.org/authors/?q=ai:su.yongSummary: This paper studies the conditional distributivity for T-uninorms over uninorms in the most general setting, transforming it into the (conditional) distributivity equation involving two uninorms.The relation between polynomial calculus, Sherali-Adams, and sum-of-squares proofshttps://zbmath.org/1491.030772022-09-13T20:28:31.338867Z"Berkholz, Christoph"https://zbmath.org/authors/?q=ai:berkholz.christophSummary: We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations.
Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree \(d\) can be transformed into a sum-of-squares refutation of degree \(2d\) and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree \(\Omega(\sqrt{n}/\log n)\) and exponential size.
A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.
For the entire collection see [Zbl 1381.68010].Proof complexity of substructural logicshttps://zbmath.org/1491.030782022-09-13T20:28:31.338867Z"Jalali, Raheleh"https://zbmath.org/authors/?q=ai:jalali.rahelehThe article is well written and organized, no typos where found.
Three results are proved; the first one, is an exponential lower bound on the length of the proof of a sequence of hard \textbf{P}-tautologies \((\mathrm{Clique}_{n,k}\), and \(\mathrm{Color}_{n,m})\), namely a sequence of \textbf{P}-provable formulas \(\{A_n\}^\infty_{n=1}\), that is, the length of the shortest \textbf{P}-proof for \(A_n\) is exponential in \(|A_n|\), where \textbf{P} is a proof system as strong as full Lambek calculus (\textbf{FL}), and it is polynomially simulated by \textbf{eF} (extended Frege) for some superintuitionistic logic of \(\infty\)-branching, denoted by L-\textbf{EF}.
The second result is a similar proof of the previous one but for a proof system and logic extending Visser's basic propositional calculus (\textbf{BPC}).
And the third result is in the classical setting, again, an exponential lower bound on the number of proof line of any proof system polynomially simulated by \textbf{CLF}\(^{-}_{ew}\).
Reviewer: Ariel Germán Fernández (Buenos Aires)Logics of intuitionistic Kripke-Platek set theoryhttps://zbmath.org/1491.030792022-09-13T20:28:31.338867Z"Iemhoff, Rosalie"https://zbmath.org/authors/?q=ai:iemhoff.rosalie"Passmann, Robert"https://zbmath.org/authors/?q=ai:passmann.robertA theorem of De Jongh's says that intuitionistic logic is maximal with respect to Heyting arithmetic, which is to say that the arithmetical axioms do not affect the logic. One of the central results of the present paper is that the intuitionistic Kripke-Platek set theory coined by Lubarsky behaves equally well, and so do certain extensions even with the axiom of choice. This is to be contrasted with the situation for intuitionistic Zermelo-Fraenkel set theory, which Diaconescu has proved to yield its classical forerunner once one adds the axiom of choice.
Reviewer: Peter M. Schuster (Verona)Co-quasiordered residuated systems: an introductionhttps://zbmath.org/1491.030802022-09-13T20:28:31.338867Z"Romano, Daniel A."https://zbmath.org/authors/?q=ai:romano.daniel-abrahamThe logic induced by effect algebrashttps://zbmath.org/1491.030812022-09-13T20:28:31.338867Z"Chajda, Ivan"https://zbmath.org/authors/?q=ai:chajda.ivan"Halaš, Radomír"https://zbmath.org/authors/?q=ai:halas.radomir"Länger, Helmut"https://zbmath.org/authors/?q=ai:langer.helmut-mSummary: Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras \({\mathbf{E}} \), we investigate a natural implication and prove that the implication reduct of \({\mathbf{E}}\) is term equivalent to \({\mathbf{E}} \). Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.Monadic classes of quantum B-algebrashttps://zbmath.org/1491.030822022-09-13T20:28:31.338867Z"Ciungu, Lavinia Corina"https://zbmath.org/authors/?q=ai:ciungu.lavinia-corinaSummary: The aim of this paper is to define the monadic quantum B-algebras and to investigate their properties. If the monadic operators are isotone, we show that they form a residuated pair. Special properties are studied for the particular case of monadic quantum B-algebras with pseudo-product, and a representation theorem for monadic quantum B-algebras with pseudo-product is proved. The monadic filters of monadic quantum B-algebras are defined, and their properties are studied. We prove that there is an isomorphism between the lattice of all filters of a monadic quantum B-algebra and the lattice of all filters of its subalgebra of fixed elements. Monadic operators on unital quantales are introduced, and the functional monadic quantale is constructed.Results on topological lattice effect algebrashttps://zbmath.org/1491.030832022-09-13T20:28:31.338867Z"Rakhshani, M. R."https://zbmath.org/authors/?q=ai:rakhshani.m-r"Rezaei, G. R."https://zbmath.org/authors/?q=ai:rezaei.g-r"Borzooei, R. A."https://zbmath.org/authors/?q=ai:borzooei.rajab-aliSummary: In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a \(T_0,T_1\), and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.A constructive sequence algebra for the calculus of indicationshttps://zbmath.org/1491.030842022-09-13T20:28:31.338867Z"Gangle, Rocco"https://zbmath.org/authors/?q=ai:gangle.rocco"Caterina, Gianluca"https://zbmath.org/authors/?q=ai:caterina.gianluca"Tohmé, Fernando"https://zbmath.org/authors/?q=ai:tohme.fernando-aSummary: In this paper, we investigate some aspects of Spencer-Brown's Calculus of Indications. Drawing from earlier work by Kauffman and Varela, we present a new categorical framework that allows to characterize the construction of infinite arithmetic expressions as sequences taking values in grossone.
Editorial remark: For more information on the notion of grossone, introduced by \textit{Y. D. Sergeyev}, see [Arithmetic of infinity. Cosenza: Edizioni Orizzonti Meridionali (2003; Zbl 1076.03048); EMS Surv. Math. Sci. 4, No. 2, 219--320 (2017; Zbl 1390.03048)]; see also [\textit{A. E. Gutman} and \textit{S. S. Kutateladze}, Sib. Mat. Zh. 49, No. 5, 1054--1076 (2008; Zbl 1224.03045); translation in Sib. Math. J. 49, No. 5, 835--841 (2008)].Picture fuzzy set theory applied to UP-algebrashttps://zbmath.org/1491.030852022-09-13T20:28:31.338867Z"Kankaew, Pimwaree"https://zbmath.org/authors/?q=ai:kankaew.pimwaree"Yuphaphin, Sunisa"https://zbmath.org/authors/?q=ai:yuphaphin.sunisa"Lapo, Nattacha"https://zbmath.org/authors/?q=ai:lapo.nattacha"Chinram, Ronnason"https://zbmath.org/authors/?q=ai:chinram.ronnason"Iampan, Aiyared"https://zbmath.org/authors/?q=ai:iampan.aiyaredSummary: The concept of picture fuzzy sets was first considered by \textit{B. C. Cuong} and \textit{V. Kreinovich} [``Picture fuzzy sets -- a new concept for computational intelligence problems'', in: Proceedings of the 2013 third world congress on information and communication technologies (WICT 2013), Hanoi, Vietnam, December 15--18, 2013. Piscataway, NJ: IEEE. 1--6 (2013; 10.1109/WICT.2013.7113099)], which are direct extensions of fuzzy sets and intuitionistic fuzzy sets. In this paper, we applied the concept of picture fuzzy sets in UP-algebras to introduce the eight new concepts of picture fuzzy sets: picture fuzzy UP-subalgebras, picture fuzzy near UP-filters, picture fuzzy UP-filters, picture fuzzy implicative UP-filters, picture fuzzy comparative UP-filters, picture fuzzy shift UP-filters, picture fuzzy UP-ideals, and picture fuzzy strong UP-ideals. The link between the eight new concepts of picture fuzzy sets in UP-algebras is also discussed.Erratum to: ``Some result on simple hyper \(K\)-algebras''https://zbmath.org/1491.030862022-09-13T20:28:31.338867Z"Madadi-Dargahi, Soodabeh"https://zbmath.org/authors/?q=ai:madadi-dargahi.soodabeh"Nasr-Azadani, Mohammad Ali"https://zbmath.org/authors/?q=ai:nasr-azadani.mohammad-aliSummary: In this manuscript, we show that the Theorem 3.28 [\textit{T. Roudbari} and \textit{M. M. Zahedi}, ibid. 3, No. 2, 29--48 (2008; Zbl 1301.03068)] is not correct and modify it.State theory on bounded hyper EQ-algebrashttps://zbmath.org/1491.030872022-09-13T20:28:31.338867Z"Xin, Xiao Long"https://zbmath.org/authors/?q=ai:xin.xiaolongSummary: In a hyper structure \((X,\star)\), \(x\star y\) is a non-empty subset of \(X\). For a state \(s\), \(s(x\star y)\) need not be well defined. In this paper, by defining \(s^*(x\star y)=\sup\{s(z)\mid z\in x\star y\}\), we introduce notions of sup-Bosbach states, state-morphisms and sup-Riečan states on a bounded hyper EQ-algebra and discuss the related properties. The states on bounded hyper EQ-algebras are the generalization of states on EQ-algebras. Then we discuss the relations among sup-Bosbach states, state-morphisms and sup-Riečan states on bounded hyper EQ-algebras. By giving a counter example, we show that a sup-Bosbach state may not be a sup-Riečan state on a hyper EQ-algebra. We give conditions in which each sup-Bosbach state becomes a sup-Riečan state on bounded hyper EQ-algebras. Moreover, we introduce several kinds of congruences on bounded hyper EQ-algebras, by which we construct the quotient hyper EQ-algebras. By use of the state \(s\) on a bounded hyper EQ-algebra \(H\), we set up a state \(\bar{s}\) on the quotient hyper EQ-algebra \(H/\theta \). We also give the condition, by which a bounded hyper EQ-algebra admits a sup-Bosbach state.Logics of left variable inclusion and Płonka sums of matriceshttps://zbmath.org/1491.030882022-09-13T20:28:31.338867Z"Bonzio, S."https://zbmath.org/authors/?q=ai:bonzio.stefano"Moraschini, T."https://zbmath.org/authors/?q=ai:moraschini.tommaso"Pra Baldi, M."https://zbmath.org/authors/?q=ai:pra-baldi.mThe paper is dedicated to the study of logics with left variable inclusion condition. With each substitution invariant logic \(\vdash\) one can associate a substitution invariant logic \(\vdash^l\) given by:
\[
\Gamma \vdash^l \varphi \iff \text{ there is } \Delta \subseteq \Gamma \text{ such that} Var(\Delta) \subseteq Var(\varphi)\text{ and } \Delta \vdash \varphi.
