MUCHNIK DEGREES AND CARDINAL CHARACTERISTICS
BENOIT MONIN, André Nies · Journal of Symbolic Logic · 2020
Abstract A 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 sequencesy(i.e., $\{0,1\}$ -valued functions) such that for each computable bit sequencex, the bit sequence $ x {\,\leftrightarrow\,} y$ has asymptotic lower density at mostp(where $x {\,\leftrightarrow\,} y$ has a $1$ in positionniff $x(n) = y(n)$ ). We show that all members of this family of mass problems parameterized by a realpwith $0 p$ for each computable setx. 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 functiongthere 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 ins