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](http://www.personal.psu.edu/t20/logic/seminar/030304.pdf) *(where the mass problem corresponding to a countable structure is the set of copies of the structure)*. [Richter showed](https://www.jstor.org/stable/2273222) 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](https://link.springer.com/article/10.1007/s11856-016-1413-7). 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?
> 
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Is self-escaping without self-dominating possible?
[Diamondstone, Greenberg and I](http://homepages.ecs.vuw.ac.nz/~greenberg/Papers/39-non_dominated_degree_spectrum.pdf) showed that there is a structure with spectrum precisely the array non-computable degrees. Since this contains a minimal pair, its cospectrum is just the computable reals. Every A.N.C. computes an escaping function, but they do not all compute dominating functions, so this structure is self-escaping but not self-dominating.


Edit: Prior to that, [Csima and Kalimullin showed](https://philpapers.org/rec/CSIDSA) that there's a structure with spectrum precisely the hyperimmune degrees, which will also do.