\]
As usual, matrix is a pair \(\langle \mathbf{A},F \rangle\) where \(\mathsf{A}\) is an algebra and \(F \subseteq A\).
A directed system of algebras consists of
\begin{itemize}
\item[(i)] a semilattice \(\mathbf{I} = \langle I, \lor\rangle\);
\item[(ii)] a family of algebras \(\{\mathbf{A}_i : i \in I\}\);
\item[(iii)] a homomorphism \(f_{ij}: \mathbf{A}_i \leftarrow \mathbf{A}_j\), for every \(i,j \in I\) such that \(i \leq j\);
\end{itemize}
moreover, \(f_{ii}\) is the identity map for every \(i \in I\), and if \(i \leq j \leq k\), then \(f_{ik} = f_{jk}\circ f_{ij}\).
If \(X\) is a directed system, the Plonka sum of X (in symbols \(\mathcal{P}{\textit{l}}(X)\)) is the algebra defined as follows. The universe of \(\mathcal{P}l(X)\) is the union \(\bigcup_{i \in I}\mathbf{A}_i\), and for every \(n\)-ary operation \(f\) and \(a_1,\dots,a_n \in \bigcup_{i \in I}\mathbf{A}_i\),
\[
f^{\mathcal{P}l(X)}(a_1,\dots,a_n) := f^{\mathbf{A}_j}(f_{i_1j}(a_1),\dots, f_{i_nj}(a_n)).
\]
It is proven that given a logic \(\vdash\) and a class of matrices \(M\) containing \(\langle \mathbf{1}, {1}\rangle\), where \(\mathbf{1}\) is a trivial algebra, if \(\vdash\) is complete w.r.t. \(M\), then \(\vdash^l\) is complete w.r.t. \(\mathcal{P}{\textit{l}}(M)\).
Reviewer: Alex Citkin (Warren)Computability and the game of cops and robbers on graphshttps://zbmath.org/1491.051342022-09-13T20:28:31.338867Z"Stahl, Rachel D."https://zbmath.org/authors/?q=ai:stahl.rachel-dSummary: Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal \(\alpha \), any winning strategy has complexity at least \(0^{(\alpha )}\). Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are explored. The Turing computability of a binary relation used to classify cop-win graphs is studied, and the computational difficulty of determining the winner for locally finite computable graphs is discussed.\(\gamma\)-variable first-order logic of uniform attachment random graphshttps://zbmath.org/1491.051712022-09-13T20:28:31.338867Z"Malyshkin, Y. A."https://zbmath.org/authors/?q=ai:malyshkin.yu-a"Zhukovskii, M. E."https://zbmath.org/authors/?q=ai:zhukovskiy.mikhail-e|zhukovskii.maximThis paper examines the convergence of the probability that a first-order statement holds in a uniform attachment graph. To build such a random graph, we start from a complete graph of \(m\) vertices, and, at each step, we add a new vertex with \(m\) new edges, with the endpoints chosen uniformly at random from the already existing vertices, without replacement. The authors prove that if the number of different variables in a first-order sentence is at most \(m-2\), then the probability that the sentence is satisfied for this random graph with \(n\) vertices converges. It was already known that (contrary to many other random graph models, e.g. random regular graphs) the zero-one law is not true for this graph model for first-order sentences; the current paper shows that although the limit of the probabilities is not necessarily \(0\) or \(1\), it exists if the number of different variables is at most \(m-2\).
As for the method of the proof, the authors rely on an already known connection between first-order logic statements and the winning strategy of a so-called pebble game, and certain local properties of uniform attachment graphs. Namely, in order to make use of this correspondence, the authors describe the typical position of cycles and paths of a given length in a uniform attachment graph.
Reviewer: Ágnes Backhausz (Budapest)Residuated operators in complemented posetshttps://zbmath.org/1491.060052022-09-13T20:28:31.338867Z"Chajda, Ivan"https://zbmath.org/authors/?q=ai:chajda.ivan"Länger, Helmut"https://zbmath.org/authors/?q=ai:langer.helmut-mOn residuation in paraorthomodular latticeshttps://zbmath.org/1491.060132022-09-13T20:28:31.338867Z"Chajda, I."https://zbmath.org/authors/?q=ai:chajda.ivan"Fazio, D."https://zbmath.org/authors/?q=ai:fazio.davideSummary: Paraorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution \({\mathbf{A}}\) can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever \({\mathbf{A}}\) is distributive.An equational theory for \(\sigma \)-complete orthomodular latticeshttps://zbmath.org/1491.060152022-09-13T20:28:31.338867Z"Freytes, Hector"https://zbmath.org/authors/?q=ai:freytes.hectorSummary: The condition of \(\sigma \)-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of \(\sigma \)-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to \(\sigma \)-complete orthomodular lattices is introduced and a completeness theorem is obtained.Orthomodular lattices as \(L\)-algebrashttps://zbmath.org/1491.060172022-09-13T20:28:31.338867Z"Wu, Yali"https://zbmath.org/authors/?q=ai:wu.yali"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yi-chuan|yang.yichuanSummary: We first prove that the axioms system of orthomodular \(L\)-algebra (\(O\)-\(L\)-algebras for short) as given in [\textit{W. Rump}, Forum Math. 30, No. 4, 973--995 (2018; Zbl 1443.06005)] are not independent by giving an independent axiom one. Then, two conditions for \textit{KL}-algebras to be Boolean are provided. Furthermore, some theorems of Holland are reproved using the self-similar closure of \(OM \)-\(L\)-algebras. In particular, the monoid operation of the self-similar closure is shown to be commutative.Ideals of core regular double Stone algebrahttps://zbmath.org/1491.060192022-09-13T20:28:31.338867Z"Srikanth, A. R. J."https://zbmath.org/authors/?q=ai:srikanth.ammu-r-j"Ravi Kumar, R. V. G."https://zbmath.org/authors/?q=ai:ravi-kumar.r-v-gTense De Morgan \(S4\)-algebrashttps://zbmath.org/1491.060212022-09-13T20:28:31.338867Z"Segura, Cecilia"https://zbmath.org/authors/?q=ai:segura.ceciliaQuasi-pseudo-hoops: an extension to pseudo-hoopshttps://zbmath.org/1491.060222022-09-13T20:28:31.338867Z"Chen, Wenjuan"https://zbmath.org/authors/?q=ai:chen.wenjuan"Chen, Zhaoying"https://zbmath.org/authors/?q=ai:chen.zhaoying"Wang, Hongkai"https://zbmath.org/authors/?q=ai:wang.hongkaiSummary: In this paper, we introduce the notion of quasi-pseudo-hoops (\(qp\)-hoops, for short) as the generalization of pseudo-hoops. First we give some new notions in order to define \(qp\)-hoops. We investigate the basic properties of \(qp\)-hoops and also prove that any \(qp\)-hoop has the Riesz Decomposition Property. Second we discuss filters of \(qp\)-hoops and show that there exists a bijective correspondence between normal filters and filter congruences on any \(qp\)-hoop. Finally, we introduce and study some subclasses of \(qp\)-hoops. The subdirect product decomposition of a bounded \(qp\)-hoop is shown. We also present that bounded Wajsberg \(qp\)-hoops with additional conditions are equivalent to quasi-pseudo-MV algebras and bounded basic \(qp\)-hoops with additional conditions are equivalent to quasi-pseudo-BL algebras.An approach to stochastic processes via non-classical