Medvedev degrees of two-dimensional subshifts of finite type

Abstract

AbstractIn this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two setsXandYare said to beMedvedev equivalentif there exist partial computable functionals fromXintoYand vice versa. TheMedvedev degreeofXis the equivalence class ofXunder Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices ℰsand ℰwof Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰsand 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.