Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-5bf98f6d76-rtbc9 Total loading time: 0.31 Render date: 2021-04-20T17:52:01.167Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Combinatorics, Probability and Computing Combinatorics, Probability and Computing

Article contents

Three Thresholds for a Liar

Published online by Cambridge University Press:  12 September 2008

Joel Spencer  and
Peter Winkler
Joel Spencer
Affiliation:
NYU-Courant Institute, 251 Mercer St., New York, NY 10012
Peter Winkler
Affiliation:
Bellcore, 445 South St., Morristown, NJ 07962-1910
Get access
Rights & Permissions[Opens in a new window]

Abstract

Motivated by the problem of making correct computations from partly false information, we study a corruption of the classic game “Twenty Questions” in which the player who answers the yes-or-no questions is permitted to lie up to a fixed fraction r of the time. The other player is allowed q arbitrary questions with which to try to determine, with certainty, which of n objects his opponent has in mind; he “wins” if he can always do so, and “wins quickly” if he can do so using only O(log n) questions.

It turns out that there is a threshold value for r below which the querier can win quickly, and above which he cannot win at all. However, the threshold value varies according to the precise rules of the game. Our “three thresholds theorem” says that when the answerer is forbidden at any point to have answered more than a fraction r of the questions incorrectly, then the threshold value is r = ½; when the requirement is merely that the total number of lies cannot exceed rq, the threshold is ⅓; and finally if the answerer gets to see all the questions before answering, the threshold drops to ¼.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 1 , March 1992 , pp. 81 - 93
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

[1] Aslam, Javed and Dhagat, Aditi. Searching in the presence of linearly bounded errors. In Proceedings of the 23rd Annual ACM Symposium of Theory of Computing, 1991.Google Scholar
[2] Berlekamp, E.. Block Coding with Noiseless Feedback. PhD thesis, M.I.T., 1964.Google Scholar
[3] Feige, U., Peleg, D., Raghavan, P., and Upfal, E.. Computing with unreliable information. In Symposium on Theory of Computing, pages 128–137, 1990.Google Scholar
[4] Frazier, Michael. Searching with a non-constant number of lies. 1990.Google Scholar
[5] Hall, Marshall. Combinatorial Theory (2nd ed.). Wiley, 1986.Google Scholar
[6] Kenyon, Claire and Yao, Andrew C.. On evaluation of boolean functions with unreliable tests. International Journal of Foundations of Computer Science, 1(1): 1–10, 1990.CrossRefGoogle Scholar
[7] Pelc, Andrzej. Solution of ulam's problem on searching with a lie. Journal of Combinatorial Theory, Ser. A, 44: 129–140, 1987.CrossRefGoogle Scholar
[8] Pelc, Andrzej. Searching with known error probability. Theoretical Computer Science, 63: 185–202, 1989.CrossRefGoogle Scholar
[9] Rivest, R. L., Meyer, A. R., Kleitman, D. J., Winklmann, K., and Spencer, J.. Coping with errors in binary search procedures. Journal of Computer and System Sciences, 20: 396–404, 1980.CrossRefGoogle Scholar
[10] Spencer, Joel. Guess a number - with lying. Mathematics Magazine, 57(2): 105–108, 1984.Google Scholar
[11] Spencer, Joel. Balancing vectors in the max norm. Combinatorica, 6: 55–65, 1986.CrossRefGoogle Scholar
[12] Ulam, S. M.. Adventures of a Mathematician. Charles Scribner's Sons, 1976.Google Scholar
[13] Lint, J. H. van. Introduction to Coding Theory. Springer-Verlag, 1982.CrossRefGoogle Scholar

Altmetric attention score

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 69 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 20th April 2021. This data will be updated every 24 hours.

29
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

Three Thresholds for a Liar
Available formats
×

Send article to Dropbox

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Three Thresholds for a Liar
Available formats
×

Send article to Google Drive

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Three Thresholds for a Liar
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *