In this paper we first consider hyperfinite Borel equivalence relations with a pair of Borel $\mathbb {Z}$-orderings. We define a notion of compatibility between such pairs, and prove a dichotomy theorem which characterizes exactly when a pair of Borel $\mathbb {Z}$-orderings are compatible with each other. We show that, if a pair of Borel $\mathbb {Z}$-orderings are incompatible, then a canonical incompatible pair of Borel $\mathbb {Z}$-orderings of $E_0$ can be Borel embedded into the given pair. We then consider hyperfinite-over-finite equivalence relations, which are countable Borel equivalence relations admitting Borel $\mathbb {Z}^2$-orderings. We show that if a hyperfinite-over-hyperfinite equivalence relation E admits a Borel $\mathbb {Z}^2$-ordering which is self-compatible, then E is hyperfinite.

We study the computational problem of rigorously describing the asymptotic behavior of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and we prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constraints and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non-computable.

We investigate Steel’s conjecture in ‘The Core Model Iterability Problem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$-iterable, $1$-small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$, if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$. We will show that the conjecture fails, assuming that there is an iterable premouse M which models KP and which has a -Woodin cardinal. On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal the conjecture holds. In the course of this we will also show that if M is a premouse which models KP with a largest, regular, uncountable cardinal $\delta $, and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$, and $g\subset \mathbb {P}$ is M-generic, then $M[g]\models \text {KP}$. Additionally, we study the preservation of admissibility under iteration maps. At last, we will prove a fact about the closure of the set of ordinals at which a weasel has the S-hull property. This answers another question implicit in remarks in [10].

The class of all $\ast $-continuous Kleene algebras, whose description includes an infinitary condition on the iteration operator, plays an important role in computer science. The complexity of reasoning in such algebras—ranging from the equational theory to the Horn one, with restricted fragments of the latter in between—was analyzed by Kozen (2002). This paper deals with similar problems for $\ast $-continuous residuated Kleene lattices, also called $\ast $-continuous action lattices, where the product operation is augmented by residuals. We prove that, in the presence of residuals, the fragment of the corresponding Horn theory with $\ast $-free hypotheses has the same complexity as the $\omega ^\omega $ iteration of the halting problem, and hence is properly hyperarithmetical. We also prove that if only commutativity conditions are allowed as hypotheses, then the complexity drops down to $\Pi ^0_1$ (i.e., the complement of the halting problem), which is the same as that for $\ast $-continuous Kleene algebras. In fact, we get stronger upper bound results: the fragments under consideration are translated into suitable fragments of infinitary action logic with exponentiation, and our upper bounds are obtained for the latter ones.

We study versions of the tree pigeonhole principle, $\mathsf {TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether $\mathsf {TT}^1$ is $\Pi ^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf {RT}^1$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike $\mathsf {RT}^1$, the problem $\mathsf {TT}^1$ is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf {TT}^1$.

We investigate natural variations of behaviourally correct learning and explanatory learning—two learning paradigms studied in algorithmic learning theory—that allow us to “learn” equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviourally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\boldsymbol {\Pi }^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

We consider equational theories based on axioms for recursively defining functions, with rules for equality and substitution, but no form of induction—we denote such equational theories as PETS for pure equational theories with substitution. An example is Cook’s system PV without its rule for induction. We show that the Bounded Arithmetic theory $\mathrm {S}^{1}_2$ proves the consistency of PETS. Our approach employs model-theoretic constructions for PETS based on approximate values resembling notions from domain theory in Bounded Arithmetic, which may be of independent interest.

In this article, we propose a new classification of $\Sigma ^0_2$ formulas under the realizability interpretation of many-one reducibility (i.e., Levin reducibility). For example, $\mathsf {Fin}$, the decision of being eventually zero for sequences, is many-one/Levin complete among $\Sigma ^0_2$ formulas of the form $\exists n\forall m\geq n.\varphi (m,x)$, where $\varphi $ is decidable. The decision of boundedness for sequences $\mathsf {BddSeq}$ and for width of posets $\mathsf {FinWidth}$ are many-one/Levin complete among $\Sigma ^0_2$ formulas of the form $\exists n\forall m\geq n\forall k.\varphi (m,k,x)$, where $\varphi $ is decidable. However, unlike the classical many-one reducibility, none of the above is $\Sigma ^0_2$-complete. The decision of non-density of linear orders $\mathsf {NonDense}$ is truly $\Sigma ^0_2$-complete.

