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Mathematical Logic and Its Applications 2020

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A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (15 November 2020) | Viewed by 14486

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Special Issue Editors

Prof. Dr. Vassily Lyubetsky

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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 with a wide range of various applications.

The volume accepts high-quality papers with original research in mathematical logic and applications, with the main (but not only) focus on descriptive set theory, definability, and forcing, on the one hand, and algorithmic and combinatorial optimization, including optimization by linear (or near-linear complexity) algorithms, as well as exact algorithms of low computational complexity or, by contrast, NP-hardness of the corresponding problem, on the other hand.

We gladly invite papers to this Special Issue related to applications of mathematical logic to mathematical physics, mathematical biology, and theoretical medicine. These topics, both fundamental and applied, cover many important modern trends.

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 2600 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
definability
forcing
algorithmic optimization
combinatorial optimization
exact algorithms of low computational complexity
NP-hardness
Kolmogorov complexity

Published Papers (6 papers)

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Research

8 pages, 1372 KiB

Open AccessCommunication

Physiological Balance of the Body: Theory, Algorithms, and Results

by Irina Alchinova and Mikhail Karganov

Mathematics 2021, 9(3), 209; https://doi.org/10.3390/math9030209 - 20 Jan 2021

Cited by 5 | Viewed by 2091

Abstract

Aim: To confirm algorithm of determination of risk groups with physiological imbalance in the population exposed to unfavorable anthropogenic influences. Methods: The testing included such functional systems as constitution, myocardial contractility, autonomic regulation of the heart rate, regulation of peripheral circulation, psychomotor regulation, respiratory regulation and metabolism. Monitoring is carried out using computerized measurement instrumentation and data processing systems. Results: A risk group with pronounced shifts in the physiological balance was identified, which made up 38% of the surveyed population. The greatest contribution to the imbalance was made by the psychomotor system. Conclusion: We analyzed two different components of organism’s adaptation: resistance and resilience. Physiological systems experiencing increasing load attain a tipping points, where even a weak disturbing influence can induce transition to a qualitatively different state. This transition can result in either recovery of the regulatory stability of the system, or its transition to a lower level (dysregulation) with further development of a pathology. In this regard, of paramount importance is early detection of the signals about approaching the tipping points, one of these is the slowing down phenomenon during functional tests. In view of intricate interaction of physiological systems, recording of as much indicators as possible is advisable. The method of partial correlations is effective for evaluation of adaptive interaction of systems. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)

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36 pages, 703 KiB

Open AccessArticle

On the ‘Definability of Definable’ Problem of Alfred Tarski

by Vladimir Kanovei and Vassily Lyubetsky

Mathematics 2020, 8(12), 2214; https://doi.org/10.3390/math8122214 - 14 Dec 2020

Cited by 6 | Viewed by 2250

Abstract

In this paper we prove that for any

m \geq 1

there exists a generic extension of

L

, the constructible universe, in which it is true that the set of all constructible reals (here subsets of

ω

) is equal to the [...] Read more.

In this paper we prove that for any

m \geq 1

there exists a generic extension of

L

, the constructible universe, in which it is true that the set of all constructible reals (here subsets of

ω

) is equal to the set

D_{1 m}

of all reals definable by a parameter free type-theoretic formula with types bounded by m, and hence the Tarski ‘definability of definable’ sentence

D_{1 m} \in D_{2 m}

(even in the form

D_{1 m} \in D_{21}

) holds for this particular m. This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)

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30 pages, 2293 KiB

Open AccessArticle

Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions

by Konstantin Gorbunov and Vassily Lyubetsky

Mathematics 2020, 8(11), 2001; https://doi.org/10.3390/math8112001 - 10 Nov 2020

Cited by 3 | Viewed by 1725

Abstract

We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)

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33 pages, 451 KiB

Open AccessArticle

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

by Alexei Kanel-Belov, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie-Tai Yu and Wenchao Zhang

Mathematics 2020, 8(10), 1694; https://doi.org/10.3390/math8101694 - 2 Oct 2020

Viewed by 2133

Abstract

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)

30 pages, 606 KiB

Open AccessArticle

On the Δ n 1 Problem of Harvey Friedman

by Vladimir Kanovei and Vassily Lyubetsky

Mathematics 2020, 8(9), 1477; https://doi.org/10.3390/math8091477 - 1 Sep 2020

Cited by 8 | Viewed by 2116

Abstract

In this paper, we prove the following. If

n \geq 3

, then there is a generic extension of

L

, the constructible universe, in which it is true that the set

P (ω) \cap L

of all constructible reals (here—subsets [...] Read more.

