Asymptotic density and the coarse computability bound

Abstract

Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n : C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r : A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibility. For example, we show that if r ∈ (0, 1] there are sets A 0 , A 1 such that γ(A 0 ) = γ(A 1 ) = r where A 0 is coarsely computable at density r while A 1 is not coarsely computable at density r. We show that a real r ∈ [0, 1] is equal to γ(A) for some c.e. set A if and only if r is left-Σ 0 3 . A surprising result is that if G is a ∆ 0 2 1-generic set, and A T G with γ(A) = 1, then A is coarsely computable at density 1.

Dense computability, upper cones, and minimal pairs

“…For the next result, we will use the following lemma from [9]. Let J k be the interval [2 k − 1, 2 k+1 − 1).…”

“…Generic computability and coarse computability, which have played significant roles in several recent papers such as [1,2,3,5,6,8,9,10,11,13], both capture the idea of computing a function on "almost all" inputs, where "almost all" is defined in terms of asymptotic density. The difference between the two notions is that generic computability permits errors of omission (i.e., divergent computations), while coarse computability permits errors of commission (i.e., incorrect answers).…”

Asymptotic Density and the Theory of Computability: A Partial Survey

“…In analogy with partial computability at densities less than 1, Hirschfeldt, Jockusch, McNicholl and Schupp [16] introduced the analogous concepts for coarse computability. We define A▽C = {n : A(n) = C(n)} and call A▽C the symmetric agreement of A and C. Of course, the symmetric agreement of A and C is the complement of the symmetric difference of A and C. The coarse computability bound of every 1-random set A is 1/2.…”

On unstable and unoptimal prediction

“…This work may be contrasted with recent results on bounds for coarse computability (Cf., e.g., ). In coarse computability one asks about the (lower or upper) densities of where f is a computable function and A may be arbitrary.…”