Periodic Reporting for period 2 - ACOSE (Algorithmic complexity of structures and their equivalence relations)
Berichtszeitraum: 2023-09-01 bis 2024-08-31
We investigate the algorithmic complexity of mathematical structures and its connection with notions of complexity
studied in descriptive set theory. The main subject area of the planned research is computable structure theory — an area
of logic concerning itself with the computational complexity of countable mathematical structures. Mathematicians usually
consider structures up to some equivalence relation. For example, a number theorist works in the standard model of
arithmetic, the natural numbers with addition and multiplication, but it is of little interest to him whether he works in the
canonic representation or in some isomorphic copy as this does not impact his work.
However, for computational matters, the choice of representation is highly important. Therefore one usually measures the
algorithmic complexity of a structure by its degree spectrum, the set of Turing degrees of structure equivalent to the structure
under a given equivalence relation.
Degree spectra are the main subject of investigation in computable structure theory. A natural way to think of degree spectra
is as sets of subsets of natural numbers, and these sets are studied in descriptive set theory.
So far the relation between the descriptive complexity of a set and whether it can be a degree spectrum under a given
equivalence relation has been overlooked. The goal of this project is to relate the descriptive complexity of sets with their
realizability as degree spectra under some equivalence relation.
We plan to obtain new results and develop new techniques which will be beneficial to both descriptive set theory and
computable structure theory and hope to form a lasting connection between those fields.
studied in descriptive set theory. The main subject area of the planned research is computable structure theory — an area
of logic concerning itself with the computational complexity of countable mathematical structures. Mathematicians usually
consider structures up to some equivalence relation. For example, a number theorist works in the standard model of
arithmetic, the natural numbers with addition and multiplication, but it is of little interest to him whether he works in the
canonic representation or in some isomorphic copy as this does not impact his work.
However, for computational matters, the choice of representation is highly important. Therefore one usually measures the
algorithmic complexity of a structure by its degree spectrum, the set of Turing degrees of structure equivalent to the structure
under a given equivalence relation.
Degree spectra are the main subject of investigation in computable structure theory. A natural way to think of degree spectra
is as sets of subsets of natural numbers, and these sets are studied in descriptive set theory.
So far the relation between the descriptive complexity of a set and whether it can be a degree spectrum under a given
equivalence relation has been overlooked. The goal of this project is to relate the descriptive complexity of sets with their
realizability as degree spectra under some equivalence relation.
We plan to obtain new results and develop new techniques which will be beneficial to both descriptive set theory and
computable structure theory and hope to form a lasting connection between those fields.
This project consisted of a two-year outgoing phase at the University of California, Berkeley, and a one-year return phase at Technische Universität Wien.
The scientific results of this project were published in 10 peer-reviewed research articles. As of now, all of these articles have been submitted, with several of them already accepted for publication.
Additionally, the principal investigators gave 20 talks advertising the results obtained in this project to the scientific community.
The four highlight results obtained in the project are:
1. A characterization of the descriptive complexity of the set of models of a first-order theory in terms of the quantifier complexity of axiomatizations of the theory. This result, obtained with Andrews, Gonzalez, Lempp, and Zhu, shows that in order to get a grasp of the descriptive complexity of a theory's models, one just has to look at the quantifier complexity of the theory's simplest axiomatization, which is usually much easier to determine.
2. A result showing that elementary bi-embeddability is an analytic complete equivalence relation and, thus among the most complicated analytic equivalence relations. The PI showed this by reducing bi-embeddability to elementary bi-embeddability. Their reduction shows that the bi-embeddability relation and the elementary bi-embeddability relation not only have the same descriptive complexity but while their degree spectra are not the same, they are at most one jump apart. Intuitively, this means that the computational content of a structure relative to the bi-embeddability and the elementary bi-embeddability relation can be different, but that this difference is very small.
3. An exhaustive Scott analysis of Peano arithmetic and linear orderings. With Montalbán the principal investigator investigated the possible Scott ranks of models of arithmetic, that is, models of the first-order theory of the natural numbers with addition and multiplication. It follows from Gödel's completeness theorem that this theory has not only
the standard model of natural numbers but many more models which extend the natural numbers by containing infinite numbers. The authors gave precise calculations of the possible Scott ranks of such models, giving a precise idea of how complicated the possible degree spectra of such structures can be and, thus, the computational complexity of such structures important to the foundations of mathematics. With Gonzalez, the principal investigator did a similar Scott analysis of models of the theory of linear orderings. Linear orderings are of interest as they are complete for Borel reducibility but not for any stronger known reducibilities. They gave a full characterization of the possible Scott sentence complexities and showed that the "incompleteness" phenomenon present in linear orderings is not reflected in their Scott analysis.
During the return phase at Technische Universität Wien, the principal investigator started several projects following up on the above highlights with local researchers and researchers across Europe. In order to attract the interest of the local scientific community they gave several talks at seminars and organized a workshop at Technische Universität Wien in the summer of 2024. For three days, 12 international researchers presented their work at this workshop titled "Computable Structure Theory and Interactions" and collaborated in Vienna.
