On the Muchnik degrees of 2-dimensional subshifts of finite type
نویسنده
چکیده
We apply some concepts and results from mathematical logic in order to obtain an apparently new counterexample in symbolic dynamics. Two sets are said to be Muchnik equivalent if any point of either set can be used as a Turing oracle to compute a point of the other set. The Muchnik degree of a set is its Muchnik equivalence class. There is an extensive recursion-theoretic literature on the lattice Pw of Muchnik degrees of nonempty, recursively closed subsets of the Cantor space. It is known that Pw contains many specific, interesting, Muchnik degrees related to various topics in the foundations of mathematics and the foundations of computer science. Moreover, the lattice-theoretical structure of Pw is fairly well understood. We prove that Pw consists precisely of the Muchnik degrees of 2-dimensional subshifts of finite type. We use this result to obtain an infinite collection of 2-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.
منابع مشابه
Degrees of unsolvability of 2-dimensional subshifts of finite type
We apply some concepts and results from mathematical logic in order to obtain an apparently new counterexample in 2-dimensional symbolic dynamics. A set X is said to be Muchnik reducible to a set Y if each point of Y can be used as a Turing oracle to compute a point of X. The Muchnik degree of X is the equivalence class of X under the equivalence relation of mutual Muchnik reducibility. There i...
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