Description du projet
Approches novatrices d’importantes questions ouvertes en mathématiques et en informatique
La logique monadique du second ordre (MSO), ou théorie MSO, est un type de logique mathématique particulièrement important pour l’expression des spécifications formelles des propriétés des graphes et pour la théorie des automates décrivant des machines autopropulsées abstraites et réelles. Au cours du dernier demi-siècle, les scientifiques ont accompli de nombreuses avancées dans la théorie MSO, mais d’importantes questions demeurent toutefois ouvertes. Avec le soutien du programme Actions Marie Skłodowska-Curie, le projet FINTOINF se penche sur deux d’entre elles qui sont restées des énigmes malgré un intérêt et des efforts intenses: la conjecture de Shelah sur la théorie monadique de l’ordre, et les limites combinatoires dénombrables.
Objectif
The project will concentrate on two main directions (MD), which are connected through them both relying on Monadic Second Order (MSO) and its variants. (MD 1) Shelah's conjecture. In his celebrated 1975 paper Shelah proved that the monadic second order theory (MSO) of the real order is undecidable. He conjectured in his Conjecture 7B that Conjecture: MSO of the real order where the second order quantifier ranges only over Borel sets, is decidable. In spite of important efforts on this question in both mathematics and computer science community, the conjecture is still open. Many strategies, including the one suggested by Shelah in his paper (to use Borel determinacy) have been tried. We propose to study this question using the recent methods of the generalised descriptive set theory and the generalised automata that we intend to develop. This is novel and might lead to important advances and the solution. (MD2) Countable combinatorial limits. Since the work of Lovasz and others in his group around 2006, a new area of discrete mathematics emerged: the combinatorial limits. This fast growing area aroused much interest and found many applications Its first development was that of a graphon, which is an uncountable limit of a sequence of finite graphs, but there have been several others. It is always important to understand the transfer properties of statements between the sequence forming the limit and the limit itself. The question has been considered through ultrapowers and through topology and Stone's pairings. None suffices for the transfer of MSO sentence. We propose to study that transfer through the novel notion of a countable model where the notion of satisfaction has been changed so that the countable model reflects the structure of the sequence of finite models that were used to obtain the uncountable combinatorial limit. In this sense we obtain a countable combinatorial limit which we study using the methods of finite model theory.
Champ scientifique (EuroSciVoc)
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CORDIS classe les projets avec EuroSciVoc, une taxonomie multilingue des domaines scientifiques, grâce à un processus semi-automatique basé sur des techniques TLN. Voir: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
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Mots‑clés
Programme(s)
Appel à propositions
(s’ouvre dans une nouvelle fenêtre) H2020-MSCA-IF-2020
Voir d’autres projets de cet appelRégime de financement
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinateur
75794 Paris
France