Recent Aspects of Mass Problems:

Symbolic Dynamics and Intuitionism

Stephen G. Simpson

Department of Mathematics

Pennsylvania State University

`http://www.math.psu.edu/simpson/`

September 12, 2007

Symbolic Dynamics and Intuitionism

Stephen G. Simpson

Department of Mathematics

Pennsylvania State University

September 12, 2007

This note consists of an abstract and references for a talk to be given at an NSF-sponsored workshop on algorithmic randomness, at the University of Chicago, September 15-19, 2007.

A set
may be viewed as a *mass problem*,
i.e., a decision problem with more than one solution. By definition,
the *solutions* of are the elements of . A mass problem
is said to be *solvable* if at least one of its solutions is
recursive. A mass problem is said to be *Muchnik reducible*
to a mass problem if for each solution of there exists a
solution of which is Turing reducible to the given solution of
. A *Muchnik degree* is an equivalence class of mass problems
under mutual Muchnik reducibility.

A set
is said to be if it is
*effectively closed*, i.e., it is the complement of the union of
a recursive sequence of basic open sets. The lattice of Muchnik
degrees of mass problems associated with nonempty subsets of
has been investigated by the speaker and others. It is
known that 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 be a finite set of symbols. The *full two-dimensional
shift* on is the dynamical system consisting of the natural
action of the group
on the compact space
. A
*two-dimensional subshift* is a nonempty closed subset of
which is invariant under the action of
. A
two-dimensional subshift is said to be *of finite type* if it is
defined by a finite set of excluded 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 subset of , hence its Muchnik degree belongs to . Conversely, we prove that every Muchnik degree in 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.

Historically, the study of mass problems originated from intuitionistic considerations. Kolmogorov 1932 proposed to view intuitionism as a ``calculus of problems.'' Muchnik 1963 introduced Muchnik degrees as a rigorous elaboration of Kolmogorov's proposal. As noted by Muchnik, the lattice of all Muchnik degrees is Brouwerian.

The question arises, is the sublattice Brouwerian? We prove that it is not. The proof uses our adaptation of a technique of Posner and Robinson.

Joshua A. Cole and Stephen G. Simpson, Mass problems and hyperarithmeticity, 19 pages, 2006, submitted for publication.

Andrei N. Kolmogorov, Zur Deutung der intuitionistischen Logik,
*Mathematische Zeitschrift*, 35, 1932, 58-65.

Albert A. Muchnik, On strong and weak reducibilities of
algorithmic problems, *Sibirskii Matematicheskii Zhurnal*, 4,
1963, 1328-1341, in Russian.

David B. Posner and Robert W. Robinson, Degrees joining to ,
*Journal of Symbolic Logic*, 46, 1981, 714-722.

Stephen G. Simpson, Mass problems and randomness, *Bulletin
of Symbolic Logic*, 11, 2005, 1-27.

Stephen G. Simpson, An extension of the recursively enumerable
Turing degrees, *Journal of the London Mathematical Society*,
75, 2007, 287-297.

Stephen G. Simpson, Mass problems and almost everywhere
domination, *Mathematical Logic Quarterly*, 53, 483-492, 2007.

Stephen G. Simpson, Medvedev degrees of 2-dimensional subshifts of finite type, 8 pages, 2007, submitted for publication.

Stephen G. Simpson, Mass problems and intuitionism, 9 pages, 2007, submitted for publication.

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