Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-npccv Total loading time: 0.4 Render date: 2022-09-29T07:34:50.834Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Benign cost functions and lowness properties

Published online by Cambridge University Press:  12 March 2014

Noam Greenberg
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, Wellington, New Zealand, E-mail: greenberg@msor.vuw.ac.nz
André Nies
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand, E-mail: andre@cs.auckland.ac.nz

Abstract

We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ambos-Spies, Klaus, Jockusch, Carl G. Jr., Shore, Richard A., and Soare, Robert I., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), no. 1, pp. 109128.CrossRefGoogle Scholar
[2]Beigel, Richard, Buhrman, Harry, Fejer, Peter, Fortnow, Lance, Grabowski, Piotr, Longpre, Luc, Muchnik, Andrej, Stephan, Frank, and Torenvliet, Leen, Enumerations of the Kolmogorov function, this Journal, vol. 71 (2006), no. 2, pp. 501528.Google Scholar
[3]Cholak, Peter, Downey, Rod, and Greenberg, Noam, Strong jump-traceabilty. I. The computably enumerable case, Advances in Mathematics, vol. 217 (2008), no. 5, pp. 20452074.CrossRefGoogle Scholar
[4]Cholak, Peter, Groszek, Marcia, and Slaman, Theodore, An almost deep degree, this Journal, vol. 66 (2001), no. 2, pp. 881901.Google Scholar
[5]Diamondstone, David, Promptness does not imply superlow cuppability, this Journal, vol. 74 (2009), no. 4, pp. 12641272.Google Scholar
[6]Downey, Rod G., Hirschfeldt, Denis R., Nies, André, and Stephan, Frank, Trivial Reals, Proceedings of the 7th and 8th Asian Logic Conferences, Singapore University Press, Singapore, 2003, pp. 103131.CrossRefGoogle Scholar
[7]Figueira, Santiago, Nies, André, and Stephan, Frank, Lowness properties and approximations of the jump, Annals of Pure and Applied Logic, vol. 152 (2008), no. 1–3, pp. 5166.CrossRefGoogle Scholar
[8]Franklin, Johanna N.Y., Greenberg, Noam, Stephan, Frank, and Wu, Guohua, Reducibilities with tiny use, In preparation.Google Scholar
[9]Greenberg, Noam, Hirschfeldt, Denis R., and Nies, André, Characterizing the strongly jump traceable sets via randomness, To appear.Google Scholar
[10]Hirschfeldt, Denis R., Nies, André, and Stephan, Frank, Using random sets as oracles, Journal of the London Mathematical Society. Second Series, vol. 75 (2007), no. 3, pp. 610622.CrossRefGoogle Scholar
[11]Ishmukhametov, Shamil, Weak recursive degrees and a problem ofSpector, Recursion theory and complexity (Kazan, 1997), de Gruyter Series in Logic and its Applications, vol. 2, de Gruyter, Berlin, 1999, pp. 8187.Google Scholar
[12]Kučera, Antonín and Terwijn, Sebastiaan A., Lowness for the class of random sets, this Journal, vol. 64 (1999), no. 4, pp. 13961402.Google Scholar
[13]Miller, Joseph S. and Nies, André, Randomness and computability: open questions, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 390410.CrossRefGoogle Scholar
[14]Ng, Keng Meng, On strongly jump traceable reals, Annals of Pure and Applied Logic, vol. 154 (2008), no. 1, pp. 5169.CrossRefGoogle Scholar
[15]Ng, Keng Meng, Beyond strong jump-traceability, Proceedings of the London Mathematical Society, To appear.Google Scholar
[16]Ng, Keng Meng, Almost superdeep degrees, In preparation.Google Scholar
[17]Nies, André, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274305.CrossRefGoogle Scholar
[18]Nies, André, Reals which compute little, Logic Colloquium '02, Lecture Notes in Logic, vol. 27, Association of Symbolic Logic, La Jolla, CA, 2006, pp. 261275.Google Scholar
[19]Nies, André, Computability and randomness, Oxford University Press, 2009.CrossRefGoogle Scholar
[20]Nies, André, Being below Demuth random sets, In preparation.Google Scholar
[21]Nies, André, Calculus of cost functions, In preparation.Google Scholar
[22]Simpson, Stephen G., Almost everywhere domination and superhighness, Mathematical Logic Quarterly, vol. 53 (2007), no. 4–5, pp. 462482.CrossRefGoogle Scholar
[23]Nies, André, Mass problems andalmost everywhere domination, Mathematical Logic Quarterly, vol. 53 (2007), no. 4–5, pp. 483492.Google Scholar
[24]Terwijn, Sebastiaan A. and Zambella, Domenico, Computational randomness and lowness, this Journal, vol. 66 (2001), no. 3, pp. 11991205.Google Scholar
12
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Benign cost functions and lowness properties
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Benign cost functions and lowness properties
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Benign cost functions and lowness properties
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *