The logic of Ulam's games with lies
Daniele Mundici · Cambridge University Press eBooks · 1992
Someone thinks of a number between one and one million (which is just less than 2 20 ). 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 log 2 (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, …, 2 n – 1}. Writing numbers in binary notation, S is more conveniently represented by the n -cube {0, l] n .