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Generalised Tree Automata, Monadic Second Order Logic and Transfer Principles in Combinatorial Limits

Project description

Novel approaches to important open questions in mathematics and computer science

Monadic second order (MSO) logic, or MSO theory, is a type of mathematical logic that is particularly important for expressing the formal specifications of graph properties and for automata theory describing both abstract and real self-propelled machines. Over the last half-century, scientists have accomplished numerous advances in MSO theory, yet important open questions still remain. With the support of the Marie Skłodowska-Curie Actions programme, the FINTOINF project is tackling two of these that have remained enigmas despite intense interest and effort: Shelah's conjecture on the monadic theory of order, and countable combinatorial limits.

Objective

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.

Fields of science

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Coordinator

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Net EU contribution
€ 196 707,84
Address
RUE MICHEL ANGE 3
75794 Paris
France

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Region
Ile-de-France Ile-de-France Paris
Activity type
Research Organisations
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Total cost
€ 196 707,84