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Mathematics
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Mathematical Logic, Algorithms and Applications
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Mathematical Logic, Algorithms and Applications

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 2023) | Viewed by 9253

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

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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
Prof. Dr. Su Gao

E-Mail Website
Guest Editor
School of Mathematical Science, Nankai University, Tianjin 300071, China
Interests: mathematical logic and foundations of mathematics; descriptive set theory; topology and topological groups; dynamical systems

Special Issue Information

Dear Colleagues,

Mathematical logic and algorithms is a thriving field of mathematics with a considerable variety of applications.

The volume accepts high-quality papers presenting original research on mathematical logic and algorithms 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 (linear or close to them ) 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 invite papers related to applications of mathematical logic and algorithms to mathematical biology, bioinformatics, theoretical medicine, mathematical physics, to problems of artificial intelligence (knowledge representation, pattern recognition, image analysis and understanding, etc.), and to challenges of finding information quickly and efficiently by accessing big data repositories.

Papers devoted to the conceptual and philosophical aspects of these fundamental and applied problems are also welcome.

Papers devoted to theoretical and applied aspects of programming (especially for distributed and supercomputer systems) and heuristic techniques that ensure efficient computing and work with big data will be met with special attention and interest. Finally, examples of large projects carried out using supercomputer computing are also included in the area of interest.

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

This Special Issue will be a continuation of the Special Issues "Mathematical Logic and Its Applications 2020" and "Mathematical Logic and Its Applications 2021".

Prof. Dr. Vassily Lyubetsky
Prof. Dr. Vladimir Kanovei
Prof. Dr. Su Gao
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
  • 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