This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. In the framework of reverse mathematics, these theorems are equivalent, respectively, to $\mathsf {ATR}_0$ and $\boldsymbol {\Pi }^1_1{-}\mathsf{CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbol {\Pi }^1_1{-}\mathsf{CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.

This paper studies the conjecture of Hirschfeldt, Miller, and Podzorov in [13] on the complexity of order-computable sets, where a set A is order-computable if there is a computable copy of the structure $(\mathbb {N}, <,A)$ in the language of linear orders together with a unary predicate. The class of order-computable sets forms a subclass of $\Delta ^{0}_{2}$ sets. Firstly, we study the complexity of computably enumerable (c.e.) order-computable sets and prove that the index set of c.e. order-computable sets is $\Sigma ^{0}_{4}$-complete. Secondly, as a corollary of the main result on c.e. order-computable sets, we obtain that the index set of general order computable sets is $\Sigma ^{0}_{4}$-complete within the index set of $\Delta ^{0}_{2}$ sets. Finally, we continue to study the complexity of more general $\Delta ^{0}_{2}$ sets and prove that the index set of $\Delta ^{0}_{2}$ sets is $\Pi ^{0}_{3}$-complete.

We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman–Stanley embedding on Scott sentence complexity and show that it only preserves $\Pi ^{\mathrm {in}}_{\alpha }$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except $\Sigma ^{\mathrm {in}}_{3}$ and $\Sigma ^{\mathrm {in}}_{\lambda +1}$ for $\lambda $ a limit ordinal. We show that the former cannot be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.

We give a short proof of a theorem of Beleznay asserting that the set $L2$ of reals coding linear orders of the form $I + I$ is complete analytic.

The thesis uses various approaches to explore the algorithmic complexity of families of subsets of natural numbers. One of these approaches involves investigating upper semilattices of computable numberings of a given family and their complexity in different hierarchies. These semilattices, known as Rogers semilattices, can help distinguish different structural properties of families of partial computable functions and computably enumerable sets. As a result, by using Rogers semilattices of computable numberings, we can measure the algorithmic complexity of the corresponding family.

In the first part of thesis, we focus on limitwise monotonic numberings for families of limitwise monotonic sets and define their Rogers semilattices. The chapter investigates global invariants that show differences in the algebraic and elementary properties of the Rogers semilattices of families of sets from arithmetical hierarchy and Rogers semilattices of limitwise monotonic numberings. Such invariants include cardinality, laticeness, and types of isomorphism.

Within the second part of thesis, we explore the different forms of isomorphism exhibited by Rogers semilattices of families of sets in the analytical hierarchy. Additionally, we take into account various set-theoretic assumptions. Our research demonstrates that, when set-theoretic assumption known as Projective Determinacy is assumed, there exist an infinite number of non-isomorphic Rogers semilattices at each $\Sigma _n^1$-level of the analytical hierarchy.

I would also like to acknowledge the Grant of Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (AP19676989) and the Nazarbayev University Faculty Development Grant (N021220FD3851) for funding this research.

Abstract prepared by Zhansaya Tleuliyeva

zhansaya.tleuliyeva@nu.edu.kz.

URL: http://nur.nu.edu.kz/handle/123456789/7769.

We show that there exists a recursive coloring function c such that any $\Pi ^0_3$ set is not a solution to c for Hindman’s theorem. We also show that there exists a recursive coloring function c such that any $\Delta ^0_3$ set is not a solution to c for Hindman’s theorem restricted to sums of at most three numbers.

Using tools from computable analysis, we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural systems one can think of are effective in this sense, including some group rotations, affine actions on the torus and finitely presented algebraic actions. We show that for finitely generated and recursively presented groups, every effective dynamical system is the topological factor of a computable action on an effectively closed subset of the Cantor space. We then apply this result to extend the simulation results available in the literature beyond zero-dimensional spaces. In particular, we show that for a large class of groups, many of these natural actions are topological factors of subshifts of finite type.

Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A. We show that for any n, $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K. It was already known that 2-randomness is definable in terms of K (and plain complexity C) as those reals which infinitely often have maximal complexity. We can use our characterization to show that n-randomness is definable purely in terms of K. To do this we extend a certain “limsup” formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta _2^0$ sets of minimal descriptions.

In this thesis, we study the complexity of some mathematical problems: in particular, those arising in computable analysis and algorithmic learning theory for algebraic structures. Our study is not limited to these two areas: indeed, in both cases, the results we obtain are tightly connected to ideas and tools coming from different areas of mathematical logic, including for example descriptive set theory and reverse mathematics.

After giving the necessary preliminaries, we first study the uniform computational strength of the Cantor–Bendixson theorem in the Weihrauch lattice. This work falls into the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. We concentrate on problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. As far as we know, this is the first systematic study of problems at the level of $\mathbf {\Pi }^1_1\mathsf {-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.

We continue considering problems related to (induced) subgraphs. We provide results on the (effective) Wadge complexity of sets of graphs, that are also used to determine the Weihrauch degree of certain decision problems. The decision problems we consider are defined for a fixed graph G, and they take as input a graph H, answering whether G is an (induced) subgraph of H: we also consider the opposite problem (i.e., answering whether H is an induced subgraph of G). We conclude this part on (induced) subgraphs considering the Weihrauch degree of “search problems.”These problems are defined for a fixed graph G, and they take as input a graph H such that G is an (induced) subgraph H: the output is a copy of G in H. In both cases, we highlight differences and analogies between the subgraph and the induced subgraph relation.

We then move our attention to algorithmic learning theory, and we present the framework we use to study the learnability of families of algebraic structures: here, given a countable family of pairwise nonisomorphic structures $\mathfrak {K}$, a learner receives larger and larger pieces of an arbitrary copy of a structure in $\mathfrak {K}$ and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. We say that $\mathfrak {K}$ is learnable if there exists a learner which eventually stabilizes to a correct guess. The framework was lacking a method for comparing the complexity of nonlearnable families, and so we propose a solution to this problem using tools coming from invariant descriptive set theory. To do so, we first prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on infinite binary sequences and then, replacing $E_0$ with Borel equivalence relations of higher complexity, we obtain a new hierarchy of learning problems. This leads to the notion of E-learnability, where a family of structures $\mathfrak {K}$ is E-learnable, for a Borel equivalence relation E, if there is a continuous reduction from the isomorphism relation associated with $\mathfrak {K}$ to E. It is then natural to ask how the notion of E-learnability interacts with “classical” learning paradigms.

Finally, we study the number of mind changes that a learner needs to learn a given family, both from a topological and a combinatorial point of view, and we consider how the complexity of a learner (in terms of Turing reducibility) affects the number of mind changes for learning a given family.

Abstract prepared by Vittorio Cipriani.

E-mail: vittorio.cipriani17@gmail.com.

URL: https://air.uniud.it/handle/11390/1252226.

We study the effective versions of several notions related to incompleteness, undecidability, and inseparability along the lines of Pour-El’s insights. Firstly, we strengthen Pour-El’s theorem on the equivalence between effective essential incompleteness and effective inseparability. Secondly, we compare the notions obtained by restricting that of effective essential incompleteness to intensional finite extensions and extensional finite extensions. Finally, we study the combination of effectiveness and hereditariness, and prove an adapted version of Pour-El’s result for this combination.

In the study of the arithmetic degrees the $\omega \text {-REA}$ sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the $\omega \text {-REA}$ sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of an $\omega \text {-REA}$ set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e. sets fail to show that no $\omega \text {-REA}$ set is arithmetically minimal. Finally, it constructs a $\prod ^0_{2}$ singleton of minimal arithmetic degree. Not only is this a result of considerable interest in its own right, constructions of $\prod ^0_{2}$ singletons often pave the way for constructions of $\omega \text {-REA}$ sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.

We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff–Kuratowski and Wadge theorems on the structure of $\mathbf {\Delta }^0_\xi $, Louveau and Saint Raymond’s separation theorem, and Louveau’s separation theorem.