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- W2771979056 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." @default.
- W2771979056 created "2017-12-22" @default.
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- W2771979056 date "2020-09-04" @default.
- W2771979056 modified "2023-10-16" @default.
- W2771979056 title "MUCHNIK DEGREES AND CARDINAL CHARACTERISTICS" @default.
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- W2771979056 doi "https://doi.org/10.1017/jsl.2020.1" @default.
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