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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

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.

ABSTRACT

Mass Problems:

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.

Symbolic Dynamics:

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 x Z 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
Z x Z.  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
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.

Intuitionism:

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 P_w Brouwerian?  We prove that
it is not.  The proof uses our adaptation of a technique of Posner and
Robinson.

REFERENCES

Joshua A. Cole and Stephen G. Simpson, Mass problems and
hyperarithmeticity, 20 pages, 28 November 2006, submitted for
publication.

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

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

David B. Posner and Robert W. Robinson, Degrees joining to 0', Journal
of Symbolic Logic, 46, 714-722, 1981.

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

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

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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