logichttps://zbmath.org/1491.060232022-09-13T20:28:31.338867Z"Di Nola, Antonio"https://zbmath.org/authors/?q=ai:di-nola.antonio"Dvurečenskij, Anatolij"https://zbmath.org/authors/?q=ai:dvurecenskij.anatolij"Lapenta, Serafina"https://zbmath.org/authors/?q=ai:lapenta.serafinaSummary: Within the infinitary variety of \(\sigma \)-complete Riesz MV-algebras \(\mathbf{RMV}_\sigma \), we introduce the algebraic analogue of a random variable as a homomorphism defined on the free algebra in \(\mathbf{RMV}_\sigma \). After a preliminary study of the proposed notion, we use it to define stochastic processes in the framework of non-classical logic (Łukasiewicz logic, more precisely) and we define stochastic independence.Hyper commutative basic algebras, hyper MV-algebras, and stateshttps://zbmath.org/1491.060322022-09-13T20:28:31.338867Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.18"Wu, Yali"https://zbmath.org/authors/?q=ai:wu.yali"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yi-chuan|yang.yichuanSummary: In this paper, we introduce hyper commutative basic algebras. We first prove that every hyper MV-algebra is a hyper commutative basic algebra. Then we show that a hyper commutative basic algebra of cardinality 2 is a hyper MV-algebra, which is similar to the result of Botur and Halaš that finite commutative basic algebras coincide with MV-algebras. However, we find a hyper commutative basic algebra of cardinality 3 which is not a hyper MV-algebra. Finally, we study two types of states on hyper commutative basic algebras.Basic algebras and L-algebrashttps://zbmath.org/1491.060332022-09-13T20:28:31.338867Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.18"Wu, Yali"https://zbmath.org/authors/?q=ai:wu.yali"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yi-chuan|yang.yichuanSummary: In this paper, we study the relation between L-algebras and basic algebras. In particular, we construct a lattice-ordered effect algebra which improves an example of \textit{I. Chajda} et al. [Algebra Univers. 60, No. 1, 63--90 (2009; Zbl 1219.06013)].Ehoopshttps://zbmath.org/1491.060342022-09-13T20:28:31.338867Z"Xie, Fei"https://zbmath.org/authors/?q=ai:xie.fei|xie.fei.1"Liu, Hongxing"https://zbmath.org/authors/?q=ai:liu.hongxing.1|liu.hongxingSummary: In this paper, we introduce the notion of Ehoops, which are generalizations of hoops. Unlike the hoop, an Ehoop does not necessarily have a top element. The notions of ideals and filters in Ehoops are defined. In Ehoops, both ideals and filters can be used to define the congruences. It is proved that if an Ehoop A satisfies the double negation property, there is a one-to-one correspondence between the set of all ideals of A and the set of all congruences on A. Prime ideal theorem on Ehoops is also given. In addition, we define the notions of implicative filters and positive implicative filters of Ehoops and investigate the quotient algebras induced by (positive) implicative filters.Intuitionistic fuzzy congruences on product latticeshttps://zbmath.org/1491.060362022-09-13T20:28:31.338867Z"Rasuli, Rasul"https://zbmath.org/authors/?q=ai:rasuli.rasulSummary: In this work, the concept of intuitionistic fuzzy congruences on lattice \(X\) was introduced and was defined direct product between them. Also some characterizations of them were established. Finally, isomorphism between factor lattices of similarity classes was investigated.Semidistributivity and whitman property in implication zroupoidshttps://zbmath.org/1491.060372022-09-13T20:28:31.338867Z"Cornejo, Juan M."https://zbmath.org/authors/?q=ai:cornejo.juan-manuel"Sankappanavar, Hanamantagouda P."https://zbmath.org/authors/?q=ai:sankappanavar.hanamantagouda-pSummary: In 2012, the second author [Sci. Math. Jpn. 75, No. 1, 21--50 (2012; Zbl 1279.06009)] introduced, and initiated the investigations into, the variety \(\mathcal{I}\) of implication zroupoids that generalize De Morgan algebras and \(\vee\)-semilattices with 0. An algebra \(\mathbf{A}=\langle A,\rightarrow,0\rangle\), where \(\rightarrow\) is binary and 0 is a constant, is called an \textit{implication zroupoid} (\(\mathcal{I}\)-zroupoid, for short) if \textbf{A} satisfies: \((x\rightarrow y)\rightarrow z\approx [(z'\rightarrow x)\rightarrow (y\rightarrow z)']'\), where \(x':=x\rightarrow 0\), and \(0''\approx 0\). Let \(\mathcal{I}\) denote the variety of implication zroupoids and \(\mathbf{A}\in\mathcal{I}\). For \(x,y\in\mathbf{A}\), let \(x\wedge y:=(x\rightarrow y')'\) and \(x\vee y:=((x'\wedge y')'\). In an earlier paper, we had proved that if \(\mathbf{A}\in\mathcal{I}\), then the algebra \(\mathbf{A}_{mj}=\langle A,\vee,\wedge\rangle\) is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every \(\mathbf{A}\in\mathcal{I}\), the bisemigroup \(\mathbf{A}_{mj}\) is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety \(\mathcal{MEJ}\) of \(\mathcal{I}\), defined by the identity: \(x\wedge y \approx x\vee y\), satisfies the Whitman Property. We conclude the paper with two open problems.Isomorphism theorems in generalized \(d\)-algebrashttps://zbmath.org/1491.060462022-09-13T20:28:31.338867Z"Chaudhry, Muhammad Anwar"https://zbmath.org/authors/?q=ai:chaudhry.muhammad-anwar"Qureshi, Muhammad Imran"https://zbmath.org/authors/?q=ai:qureshi.muhammad-imran.1|qureshi.muhammad-imran"Fahad, Asfand"https://zbmath.org/authors/?q=ai:fahad.asfand"Bashir, Muhammad Sajjad"https://zbmath.org/authors/?q=ai:bashir.muhammad-sajjad(no abstract)\(t\)-derivations on complicated subtraction algebrashttps://zbmath.org/1491.060482022-09-13T20:28:31.338867Z"Jana, Chiranjibe"https://zbmath.org/authors/?q=ai:jana.chiranjibe"Senapati, Tapan"https://zbmath.org/authors/?q=ai:senapati.tapan"Pal, Madhumangal"https://zbmath.org/authors/?q=ai:pal.madhumangal(no abstract)Fuzzy set theoretic approach to generalized ideals in BCK/BCI-algebrashttps://zbmath.org/1491.060512022-09-13T20:28:31.338867Z"Muhiuddin, G."https://zbmath.org/authors/?q=ai:muhiuddin.ghulam"Alam, N."https://zbmath.org/authors/?q=ai:alam.noor"Obeidat, S."https://zbmath.org/authors/?q=ai:obeidat.sofian"Khan, N. M."https://zbmath.org/authors/?q=ai:khan.noor-mohammad"Zaidi, H. N."https://zbmath.org/authors/?q=ai:zaidi.hasan-nihal"Kirmani, S. A. K."https://zbmath.org/authors/?q=ai:kirmani.s-a-k"Altaleb, A."https://zbmath.org/authors/?q=ai:altaleb.anas"Aqib, J. M."https://zbmath.org/authors/?q=ai:aqib.j-m(no abstract)An overview of cubic intuitionistic \(\beta\)-subalgebrashttps://zbmath.org/1491.060522022-09-13T20:28:31.338867Z"Muralikrishna, P."https://zbmath.org/authors/?q=ai:muralikrishna.prakasam"Borumand Saeid, A."https://zbmath.org/authors/?q=ai:borumand-saeid.arsham"Vinodkumar, R."https://zbmath.org/authors/?q=ai:vinodkumar.r"Palani, G."https://zbmath.org/authors/?q=ai:palani.g-sThe basic concepts of cubic intuitionistic sets are adopted to \(\beta\)-subalgebras. The obtained results are typical for this theory.
Reviewer: Wiesław A. Dudek (Wrocław)Construction of an HV-BE-algebra from a BE-algebra based on ``begins lemma''https://zbmath.org/1491.060532022-09-13T20:28:31.338867Z"Naghibi, R."https://zbmath.org/authors/?q=ai:naghibi.razieh"Anvariyeh, S. M."https://zbmath.org/authors/?q=ai:anvariyeh.said-m|anvariyeh.seid-mohammad"Mirvakili, S."https://zbmath.org/authors/?q=ai:mirvakili.saeedSummary: In this paper, first we introduce the new class of HV-BE-algebra as a generalization of a (hyper) BE-algebra and prove some basic results and present several examples. Then, we construct the HV-BE-algebra associated to a BE-algebra (namely BL-BE-algebra) based on ``Begins lemma'' and investigate it.Ideals of transitive BE-algebrashttps://zbmath.org/1491.060542022-09-13T20:28:31.338867Z"Prabhakar, M. Bala"https://zbmath.org/authors/?q=ai:prabhakar.m-bala"Vali, S. Kalesha"https://zbmath.org/authors/?q=ai:vali.s-kalesha"Rao, M. Sambasiva"https://zbmath.org/authors/?q=ai:rao.m-sambasiva|rao.mukkamala-sambasivaSummary: The notion of ideals is introduced in transitive BE-algebras. Some characterization theorems of ideals of transitive BE-algebras are derived. The notion of semi-ideals is introduced and studied a relationship between semi-ideals and ideals. Properties of ideals are studied with the help of homomorphisms and congruences.Generalized lower sets of transitive BE-algebrashttps://zbmath.org/1491.060552022-09-13T20:28:31.338867Z"Prabhakar, M. Bala"https://zbmath.org/authors/?q=ai:prabhakar.m-bala"Vali, S. Kalesha"https://zbmath.org/authors/?q=ai:vali.s-kalesha"Rao, M. Sambasiva"https://zbmath.org/authors/?q=ai:rao.m-sambasiva|rao.mukkamala-sambasivaSummary: The notion of generalized lower sets is introduced in transitive BE-algebras. Some properties of generalized lower sets are investigated in transitive BE-algebras. Furthermore, a sufficient condition is derived for every generalized lower set BE-algebra to become an ideal.Isomorphism theorems on weak AB-algebrashttps://zbmath.org/1491.060572022-09-13T20:28:31.338867Z"Sriponpaew, Boonyong"https://zbmath.org/authors/?q=ai:sriponpaew.boonyong"Sassanapitax, Lee"https://zbmath.org/authors/?q=ai:sassanapitax.leeSummary: In this paper, we introduce the notion of weak AB-algebras, which are a generalization of BCC-algebras. The concepts of congruences and formation of quotients of these algebras are demonstrated. In addition, we prove the fundamental theorems of isomorphism for weak AB-algebras.On some intervals of partial cloneshttps://zbmath.org/1491.080042022-09-13T20:28:31.338867Z"Alekseev, Valeriy B."https://zbmath.org/authors/?q=ai:alekseev.valerii-bSummary: This paper deals with clones, i.e. sets of functions containing all projections and closed under compositions. If \(A\) is any clone from the \(k\)-valued logic \(P_k\), then \(Str(A)\) is the set of all functions from the partial \(k\)-valued logic \(P_k^\ast\), which can be expanded to a function from \(A\). For any clone \(A\) from \(P_k\), the set \(Int(A)\) of all partial clones in \(P_k^\ast\) lying between \(A\) and \(Str(A)\) is investigated. We define a special family \(Z(A)\) of sets of predicates and prove that the lattice of partial clones in \(Int(A)\) (according to inclusion) is isomorphic to the lattice of sets in \(Z(A)\) (according to inclusion). For the set \(J_k\) of all projections in \(P_k\), we prove that the cardinality of \(Int(J_k)\) is continuum. For the set \(Pol_k\) of all polynomials in \(P_k\) where \(k\) is a product of two different prime numbers, we prove that \(Int(Pol_k)\) consists of 7 partial clones which are completely described.Adequate predimension inequalities in differential fieldshttps://zbmath.org/1491.120032022-09-13T20:28:31.338867Z"Aslanyan, Vahagn"https://zbmath.org/authors/?q=ai:aslanyan.vahagn-aThe notion of a ``predimension inequality'' (in the context of so-called \textit{Fraïssé construction}) has been introduced by Hrushovski in 1990s. In short, if there is a good predimension notion on a certain category of finite structures, then one can construct its Fraïssé limit (a special kind of a direct limit), which has good model-theoretical properties.
In the paper under review, the author considers predimension inequalities in differential fields and formalizes Zilber's notion of \textit{adequacy} of such an inequality. The main examples of predimension inequalities in this context are the Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the \(j\)-function (established by Pila and Tsimerman). The author shows (Theorem 1.3) that the Ax-Schanuel inequality for the \(j\)-function is adequate. Using this result, the author performs Hrushovski's construction and obtains a natural candidate for the first-order theory of the differential equation of the \(j\)-function.
Reviewer: Piotr Kowalski (Wroclaw)Dynamic evaluation of integrity and the computational content of Krull's lemmahttps://zbmath.org/1491.130112022-09-13T20:28:31.338867Z"Schuster, Peter"https://zbmath.org/authors/?q=ai:schuster.peter-michael"Wessel, Daniel"https://zbmath.org/authors/?q=ai:wessel.daniel"Yengui, Ihsen"https://zbmath.org/authors/?q=ai:yengui.ihsenIn the paper under review, the authors suggest a constructive procedure to determine the Krull dimension of a commutative ring. This procedure has been mostly developed in a previous paper by \textit{G. Kemper} and \textit{I. Yengui} [J. Algebra 557, 278--288 (2020; Zbl 1440.13112)]. Such paper deals with a constructive characterisation of the more general valuative dimension of a domain and contains only one non-constructive step: a reduction from the general case to the integral case. In the paper under review, the authors present a constructive argument for this reduction step via a dynamical solution.
Reviewer: Paolo Lella (Trento)An introduction to abstract algebra. Sets, groups, rings, and fieldshttps://zbmath.org/1491.200012022-09-13T20:28:31.338867Z"Weintraub, Steven H."https://zbmath.org/authors/?q=ai:weintraub.steven-hPublisher's description: This book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper undergraduate or beginning graduate level.
It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates with a treatment of Dedekind rings, including rings of algebraic integers.
In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.
As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear, and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.
Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate the theory while others are general results that develop the theory further.Knapsack problems for wreath productshttps://zbmath.org/1491.200782022-09-13T20:28:31.338867Z"Ganardi, Moses"https://zbmath.org/authors/?q=ai:ganardi.moses"König, Daniel"https://zbmath.org/authors/?q=ai:konig.daniel"Lohrey, Markus"https://zbmath.org/authors/?q=ai:lohrey.markus"Zetzsche, Georg"https://zbmath.org/authors/?q=ai:zetzsche.georgSummary: In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group \(G\), knapsack (as well as the related subset sum problem) for the wreath product \(G\wr\mathbb{Z}\) is \textsf{NP}-complete.
For the entire collection see [Zbl 1381.68010].Isomorphism theorems for basic constructive algebraic structures with special emphasize on constructive semigroups with apartness -- an overviewhttps://zbmath.org/1491.201152022-09-13T20:28:31.338867Z"Mitrović, Melanija"https://zbmath.org/authors/?q=ai:mitrovic.melanija-s"Silvestrov, Sergei"https://zbmath.org/authors/?q=ai:silvestrov.sergei-dSummary: This overview is an introduction to the basic constructive algebraic structures with apartness with special emphasises on a set and semigroup with apartness. The main purpose of this paper, inspired by \textit{A. Bauer} [Bull. Am. Math. Soc., New Ser. 54, No. 3, 481--498 (2017; Zbl 1469.03171)], is to make some sort of understanding of constructive algebra in Bishop's style position for those (classical) algebraists as well as for the ones who apply algebraic knowledge who might wonder what is constructive algebra all about. Every effort has been made to produce a reasonably prepared text with such definite need. In the context of basic constructive algebraic structures constructive analogous of isomorphism theorems will be given. Following their development, two points of view on a given subject: classical and constructive will be considered. This overview is not, of course, a comprehensive one.
