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Special Issue "Mathematical Logic and Its Applications 2021"

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 July 2022) | Viewed by 8931

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Special Issue Editors

Prof. Dr. Vassily Lyubetsky

E-Mail Website
Guest Editor

Institute for Information Transmission Problems of the Russian Academy of Sciences, Department of Mechanics and Mathematics of Moscow Lomonosov State University, Moscow, Russia
Interests: descriptive set theory; forcing; nonstandard analysis; discrete optimization; mathematical biology
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Vladimir Kanovei

E-Mail Website
Guest Editor

Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia
Interests: descriptive set theory; forcing; nonstandard analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical logic is a thriving field of mathematics having a considerable variety of applications.

The volume accepts high-quality papers presenting original researches in mathematical logic and applications that focus mainly (but not only) on descriptive set theory, non-standard analysis, definability, and forcing, as well as discrete optimization, including optimization by exact algorithms of polynomial computational complexity or, by contrast, NP-hardness of the corresponding problem. Papers on the complexity of objects (words and graphs) and of the distance between them, on the complexity of computations, on proof theory (in particular as generalized computations), on non-classical and modal logic and (especially) related theories, including intuitionistic set theory, are also welcome.

We gladly invite papers to this Special Issue related to applications of mathematical logic to mathematical physics, mathematical biology, bioinformatics, theoretical medicine, to problems of artificial intelligence (knowledge representation, pattern recognition, image analysis and understanding, et cetera), and to challenges of finding information quickly and efficiently by accessing big data repositories.

These topics, both fundamental and applied, cover many important modern trends.

This special issue will be a continuation of the special issue "Mathematical Logic and Its Applications 2020", published at https://www.mdpi.com/journal/mathematics/special_issues/math-logic-2020 and also as a separate reprint (ISBN TBA).

Prof. Dr. Vassily Lyubetsky
Prof. Dr. Vladimir Kanovei
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2100 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

descriptive set theory
non-standard analysis
definability
forcing
algorithmic optimization
discrete optimization
exact algorithms of polynomial complexity
NP-hardness
computational complexity
proof theory
non-classical logic
modal logic
intuitionistic set theory
mathematical biology
artificial intelligence

Published Papers (6 papers)

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Research

Open AccessArticle

The Alpha-Beta Family of Filters to Solve the Threshold Problem: A Comparison

Bogdan Ilie Sighencea

Rareș Ion Stanciu

Ciprian Șorândaru

and

Cătălin Daniel Căleanu

Mathematics 2022, 10(6), 880; https://doi.org/10.3390/math10060880 - 10 Mar 2022

Cited by 3 | Viewed by 1716

Abstract

Typically, devices work to improve life quality, measure parameters, and make decisions. They also signalize statuses, and take actions accordingly. When working, they measure different values. These are to be compared against thresholds. Some time ago, vision systems came into play. They use camera(s) to deliver(s) images to a processor module. The received images are processed to perform detections (typically, they focus to detect objects, pedestrians, mopeds, cyclists, etc.). Images are analyzed and thresholds are used to compare the computed values. The important thing is that images are affected by noise. Therefore, the vision system performance can be affected by weather in some applications (for example, in automotive). An interesting case in this domain is when the measured/computed values show small variations near the threshold (not exceeding) but very close to it. The system is not able to signalize/declare a state in this case. It is also important to mention that changing the threshold does not guarantee solving the problem in any future case, since this may happen again. This paper proposes the Alpha-Beta family of filters as a solution to this problem. The members can track a signal based on measured values. This reveals errors when the tracked-signal’s first derivative changes sign. These errors are used in this paper to bypass the threshold problem. Since these errors appear in both situations (when the first derivative decreases from positive to negative and increases from negative to positive), the proposed method works when the observed data are in the vicinity of the threshold but above it. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)

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Figure 1

Open AccessArticle

A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

Vladimir Kanovei

and

Vassily Lyubetsky

Mathematics 2022, 10(3), 491; https://doi.org/10.3390/math10030491 - 03 Feb 2022

Cited by 2 | Viewed by 909

Abstract

The notion of ordinal definability and the related notions of ordinal definable sets (class

OD

) and hereditarily ordinal definable sets (class

HOD

) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still [...] Read more.

The notion of ordinal definability and the related notions of ordinal definable sets (class

OD

) and hereditarily ordinal definable sets (class

HOD

) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still based on the notion of ordinal definability, but in a more blurry way. In particular, Tzouvaras has recently introduced the notion of sets nontypical in the Russell sense, so that a set x is nontypical if it belongs to a countable ordinal definable set. Tzouvaras demonstrated that the class

HNT

of all hereditarily nontypical sets satisfies all axioms of

ZF

and satisfies

HOD \subseteq HNT

. In view of this, Tzouvaras proposed a problem—to find out whether the class

HNT

can be separated from

HOD

by the strict inclusion

HOD ⫋ HNT

, and whether it can also be separated from the universe

V

of all sets by the strict inclusion

HNT ⫋ V

, in suitable set theoretic models. Solving this problem, a generic extension

L [a, x]

of the Gödel-constructible universe

L

, by two reals

a, x

, is presented in this paper, in which the relation

L = HOD ⫋ L [a] = HNT ⫋ L [a, x] = V

is fulfilled, so that

HNT

is a model of

ZFC

strictly between

HOD

and the universe. Our result proves that the class

HNT

is really a new rich class of sets, which does not necessarily coincide with either the well-known class

HOD

or the whole universe

V

. This opens new possibilities in the ongoing study of the consistency and independence problems in modern set theory. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)

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Figure 1

Open AccessArticle

Blurry Definability

Gunter Fuchs

Mathematics 2022, 10(3), 452; https://doi.org/10.3390/math10030452 - 30 Jan 2022

Cited by 4 | Viewed by 1335

Abstract

I begin the study of a hierarchy of (hereditarily)

< κ

-blurrily ordinal definable sets. Here for a cardinal

κ

, a set is

< κ

-blurrily ordinal definable if it belongs to an

OD

set of cardinality less than

κ

, and [...] Read more.

