# Many-one degrees of some particular sets

Let $$W_0, W_1, W_2,\dotsc$$ be an effective numbering of r.e. sets.

Consider sets $$\text{Emp}=\{x\mid W_x=\emptyset\}$$, $$\text{Tot}=\{x\mid W_x=\mathbb{N}\}$$ and $$S_n=\{x\mid W_x=W_n\}$$ (for some fixed $$n$$).

It is easy to show that $$\text{Emp}$$ is many-one reducible to $$\text{Tot}$$, but $$\text{Tot}$$ is not many-one reducible to $$\text{Emp}$$. Moreover, for all $$n$$, $$S_n$$ is many-one reducible to $$\text{Tot}$$.

I was able to show that if $$W_n$$ is infinite, then $$S_n$$ and $$\text{Tot}$$ are many-one equivalent, i.e. have the same m-degree.

But what if $$W_n$$ is nonempty but finite? I have shown that the m-degree of $$S_n$$ is in such a case strictly greater than the m-degree of $$\text{Emp}$$. But is it strictly smaller then the m-degree of $$\text{Tot}$$, or maybe they are equal?

In fact if $$W_n$$ is finite then $$S_n$$ is weaker than $$\mathsf{Tot}$$. This is because, in this case, $$S_n$$ is $$\Sigma^0_1\wedge\Pi^0_1$$-definable (or equivalently, is the intersection of a c.e. set and a co-c.e. set), while $$\mathsf{Tot}$$ is $$\Pi^0_2$$-complete.
Specifically, fix $$n$$ such that $$W_n$$ is finite, let $$A=\{m: W_m\supseteq W_n\}$$ and let $$B=\{m: W_m\subseteq W_n\}$$. Since $$W_n$$ is finite we have that $$A$$ is c.e. (if $$W_n$$ were infinite then $$A$$ would be merely $$\Pi^0_2$$) and $$B$$ is co-c.e. (Note that without an assumption on $$W_n$$, the most we could say is that $$A$$ and $$B$$ are each $$\Pi^0_2$$.) We clearly have $$S_n=A\cap B$$.
With this question we've stepped into the difference (Ershov) hierarchy; see the first section of Stephan/Yang/Yu for a good survey of its basic properties. Note that every set in the difference hierarchy is Turing-reducible to the Halting Problem, while $$\mathsf{Tot}$$ is strictly above the Halting Problem even with respect to Turing reducibility.