## Final Report Summary - STRIDE (The Spectrum of Relative Definability)

The mathematical analysis of the notion of Definability is one of the principal objectives of Mathematical Logic. In particular, we wish to understand relative definability between real numbers. Real numbers can be represented as sets of natural numbers, so our goal becomes to understand how one subset of the natural numbers can be used to specify another. This specification is called a reducibility, it can be computational, or arithmetic, or even by the application of a countable sequence of Borel operations. In this way, definability notions are parameterized by the fineness of their ingredients and we obtain a whole spectrum of relative definability. Each reducibility has a natural structural representation as a partial order, it's degree structure. We say that two sets are equivalent when each is reducible to the other. A reducibility thus induces a natural order on the resulting equivalence classes. We have the many-one, the Turing, the enumeration and the hyperarithmetical degrees. At the two endpoints, many-one and hyperarithmetical, we do have complete accounts and those accounts are completely different.The partial ordering of the many-one degrees was characterized algebraically by Ershov and Paliutin. From this characterization it follows that the automorphism group of this structure has maximal cardinality: the cardinality of the powerset of the continuum and every element of the structure other than its least one has a nontrivial orbit. Thus, the least element is the only element of the structure of the many-one degrees which can be defined by its order-theoretic properties.The partial ordering of the hyperarithmetical degrees was characterized model theoretically by Slaman and Woodin. It is logically identical to the structure of second-order arithmetic. It follows that there is no nontrivial automorphism of the structure of the hyperarithmetical degrees, it is rigid. Further, a set of degrees is order-theoretically definable within the structure if and only if it corresponds to a degree-invariant relation on sets of natural numbers which is definable in Second Order Arithmetic. So there are many hyperarithmetical degrees which can be defined by their order-theoretic properties.

We have only partial understanding of the spectrum between these two extremes and many important questions remain open. The Biinterpretibility Conjecture of Slaman and Woodin puts forth the proposal that the partial order of the Turing degrees is characterized in the same way as that of the hyperarithmetical degrees. While their techniques do not establish this conjecture, they do provide contextual evidence for it. For example, they show that the Biinterpretibility Conjecture holds if and only if the structure of the Turing degrees is rigid, reducing the problem of understanding the structure to that of understanding its automorphism group. Further, they show that the group is countable. The main open problem for the structure of Turing degrees is therefore whether this group has a nontrivial element. For the structure of the enumeration degrees less was known and the situation is extremely interesting. Slaman and Woodin showed that there is a representation of Second Order Arithmetic within this structure, hence the theory of the enumeration degrees is computably isomorphic to the theory of second order arithmetic. There is also a natural structure preserving embedding of the Turing degrees into the enumeration degrees. We say that an enumeration degree is total if it is the image of a Turing degree under this embedding. The main question concerning the structure of the enumeration degrees after its own Biinterpretibility Conjecture, was considered to be whether the total degrees are order-theoretically definable in the structure of the enumeration degrees. This question was first set by Rogers in 1967. The primary objective of this project is to provide training for the researcher Mariya Soskova, within an ambitious research project, so that the researcher can expand her knowledge to a much wider field. The main scientific objective of this project is to study the spectrum of relative definability further, focusing in particular on the pair of the structure of the enumeration degrees and the structure of the Turing degrees.

During the outgoing phase os this project the researcher Dr. Mariya I. Soskova spent two years as visiting scholar at the University of California at Berkeley. In these two academic years, the researcher participated in many training activities, such as courses, teaching seminars, research seminars, conferences, workshops and other scientific events. She has mastered methods from other sectors in mathematical logic, including Set Theory, Descriptive Set Theory, Borel Combinatorics, Reverse Mathematics and Second Order Arithmetic. During the return phase, the researcher focused on the transfer of knowledge and skills acquired. The researcher gave courses and seminars on the new topics that had not been introduced at Sofia University, she mentored students, she was made responsible for the Master's Programme "Mathematical Logic and its Applications".

