Special Issue "Mathematical Logic and Its Applications 2021"

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

Deadline for manuscript submissions: closed (15 January 2022).

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 papers will be 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 1800 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 (3 papers)

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Research

Article
Nonstandard Hulls of C*-Algebras and Their Applications
Mathematics 2021, 9(20), 2598; https://doi.org/10.3390/math9202598 - 15 Oct 2021
Viewed by 170
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.
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 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)
Article
Multiplicatively Exact Algorithms for Transformation and Reconstruction of Directed Path-Cycle Graphs with Repeated Edges
Mathematics 2021, 9(20), 2576; https://doi.org/10.3390/math9202576 - 14 Oct 2021
Viewed by 278
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 [...] Read more.
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 nO(ε2.6), 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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Article
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom
Mathematics 2021, 9(14), 1670; https://doi.org/10.3390/math9141670 - 15 Jul 2021
Viewed by 957
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 Π21 equivalence relation [...] Read more.
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 Π21 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 Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class. Full article
(This article belongs to the Special Issue Mathematical Logic and Its Applications 2021)
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