For the entire collection see [Zbl 1467.16001].Homogeneity of inverse semigroupshttps://zbmath.org/1491.201242022-09-13T20:28:31.338867Z"Quinn-Gregson, Thomas"https://zbmath.org/authors/?q=ai:quinn-gregson.thomasBoundary value problem: weak solutions induced by fuzzy partitionshttps://zbmath.org/1491.340342022-09-13T20:28:31.338867Z"Nguyen, Linh"https://zbmath.org/authors/?q=ai:nguyen-viet-linh.|nguyen.linh-thi-hoai|nguyen.linh-h|nguyen.linh-tuan|nguyen.linh-trung|nguyen.linh-ngoc|nguyen.linh-viet|nguyen.linh-anh"Perfilieva, Irina"https://zbmath.org/authors/?q=ai:perfilieva.irina-g"Holčapek, Michal"https://zbmath.org/authors/?q=ai:holcapek.michalSummary: The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called ``two dimensional'' approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.Stability in a grouphttps://zbmath.org/1491.370122022-09-13T20:28:31.338867Z"Conant, Gabriel"https://zbmath.org/authors/?q=ai:conant.gabrielSummary: We develop local stable group theory directly from topological dynamics, and extend the main tools in this subject to the setting of stability ``in a model.'' Specifically, given a group \(G\), we analyze the structure of sets \(A \subseteq G\) such that the bipartite relation \(xy\in A\) omits infinite half-graphs. Our proofs rely on the characterization of model-theoretic stability via Grothendieck's ``double-limit'' theorem (as shown by \textit{I. Ben Yaacov} [Bull. Symb. Log. 20, No. 4, 491--496 (2014; Zbl 1345.03058)]), and the work of \textit{R. Ellis} and \textit{M. Nerurkar} [Trans. Am. Math. Soc. 313, No. 1, 103--119 (1989; Zbl 0674.54026)]
on weakly almost periodic \(G\)-flows.Effective rates of convergence for the resolvents of accretive operatorshttps://zbmath.org/1491.470522022-09-13T20:28:31.338867Z"Koutsoukou-Argyraki, Angeliki"https://zbmath.org/authors/?q=ai:koutsoukou-argyraki.angelikiSummary: We extract explicit, computable, and highly uniform rates for the strong convergence of the resolvents of set-valued, \(m\)-accretive, and uniformly accretive at zero/\(\phi\)-expansive operators on general real Banach spaces to the zero of each operator. This is achieved through proof mining on the proof of a theorem by \textit{J. García Falset} [in: Proceedings of the 7th international conference on fixed-point theory and its applications, Guanajuato, Mexico, July 17--23, 2005. Yokohama: Yokohama Publishers. 87--94 (2006; Zbl 1115.47039)] the motivation of which originates from a classical work by \textit{S. Reich} [J. Math. Anal. Appl. 75, 287--292 (1980; Zbl 0437.47047)]. For the bound extraction we make use of a modulus of accretivity at zero, a~notion introduced recently by \textit{U. Kohlenbach} and the author [J. Math. Anal. Appl. 423, No. 2, 1089--1112 (2015; Zbl 1300.47070)], as well as a modulus of \(\phi\)-expansivity, a~notion introduced analogously here.Optimization of the Bolza problem with higher-order differential inclusions and initial point and state constraintshttps://zbmath.org/1491.490202022-09-13T20:28:31.338867Z"Mahmudov, Elimhan N."https://zbmath.org/authors/?q=ai:mahmudov.elimhan-nadirSummary: This paper is devoted to the duality of the Bolza problem with higher order differential inclusions and constraints on the initial point and state, which can make a significant contribution to the theory of optimal control. To this end in the form of Euler-Lagrange type inclusions and transversality conditions, sufficient optimality conditions are derived. It is remarkable that in a particular case the Euler-Lagrange inclusion coincides with the classical Euler-Poisson equation of the calculus of variations. The main idea of obtaining optimal conditions is locally conjugate mappings. It turns out that inclusions of the Euler-Lagrange type for both direct and dual problems are ``duality relations''. To implement this approach, sufficient optimality conditions and duality theorems are proved in the Mayer problem with a second-order linear optimal control problem and third-order polyhedral differential inclusions, reflecting the special features of the variational geometry of polyhedral sets.Decomposition of soft continuity via soft locally \(b\)-closed sethttps://zbmath.org/1491.540012022-09-13T20:28:31.338867Z"Demirtaş, Naime"https://zbmath.org/authors/?q=ai:demirtas.naime"Ergül, Zehra Güzel"https://zbmath.org/authors/?q=ai:ergul.zehra-guzelSummary: In this paper, we introduce soft locally \(b\)-closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters and study some of their properties. We investigate their relationships with different types of subsets of soft topological spaces with the help of counterexamples. Also, the concept of soft locally b-continuous functions is presented. Finally, a decomposition of soft continuity is obtained.On some properties of intuitionistic fuzzy soft boundaryhttps://zbmath.org/1491.540082022-09-13T20:28:31.338867Z"Hussain, Sabir"https://zbmath.org/authors/?q=ai:hussain.sabirSummary: The concept of intuitionistic fuzzy soft sets in a decision making problem and the problem is solved with the help of 'similarity measurement' technique. The purpose of this paper is to initiate the concept of Intuitionistic Fuzzy(IF) soft boundary. We discuss and explore the characterizations and properties of IF soft boundary in general as well as in terms of IF soft interior and IF soft closure. Examples and counter examples are also presented to validate the discussed results.Regularity of the extensions of a double fuzzy topological spacehttps://zbmath.org/1491.540142022-09-13T20:28:31.338867Z"Vivek, S."https://zbmath.org/authors/?q=ai:vivek.srinivas|vivek.s-sree"Mathew, Sunil C."https://zbmath.org/authors/?q=ai:mathew.sunil-cThe extension of a double fuzzy topological space is a concept introduced by the authors in their previous paper [J. Adv. Stud. Topol. 9, No. 1, 75--93 (2018; Zbl 1395.54008)], and studied in other papers by them. In this new paper they study the regularity of the extension of a regular double fuzzy topological space. Though the extensions do not preserve regularity in general, some conditions which ensure the regularity of the extended space are obtained. Moreover, certain families of closed sets in a double fuzzy topological space and its extensions are investigated and some types of extensions under which these families remain unchanged are identified.
Reviewer: Francisco Gallego Lupiáñez (Madrid)Towards a general class of parametric probability weighting functionshttps://zbmath.org/1491.600062022-09-13T20:28:31.338867Z"Dombi, József"https://zbmath.org/authors/?q=ai:dombi.jozsef-daniel|dombi.jozsef"Jónás, Tamás"https://zbmath.org/authors/?q=ai:jonas.tamasSummary: In this study, we present a novel methodology that can be used to generate parametric probability weighting functions, which play an important role in behavioral economics, by making use of the Dombi modifier operator of continuous-valued logic. Namely, we will show that the modifier operator satisfies the requirements for a probability weighting function. Next, we will demonstrate that the application of the modifier operator can be treated as a general approach to create parametric probability weighting functions including the most important ones such as the Prelec and the Ostaszewski, Green and Myerson (Lattimore, Baker and Witte) probability weighting function families. Also, we will show that the asymptotic probability weighting function induced by the inverse of the so-called epsilon function is none other than the Prelec probability weighting function. Furthermore, we will prove that, by using the modifier operator, other probability weighting functions can be generated from the dual generator functions and from transformed generator functions. Finally, we will show how the modifier operator can be used to generate strictly convex (or concave) probability weighting functions and introduce a method for fitting a generated probability weighting function to empirical data.Fundamentals of Bayesian epistemology 1. Introducing credenceshttps://zbmath.org/1491.620042022-09-13T20:28:31.338867Z"Titelbaum, Michael G."https://zbmath.org/authors/?q=ai:titelbaum.michael-gPublisher's description: Bayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists' use of Bayesian probability mathematics to represent degrees of belief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implications
Volume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov's three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes over time. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five.
Volume 2 gives arguments for the five core rules introduced in Volume 1, then considers challenges to Bayesian epistemology. It begins by detailing Bayesianism's successful applications to confirmation and decision theory. Then it describes three types of arguments for Bayesian rules, based on representation theorems, Dutch Books, and accuracy measures. Finally, it takes on objections to the Bayesian approach and alternative formalisms, including the statistical approaches of frequentism and likelihoodism.
For Volume 2 see [Zbl 1491.62005].Fundamentals of Bayesian epistemology 2. Arguments, challenges, alternativeshttps://zbmath.org/1491.620052022-09-13T20:28:31.338867Z"Titelbaum, Michael G."https://zbmath.org/authors/?q=ai:titelbaum.michael-gPublisher's description: Bayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists' use of Bayesian probability mathematics to represent degrees of belief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implications
Volume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov's three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes over time. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five.
Volume 2 gives arguments for the five core rules introduced in Volume 1, then considers challenges to Bayesian epistemology. It begins by detailing Bayesianism's successful applications to confirmation and decision theory. Then it describes three types of arguments for Bayesian rules, based on representation theorems, Dutch Books, and accuracy measures. Finally, it takes on objections to the Bayesian approach and alternative formalisms, including the statistical approaches of frequentism and likelihoodism.
For Volume 1 see [Zbl 1491.62004].The 16th meeting on the mathematics of language, MOL 2019. Proceedings, University of Toronto, Toronto, Canada, July 18--19, 2019https://zbmath.org/1491.680162022-09-13T20:28:31.338867ZFrom the introduction: These are the proceedings of the 16th Meeting on the Mathematics of Language (MOL 2019), held at the
University of Toronto, on July 18--19, 2019.
The volume contains ten regular papers, which have been selected from a total of eighteen submissions,
using the EasyChair conference management system.
The conference benefited from the financial support of the University of Toronto and of the Natural Sciences and Engineering Research Council of Canada, which we gratefully acknowledge.
