We show that there is a Π^0_2 subset B of the Euclidean plane, of Hausdorff dimension 1, such that for every line ℓ through the origin with hyperarithmetic direction, the projection of B onto ℓ has Hausdorff dimension 0, thus is exceptional for Marstrand's projection theorem. It follows that for any computable ordinal α, being α-random does not guarantee the “almost-all” statement of the projection theorem.

These open problems were presented in the Problem Sessions held during the Tianyuan Workshop on Definability and Computation, June 22-26, 2026. The problems are organized into sections named after their contributors, in the order of their presentations during the workshop. Notes were taken and compiled by Wei Dai, Xiangxi Hu, Yingying Jiang, Ruiwen Li, Tianhao Wang, Xu Wang, and Jie Zou.

We show how to extend Selivanov's fine hierarchy using descriptions of Borel Wadge classes. We give a game characterisation of containment between classes. We show that every class in the extended fine hierarchy has an admissible description, and use this to calculate heights in the hierarchy.

Abstract SJT reducibility between sets A , B ⊆ N $A,B \subseteq \mathbb N$ upper A comma upper B subset of or equal to double struck upper N is defined by A ≤ S J T B $A \le _{SJT} B$ upper A less than or equals Subscript upper S upper J upper T Baseline upper B if for each computable function h that is unbounded and nondecreasing, there is an h -bounded uniformly B -c.e. trace ( T n ) n ∈ N $(T_n)_{n \in \mathbb N} $ left parenthesis upper T Subscript n Baseline right parenthesis Subscript n element of double struck upper N such that for each n , the value J A ( n ) $J^A(n)$ upper J Superscript upper A Baseline left parenthesis n right parenthesis of the jump is in T n $T_n$ upper T Subscript n , if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the K -trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.

Normal numbers were introduced by Borel. Normality is certainly a weak notion of randomness; for instance, there are computable numbers which are absolutely normal. In the present paper, we introduce a relativization of normality to a fixed representation system. When we require normality with respect to large sets of such systems, we find variants of normality that imply randomness notions much stronger than absolute normality. The primary classes of numbers investigated in this paper are the supernormal numbers and the highly normal numbers, which we will define. These are relativizations of normality which are robust to all reasonable changes of representation. Among other results, we give a proof that the highly normal numbers are exactly those of computable dimension 1, which we think gives a more natural characterization than was previously known of this interesting class.

We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces and apply it to attack classification problems concerning the Banach space C[0,1] of real-valued continuous functions on the unit interval. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipshitz functions is $\Sigma^0_2$-complete. We show that a function $f\colon [0,1] \rightarrow \mathbb{R}$ is (binary) transducer if and only if it is continuous regular; interestingly, this peculiar and nontrivial fact was overlooked by experts in automata theory. As one of many consequences, our $\Sigma^0_2$-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space $C[0,1]$ of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.

We investigate what collections of c.e. Turing degrees can be realised as the collection of elements of a separating Π^0_1 class of c.e. degree. We show that for every c.e. degree 𝐜, the collection {𝐜, 0'} can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e. degree. For example, we show that there is no super-maximal pair: disjoint c.e. sets A and B whose separating class is infinite, but every separator of c.e. degree is a finite variant of either A or B.

Let $K$ denote prefix-free Kolmogorov Complexity, and $K^A$ 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 mimimal descriptions.

We show that the existence of hyperarithmetic isomorphisms between computable structures is complete for Π11 equivalence relations under computable reductions. This uses Montalbán's true stages machinery for iterated priority arguments, of which we give a new development.

We show that structures with only one binary function symbol are universal for “online” (punctual) computable structures. In contrast, we give a description of punctually categorical graphs which implies that graphs are not universal for online computability.

Abstract We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits. As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

In this paper we study various properties of algorithmically random infinite structures. Our results address the following questions. How would one define algorithmic randomness for infinite structures? Could algorithmically random structures be computable? What are the similarities and differences between algorithmically random structures and algorithmically random infinite strings? What are the possible Turing degrees of algorithmically random structures? Are there algorithmically random infinite groups? For instance, we prove the following in this paper: (1) there are classes which contain algorithmically random yet computable structures, (2) there exist algorithmically random universal algebras with co-computably enumerable as well as computably enumerable word problems, (3) there are natural classes of structures in which the Turing degrees of algorithmically random structures can only be either computable or equivalent to the halting set, and (4) there are examples of algorithmically random groups. The first result shows a dramatic difference between algorithmically random strings and algorithmically random structures. The second result significantly improves the known theorem that algorithmically random structures computable in the halting set exist; these examples of algebras are sharp in terms of arithmetical hierarchy of the word problem for random algebras. The third result is a dichotomy theorem that characterises all possible Turing degrees of algorithmically random structures. Finally, the fourth result answers a nontrivial open question about the existence of algorithmically random groups.

