# Is self-escaping without self-dominating possible?

For a countable structure $$\mathcal{S}$$, let the cospectrum of $$\mathcal{S}$$ be the set $$CS(\mathcal{S})$$ of reals (non-uniformly) computable in every copy of $$\mathcal{S}$$ (we can also make sense of cospectra for uncountable structures, via forcing).

The cospectrum is clearly an invariant with respect to Muchnik reducibility (where the mass problem corresponding to a countable structure is the set of copies of the structure). Richter showed that in many cases the cospectrum is a silly invariant; for example, any two linear orders have the same cospectra (namely the set of computable reals). However, "higher-order" cospectra turn out to have more content. Namely, say that a structure $$\mathcal{S}$$ is:

• self-escaping if for every copy $$S$$ of $$\mathcal{S}$$ there is some $$f\le_TS$$ such that $$f$$ escapes every $$g\in CS(\mathcal{S}$$.

• self-dominating if for every copy $$S$$ of $$\mathcal{S}$$ there is some $$f\le_TS$$ such that $$f$$ dominates every $$g\in CS(\mathcal{S}$$.

It looks plausible that self-sufficiency considerations could be used to provide interesting separations with respect to Muchnik reducibility (and variants of self-sufficiency could be useful for other reducibilities); see e.g. Downey/Greenberg/Miller. However, I believe there is still very little known about this. In particular, I can't seem to separate the two kinds of "self-sufficiency" above (which seems a good test problem: if it's truly hard, then this is probably a silly notion to consider).

Question: Is there a structure which is self-$$\mathfrak{E}$$-sufficient but not self-$$\mathfrak{D}$$-sufficient?

• Next time, your structure is $I$, then you have $CS(I)$. – Asaf Karagila Oct 11 '19 at 19:11
• In particular it is now known - work of Andrews/Miller/S./Soskova, unpublished - that the non-self-dominating structure from DGM is also non-self-escaping. By way of motivation, this question grew out of considerations of cardinal characteristics of the continuum and their effective analogues; since "$\mathfrak{b}$ vs. $\mathfrak{d}$" is basically the easiest CCC question to resolve in both contexts, it seems a good one to start with here, even leaving aside the fact that it's already arisen implicitly in DGM. – Noah Schweber Oct 11 '19 at 20:01