6 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…

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…

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…

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 $(\omega,<)$.