Muchnik degrees and cardinal characteristics

Abstract

A mass problem is a set of functions on ω. For mass problems C, D, one says that C is Muchnik reducible to D if each function in C is computed by a function in 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.Let 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 ↔ y has asymptotic lower density is at most p (where x ↔ 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 D(p) with the mass problem IOE(2 2 n ); here for an order function h, the mass problem 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 first author's affirmative answer to the "Gamma question", whether Γ(A) < 1/2 implies Γ(A) = 0 for each Turing oracle A.Dual to the problem D(p), let B(p), for 0 ≤ p < 1/2, be the set of bit sequences y such that ρ(x ↔ y) > p for each computable set x. We prove that the Medvedev (and hence Muchnik) complexity of the B(p) is the same for all p ∈ (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 we obtain a proper hierarchy: together with Joseph Miller, we show that for any order function g there exists a faster growing order function h such that IOE(h) is strictly Muchnik above IOE(g).We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance, d(p) is the least size of a set G of bit sequences so that for each bit sequence x there is a bit sequence y in G so that ρ(x ↔ y) > p. We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above. Contents BENOIT MONIN AND ANDR É NIES 2. Defining mass problems based on relations 6 3. Coincidences of Muchnik degrees 7 The ∆-value of a Turing oracle 13 4. Analog of Theorem 3.5 for cardinal characteristics 13 5. A proper hierarchy of problems IOE(h) in the weak degrees 18 6. Some open questions 26 References 28

Martin-Löf random quantum states

“…Such notions are studied for instance in the books [9,25]. In recent work, the algorithmic theory of randomness has been connected to mathematical fields such as ergodic theory and set theory [23,15,20,21].…”