Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

Generic computability, Turing degrees, and asymptotic density

Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically computable if there is a partial computable function which agrees with the characteristic function of A on its domain D, and furthermore D has density 1, i.e. limn→∞ |{k < n : k ∈ D}|/n = 1. A set A is called co...

Reverse mathematics, computability, and partitions of trees

Weexamine the reversemathematics and computability theory of a formofRamsey’s theorem in which the linear n-tuples of a binary tree are colored. Let 2<N denote the full binary tree of height!. We identify nodes of the tree with finite sequences of zeros and ones, and refer to any subset of the nodes as a subtree. For positive integers n, let [2<N]n denote the set of all linearly ordered n-tuple...

Asymptotic density and the coarse computability bound

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 A0, A1 such that γ(A0) = γ(A1) = r where A0 is co...

Computability , Reverse Mathematics and Combinatorics : Open Problems

Fix a finite set L and an infinite list of variables v0, v1, v2, . . . , vn, . . .. For m,n ≤ ω, W (L,m, n) is the set of sequences w of elements of L ∪ {vi | i < m} of length n with the property that vi occurs in w for each i < m and the first occurrence of vi is before the first occurrence of vj whenever i < j < m. When m ∈ ω, W (L,m) is ⋃ n∈ωW (L,m, n). W (L) is ⋃ m∈ωW (L,m). When w ∈ W (L,m...

Medvedev Degrees, Muchnik Degrees, Subsystems of Z2 and Reverse Mathematics