In this paper, we prove the following. If

n \geq 3

, then there is a generic extension of

L

, the constructible universe, in which it is true that the set

P (ω) \cap L

of all constructible reals (here—subsets of

ω

) is equal to the set

P (ω) \cap Δ_{n}^{1}

of all (lightface)

Δ_{n}^{1}

reals. The result was announced long ago by Leo Harrington, but its proof has never been published. Our methods are based on almost-disjoint forcing. To obtain a generic extension as required, we make use of a forcing notion of the form

Q = C ℂ \times \prod_{ν} Q_{ν}

L

, where

C

adds a generic collapse surjection b from

ω

onto

P (ω) \cap L

, whereas each

Q_{ν}

ν < ω_{2}^{L}

, is an almost-disjoint forcing notion in the

ω_{1}

-version, that adjoins a subset

S_{ν}

ω_{1}^{L}

. The forcing notions involved are independent in the sense that no

Q_{ν}

-generic object can be added by the product of

C

and all

Q_{ξ}

ξ \neq ν

. This allows the definition of each constructible real by a

Σ_{n}^{1}

formula in a suitably constructed subextension of the

Q

-generic extension. The subextension is generated by the surjection b, sets

S_{ω \cdot k + j}

with

j \in b (k)

, and sets

S_{ξ}

with

ξ \geq ω \cdot ω

. A special character of the construction of forcing notions

Q_{ν}

L

, which depends on a given

n \geq 3

, obscures things with definability in the subextension enough for vice versa any

Δ_{n}^{1}

real to be constructible; here the method of hidden invariance is applied. A discussion of possible further applications is added in the conclusive section. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)

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46 pages, 794 KiB

Open AccessArticle

Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

by Vladimir Kanovei and Vassily Lyubetsky

Mathematics 2020, 8(6), 910; https://doi.org/10.3390/math8060910 - 3 Jun 2020

Cited by 6 | Viewed by 2906

Abstract

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that

n \geq 2

. Then: 1. If it holds in the constructible universe

L

that [...] Read more.

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that

n \geq 2

. Then: 1. If it holds in the constructible universe

L

that

a \subseteq ω

and

a \notin Σ_{n}^{1} \cup Π_{n}^{1}

, then there is a generic extension of

L

in which

a \in Δ_{n + 1}^{1}

but still

a \notin Σ_{n}^{1} \cup Π_{n}^{1}

, and moreover, any set

x \subseteq ω

x \in Σ_{n}^{1}

, is constructible and

Σ_{n}^{1}

L

. 2. There exists a generic extension

L

in which it is true that there is a nonconstructible

Δ_{n + 1}^{1}

set

a \subseteq ω

, but all

Σ_{n}^{1}

sets

x \subseteq ω

are constructible and even

Σ_{n}^{1}

L

, and in addition,

V = L [a]

in the extension. 3. There exists an generic extension of

L

in which there is a nonconstructible

Σ_{n + 1}^{1}

set

a \subseteq ω

, but all

Δ_{n + 1}^{1}

sets

x \subseteq ω

are constructible and

Δ_{n + 1}^{1}

L

. Thus, nonconstructible reals (here subsets of

ω

) can first appear at a given lightface projective class strictly higher than

Σ_{2}^{1}

, in an appropriate generic extension of

L

. The lower limit

Σ_{2}^{1}

is motivated by the Shoenfield absoluteness theorem, which implies that all

Σ_{2}^{1}

sets

a \subseteq ω

are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to

L

, which are very similar at a given projective level n but discernible at the next level

n + 1

. Full article

(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)

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