Open Problems in Mathematical Logic — Problem 7

Definitions and mathematical statement

The supplied question asks whether strong jump-traceability reducibility, denoted by $\le_{SJT}$, implies the relation $\le_{LR}$. The excerpt does not define $\le_{SJT}$, so a complete mathematical formulation cannot be recovered from this record alone.

For clarity, under the usual Cantor-space convention, an oracle is a set $A\subseteq{\mathbb{N}}$, identified with its characteristic sequence in $2^{{\mathbb{N}}}$. Equip $2^{{\mathbb{N}}}$ with fair-coin product measure $\mu$. A Martin--Löf test relative to $A$ is a uniformly $A$-effectively open sequence $(U_k)_{k\in{\mathbb{N}}}$ with $\mu(U_k)\leq 2^{-k}$. A sequence $X\in 2^{{\mathbb{N}}}$ is Martin--Löf random relative to $A$ if, for every such test, $X\notin\bigcap_{k\in{\mathbb{N}}}U_k$. Write $\mathrm{MLR}^{A}$ for the class of these sequences. The supplied definition is \[ A\le_{LR}B\quad\Longleftrightarrow\quad \mathrm{MLR}^{B}\subseteq\mathrm{MLR}^{A}. \] The Cantor-space convention above is an explicit conventional interpretation; the supplied excerpt does not state its oracle or randomness domains.

The other supplied formula is \[ J^{A}(n)\downarrow\quad\Longrightarrow\quad J^{A}(n)\in T_n. \] Here $\downarrow$ denotes convergence. Read as a trace condition, this requires membership whenever the indicated computation converges, and imposes no membership requirement at an input where it diverges. However, the excerpt does not define $J^{A}$, specify the sets $T_n$, state their size bounds or effectiveness requirements, or explain their dependence on $B$.

Under the conventional oracle domain above, the intended question has the schematic form \[ \text{Is it true that }\quad \forall A,B\subseteq{\mathbb{N}},\qquad A\le_{SJT}B\ \Longrightarrow\ A\le_{LR}B? \] This remains an unresolved formulation: the missing definition and quantifiers for $\le_{SJT}$ must be recovered from the source before the display can serve as a complete precise statement. The supplied registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment.

Short English statement

Does strong jump-traceability reducibility imply inclusion of relative Martin--Löf randomness classes in the direction $\mathrm{MLR}^{B}\subseteq\mathrm{MLR}^{A}$? The supplied excerpt omits the definition of $\le_{SJT}$, including the trace bounds and effectiveness conditions, so the proposed universal statement requires source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800995$, exact source problem code catalogue entry, titled Open Problems in Mathematical Logic --- Problem 7. The registry attributes the record to Barmpalias, Gao et al., gives proposed year 2026, and labels its status open.
• Supplied source attribution: Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 7, https://arxiv.org/abs/2608.26628. The record reports access on 2026-10-16 and extraction by arxiv_latex. These provenance details are supplied metadata; the source was not independently consulted or verified for this draft.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record supplies a source citation, but its extracted statement is insufficient to verify the full intended definition of $\le_{SJT}$.