Periodic Reporting for period 1 - FINTOINF (Generalised Tree Automata, Monadic Second Order Logic and Transfer Principles in Combinatorial Limits)
Reporting period: 2021-04-01 to 2023-03-31
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.
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.
The results of the project are all described in the next section. This is the concluding report
This report is written with following results to report.
I ) Publications published or to appear
II) Publications submitted
III) Conferences and events organised
All the results are transversal to the work packages.
We have achieved the scientific production and outreach activities as proposed. The objectives of (MD2) have been largely achieved, opening new directions of research. The objectives of (MD1), which was a long term project, have been started and some important ideas obtained.
I) Publications published or to appear
(1) Mirna Dzamonja, Angeliki Koutsoukou-Argyraki and Lawrence C. Paulson FRS, Formalising Ordinal
Partition Relations Using Isabelle/HOL, Experimental Mathematics vol. 31 (2022) No.2 pg. 383-400
(2) Mirna Dzamonja and Ivan Tomaic, Graphons arising from graphs definable over finite fields, Colloquium Mathematicum}, vol. 169 (2022) No. 2 pg. 269-306.
(3) David Buhagiar and Mirna Dzamonja, On middle box products and paracompact cardinals, to appear in the Annals of Pure and Applied Logic, preprint at http://arxiv.org/abs/2211.12936(opens in new window) HAL: hal- 03913664, v1
(4) Dana Bartosova, Mirna Dzamonja, Rehana Patel and Lynn Scow, Big Ramsey degrees in ultraproducts of finite structures, to appear in the Annals of Pure and Applied Logic, preprint at http://arxiv.org/abs/2211.12936(opens in new window) HAL: hal- 03913664, v1
(5) Mirna Dzamonja, Multiverse and the Society, a chapter in ``Handbook of the History and Philosophy of Mathematical Practice" (Bharath Sriraman, ed.).HAL: hal-03019882v2 ( a philosophical article in connection with the research themes of FINTOINF)
II)
(1) Krystian Jobczyk and Mirna Dzamonja, The Lindstrom Characterizability of Abstract Logic Systems for Analytic Structures Based on Measures.
(2) Uri Abraham, Robert Bonnet, Mirna Dzamonja and Maurice Pouzet, On the ABK Conjecture, alpha-well Quasi Orders and Dress-Schiffels product, preprint arxiv 2303.11451.
III) (1) Co-organiser Séminaire Philosophie et Mathématiques, ENS Paris, 2023 -
(2) Co-organiser Research School "Discrete Mathematics and Logic", CIRM, Marseille, January 2023
(3) Co-organiser DALFI, a workshop within FLOC 22, Haifa, Israel, July 2022, moved to Paris, November 2022
(4) Organiser ``Mathematical Independence and its Limits", Seminar MAMUPHI, Paris, February 2022
I ) Publications published or to appear
II) Publications submitted
III) Conferences and events organised
All the results are transversal to the work packages.
We have achieved the scientific production and outreach activities as proposed. The objectives of (MD2) have been largely achieved, opening new directions of research. The objectives of (MD1), which was a long term project, have been started and some important ideas obtained.
I) Publications published or to appear
(1) Mirna Dzamonja, Angeliki Koutsoukou-Argyraki and Lawrence C. Paulson FRS, Formalising Ordinal
Partition Relations Using Isabelle/HOL, Experimental Mathematics vol. 31 (2022) No.2 pg. 383-400
(2) Mirna Dzamonja and Ivan Tomaic, Graphons arising from graphs definable over finite fields, Colloquium Mathematicum}, vol. 169 (2022) No. 2 pg. 269-306.
(3) David Buhagiar and Mirna Dzamonja, On middle box products and paracompact cardinals, to appear in the Annals of Pure and Applied Logic, preprint at http://arxiv.org/abs/2211.12936(opens in new window) HAL: hal- 03913664, v1
(4) Dana Bartosova, Mirna Dzamonja, Rehana Patel and Lynn Scow, Big Ramsey degrees in ultraproducts of finite structures, to appear in the Annals of Pure and Applied Logic, preprint at http://arxiv.org/abs/2211.12936(opens in new window) HAL: hal- 03913664, v1
(5) Mirna Dzamonja, Multiverse and the Society, a chapter in ``Handbook of the History and Philosophy of Mathematical Practice" (Bharath Sriraman, ed.).HAL: hal-03019882v2 ( a philosophical article in connection with the research themes of FINTOINF)
II)
(1) Krystian Jobczyk and Mirna Dzamonja, The Lindstrom Characterizability of Abstract Logic Systems for Analytic Structures Based on Measures.
(2) Uri Abraham, Robert Bonnet, Mirna Dzamonja and Maurice Pouzet, On the ABK Conjecture, alpha-well Quasi Orders and Dress-Schiffels product, preprint arxiv 2303.11451.
III) (1) Co-organiser Séminaire Philosophie et Mathématiques, ENS Paris, 2023 -
(2) Co-organiser Research School "Discrete Mathematics and Logic", CIRM, Marseille, January 2023
(3) Co-organiser DALFI, a workshop within FLOC 22, Haifa, Israel, July 2022, moved to Paris, November 2022
(4) Organiser ``Mathematical Independence and its Limits", Seminar MAMUPHI, Paris, February 2022