19 problems

Computable Scott sentence conjecture. For each even $n\geq 2$, there is a computable structure $\mathcal{M}$ with a $\Pi_n$ Scott sentence but no computable $\Sigma_{2n}$ Scott sen…

Let $\mathcal{L}$ be an infinite linear order. Suppose that $\operatorname{SC}(\mathcal{L})=\Gamma_{\alpha}$, where $\Gamma$ is one of $\Pi$, $\Sigma$, or $d$-$\Sigma$. The associa…

Let $S$ be a decidable structure, and let $d$ be a degree. The degree $d$ is a degree of decidable categoricity if it is the least degree such that $S$ is decidably $d$-categorical…

Back-and-forth complexity tradeoff conjecture. For the indicated parity conditions, there are structures with the following exact complexities:

Loquacious highness characterization. If every degree loquaciously high for isomorphism for $K$ is uniformly high for isomorphism, then the isomorphism problem for $K$ is…

Incomparability conjecture. There are block functions $f$ and $g$ on $(\omega,<)$ whose degree spectra on a cone are incomparable: neither degree spectrum contains the other.

Let $F^\omega$ denote the free group of countably infinite rank. A free-group computable simplicity conjecture. With sufficient effort, the ideas used for the free abelian group sh…

Let $H$ be a subgroup of $\mathbb{Q}$, and consider a computable group of the form … A computably simple-copy conjecture. Theorem holds for any computable group of this form. That…

A Hopfian finitely presented group is a finitely presented group for which every surjective endomorphism is an automorphism; its word problem asks whether a given word represents t…

Let $\mathsf{BaerType}$ be the principle for rank $1$ Baer categorical torsion-free abelian groups. BaerType equivalence conjecture. … while … The source leaves verification to the…

Let $\mathsf{BaerType}$ be the principle concerning rank $1$ torsion-free abelian groups that are Baer categorical, with solutions coding the divisibility set of a nonzero group el…

Degree-spectrum conjecture. If the degree spectrum of $f$ on $(\omega,<)$ is equal to all c.e. degrees, then the successor is recoverable from $f$ on $(\omega,<)$.

Computable presentation conjecture. All “natural” groups considered in the field of t.d.l.c. groups have computable presentations, and this can be shown with sufficient effort.

Non-uniformity conjecture. The implication

For a linear order $L$, write $L^{\star}$ for its reverse, and let $\leq_c$ denote computable embeddability between pairs of structures. The preceding results establish computable…

Noninterpretability conjecture. There do not exist $L_{\omega_1\omega}$ formulas that, for all directed graphs $G$, define an interpretation of $G$ in $\Phi(G)$.

Thomassé's conjecture. The number of isomorphism types in the bi-embeddability type of every relational countable structure is either $1$, $\aleph_0$, or $2^{\aleph_0}$. This is th…

Finite-character weak coarse isomorphism conjecture. The structures $\mathcal{A}$ and $\mathcal{B}$ are weakly coarsely computably isomorphic.

Non-categorical-copy conjecture. There exist computable copies $\mathcal{B}$ and $\mathcal{C}$ of $\mathcal{A}$ that are not coarsely computably isomorphic.