Recursion Theory and Symbolic Dynamics

A set P ⊆ {0,1}N may be viewed as a mass problem, i.e., a decision problem with more than one solution. By definition, the solutions of P are the elements of P . A mass problem is said to be solvable if at least one of its solutions is recursive. A mass problem P is said to be Muchnik reducible to a mass problem Q if for each solution of Q there exists a solution of P which is Turing reducible to the given solution of Q. A Muchnik degree is an equivalence class of mass problems under mutual Muchnik reducibility. A set P ⊆ {0,1}N is said to be Π1 if it is effectively closed, i.e., it is the complement of the union of a recursive sequence of basic open sets. The lattice Pw of Muchnik degrees of mass problems associated with nonempty Π1 subsets of {0,1}N has been investigated by the speaker and others. It is known that Pw contains many specific, natural Muchnik degrees which are related to various topics in the foundations of mathematics. Among these topics are algorithmic randomness, reverse mathematics, almost everywhere domination, hyperarithmeticity, resource-bounded computational complexity, Kolmogorov complexity, and subrecursive hierarchies. Let A be a finite set of symbols. The full two-dimensional shift on A is the dynamical system consisting of the natural action of the group Z× Z on