A Survey of Mučnik and Medvedev Degrees

We survey the theory of Mučnik (weak) and Medvedev (strong) degrees of subsets of ωω with particular attention to the degrees of subsets of ω2. Sections 1–6 present the major definitions and results in a uniform notation. Sections 7–16 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.

Medvedev degrees of two-dimensional subshifts of finite type

In this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial computable functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices ℰs and ℰw of Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰs and ℰwconsist precisely of the Medvedev degrees and Muchnik degrees of two-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of two-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

Coding true arithmetic in the Medvedev and Muchnik degrees

We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, the first-order theory of the closed Muchnik degrees, the first-order theory of the compact Muchnik degrees, and the second-order theory of true arithmetic. Our coding methods also prove that neither the closed Medvedev degrees nor the compact Medvedev degrees are elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees.