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Anti-Complex Sets and Reducibilities with Tiny Use

Published online by Cambridge University Press:  12 March 2014

Johanna N. Y. Franklin ,
Noam Greenberg ,
Frank Stephan  and
Guohua Wu
Johanna N. Y. Franklin
Affiliation:
Department of Mathematics, 196 Auditorium Road, University of Connecticut, U-3009, Storrs, CT 06269-3009, USA, E-mail: johanna.franklin@uconn.edu
Noam Greenberg
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New zealand, E-mail: greenberg@msor.vuw.ac.nz
Frank Stephan
Affiliation:
Department of Computer Science and Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore, E-mail: fstephan@comp.nus.edu.sg
Guohua Wu
Affiliation:
Department of Computer Science and Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore, E-mail: fstephan@comp.nus.edu.sg
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Abstract

In contrast with the notion of complexity, a set A is called anti-complex if the Kolmogorov complexity of the initial segments of A chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set. A set A is anti-complex if and only if it is reducible to another set B with tiny use, whereby we mean that the use function for reducing A to B can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1307 - 1327
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Day, Adam R., Process and truth-table characterizations of randomness, Theoretical Computer Science, vol. 452 (2012), pp. 47–55.CrossRefGoogle Scholar
[2] Downey, Rod and Greenberg, Noam, Turing degrees of reals of positive packing dimension, Information Processing Letters, vol. 108 (2008), no. 5, pp. 298–303.CrossRefGoogle Scholar
[3] Downey, Rod and Griffiths, Evan, Schnorr randomness, this Journal, vol. 69 (2004), no. 2, pp. 533–554.Google Scholar
[4] Downey, Rod, Griffiths, Evan, and Laforte, Geoffrey, On Schnorr and computable randomness, martingales, and machines, Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 613–627.CrossRefGoogle Scholar
[5] Downey, Rod and Hirschfeldt, Denis R., Algorithmic randomness and complexity, Springer, 2010.CrossRefGoogle Scholar
[6] Downey, Rod, Hirschfeldt, Denis R., Nies, André, and Terwijn, Sebastiaan A., Calibrating randomness, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 411–491.CrossRefGoogle Scholar
[7] Downey, Rod, Jockusch, Carl G. Jr., and Stob, Michael, Array nonrecursive sets and multiple permitting arguments, Recursion theory week (Oberwolfach, 1989), Lecture Notes in Mathematics, vol. 1432, Springer, Berlin, 1990, pp. 141–173.CrossRefGoogle Scholar
[8] Downey, Rod, Jockusch, Carl G. Jr., and Stob, Michael, Array nonrecursive degrees and genericity, Computability, enumerability, unsolvability: Directions in recursion theory, Cambridge University Press, 1999, pp. 93–104.Google Scholar
[9] Downey, Rod, Ng, Keng Meng, and Solomon, D. Reed, On minimal wtt-degrees and computably enumerable Turing degrees, In preparation.Google Scholar
[10] Franklin, Johanna N.Y., Schnorr trivial reals: A construction, Archive for Mathematical Logic, vol. 46 (2008), no. 7-8, pp. 665–678.CrossRefGoogle Scholar
[11] Franklin, Johanna N.Y. and Stephan, Frank, Schnorr trivial sets and truth-table reducibility, this Journal, vol. 75 (2010), no. 2, pp. 501–521.Google Scholar
[12] Greenberg, Noam, Hirschfeldt, Denis, and Nies, André, Characterizing the strong jumptraceable sets via randomness, Advances in Mathematics, vol. 231 (2012), no. 3-4, pp. 2252–2293.CrossRefGoogle Scholar
[13] Greenberg, Noam and Nies, Andre, Benign cost functions and lowness notions, this Journal, vol. 76 (2011), no. 1, pp. 289–312.Google Scholar
[14] Ishmukhametov, Shamil, Weak recursive degrees and a problem of Spector, Recursion theory and complexity (Arslanov, Marat M. and Lempp, Steffen, editors), de Gruyter Series in Logic and Its Applications, vol. 2, de Gruyter, Berlin, 1999, pp. 81–87.CrossRefGoogle Scholar
[15] Jockusch, Carl Jr., and Soare, Robert, Π0 1 classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[16] Kjos-Hanssen, Bjørn, Merkle, Wolfgang, and Stephan, Frank, Kolmogorov complexity and the recursion theorem, Transactions of the American Mathematical Society, vol. 363 (2011), pp. 5465–5480.CrossRefGoogle Scholar
[17] Kučera, Antonín, Measure, Π0 1-classes and complete extensions of PA, Recursion theory week (Ebbinghaus, Heinz-Dieter, Müller, Gert H., and Sacks, Gerald E., editors), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245–259.CrossRefGoogle Scholar
[18] Kummer, Martin, Kolmogorov complexity and instance complexity of recursively enumerable sets, SIAM Journal on Computing, vol. 25 (1996), no. 6, pp. 1123–1143.CrossRefGoogle Scholar
[19] Li, Ming and Vitányi, Paul, An introduction to Kolmogorov complexity and its applications, third ed., Springer, 2008.CrossRefGoogle Scholar
[20] Marchenkov, Sergei S., The existence of recursively enumerable minimal truth-table degrees, Algebra and Logic, vol. 14 (1975), pp. 257–261.CrossRefGoogle Scholar
[21] Martin, Donald A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.CrossRefGoogle Scholar
[22] Merkle, Wolfgang, The complexity of stochastic sequences, Journal of Computer and System Sciences, vol. 74 (2008), pp. 350–357, Preliminary version in Conference on Computational Complexity 2003 , pages 230–235, IEEE Computer Society Press, 2003.CrossRefGoogle Scholar
[23] Nies, André, Reals which compute little, Logic Colloquium 2002, Lecture Notes in Logic, vol. 27, Association for Symbolic Logic, La Jolla, CA, 2006, pp. 261–275.Google Scholar
[24] Nies, André, Weak reducibilities, Computability, complexity and randomness, Nanjing University, 05 2008.Google Scholar
[25] Nies, André, Computabiaty and randomness, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009.CrossRefGoogle Scholar
[26] Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A., Randomness, relativization and Turing degrees, this Journal, vol. 70 (2005), no. 2, pp. 515–535.Google Scholar
[27] Odifreddi, Piergiorgio, Classical recursion theory, North-Holland, Amsterdam, 1989.Google Scholar
[28] Odifreddi, Piergiorgio, Classical recursion theory, vol. II, Elsevier, Amsterdam, 1999.Google Scholar
[29] Rogers, Hartley, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[30] Sacks, Gerald E., Higher recursion theory, Perspectives in Mathematical Logic, Omega Series, Springer, Heidelberg, 1990.CrossRefGoogle Scholar
[31] Schnorr, Claus-Peter, Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie, Lecture Notes in Mathematics, vol. 218, Springer-Verlag, Berlin, 1971.Google Scholar
[32] Soare, Robert, Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer, Heidelberg, 1987.CrossRefGoogle Scholar
[33] Terwijn, Sebastiaan A. and Zambella, Domenico, Computational randomness and lowness, this Journal, vol. 66 (2001), no. 3, pp. 1199–1205.Google Scholar