This intensive training has lead to outstanding scientific achievements. The first significant achievement gave a partial solution to the problem of the definability of the total enumeration degrees. Ganchev and Soskova prove that the total enumeration degrees in the local structure of the enumeration degrees are first order definable by a very natural property. The resulting article, entitled "Definability via Kalimullin pairs in the structure of the enumeration degrees", was published in the Transactions of the American Mathematical Society. This result suggested that the answer to Roger's question is positive.

The researcher then approached the question from a different direction: she proved that Slaman and Woodin's automorphism analysis of the Turing degrees can be transformed to an analysis of the automorphism properties of the structure of the enumeration degrees. As a result she obtained that the automorphism group of the enumeration degrees is at most countable, that its members are arithmetically definable and that there is a single enumeration degree which determines the action of every automorphism - i.e. an automorphism base consisting of a single member. This analysis gives a further partial solution to Roger's question: the total enumeration degrees are definable if we allow the use of a parameter.The article containing this work is entitled "The automorphism group of the enumeration degrees" has been accepted for publication in Annals of Pure and Applied Logic.

The solution to Roger's 47 year old open question was finally found by Cai, Ganchev, Lempp, Miller and Soskova - the researcher. They extended Ganchev and Soskova's original proof of the definability of the total enumeration degrees from the local structure to the global structure. This is the most significant result obtained within this project, highly surpassing the initial expectations. It shows a strong relationship between the two investigated structures: it follows that the isomorphic copy of Turing degrees is a definable automorphism base for the structure of the enumeration degrees, thus if the Turing degrees are rigid then so are the enumeration degrees. The authors showed also that the relation on Turing degrees: "computably enumerable in", is first order definable in the enumeration degrees. The article containing this work is entitled "Defining totality in the enumeration degrees" has been accepted for publication in the Journal of the American Mathematical Society.

Building on this work, in collaboration with Theodore Slaman, the researcher then investigated how automorphism properties of the local structure of the Turing degrees reflect on the global structure. Two more significant results were shown. The first one suggests a characterization of the local structure of the Turing degrees that is similar to the global one: Biinterpretability with First order arithmetic. Slaman and Soskova show that this is equivalent to rigidity of the local structure and prove that it is true, if we allow the use of parameters. Moving to the local structure of the enumeration degrees, the methodology becomes more powerful. Combined with the definability of the relation "c.e. in" allows Slaman and Soskova to show that the image of the c.e. degrees under the standard embedding is an automorphism base for the structure of the enumeration degrees. Thus the Biinterpretability conjecture for the enumeration degrees is implied by the rigidity of a very familiar structure - that of the computably enumerable degrees.

A finer analysis of the work of Slaman and Soskova, described above revealed that there are finitely many elements in the local structure of the enumeration degrees that determine the properties of every automorphism. This naturally suggests that we should study the first order definability properties in the local structure of the enumeration degrees, as these give restrictions on the possible behavior of an automorphism. Ganchev and Soskova showed that all levels of the jump hierarchy are definable in the local structure of the enumeration degrees, thereby achieving a longstanding goal and introducing a new technology that will likely play a role for definability results in the future.

We have only partial understanding of the spectrum between these two extremes and many important questions remain open. The Biinterpretibility Conjecture of Slaman and Woodin puts forth the proposal that the partial order of the Turing degrees is characterized in the same way as that of the hyperarithmetical degrees. While their techniques do not establish this conjecture, they do provide contextual evidence for it. For example, they show that the Biinterpretibility Conjecture holds if and only if the structure of the Turing degrees is rigid, reducing the problem of understanding the structure to that of understanding its automorphism group. Further, they show that the group is countable. The main open problem for the structure of Turing degrees is therefore whether this group has a nontrivial element. For the structure of the enumeration degrees less was known and the situation is extremely interesting. Slaman and Woodin showed that there is a representation of Second Order Arithmetic within this structure, hence the theory of the enumeration degrees is computably isomorphic to the theory of second order arithmetic. There is also a natural structure preserving embedding of the Turing degrees into the enumeration degrees. We say that an enumeration degree is total if it is the image of a Turing degree under this embedding. The main question concerning the structure of the enumeration degrees after its own Biinterpretibility Conjecture, was considered to be whether the total degrees are order-theoretically definable in the structure of the enumeration degrees. This question was first set by Rogers in 1967. The primary objective of this project is to provide training for the researcher Mariya Soskova, within an ambitious research project, so that the researcher can expand her knowledge to a much wider field. The main scientific objective of this project is to study the spectrum of relative definability further, focusing in particular on the pair of the structure of the enumeration degrees and the structure of the Turing degrees.

