Muchnik degrees and cardinal characteristics

Abstract

For $p \in [0,1]$ let $\mathcal D(p)$ be the mass problem of infinite bit sequences~$y$ (i.e., $\{0,1\}$-valued functions) such that for each computable bit sequence $x$, the bit sequence $ x \leftrightarrow y$ has asymptotic lower density at most $p$ (where $x \leftrightarrow y$ has a $1$ in position $n$ iff $x(n) = y(n)$). We show that all members of this family of mass problems parameterized by a real $p$ with $0 < p<1/2 $ have the same complexity in the sense of Muchnik reducibility. We prove this by showing Muchnik equivalence of the problems $\mathcal D(p)$ with the mass problem $\mathrm{IOE}(2^ { 2^ n})$. As a dual of the problem $\mathcal D(p)$, define $\mathcal B(p)$, for $0 \le p < 1/2$, to be the set of bit sequences $y$ such that $\underline \rho (x \leftrightarrow y) > p$ for each computable set~$x$. We prove that the Medvedev (and hence Muchnik) complexity of the mass problems $\mathcal B(p)$ is the same for all $p \in (0, 1/2)$, by showing that they are Medvedev equivalent to the mass problem of functions bounded by $2^{2^ n}$ that are almost everywhere different from each computable function. Together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type $\mathrm{IOE}$: We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above.