The scientific results of this project were published in 10 peer-reviewed research articles. As of now, all of these articles have been submitted, with several of them already accepted for publication.
Additionally, the principal investigators gave 20 talks advertising the results obtained in this project to the scientific community.
The four highlight results obtained in the project are:
1. A characterization of the descriptive complexity of the set of models of a first-order theory in terms of the quantifier complexity of axiomatizations of the theory. This result, obtained with Andrews, Gonzalez, Lempp, and Zhu, shows that in order to get a grasp of the descriptive complexity of a theory's models, one just has to look at the quantifier complexity of the theory's simplest axiomatization, which is usually much easier to determine.
2. A result showing that elementary bi-embeddability is an analytic complete equivalence relation and, thus among the most complicated analytic equivalence relations. The PI showed this by reducing bi-embeddability to elementary bi-embeddability. Their reduction shows that the bi-embeddability relation and the elementary bi-embeddability relation not only have the same descriptive complexity but while their degree spectra are not the same, they are at most one jump apart. Intuitively, this means that the computational content of a structure relative to the bi-embeddability and the elementary bi-embeddability relation can be different, but that this difference is very small.
3. An exhaustive Scott analysis of Peano arithmetic and linear orderings. With Montalbán the principal investigator investigated the possible Scott ranks of models of arithmetic, that is, models of the first-order theory of the natural numbers with addition and multiplication. It follows from Gödel's completeness theorem that this theory has not only
the standard model of natural numbers but many more models which extend the natural numbers by containing infinite numbers. The authors gave precise calculations of the possible Scott ranks of such models, giving a precise idea of how complicated the possible degree spectra of such structures can be and, thus, the computational complexity of such structures important to the foundations of mathematics. With Gonzalez, the principal investigator did a similar Scott analysis of models of the theory of linear orderings. Linear orderings are of interest as they are complete for Borel reducibility but not for any stronger known reducibilities. They gave a full characterization of the possible Scott sentence complexities and showed that the "incompleteness" phenomenon present in linear orderings is not reflected in their Scott analysis.
During the return phase at Technische Universität Wien, the principal investigator started several projects following up on the above highlights with local researchers and researchers across Europe. In order to attract the interest of the local scientific community they gave several talks at seminars and organized a workshop at Technische Universität Wien in the summer of 2024. For three days, 12 international researchers presented their work at this workshop titled "Computable Structure Theory and Interactions" and collaborated in Vienna.
The results obtained in this project push the state of the art considerably by providing new insights into both the descriptive and computational complexity of equivalence relations defined on countable mathematical structures. We showed that the reason for linear orderings not being complete for strengthening of Borel reducibility is not reflected in its Scott analysis, connected the first-order complexity of theories usually studied in proof theory with the descriptive complexity of the theory's set of models, usually studied by computability and descriptive set theorists. Last but not least, our Scott analysis of Peano arithmetic showed researchers in the foundations of mathematics how modern tools from computability theory can help to understand foundational theories of mathematics.
We believe that the connections established in this project between notions from different research areas and the interests generated by scientific communities outside of the core community this project targeted is our biggest success and beyond what we imagined when we conceived this project; we know about at least two ongoing projects by researchers in the foundations of mathematics that, inspired by our work, use Scott analysis as tools for their investigations. Furthermore, the interdisciplinary character of the results obtained in this project allowed the researcher to work with people from different fields, such as proof theory, descriptive set theory, and group theory, and to obtain a bigger scientific horizon.
Our workshop this summer in Vienna brought together researcher from all over the world to collaborate and share their results. We had researchers working at institutions across Europe present their current project and internal Ph.D. students and postdoctoral fellows present their work. We hope that this event and potential follow-up events will strengthen the European research environment by bringing promising international talent to European research institutions and increasing the number of collaborations within Europe. One example of a project coming out of this fellowship is the ongoing project between the principal investigator and researchers in the foundations of mathematics in Warsaw, Poland.
We believe that the connections established in this project between notions from different research areas and the interests generated by scientific communities outside of the core community this project targeted is our biggest success and beyond what we imagined when we conceived this project; we know about at least two ongoing projects by researchers in the foundations of mathematics that, inspired by our work, use Scott analysis as tools for their investigations. Furthermore, the interdisciplinary character of the results obtained in this project allowed the researcher to work with people from different fields, such as proof theory, descriptive set theory, and group theory, and to obtain a bigger scientific horizon.
Our workshop this summer in Vienna brought together researcher from all over the world to collaborate and share their results. We had researchers working at institutions across Europe present their current project and internal Ph.D. students and postdoctoral fellows present their work. We hope that this event and potential follow-up events will strengthen the European research environment by bringing promising international talent to European research institutions and increasing the number of collaborations within Europe. One example of a project coming out of this fellowship is the ongoing project between the principal investigator and researchers in the foundations of mathematics in Warsaw, Poland.