Last but not least, we would like to express our sincere gratitude to all the reviewers for MOL 2019 and
to all the people who helped with the local organization.
The articles of this volume will be reviewed individually. For the preceding meeting see [Zbl 1376.68008].
Indexed articles:
\textit{Björklund, Henrik; Drewes, Frank; Ericson, Petter}, Parsing weighted order-preserving hyperedge replacement grammars, 1-11 [Zbl 07568231]
\textit{Graf, Thomas; De Santo, Aniello}, Sensing tree automata as a model of syntactic dependencies, 12-26 [Zbl 07568232]
\textit{Winter, Yoad}, Presupposition projection and repair strategies in trivalent semantics, 27-39 [Zbl 07568233]
\textit{Zwanziger, Colin}, Dependently-typed montague semantics in the proof assistant Agda-flat, 40-49 [Zbl 07568234]
\textit{Chandlee, Jane; Jardine, Adam}, Quantifier-free least fixed point functions for phonology, 50-62 [Zbl 07568235]
\textit{Rogers, James; Lambert, Dakotah}, Some classes of sets of structures definable without quantifiers, 63-77 [Zbl 07568236]
\textit{Burness, Phillip; McMullin, Kevin}, Efficient learning of output tier-based strictly 2-local functions, 78-90 [Zbl 07568237]
\textit{Chandlee, Jane; Eyraud, Rémi; Heinz, Jeffrey; Jardine, Adam; Rawski, Jonathan}, Learning with partially ordered representations, 91-101 [Zbl 07568238]
\textit{Shibata, Chihiro; Heinz, Jeffrey}, Maximum likelihood estimation of factored regular deterministic stochastic languages, 102-113 [Zbl 07568239]
\textit{Borbély, Gábor; Kornai, András}, Sentence length, 114-125 [Zbl 07568240]7th international conference on formal structures for computation and deduction, FSCD 2022, Haifa, Israel, August 2--5, 2022https://zbmath.org/1491.680172022-09-13T20:28:31.338867ZThe articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1465.68024].On symbolic heaps modulo permission theorieshttps://zbmath.org/1491.680502022-09-13T20:28:31.338867Z"Demri, Stéphane"https://zbmath.org/authors/?q=ai:demri.stephane-p"Lozes, Etienne"https://zbmath.org/authors/?q=ai:lozes.etienne"Lugiez, Denis"https://zbmath.org/authors/?q=ai:lugiez.denisSummary: We address the entailment problem for separation logic with symbolic heaps admitting list predicates and permissions for memory cells that are essential to express ownership of a heap region. In the permission-free case, the entailment problem is known to be in P. Herein, we design new decision procedures for solving the satisfiability and entailment problems that are parameterised by the permission theories. This permits the use of solvers dealing with the permission theory at hand, independently of the shape analysis. We also show that the entailment problem without list predicates is coNP-complete for several permission models, such as counting permissions and binary tree shares but the problem is in P for fractional permissions. Furthermore, when list predicates are added, we prove that the entailment problem is coNP-complete when the entailment problem for permission formulae is in coNP, assuming the write permission can be split into as many read permissions as desired. Finally, we show that the entailment problem for any Boolean permission model with infinite width is coNP-complete.
For the entire collection see [Zbl 1388.68010].Total search problems in bounded arithmetic and improved witnessinghttps://zbmath.org/1491.680762022-09-13T20:28:31.338867Z"Beckmann, Arnold"https://zbmath.org/authors/?q=ai:beckmann.arnold"Razafindrakoto, Jean-José"https://zbmath.org/authors/?q=ai:razafindrakoto.jean-joseSummary: We define a new class of total search problems as a subclass of Megiddo and Papadimitriou's class of total \({\mathsf{NP}}\) search problems, in which solutions are verifiable in \({\mathsf{AC}}^0\). We denote this class \(\forall\exists{\mathsf{AC}}^0\). We show that all total \({\mathsf{NP}}\) search problems are equivalent, w.r.t. \({\mathsf{AC}}^0\)-many-one reductions, to search problems in \(\forall\exists{\mathsf{AC}}^0\). Furthermore, we show that \(\forall\exists{\mathsf{AC}}^0\) contains well-known problems such as the Stable Marriage and the Maximal Independent Set problems. We introduce the class of Inflationary Iteration problems in \(\forall\exists{\mathsf{AC}}^0\), and show that it characterizes the provably total \({\mathsf{NP}}\) search problems of the bounded arithmetic theory corresponding to polynomial-time. Cook and Nguyen introduced a generic way of defining a bounded arithmetic theory \({\mathsf {VC}}\) for complexity classes \({\mathsf{C}}\) which can be obtained using a complete problem. For such \(C\) we will define a new class \({\mathsf {KPT[C]}}\) of \(\forall\exists{\mathsf{AC}}^0\) search problems based on student-teacher games in which the student has computing power limited to \({\mathsf{AC}}^0\). We prove that \({\mathsf{KPT[C]}}\) characterizes the provably total \({\mathsf{NP}}\) search problems of the bounded arithmetic theory corresponding to \({\mathsf{C}}\). All our characterizations are obtained via ``new-style'' witnessing theorems, where reductions are provable in a theory corresponding to \({\mathsf{AC}}^0\).
For the entire collection see [Zbl 1369.03021].Automating resolution is NP-hardhttps://zbmath.org/1491.680782022-09-13T20:28:31.338867Z"Atserias, Albert"https://zbmath.org/authors/?q=ai:atserias.albert"Müller, Moritz"https://zbmath.org/authors/?q=ai:muller.moritzLogic for \(\omega\)-pushdown automatahttps://zbmath.org/1491.680972022-09-13T20:28:31.338867Z"Droste, Manfred"https://zbmath.org/authors/?q=ai:droste.manfred"Dziadek, Sven"https://zbmath.org/authors/?q=ai:dziadek.sven"Kuich, Werner"https://zbmath.org/authors/?q=ai:kuich.wernerThis paper gives a characterization of \(\omega\)-context-free languages in terms of monadic second-order logic.
The authors first introduce a notion of \textit{simple \(\omega\)-pushdown automaton} (\(\omega\)SPDA), where the transitions allow only to push/pop a single symbol on the stack or leave the stack unaltered. In order to show that any \(\omega\)-context-free language, as defined in [\textit{R. S. Cohen} and \textit{A. Y. Gold}, J. Comput. Syst. Sci. 15, 169--184 (1977; Zbl 0363.68113)], can be accepted by an \(\omega\)SPDA, the authors use a result of [loc. cit.] to show that that any \(\omega\)-context-free language has a Büchi-accepting grammar in quadratic Greibach normal form. This hence shows the equivalence between \(\omega\)-context-freeness and being recognized by an \(\omega\)SPDA.
Furthermore, the authors introduce a monadic second-order logic \(\omega\)ML that uses a (non-overlapping) matching condition on pairs of positions similar to matching parentheses in a well-formed expression. Then, using the fact that their \(\omega\)SPDA are projections of the visibly pushdown automata of [\textit{R. Alur} and \textit{P. Madhusudan}, in: Proceedings of the 36th annual ACM symposium on theory of computing, STOC 2004. New York, NY: ACM Press. 202--211 (2004; Zbl 1192.68396)] and the expressive equivalence result of [\textit{R. Alur} and \textit{P. Madhusudan}, J. ACM 56, No. 3, Article No. 16, 43 p. (2009; Zbl 1325.68138)], they prove the equivalence between \(\omega\)ML-definability and \(\omega\)SPDA-recognizability.
This shows that, for an \(\omega\)-language, being \(\omega\)-context-free, \(\omega\)SPDA-recognizable, and \(\omega\)ML-definable are three equivalent notions.
Reviewer: Roger Villemaire (Montréal)Backward deterministic Büchi automata on infinite wordshttps://zbmath.org/1491.681032022-09-13T20:28:31.338867Z"Wilke, Thomas"https://zbmath.org/authors/?q=ai:wilke.thomasSummary: This paper describes how backward deterministic Büchi automata are defined, what their main features are, and how they can be applied to solve problems in temporal logic.
For the entire collection see [Zbl 1388.68010].Monitoring for silent actionshttps://zbmath.org/1491.681042022-09-13T20:28:31.338867Z"Aceto, Luca"https://zbmath.org/authors/?q=ai:aceto.luca"Achilleos, Antonis"https://zbmath.org/authors/?q=ai:achilleos.antonis"Francalanza, Adrian"https://zbmath.org/authors/?q=ai:francalanza.adrian"Ingólfsdóttir, Anna"https://zbmath.org/authors/?q=ai:ingolfsdottir.annaSummary: Silent actions are an essential mechanism for system modelling and specification. They are used to abstractly report the occurrence of computation steps without divulging their precise details, thereby enabling the description of important aspects such as the branching structure of a system. Yet, their use rarely features in specification logics used in runtime verification. We study monitorability aspects of a branching-time logic that employs silent actions, identifying which formulas are monitorable for a number of instrumentation setups. We also consider defective instrumentation setups that imprecisely report silent events, and establish monitorability results for tolerating these imperfections.
