Tohoku Mathematical Journal

Mass problems associated with effectively closed sets

Stephen G. Simpson

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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.

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

First available in Project Euclid: 6 January 2012

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Zentralblatt MATH identifier

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]

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


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.

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