Skip to main content Accessibility help
×
Home

DEEP ${\rm{\Pi }}_1^0 $ CLASSES

  • LAURENT BIENVENU (a1) and CHRISTOPHER P. PORTER (a2)

Abstract

A set of infinite binary sequences ${\cal C} \subseteq 2$ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.

We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.

Copyright

References

Hide All
[1]Bennett, Charles H., Logical depth and physical complexity, The Universal Turing Machine: A Half-century Survey, Springer, New York, 1995, pp. 207235.
[2]Bienvenu, Laurent, Hölzl, Rupert, Porter, Christopher P., and Shafer, Paul, Randomness and semi-measures, Notre Dame Journal of Formal Logic, preprint, arXiv:1310.5133, 2014.
[3]Bienvenu, Laurent and Miller, Joseph S., Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, vol. 163 (2012), no. 5, pp. 506518.
[4]Bienvenu, Laurent and Porter, Christopher P., Strong reductions in effective randomness. Theoretical Computer Science, vol. 459 (2012), pp. 5568.
[5]Day, Adam and Miller, Joseph S., Density, forcing, and the covering problem, Mathematical Research Letters, vol. 22 (2015), no. 3, submitted.
[6]de Leeuw, Karel, Moore, Edward F., Shannon, Claude, and Shapiro, Norman, Computability by probabilistic machines, Automata Studies, Princeton University Press, Princeton, NJ, 1956.
[7]Downey, Rodney, Greenberg, Noam, and Miller, Joseph S., The upward closure of a perfect thin class. Annals of Pure and Applied Logic, vol. 156 (2008), pp. 5158.
[8]Downey, Rodney and Hirschfeldt, Denis, Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.
[9]Durand, Bruno, Levin, Leonid, and Shen, Alexander, Complex tilings. The Journal of Symbolic Logic, vol. 73 (2008), no. 2, pp. 593613.
[10]Franklin, Johanna N.Y. and Ng, Keng Meng, Difference randomness. Proceedings of the American Mathematical Society, vol. 139 (2011), pp. 345360.
[11]Gács, Peter, Lecture notes on descriptional complexity and randomness, manuscript, available at http://www.cs.bu.edu/fac/gacs/recent-publ.html.
[12]Gács, Peter, Exact expressions for some randomness tests, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 26 (1980), pp. 385394.
[13]Greenberg, Noam and Miller, Joseph S., Lowness for Kurtz randomness, this Journal, vol. 74 (2009), no. 2, pp. 665678.
[14]Greenberg, Noam and Miller, Joseph S., Diagonally non-recursive functions and effective Hausdorff dimension. Bulletin of the London Mathematical Society, vol. 43 (2011), no. 4, pp. 636654.
[15]Greenberg, Noam, Miller, Joseph S., and Nies, André, PA-completeness and its weakenings, in preparation.
[16]Hölzl, Rupert and Merkle, Wolfgang, Traceable sets, IFIP TCS, IFIP Advances in Information and Communication Technology, no. 323, Springer, Springer Berlin Heidelberg, 2010, pp. 301315.
[17]Jockusch, Carl and Soare, Robert, ${\rm{\Pi }}_1^0 $classes and degrees of theories. Transaction of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[18]Kautz, Steven M., Degrees of Random Sequences, Ph.D. Thesis, Cornell University, 1991.
[19]Khan, Mushfeq, Shift-complex sequences, this Bulletin, vol. 19 (2013), no. 2, pp. 199215.
[20]Kurtz, Stuart, Randomness and genericity in the degrees of unsolvability, Ph. D dissertation, University of Illinois at Urbana, 1981.
[21]Levin, Leonid A., Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, vol. 61 (1984), no. 1, pp. 1537.
[22]Levin, Leonid A., Forbidden information. Journal of the ACM, vol. 60 (2013), no. 2, p. 9.
[23]Levin, Leonid A. and V’yugin, Vladimir V., Invariant properties of informational bulks, Lecture Notes in Computer Science, vol. 53 (1977), pp. 359364.
[24]Levin, Leonid A. and Zvonkin, Alexander K., The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms. Uspekhi Matematicheskikh Nauk, vol. 25 (1970), no. 6(156), pp. 85127.
[25]Nies, André, Computability and Randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.
[26]Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A., Randomness, relativization and Turing degrees. The Journal of Symbolic Logic, vol. 70 (2005), no. 2, pp. 515535.
[27]Rumyantsev, Andrey Yu., Everywhere complex sequences and the probabilistic method, STACS, LIPIcs, vol. 9, 2011, pp. 464471.
[28]Sacks, Gerald, Degrees of Unsolvability, Princeton University Press, Princeton, NJ, 1963.
[29]Simpson, Stephen G., Mass problems and randomness, this Bulletin, vol. 11 (2005), pp. 127.
[30]Simpson, Stephen G., An extension of the recursively enumerable Turing degrees. Journal of the London Mathematical Society, vol. 75 (2006), p. 2007.
[31]Simpson, Stephen G., Mass problems associated with effectively closed sets. Tohoku Mathematical Journal, vol. 63 (2011), pp. 489517.
[32]Stephan, Frank, Martin-Löf random and PA-complete sets, Proceedings of ASL Logic Colloquium 2002, ASL Lecture Notes in Logic, vol. 27, 2006, pp. 342348.
[33]V’yugin, Vladimir V., Algebra of invariant properties of binary sequences. Problemy Peredachi Informatsii, vol. 18 (1982), no. 2, pp. 83100.

Keywords

DEEP ${\rm{\Pi }}_1^0 $ CLASSES

  • LAURENT BIENVENU (a1) and CHRISTOPHER P. PORTER (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.