For the entire collection see [Zbl 1388.68010].Complexity of model checking MDPs against LTL specificationshttps://zbmath.org/1491.681112022-09-13T20:28:31.338867Z"Kini, Dileep"https://zbmath.org/authors/?q=ai:kini.dileep-raghunath"Viswanathan, Mahesh"https://zbmath.org/authors/?q=ai:viswanathan.maheshSummary: Given a Markov Decision Process (MDP) \(\mathcal{M}\), an LTL formula \(\varphi\), and a threshold \(\theta\in[0,1]\), the verification question is to determine if there is a scheduler with respect to which the executions of \(\mathcal{M}\) satisfying \(\varphi\) have probability greater than (or \(\ge\)) \(\theta\). When \(\theta=0\), we call it the qualitative verification problem, and when \(\theta\in(0,1]\), we call it the quantitative verification problem. In this paper we study the precise complexity of these problems when the specification is constrained to be in different fragments of LTL.
For the entire collection see [Zbl 1388.68010].VLDL satisfiability and model checking via tree automatahttps://zbmath.org/1491.681142022-09-13T20:28:31.338867Z"Weinert, Alexander"https://zbmath.org/authors/?q=ai:weinert.alexanderSummary: We present novel algorithms solving the satisfiability problem and the model checking problem for Visibly Linear Dynamic Logic (VLDL) in asymptotically optimal time via a reduction to the emptiness problem for tree automata with Büchi acceptance. Since VLDL allows for the specification of important properties of recursive systems, this reduction enables the efficient analysis of such systems. \par Furthermore, as the problem of tree automata emptiness is well-studied, this reduction enables leveraging the mature algorithms and tools for that problem in order to solve the satisfiability problem and the model checking problem for VLDL.
For the entire collection see [Zbl 1388.68010].Logic and algebra in unfolded Petri nets: on a duality between concurrency and causal dependencehttps://zbmath.org/1491.681162022-09-13T20:28:31.338867Z"Bernardinello, Luca"https://zbmath.org/authors/?q=ai:bernardinello.luca"Ferigato, Carlo"https://zbmath.org/authors/?q=ai:ferigato.carlo"Pomello, Lucia"https://zbmath.org/authors/?q=ai:pomello.luciaA partially ordered set is a couple \((P, \leqslant )\), where \(\leqslant\) is an antisymetric, transitive and reflexive relation on \(P\). If \(x \leqslant y\) or \(y \leqslant x\) we say that \(x\) and \(y\) are comparable (a causal dependence relation); otherwise they are called concurrent (denoted by \( x\, \mathbf{co}\, y\), a concurrency relation). In this way two distinct orthogonality spaces are consequently obtained. \((P, \leqslant )\) is \textit{N-dense} iff for any \( x,y,z,t \in P\) such that \(x < y\), \(x <t\), \(z<t\), \( x\, \mathbf{co}\, z\), \(y\, \mathbf{co}\, z\) and \(y\, \mathbf{co}\, t\), there exists \(k \in P\) such that \(k\, \mathbf{co}\, y\), \(k\, \mathbf{co}\, z\), \( x < k\) and \(k < t\). When the condition of N-density holds on both these orthogonality spaces, orthomodular posets formed by closed sets, defined according to Dacey, are studied. It is shown that the condition originally imposed by Dacey on the orthogonality spaces for obtaining an orthomodular poset from his closed sets is in fact equivalent to N-density. The requirement of N-density was as well fundamental in a previous work on orthogonality spaces with the concurrency relation [\textit{L. Bernardinello} et al., Fundam. Inform. 105, No. 3, 211--235 (2010; Zbl 1209.68333)]. Starting from a partially ordered set modelling a concurrent process, dual results for orthogonality spaces with the causal dependence relation with respect to orthogonality spaces with the concurrency relation are obtained.
Reviewer: Damas Gruska (Bratislava)A new linear logic for deadlock-free session-typed processeshttps://zbmath.org/1491.681202022-09-13T20:28:31.338867Z"Dardha, Ornela"https://zbmath.org/authors/?q=ai:dardha.ornela"Gay, Simon J."https://zbmath.org/authors/?q=ai:gay.simon-jSummary: The \(\pi\)-calculus, viewed as a core concurrent programming language, has been used as the target of much research on type systems for concurrency. In this paper we propose a new type system for deadlock-free session-typed \(\pi\)-calculus processes, by integrating two separate lines of work. The first is the propositions-as-types approach by Caires and Pfenning, which provides a linear logic foundation for session types and guarantees deadlock-freedom by forbidding cyclic process connections. The second is Kobayashi's approach in which types are annotated with priorities so that the type system can check whether or not processes contain genuine cyclic dependencies between communication operations. We combine these two techniques for the first time, and define a new and more expressive variant of classical linear logic with a proof assignment that gives a session type system with Kobayashi-style priorities. This can be seen in three ways: (i) as a new linear logic in which cyclic structures can be derived and a \textsc{Cycle}-elimination theorem generalises \textsc{Cut}-elimination; (ii) as a logically-based session type system, which is more expressive than Caires and Pfenning's; (iii) as a logical foundation for Kobayashi's system, bringing it into the sphere of the propositions-as-types paradigm.
For the entire collection see [Zbl 1386.68002].Modulo counting on words and treeshttps://zbmath.org/1491.681272022-09-13T20:28:31.338867Z"Bednarczyk, Bartosz"https://zbmath.org/authors/?q=ai:bednarczyk.bartosz"Charatonik, Witold"https://zbmath.org/authors/?q=ai:charatonik.witoldSummary: We consider the satisfiability problem for the two-variable fragment of the first-order logic extended with modulo counting quantifiers and interpreted over finite words or trees. We prove a small-model property of this logic, which gives a technique for deciding the satisfiability problem. In the case of words this gives a new proof of EXPSPACE upper bound, and in the case of trees it gives a 2EXPTIME algorithm. This algorithm is optimal: we prove a matching lower bound by a generic reduction from alternating Turing machines working in exponential space; the reduction involves a development of a new version of tiling games.
For the entire collection see [Zbl 1388.68010].Skolem function continuation for quantified Boolean formulashttps://zbmath.org/1491.681922022-09-13T20:28:31.338867Z"Fazekas, Katalin"https://zbmath.org/authors/?q=ai:fazekas.katalin"Heule, Marijn J. H."https://zbmath.org/authors/?q=ai:heule.marijn-j-h"Seidl, Martina"https://zbmath.org/authors/?q=ai:seidl.martina"Biere, Armin"https://zbmath.org/authors/?q=ai:biere.arminSummary: Modern solvers for quantified Boolean formulas (QBF) not only decide the satisfiability of a formula, but also return a set of Skolem functions representing a model for a true QBF. Unfortunately, in combination with a preprocessor this ability is lost for many preprocessing techniques. A preprocessor rewrites the input formula to an equi-satisfiable formula which is often easier to solve than the original formula. Then the Skolem functions returned by the solver represent a solution for the preprocessed formula, but not necessarily for the original encoding.
Our solution to this problem is to combine Skolem functions obtained from a \textsf{QRAT} trace as produced by the widely-used preprocessor \textsf{Bloqqer} with Skolem functions for the preprocessed formula. This approach is agnostic of the concrete rewritings performed by the preprocessor and allows the combination of \textsf{Bloqqer} with any Skolem function producing solver, hence realizing a smooth integration into the solving tool chain.
For the entire collection see [Zbl 1367.68007].A Hasse diagram for weighted sceptical semantics with a unique-status grounded semanticshttps://zbmath.org/1491.681962022-09-13T20:28:31.338867Z"Bistarelli, Stefano"https://zbmath.org/authors/?q=ai:bistarelli.stefano"Santini, Francesco"https://zbmath.org/authors/?q=ai:santini.francescoSummary: We provide an initial study on the Hasse diagram that represents the partial order -- w.r.t. set inclusion -- among weighted sceptical semantics in Argumentation: grounded, ideal, and eager. Being our framework based on a parametric structure of weights, we can directly compare weighted and classical approaches. We define a unique-status weighted grounded semantics, and we prove that the lattice of strongly-admissible extensions becomes a semi-lattice.