I begin the study of a hierarchy of (hereditarily)

< κ

-blurrily ordinal definable sets. Here for a cardinal

κ

, a set is

< κ

-blurrily ordinal definable if it belongs to an

OD

set of cardinality less than

κ

, and it is hereditarily so if it and each member of its transitive closure is. I show that the class of hereditarily

< κ

-blurrily ordinal definable sets is an inner model of

ZF

. It satisfies the axiom of choice iff it is a

κ

-c.c. forcing extension of

HOD

, and

HOD

is definable inside it (even if it fails to satisfy the axiom of choice). Of particular interest are cardinals

λ

such that some set is hereditarily

< λ

-blurrily ordinal definable but not hereditarily

< κ

-blurrily ordinal definable for any cardinal

κ < λ

. Such cardinals I call leaps. The main results concern the structure of leaps. For example, I show that if

λ

is a limit of leaps, then the collection of all hereditarily

< λ

-blurrily ordinal definable sets is a model of

ZF

in which the axiom of choice fails. Using forcing, I produce models exhibiting various leap constellations, for example models in which there is a (regular/singular) limit leap whose cardinal successor is a leap. Many open questions remain. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)

Open AccessArticle

Nonstandard Hulls of C^*-Algebras and Their Applications

Stefano Baratella

Mathematics 2021, 9(20), 2598; https://doi.org/10.3390/math9202598 - 15 Oct 2021

Viewed by 528

Abstract

For the sake of providing insight into the use of nonstandard techniques à la A. Robinson and into Luxemburg’s nonstandard hull construction, we first present nonstandard proofs of some known results about

C^{*}

-algebras. Then we introduce extensions of the nonstandard hull [...] Read more.

C^{*}

-algebras. Then we introduce extensions of the nonstandard hull construction to noncommutative probability spaces and noncommutative stochastic processes. In the framework of internal noncommutative probability spaces, we investigate properties like freeness and convergence in distribution and their preservation by the nonstandard hull construction. We obtain a nonstandard characterization of the freeness property. Eventually we provide a nonstandard characterization of the property of equivalence for a suitable class of noncommutative stochastic processes and we study the behaviour of the latter property with respect to the nonstandard hull construction. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)

Open AccessArticle

Multiplicatively Exact Algorithms for Transformation and Reconstruction of Directed Path-Cycle Graphs with Repeated Edges

Konstantin Gorbunov

and

Vassily Lyubetsky

Mathematics 2021, 9(20), 2576; https://doi.org/10.3390/math9202576 - 14 Oct 2021

Cited by 1 | Viewed by 828

Abstract

For any weighted directed path-cycle graphs, a and b (referred to as structures), and any equal costs of operations (intermergings and duplication), we obtain an algorithm which, by successively applying these operations to a, outputs b if the first structure contains no paralogs (i.e., edges with a repeated name) and the second has no more than two paralogs for each edge. In finding the shortest sequence of operations to be applied to pass from a to b, the algorithm has a multiplicative error of at most 13/9 + ε, where ε is any strictly positive number, and its runtime is of the order of

n^{O (ε^{- 2.6})}

, where n is the size of the input pair of graphs. In the case of no paralogs, equal sets of names in the structures, and equal operation costs, we have considered the following conditions on the transformation of a into b: all structures in them are from one cycle; all structures are from one path; all structures are from paths. For each of the conditions, we have obtained an exact (i.e., zero-error) quadratic time algorithm for finding the shortest transformation of a into b. For another list of operations (join and cut of a vertex, and deletion and insertion of an edge) over structures and for arbitrary costs of these operations, we have obtained an algorithm for the extension of structures specified at the leaves of a tree onto its interior vertices. The algorithm is exact if the tree is a star—in this case, structures in the leaves may even have unequal sets of names or paralogs. The runtime of the algorithm is of the order of nΧ + n²log(n), where n is the number of names in the leaves, and Χ is an easily computable characteristic of the structures in the leaves. In the general case, a cubic time algorithm finds a locally minimal solution. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)

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Open AccessArticle

On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom

Ali Enayat

Vladimir Kanovei

and

Vassily Lyubetsky

Mathematics 2021, 9(14), 1670; https://doi.org/10.3390/math9141670 - 15 Jul 2021

Cited by 2 | Viewed by 1665

Abstract

Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface

Π_{2}^{1}

equivalence relation [...] Read more.

Π_{2}^{1}

equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface

Π_{n}^{1}

, does not imply the existence of such a pair with the associated relation in

Σ_{n}^{1}

or in a lower class. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)

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