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

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

This is a project that borders the computer sciences and mathematics. In particular, it addressed the fascinating boundary where abstract machines are studied in order to determine decidability of mathematical theories. By decidability is meant the possibility to mechanically decide if a given statement of mathematics can be proved by the theory or not. The project concentrated 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 (Schock Prize 2020) 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 are studying this question using the recent methods of the generalised descriptive set theory and the generalised automata that we are developing. This is novel and might lead to important advances and the solution. (MD2) Countable combinatorial limits. Since the work of Lovasz (Abel Prize 2021) 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 are studying 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.

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