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.
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.
Published online by Cambridge University Press: 05 November 2011
Someone thinks of a number between one and one million
(which is just less than 220).
Another person is allowed to ask up to twenty questions,
to each of which the first person is supposed to answer only yes or no.
Obviously the number can be guessed by asking first:
Is the number in the first half million?
then again reduce the reservoir of numbers in the next question
by one-half, and so on.
Finally the number is obtained in less than log2(1000000).
Now suppose one were allowed to lie once or twice,
then how many questions would one need to get the right answer?
S. M. Ulam (1976), Adventures of a Mathematician (New York: Scribner, p. 281)PLAYING ULAM'S GAME
The questions and answers exchanged between Questioner and Responder in Ulam's game with k lies are propositions. In this chapter we show that the Lukasiewicz (k + 2)-valued sentential calculus ([14], [15]) provides a natural logic for these propositions. Throughout this chapter, the Responder is identified with Pinocchio. Author and Reader will often impersonate the Questioner. Unless otherwise stated, in this section we consider Ulam's game with at most one lie.
Initially, Pinocchio and the Questioner agree to fix a search space S = {0,1, …, 2n – 1}. Writing numbers in binary notation, S is more conveniently represented by the n-cube {0, l]n.
To save this book to your Kindle, first ensure [email protected] 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.
To save content items to your account, please 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 account. Find out more about saving content to Dropbox.
To save content items to your account, please 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 account. Find out more about saving content to Google Drive.
