Recursion Theory and Symbolic Dynamics

Stephen G. Simpson
Pennsylvania State University

Trimester on Methods of Proof Theory
Max Planck Institute for Mathematics
Bonn, Germany

May 30, 2007


A subset P of {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 subset P of {0,1}^N is said to be Pi^0_1 if it is effectively closed, i.e., it is the complement of the union of a recursive sequence of basic open sets. The lattice P_w of Muchnik degrees of mass problems associated with nonempty Pi^0_1 subsets of {0,1}^N has been investigated by the speaker and others. It is known that P_w 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 ZxZ on the compact space A^ZxZ. A two-dimensional subshift is a nonempty closed subset of A^ZxZ which is invariant under the action of ZxZ. A two-dimensional subshift is said to be of finite type if it is defined by a finite set of forbidden configurations. The two-dimensional subshifts of finite type are known to form an important class of dynamical systems, with connections to mathematical physics, etc.

Clearly every two-dimensional subshift of finite type is a nonempty Pi^0_1 subset of A^ZxZ, hence its Muchnik degree belongs to P_w. Conversely, we prove that every Muchnik degree in P_w is the Muchnik degree of a two-dimensional subshift of finite type. The proof of this result uses tilings of the plane. We present an application of this result to symbolic dynamics. Our application is stated purely in terms of two-dimensional subshifts of finite type, with no mention of Muchnik degrees.