19 pages, 432 KiB  
Article
The Logic of Homophily Dynamics in Heterogeneous Networks: Axiomatization, Model Checking and Validity Checking
by Xiling Luo
Mathematics 2023, 11(16), 3484; https://doi.org/10.3390/math11163484 - 11 Aug 2023
Cited by 1 | Viewed by 667
Abstract
Social networks have received considerable attention from the modal logic community. In this article, we study and characterize one of the most important principles in the field of social networks. Homophily, which means similarity breeds association, reveals the nature of social organization. In [...] Read more.
Social networks have received considerable attention from the modal logic community. In this article, we study and characterize one of the most important principles in the field of social networks. Homophily, which means similarity breeds association, reveals the nature of social organization. In order to be able to express similarity and association together, we generalize the basic network and then define the heterogeneous network. The heterogeneous network is also defined to provide a good foundation for the use of logical approaches. The Logic of Homophily LHG,M that we propose in this article is based on Computation Tree Logic and Formal Concept Analysis. LHG,M describes the homophily dynamics of the heterogeneous networks at a specified similarity coefficient. Furthermore, we not only axiomatize the LHG,M and prove that the axiom system LHG,Mn is sound and complete, but we also prove that the model checking and the validity checking for LHG,M are both PSPACE-complete. Full article
(This article belongs to the Special Issue Mathematical Logic, Algorithms and Applications)
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8 pages, 263 KiB  
Article
Borel Chain Conditions of Borel Posets
by Ming Xiao
Mathematics 2023, 11(15), 3349; https://doi.org/10.3390/math11153349 - 31 Jul 2023
Viewed by 546
Abstract
We study the coarse classification of partial orderings using chain conditions in the context of descriptive combinatorics. We show that (unlike the Borel counterpart of many other combinatorial notions), we have a strict hierarchy of different chain conditions, similar to the classical case. [...] Read more.
We study the coarse classification of partial orderings using chain conditions in the context of descriptive combinatorics. We show that (unlike the Borel counterpart of many other combinatorial notions), we have a strict hierarchy of different chain conditions, similar to the classical case. Full article
(This article belongs to the Special Issue Mathematical Logic, Algorithms and Applications)
16 pages, 406 KiB  
Article
Axiomatization of Blockchain Theory
by Sergey Goncharov and Andrey Nechesov
Mathematics 2023, 11(13), 2966; https://doi.org/10.3390/math11132966 - 03 Jul 2023
Cited by 1 | Viewed by 1976
Abstract
The increasing use of artificial intelligence algorithms, smart contracts, the internet of things, cryptocurrencies, and digital money highlights the need for secure and sustainable decentralized solutions. Currently, the blockchain technology serves as the backbone for most decentralized systems. However, the question of axiomatization [...] Read more.
The increasing use of artificial intelligence algorithms, smart contracts, the internet of things, cryptocurrencies, and digital money highlights the need for secure and sustainable decentralized solutions. Currently, the blockchain technology serves as the backbone for most decentralized systems. However, the question of axiomatization of the blockchain theory in the first-order logic has been open until today, despite the efficient computational implementations of these systems. This did not allow one to formalize the blockchain structure, as well as to model and verify it using logical methods. This work introduces a finitely axiomatizable blockchain theory T that defines a class of blockchain structures K using the axioms of the first-order logic. The models of the theory T are well-known blockchain implementations with the proof of work consensus algorithm, including Bitcoin, Ethereum (PoW version), Ethereum Classic, and some others. By utilizing mathematical logic, we can study these models and derive new theorems of the theory T through automatic proofs. Also, the axiomatization of blockchain opens up new opportunities to develop blockchain-based systems that can help solve some of the open problems in the fields of artificial intelligence, robotics, cryptocurrencies, etc. Full article
(This article belongs to the Special Issue Mathematical Logic, Algorithms and Applications)
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28 pages, 538 KiB  
Article
A Comprehensive Formalization of Propositional Logic in Coq: Deduction Systems, Meta-Theorems, and Automation Tactics
by Dakai Guo and Wensheng Yu
Mathematics 2023, 11(11), 2504; https://doi.org/10.3390/math11112504 - 29 May 2023
Cited by 1 | Viewed by 2058
Abstract
The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental [...] Read more.
The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental aspect of mathematical reasoning and proof construction. We construct four Hilbert-style axiom systems and a natural deduction system for propositional logic, and establish their equivalences through meticulous proofs. Moreover, we provide formal proofs for essential meta-theorems in propositional logic, including the Deduction Theorem, Soundness Theorem, Completeness Theorem, and Compactness Theorem. Importantly, we present an exhaustive formal proof of the Completeness Theorem in this paper. To bolster the proof of the Completeness Theorem, we also formalize concepts related to mappings and countability, and deliver a formal proof of the Cantor–Bernstein–Schröder theorem. Additionally, we devise automated Coq tactics explicitly designed for the propositional logic inference system delineated in this study, enabling the automatic verification of all tautologies, all internal theorems, and the majority of syntactic and semantic inferences within the system. This research contributes a versatile and reusable Coq library for propositional logic, presenting a solid foundation for numerous applications in mathematics, such as the accurate expression and verification of properties in software programs and digital circuits. This work holds particular importance in the domains of mathematical formalization, verification of software and hardware security, and in enhancing comprehension of the principles of logical reasoning. Full article
(This article belongs to the Special Issue Mathematical Logic, Algorithms and Applications)
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39 pages, 10218 KiB  
Article
Constructing an Evolutionary Tree and Path–Cycle Graph Evolution along It
by Konstantin Gorbunov and Vassily Lyubetsky
Mathematics 2023, 11(9), 2024; https://doi.org/10.3390/math11092024 - 24 Apr 2023
Cited by 1 | Viewed by 1106
Abstract
The paper solves the problem of constructing an evolutionary tree and the evolution of structures along it. This problem has long been posed and extensively researched; it is formulated and discussed below. As a result, we construct an exact cubic-time algorithm which outputs [...] Read more.
The paper solves the problem of constructing an evolutionary tree and the evolution of structures along it. This problem has long been posed and extensively researched; it is formulated and discussed below. As a result, we construct an exact cubic-time algorithm which outputs a tree with the minimum cost of embedding into it and of embedding it into a given network (Theorem 1). We construct an algorithm that outputs a minimum embedding of a tree into a network, taking into account incomplete linear sorting; the algorithm depends linearly on the number of nodes in the network and is exact if the sorting cost is not less than the sum of the duplication cost and the loss cost (Theorem 3). We construct an exact approximately quadratic-time algorithm which, for arbitrary costs of SCJ operations, solves the problem of reconstruction of given structures on any two-star tree (Theorem 4). We construct an exact algorithm which reduced the problem of DCJ reconstruction of given structures on any star to a logarithmic-length sequence of SAT problems, each of them being of approximately quadratic size (Theorem 5). The theorems have rigorous and complete proofs of correctness and complexity of the algorithms, and are accompanied by numerical examples and numerous explanatory illustrations, including flowcharts. Full article
(This article belongs to the Special Issue Mathematical Logic, Algorithms and Applications)
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17 pages, 300 KiB  
Article
Fullness and Decidability in Continuous Propositional Logic
by Xuanzhi Ren
Mathematics 2022, 10(23), 4455; https://doi.org/10.3390/math10234455 - 25 Nov 2022
Viewed by 1736
Abstract
In this paper we consider general continuous propositional logics and prove some basic properties about them. First, we characterize full systems of continuous connectives of the form {¬,∸,f} where f is a unary connective. We also show [...] Read more.
In this paper we consider general continuous propositional logics and prove some basic properties about them. First, we characterize full systems of continuous connectives of the form {¬,∸,f} where f is a unary connective. We also show that, in contrast to the classical propositional logic, a full system of continuous propositional logic cannot contain only one continuous connective. We then construct a closed full system of continuous connectives without any constants. Such a system does not have any tautologies. For the rest of the paper we consider the standard continuous propositional logic as defined by Yaacov, I.B and Usvyatsov, A. We show that Strong Compactness and Craig Interpolation fail for this logic, but approximated versions of Strong Compactness and Craig Interpolation hold true. In the last part of the paper, we introduce various notions of satisfiability, falsifiability, tautology, and fallacy, and show that they are either NP-complete or co-NP-complete. Full article
(This article belongs to the Special Issue Mathematical Logic, Algorithms and Applications)
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