Mass Problems Associated with Effectively Closed Sets
نویسنده
چکیده
The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov’s non-rigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let Ew be this lattice. We show that Ew provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in Ew which are associated with such problems. In addition, we present some structural results concerning the lattice Ew. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how Ew can be applied in symbolic dynamics, toward the classification of tiling problems and Z-subshifts of finite type. 2000 Mathematics Subject Classification: Primary 03D30; Secondary 03D20, 03D25, 03D28, 03D32, 03D55, 03D80, 37B10.
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