For the entire collection see [Zbl 1367.68005].A two-tiered propositional framework for handling multisource inconsistent informationhttps://zbmath.org/1491.681992022-09-13T20:28:31.338867Z"Ciucci, Davide"https://zbmath.org/authors/?q=ai:ciucci.davide"Dubois, Didier"https://zbmath.org/authors/?q=ai:dubois.didierSummary: This paper proposes a conceptually simple but expressive framework for handling propositional information stemming from several sources, namely a two-tiered propositional logic augmented with classical modal axioms (BC-logic), a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set-functions called Boolean capacities. We present a theorem-preserving translation of Belnap logic in this setting. As special cases, we can recover previous translations of three-valued logics such as Kleene and Priest logics. Our translation bridges the gap between Belnap logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various inconsistency handling methods.
For the entire collection see [Zbl 1367.68004].On Boolean algebras of conditionals and their logical counterparthttps://zbmath.org/1491.682032022-09-13T20:28:31.338867Z"Flaminio, Tommaso"https://zbmath.org/authors/?q=ai:flaminio.tommaso"Godo, Lluis"https://zbmath.org/authors/?q=ai:godo.lluis"Hosni, Hykel"https://zbmath.org/authors/?q=ai:hosni.hykelSummary: This paper sheds a novel light on the longstanding problem of investigating the logic of conditional events. Building on the framework of Boolean algebras of conditionals previously introduced by the authors, we make two main new contributions. First, we fully characterise the atomic structure of these algebras of conditionals. Second, we introduce the logic of Boolean conditionals (LBC) and prove its completeness with respect to the natural semantics induced by the structural properties of the atoms in a conditional algebra as described in the first part. In addition we outline the close connection of LBC with preferential consequence relations, arguably one of the most appreciated systems of non-monotonic reasoning.
For the entire collection see [Zbl 1367.68004].A monotonic view on reflexive autoepistemic reasoninghttps://zbmath.org/1491.682102022-09-13T20:28:31.338867Z"Su, Ezgi Iraz"https://zbmath.org/authors/?q=ai:su.ezgi-irazSummary: This paper introduces a novel monotonic modal logic, able to characterise reflexive autoepistemic reasoning of the nonmonotonic variant of modal logic SW5: we add a second new modal operator into the original language of SW5, and show that the resulting formalism called MRAE* is strong enough to capture the minimal model notion underlying some major forms of nonmonotonic logic among which are autoepistemic logic, default logic, and nonmonotonic logic programming. The paper ends with a discussion of a general strategy, naturally embedding several nonmonotonic logics of similar kinds.
For the entire collection see [Zbl 1367.68005].Definitions of concepts and imprecisionhttps://zbmath.org/1491.682202022-09-13T20:28:31.338867Z"Reformat, Marek Z."https://zbmath.org/authors/?q=ai:reformat.marek-z"Yager, Ronald R."https://zbmath.org/authors/?q=ai:yager.ronald-r"Chen, Jesse X."https://zbmath.org/authors/?q=ai:chen.jesse-xSummary: Knowledge graphs become an important form of representing data and information. Their intrinsic ability to express semantics via relations enables development of novel methods of processing data and building data models. In the paper, we propose a methodology for generating definitions of concepts and constructing their hierarchy. It is a fully data-driven process that uses information about multiple entities represented in a form of a knowledge graph. In this work, we state that a concept is defined via relations between the concept and other concepts. We perform a thorough analysis of relations and determine their levels of importance and degrees of their contributions to the definitions. This allows us to include impression reflecting the dependence of the construction process on the context in which it is performed -- a limited amount of available data in our case. We provide details of the proposed approach, and illustrate its performance presenting a case study using a set of facts from \url{https://dbpedia.org}. In the study, we construct a structure of concepts and investigate how importance of relations between them changes when levels of concept abstractions change.Pregroup grammars, their syntax and semanticshttps://zbmath.org/1491.682522022-09-13T20:28:31.338867Z"Sadrzadeh, Mehrnoosh"https://zbmath.org/authors/?q=ai:sadrzadeh.mehrnooshThis paper is dedicated to Lambek's pregroup grammars [\textit{J. Lambek}, J. Logic Lang. Inf. 17, No. 2, 141--160 (2008; Zbl 1162.68721)], the trouble with which has always been their semantics, or lack thereof. A cut-free sequent calculus has been developed for pregroups by \textit{W. Buszkowski} [Math. Log. Q. 49, No. 5, 467--474 (2003; Zbl 1036.03046)], who has also shown that the expressive power of pregroup grammars, similar to that of the syntactic calculus [\textit{J. Lambek}, Am. Math. Mon. 65, 154--170 (1958; Zbl 0080.00702)], is context-free [\textit{W. Buszkowski}, Z. Math. Logik Grundlagen Math. 31, 369--384 (1985; Zbl 0559.68063)]. The set-theoretic semantics is, however, ambiguous, as a pregroup term \(abc^{l}\) has two interpretations, namely, \(A\times C^{B}\) and \(C^{A\times B}\).
This article studies the semantics of pregroup grammars, surveying recent advances in vector space modelling in natural language processing. Following a suggestion of Lambek, the author addresses finite-dimensional vector space semantics for pregroups, in which the adjoint types are to be interpreted as dual spaces. The author builds semantic vector representations for some exemplary words, phrases and sentences of language, showing how compositionality of vector semantics disambiguates meaning. Finally, the paper presents a vector semantics and analysis of questions, demonstrating how their representations relate to the sentences they are asked about.
For the entire collection see [Zbl 1470.03008].
Reviewer: Hirokazu Nishimura (Tsukuba)Decision making using new category of similarity measures and study their applications in medical diagnosis problemshttps://zbmath.org/1491.900952022-09-13T20:28:31.338867Z"Khalil, Shuker Mahmood"https://zbmath.org/authors/?q=ai:khalil.shuker-mahmoodSummary: The aim of this paper is to propose a new category of similarity measures, we begin to introduce the concept of effect matrix \(\overset{\wedge}{T} (R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v)\) of type 3-tuple and study some of their properties. Moreover, from the soft set we find the pictures of type regular and irregular using effect matrix \(\overset{\wedge}{T}(R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v)\) of type 3-tuple are found. An effect matrix \(\overset{\wedge}{T} (R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v )\) of type 3-tuple is better than effect matrix \(\overset{\wedge}{T}(\mathfrak{R}_1 \times \mathfrak{R}_2 ,p_n ,q_m)\) of type 2-tuple, because we can deal with three different sets \(R_1\) (a family of objectives), \(R_2\) (a family of parameters), \(R_3\) (a family of second parameters) in the same problem. This burden can be alleviated by application of type 3-tuple. Some applications of soft effect matrix of type 3-tuple \(\overset{\wedge}{T} (R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v)\) in decision making problems are studied and explained. In this work we deal with pictures. Moreover, the similarity measure between two different soft sets under the same universal soft set \((R_1 \times R_2 \times R_3)\) can be studied and explained its applications in medical diagnosis problems.An epistemic generalization of rationalizabilityhttps://zbmath.org/1491.910292022-09-13T20:28:31.338867Z"Parikh, Rohit"https://zbmath.org/authors/?q=ai:parikh.rohitSummary: Rationalizability, originally proposed by Bernheim and Pearce, generalizes the notion of Nash equilibrium. Nash equilibrium requires common knowledge of strategies. Rationalizability only requires common knowledge of rationality. However, their original notion assumes that the payoffs are common knowledge.
I.e. agents do know what world they are in, but may be ignorant of what other agents are playing.
We generalize the original notion of rationalizability to consider situations where agents do not know what world they are in, or where some know but others do not know. Agents who know something about the world can take advantage of their superior knowledge. It may also happen that both Ann and Bob know about the world but Ann does not know that Bob knows. How might they act?
We will show how a notion of rationalizability in the context of partial knowledge, represented by a Kripke structure, can be developed.
For the entire collection see [Zbl 1369.03021].Development of harmonic aggregation operator with trapezoidal Pythagorean fuzzy numbershttps://zbmath.org/1491.910472022-09-13T20:28:31.338867Z"Aydin, Serhat"https://zbmath.org/authors/?q=ai:aydin.serhat"Kahraman, Cengiz"https://zbmath.org/authors/?q=ai:kahraman.cengiz"Kabak, Mehmet"https://zbmath.org/authors/?q=ai:kabak.mehmetSummary: Pythagorean fuzzy sets are one of the extensions of ordinary fuzzy sets and allow a larger domain to be utilized by decision makers with respect to other extensions. Pythagorean fuzzy sets have been often used as an effective tool for handling the vagueness of multi-criteria decision making problems. Aggregation operators are a useful tool in order to collect different information provided by different sources. The objective of this paper is to develop harmonic aggregation operators for trapezoidal Pythagorean fuzzy numbers. We developed trapezoidal Pythagorean fuzzy weighted harmonic mean operator, trapezoidal Pythagorean fuzzy ordered weighted harmonic mean operator, and trapezoidal Pythagorean fuzzy hybrid harmonic mean operator. We proved some theorems for the developed operators. Finally, we presented an illustrative example using the proposed aggregation operators in order to rank the alternatives.