Stephen G. Simpson
Department of Mathematics
Pennsylvania State University

December 18, 2008

This is the abstract of my invited 50-minute talk at the Very Informal
Gathering of Logicians at UCLA, January 30 to February 1, 2009, in
honor of the 60th birthday of John Steel.


Kolmogorov 1932 proposed to view intuitionistic logic as a ``calculus
of problems'' (Aufgabenrechnung).  This is essentially the famous BHK
interpretation of intuitionism.  Medvedev 1955 introduced mass
problems as a rigorous elaboration of Kolmogorov's proposal.  A mass
problem is a set of reals.  If P is a mass problem, the solutions of P
are the elements of P.  We say that P is solvable if there exists a
computable solution of P.  We say that P is weakly reducible to Q if
each solution of Q can be used as a Turing oracle to compute some
solution of P.  A weak degree is an equivalence class of mass problems
under mutual weak reducibility.  Let D_w be the lattice of weak
degrees.  There is an obvious, natural embedding of the Turing degrees
into D_w, obtained by identifying the Turing degree of a real with the
weak degree of the singleton set consisting of that real.  Muchnik
1963 observed that D_w is a model of intuitionistic propositional
calculus.  Since 1999 I have been studying the sublattice P_w
consisting of the weak degrees of nonempty effectively closed sets in
Euclidean space.  I discovered a natural embedding of the recursively
enumerable Turing degrees into P_w.  Moreover, I discovered that P_w
contains a variety of specific, natural, weak degrees which are
closely related to various foundationally interesting topics.  Among
these topics are reverse mathematics, algorithmic randomness,
algorithmic information theory, hyperarithmeticity, diagonal
nonrecursiveness, almost everywhere domination, subrecursive
hierarchies, resource-bounded computational complexity, effective
Hausdorff dimension, and Kolmogorov complexity.  Recently I applied
P_w to the study of 2-dimensional symbolic dynamics.  The purpose of
this talk is to introduce P_w and survey what is known about it.