MUCHNIK DEGREES AND CARDINAL CHARACTERISTICS

Abstract

AbstractA mass problem is a set of functions $\omega \to \omega $ . For mass problems ${\mathcal {C}}, {\mathcal {D}}$ , one says that ${\mathcal {C}}$ is Muchnik reducible to ${\mathcal {D}}$ if each function in ${\mathcal {C}}$ is computed by a function in ${\mathcal {D}}$ . In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.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 $\text {IOE}({2^{2}}^{n})$ ; here for an order function h, the mass problem $\text {IOE}(h)$ consists of the functions f that agree infinitely often with each computable function bounded by h. This result also yields a new version of the proof to of the affirmative answer to the “Gamma question” due to the first author: $\Gamma (A)< 1/2$ implies $\Gamma (A)=0$ for each Turing oracle A.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.Next, together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type $\text {IOE}$ : we show that for any order function g there exists a faster growing order function $h $ such that $\text {IOE}(h)$ is strictly above $\text {IOE}(g)$ in the sense of Muchnik reducibility.We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance, ${\mathfrak {d}} (p)$ is the least size of a set G of bit sequences such that for each bit sequence x there is a bit sequence y in G so that $\underline \rho (x {\,\leftrightarrow\,} y)>p$ . We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above.