Tohoku Mathematical Journal

Mass problems associated with effectively closed sets

Stephen G. Simpson

Full-text: Open access

Abstract

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 $\mathcal{E}_\mathrm{w}$ be this lattice. We show that $\mathcal{E}_\mathrm{w}$ provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in $\mathcal{E}_\mathrm{w}$ which are associated with such problems. In addition, we present some structural results concerning the lattice $\mathcal{E}_\mathrm{w}$. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how $\mathcal{E}_\mathrm{w}$ can be applied in symbolic dynamics, toward the classification of tiling problems and $\boldsymbol{Z}^d$-subshifts of finite type.

Article information

Source
Tohoku Math. J. (2) Volume 63, Number 4 (2011), 489-517.

Dates
First available in Project Euclid: 6 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1325886278

Digital Object Identifier
doi:10.2748/tmj/1325886278

Mathematical Reviews number (MathSciNet)
MR2872953

Zentralblatt MATH identifier
1246.03064

Subjects
Primary: 03D30: Other degrees and reducibilities
Secondary: 03D20: Recursive functions and relations, subrecursive hierarchies 03D25: Recursively (computably) enumerable sets and degrees 03D28: Other Turing degree structures 03D32: Algorithmic randomness and dimension [See also 68Q30] 03D55: Hierarchies 03D80: Applications of computability and recursion theory 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Keywords
Mass problems unsolvable problems degrees of unsolvability Muchnik degrees algorithmic randomness Kolmogorov complexity resourse-bounded computational complexity recursively enumerable degrees hyperarithmetical hierarchy intuitionism proof theory

Citation

Simpson, Stephen G. Mass problems associated with effectively closed sets. Tohoku Math. J. (2) 63 (2011), no. 4, 489--517. doi:10.2748/tmj/1325886278. https://projecteuclid.org/euclid.tmj/1325886278


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