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Mathematics
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Decidability of Logics and Their Theories and Combinations
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Decidability of Logics and Their Theories and Combinations

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

Deadline for manuscript submissions: closed (31 May 2022) | Viewed by 5451

Special Issue Editors

Prof. Dr. Cristina Sernadas

E-Mail Website
Guest Editor
Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Interests: logics and theories and their combination (e.g., meet, importing, fibring); properties of logics and their preservation, including decidability; deductive systems; evidence, probability, and quantum logics
Special Issues, Collections and Topics in MDPI journals
Dr. João Rasga

E-Mail Website
Guest Editor
Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Interests: logics and theories and their combination (e.g., meet, importing, fibring); preservation of properties of logics and theories under combination; proof theory; evidence, probability, and quantum logics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The decidability problem in the context of mathematical theories was clearly stated by David Hilbert at the International Congress of Mathematicians held in Paris in 1900. From then on, analyzing whether a logic is decidable or not became part of the list of properties worth exploring. Decidability can be discussed in a pure theoretical way, but it is also an important requirement when introducing new logics for coping with the increasing number of sophisticated applications. Usually, decidability is either proved directly, or by reduction (via an adequate computable map) to decidability of another logic or theory or by using other logical properties that imply decidability. Since combinations of logics and theories are also very important, due to the many applications where several logics must be put together, it is fundamental to discuss preservation of decidability if not in general, at least by providing sufficient conditions for the preservation.

This Special Issue collects original research papers with the aim to uncover and exploit theoretical aspects of decidability but also to show decidability at work for new logics and theories. Furthermore, papers about preservation of decidability are also welcome either in the form of general results or specific cases for logics that are involved in significant applications.

Prof. Dr. Cristina Sernadas
Prof. Dr. João Rasga
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

  • Decidability of logics
  • Techniques for proving decidability
  • Decidability of theories
  • Preservation of decidability by combination
  • Decidability of new logics for challenging applications
  • Relationship between decidability and other logical properties

Published Papers (4 papers)

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Research

19 pages, 432 KiB  
Article
Decidability Preservation and Complexity Bounds for Combined Logics
by Carlos Caleiro and Sérgio Marcelino
Mathematics 2022, 10(19), 3481; https://doi.org/10.3390/math10193481 - 23 Sep 2022
Viewed by 879
Abstract
Transfer theorems for combined logics provide essential tools and insight for reasoning about complex logical systems. In this paper, we present the first sufficient criterion (contextual extensibility) for decidability to be preserved through combination of propositional logics, and we study the complexity [...] Read more.
Transfer theorems for combined logics provide essential tools and insight for reasoning about complex logical systems. In this paper, we present the first sufficient criterion (contextual extensibility) for decidability to be preserved through combination of propositional logics, and we study the complexity upper bounds induced by the method. In order to assess the scope and usability of our criterion, we illustrate its use in re-obtaining two standard important (though partial) results of the area: the preservation of decidability for disjoint combinations of logics, and the preservation of decidability for fusions of modal logics. Due to the very abstract nature and generality of the idea underlying contextual extensibility, we further explore its applicability beyond propositional logics. Namely, we explore the particular case of 2-deductive systems, and as a byproduct, we obtain the preservation of decidability for disjoint combinations of equational logics and discuss the relationship of this result and of our criterion with several related results with meaningful applications in satisfiability modulo theories. Full article
(This article belongs to the Special Issue Decidability of Logics and Their Theories and Combinations)
22 pages, 405 KiB  
Article
Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories
by Silvio Ghilardi and Alessandro Gianola
Mathematics 2022, 10(3), 461; https://doi.org/10.3390/math10030461 - 31 Jan 2022
Cited by 3 | Viewed by 2128
Abstract
In this survey, we report our recent work concerning combination results for interpolation and uniform interpolation in the context of quantifier-free fragments of first-order theories. We stress model-theoretic and algebraic aspects connecting this topic with amalgamation, strong amalgamation, and model-completeness. We give sufficient [...] Read more.
In this survey, we report our recent work concerning combination results for interpolation and uniform interpolation in the context of quantifier-free fragments of first-order theories. We stress model-theoretic and algebraic aspects connecting this topic with amalgamation, strong amalgamation, and model-completeness. We give sufficient (and, in relevant situations, also necessary) conditions for the transfer of the quantifier-free interpolation property to combined first-order theories; we also investigate the non-disjoint signature case under the assumption that the shared theory is universal Horn. For convex, strong-amalgamating, stably infinite theories over disjoint signatures, we also provide a modular transfer result for the existence of uniform interpolants. Model completions play a key role in the whole paper: They enter into transfer results in the non-disjoint signature case and also represent a semantic counterpart of uniform interpolants. Full article
(This article belongs to the Special Issue Decidability of Logics and Their Theories and Combinations)
22 pages, 342 KiB  
Article
Schema Complexity in Propositional-Based Logics
by Jaime Ramos, João Rasga and Cristina Sernadas
Mathematics 2021, 9(21), 2671; https://doi.org/10.3390/math9212671 - 21 Oct 2021
Viewed by 925
Abstract
The essential structure of derivations is used as a tool for measuring the complexity of schema consequences in propositional-based logics. Our schema derivations allow the use of schema lemmas and this is reflected on the schema complexity. In particular, the number of times [...] Read more.
The essential structure of derivations is used as a tool for measuring the complexity of schema consequences in propositional-based logics. Our schema derivations allow the use of schema lemmas and this is reflected on the schema complexity. In particular, the number of times a schema lemma is used in a derivation is not relevant. We also address the application of metatheorems and compare the complexity of a schema derivation after eliminating the metatheorem and before doing so. As illustrations, we consider a propositional modal logic presented by a Hilbert calculus and an intuitionist propositional logic presented by a Gentzen calculus. For the former, we discuss the use of the metatheorem of deduction and its elimination, and for the latter, we analyze the cut and its elimination. Furthermore, we capitalize on the result for the cut elimination for intuitionistic logic, to obtain a similar result for Nelson’s logic via a language translation. Full article
(This article belongs to the Special Issue Decidability of Logics and Their Theories and Combinations)
9 pages, 243 KiB  
Article
On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers
by S. Subburam, Lewis Nkenyereye, N. Anbazhagan, S. Amutha, M. Kameswari, Woong Cho and Gyanendra Prasad Joshi
Mathematics 2021, 9(15), 1813; https://doi.org/10.3390/math9151813 - 30 Jul 2021
Viewed by 1866
Abstract
Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In [...] Read more.
Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n. Full article
(This article belongs to the Special Issue Decidability of Logics and Their Theories and Combinations)
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