Computability-theoretic investigation of algorithmic complexity of isomorphisms between countable structures is a key topic in computable structure theory since Frohlich and Shepherdson, Mal'cev, and Metakides and Nerode. A computable structure A is called computably categorical if for every computable isomorphic B, there is a computable isomorphism from A onto B. By relativizing the notion of computable categoricity to a Turing degree d, we obtain the notion of d-computable categoricity. For the case when d is 0((n-1)), we also speak about Delta(0)(n)-categoricity, for n >= 1. More generally, A is relatively Delta(0)(n)-categorical if for every isomorphic B, there is an isomorphism that is Delta(0)(n) relative to the atomic diagram of B. Equivalently, A is relatively Delta(0)(n)-categorical if and only if A has a computably enumerable Scott family of computable (infinitary) Sigma(n) formulas. Relative Delta(0)(n)-categoricity implies Delta(0)(n)-categoricity, but not vice versa. In this paper, we present an example of a computable Fraisse limit that is computably categorical (that is, Delta(0)(1)-categorical) but not relatively computably categorical. We also present examples of Delta(0)(2)-categorical but not relatively Delta(0)(2)-categorical structures in natural classes such as trees of finite and infinite heights, and homogeneous, completely decomposable, abelian groups. It is known that for structures from these classes computable categoricity and relative computable categoricity coincide. The categoricity spectrum of a computable structure M is the set of all Turing degrees d such that M is d-computably categorical. The degree of categoricity of M is the least degree in the categoricity spectrum of M, if such a degree exists. It provides the exact level of categoricity of the structure. In this paper, we compute degrees of categoricity for relatively Delta(0)(2)-categorical abelian p-groups and for relatively Delta(0)(3)-categorical Boolean algebras. (C) 2019 Elsevier B.V. All rights reserved.

We give two new characterizations of K-triviality. We show that if for all Y such that Omega is Y-random, Omega is (Y circle plus A)-random, then A is K-trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of K-triviality and answering a question of Yu. We also prove that if A is K-trivial, then for all Y such that Omega is Y-random, (Y circle plus A) (LR) Y. This answers a question of Merkle and Yu. The other direction is immediate, so we have the second characterization of K-triviality. The proof of the first characterization uses a new cupping result. We prove that if A not less than or equal to(LR) B, then for every set X there is a B-random set Y such that X is computable from Y circle plus A.

We apply methods of computable structure theory to study effectively closed subgroups of S ∞ S_\infty . The main result of the paper says that there exists an effectively closed presentation of Z 2 \mathbb {Z}_2 which is not the automorphism group of any computable structure M M . In contrast, we show that every effectively closed discrete group is topologically isomorphic to A u t ( M ) \rm {Aut}(M) for some computable structure M M . We also prove that there exists an effectively closed compact (thus, profinite) subgroup of S ∞ S_\infty that has no computable Polish presentation. In contrast, every profinite computable Polish group is topologically isomorphic to an effectively closed subgroup of S ∞ S_\infty . We also look at oligomorphic subgroups of S ∞ S_\infty ; we construct a Σ 1 1 \Sigma ^1_1 closed oligomorphic group in which the orbit equivalence relation is not uniformly HYP. Our proofs rely on methods of computable analysis, techniques of computable structure theory, elements of higher recursion theory, and the priority method.

This article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [ID. The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of R(n )contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality kappa using kappa-computability.

Martin-Löf (ML)-reducibility compares K-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to capture ML-random sequences. We show that for every cost function there is a c.e. set ML-above the sets obeying it (called a “smart” set for the cost function). We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω. This leads to a new characterisation of strong jump-traceability.

Abstract Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set $\omega /E$ (or equivalently, the relation E) realizes a linearly ordered set ${\cal L}$ if there exists a c.e. relation ⊴ respecting E such that the induced structure ( $\omega /E$ ; ⊴) is isomorphic to ${\cal L}$ . Thus, one can consider the class of all linearly ordered sets that are realized by $\omega /E$ ; formally, ${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$ . In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order $ \le _{lo} $ on the class of all c.e. equivalence relations: $E_1 \le _{lo} E_2 $ if every linear order realized by E 1 is also realized by E 2. Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order $ \le _{lo} $ . We study the partially ordered set of lo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among the lo-degrees.

AbstractWe study the computable structure theory of linear orders of size $\aleph _1 $ within the framework of admissible computability theory. In particular, we study degree spectra and the successor relation.

Abstract We study the computable structure theory of linear orders of size $\aleph _1 $ within the framework of admissible computability theory. In particular, we characterize which of these linear orders are computably categorical.