During the outgoing phase os this project the researcher Dr. Mariya I. Soskova spent two years as visiting scholar at the University of California at Berkeley. In these two academic years, the researcher participated in many training activities, such as courses, teaching seminars, research seminars, conferences, workshops and other scientific events. She has mastered methods from other sectors in mathematical logic, including Set Theory, Descriptive Set Theory, Borel Combinatorics, Reverse Mathematics and Second Order Arithmetic. During the return phase, the researcher focused on the transfer of knowledge and skills acquired. The researcher gave courses and seminars on the new topics that had not been introduced at Sofia University, she mentored students, she was made responsible for the Master's Programme "Mathematical Logic and its Applications".

This intensive training has lead to outstanding scientific achievements. The first significant achievement gave a partial solution to the problem of the definability of the total enumeration degrees. Ganchev and Soskova prove that the total enumeration degrees in the local structure of the enumeration degrees are first order definable by a very natural property. The resulting article, entitled "Definability via Kalimullin pairs in the structure of the enumeration degrees", was published in the Transactions of the American Mathematical Society. This result suggested that the answer to Roger's question is positive.

The researcher then approached the question from a different direction: she proved that Slaman and Woodin's automorphism analysis of the Turing degrees can be transformed to an analysis of the automorphism properties of the structure of the enumeration degrees. As a result she obtained that the automorphism group of the enumeration degrees is at most countable, that its members are arithmetically definable and that there is a single enumeration degree which determines the action of every automorphism - i.e. an automorphism base consisting of a single member. This analysis gives a further partial solution to Roger's question: the total enumeration degrees are definable if we allow the use of a parameter.The article containing this work is entitled "The automorphism group of the enumeration degrees" has been accepted for publication in Annals of Pure and Applied Logic.

The solution to Roger's 47 year old open question was finally found by Cai, Ganchev, Lempp, Miller and Soskova - the researcher. They extended Ganchev and Soskova's original proof of the definability of the total enumeration degrees from the local structure to the global structure. This is the most significant result obtained within this project, highly surpassing the initial expectations. It shows a strong relationship between the two investigated structures: it follows that the isomorphic copy of Turing degrees is a definable automorphism base for the structure of the enumeration degrees, thus if the Turing degrees are rigid then so are the enumeration degrees. The authors showed also that the relation on Turing degrees: "computably enumerable in", is first order definable in the enumeration degrees. The article containing this work is entitled "Defining totality in the enumeration degrees" has been accepted for publication in the Journal of the American Mathematical Society.

Building on this work, in collaboration with Theodore Slaman, the researcher then investigated how automorphism properties of the local structure of the Turing degrees reflect on the global structure. Two more significant results were shown. The first one suggests a characterization of the local structure of the Turing degrees that is similar to the global one: Biinterpretability with First order arithmetic. Slaman and Soskova show that this is equivalent to rigidity of the local structure and prove that it is true, if we allow the use of parameters. Moving to the local structure of the enumeration degrees, the methodology becomes more powerful. Combined with the definability of the relation "c.e. in" allows Slaman and Soskova to show that the image of the c.e. degrees under the standard embedding is an automorphism base for the structure of the enumeration degrees. Thus the Biinterpretability conjecture for the enumeration degrees is implied by the rigidity of a very familiar structure - that of the computably enumerable degrees.

A finer analysis of the work of Slaman and Soskova, described above revealed that there are finitely many elements in the local structure of the enumeration degrees that determine the properties of every automorphism. This naturally suggests that we should study the first order definability properties in the local structure of the enumeration degrees, as these give restrictions on the possible behavior of an automorphism. Ganchev and Soskova showed that all levels of the jump hierarchy are definable in the local structure of the enumeration degrees, thereby achieving a longstanding goal and introducing a new technology that will likely play a role for definability results in the future.