AXIOMATIC METHODS IN NON-CLASSICAL MODEL THEORY

project PN-III-P4-ID-PCE-2020-0446, funded by UEFISCDI under contract PCE 6 / 2021


Domain PE. PHYSICAL SCIENCES AND ENGINEERING
Subdomain PE1. Mathematics: All areas of mathematics, pure and applied, plus mathematical foundations of computer science, mathematical physics and statistics
Area PE1_1. Logic and foundations

Project summary

In this project, we develop an axiomatic study of the model theory of non-classical logics. The mathematical underpinnings of our work is Goguen and Burstall's theory of institutions, which is the only fully axiomatic approach to model theory. Institutions are structurally rooted within category theory, and they have already proved very successful in addressing in-depth classical model-theoretic problems by axiomatic means. Here we take one step further by extending institutional model theory to address non-classical issues, including models with states (as in modal logic, but not only), many-valued truth, and the partiality of the language translations. The bases of these extensions are corresponding refinements of the categorical structure of institution. To that end, several important model theoretic methods will be developed, such as diagrams, ultraproducts, or logic-by-translation techniques. We target results on interpolation, definability, quasi-varieties, model amalgamation, bisimulation, etc. Besides the usual concrete inspirational examples, we will explore new unconventional interpretations of our abstract developments. This project will advance the understanding of non-classical model theory on an axiomatic base that implies structurally clean causality and relativization and will also have a foundational impact on areas from computing science and artificial intelligence.

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Members

Răzvan Diaconescu
CS1, project director (Simion Stoilow Institute of Mathematics of the Romanian Academy – IMAR)
Ionuț Țuțu
CS3, member (Simion Stoilow Institute of Mathematics of the Romanian Academy – IMAR)
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Publications

  1. R. Diaconescu: Translation structures for fuzzy model theory, Fuzzy Sets and Systems 408:108866 Elsevier, (2024)
  2. R. Diaconescu: Partialising Institutions, Applied Categorical Structures 31(6):46, Springer, (2023).
  3. R. Diaconescu: Concepts of interpolation in stratified institutions, Logics 1(2):80--96, MDPI (2023).
  4. R. Diaconescu: Generalised graded interpolation, International Journal of Approximate Reasoning 152:236-261, Elsevier (2023).
  5. R. Diaconescu: Preservation in many-valued truth institutions, Fuzzy Sets and Systems 456:38-71, Elsevier (2023).
  6. R. Diaconescu: Decompositions of Stratified Institutions, Journal of Logic and Computation33(7):1625-1664, Oxford University Press (2023).
  7. R. Diaconescu: The axiomatic approach to non-classical model theory, Mathematics 10(19):3428, MDPI (2022).
  8. R. Diaconescu: Representing 3/2-Institutions as Stratified Institutions, Mathematics 10(9):1507, MDPI (2022).

    9. Currently there are three other articles under evaluation for publication in journals:

  9. R. Diaconescu: Quasi-varieties and initial semantics in stratified institutions, Manuscript submitted for publication.
  10. I. Țuțu: Bisimulations in an arbitrary stratified institution, Manuscript submitted for publication.
  11. I. Țuțu: Transporting connectives along parchments addenda, Manuscript submitted for publication.

    12. Two book chapters of the second edition of the monograph Institution-independent Model Theory, which is now under preparation, are mostly based on results from this project.

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Popular explanation of results

DISCLAIMER: This explanation is included in the web page at the request of the funding agency and it does not reflect in any way the position of the project director on how scientific results should be disseminated.

This project is about `model theory', a branch of mathematical logic concerned mainly with interpretations of logical systems. Logical systems are mathematical formalisations of various ways / techniques of reasoning. Model theory is traditionally well connected to various branches of mainstream mathematics as well as to computing science. There are two defining characteristics of this project: on the one hand, we approach model theory axiomatically by using high mathematical abstraction, and, on the other hand, we study model-theoretic phenomena that comes from `non-classical' logics, which means logical systems that go beyond the common logics. Through the former aspect we are able to achieve greater conceptual clarity and a uniform treatment across a broad spectrum of logical systems, while through the latter aspect we are able to make our results useful to computing science. The research in this project has been driven both by genuine mathematical questions but also by computing science applications. Almost always these two are inter-related, although as it is often the case, the mathematics-motivated developments may precede temporally the applications. The actual results can be explained as follows:
  1. Mathematical theories supporting the foundations of the dynamics of logic-based systems.
  2. We have advanced the axiomatic general study of logical systems which are based on non-binary truth. These provide foundations in many areas of engineering, called `fuzzy engineering' and also in current AI which is based on machine learning.
  3. We have advanced mathematical theories supporting the creation of concepts by `intelligent' systems. This is a topic which is very difficult to deal with by the current AI technologies and it may be one of the most drastic limiting factors in machine learning based AI.
  4. We have also advanced the mathematical theory of `logic translations', whose ultimate practical consequence is to support co-operation between computing systems that are based on different logical formalisms.
  5. Based on the results of this project it is possible to develop computer languages that would assist system development in the above mentioned areas.
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Scientific and evaluation reports

  1. Scientific Report of the project for 2021
  2. Scientific Report of the project for 2022
  3. Final Scientific Report of the project (2023)
  4. Final Evaluation Report (in Romanian)
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Events

  1. visit by the project director to the λ-Form Group, Laboratory of Algorithmic Applications and Logic, National Technical University Athens (NTUA), Greece, 11 September - 6 October 2023.
  2. talk by the project director to National Technical University Athens, Greece, 27 September 2023.
  3. visit by the project director to the λ-Form Group, Laboratory of Algorithmic Applications and Logic, National Technical University Athens (NTUA), Greece, 11-30 September 2022.
  4. invited talk by the project director to Logic and Formal Methods (a UNILOG 2002 workshop), Athens, Greece, 22 September 2022.
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