TY - JOUR

T1 - Minimum Guesswork with an Unreliable Oracle

AU - Ardimanov, Natan

AU - Shayevitz, Ofer

AU - Tamo, Itzhak

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2020/12

Y1 - 2020/12

N2 - We study a guessing game where Alice holds a discrete random variable $X$ , and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole's answer is however unreliable, and is incorrect with probability $\epsilon $. We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities - this question simultaneously minimizes all the guessing moments for any value of $\epsilon $. In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$ -ary question, and then observe Carole's answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob's question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$ ) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$ -ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$.

AB - We study a guessing game where Alice holds a discrete random variable $X$ , and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole's answer is however unreliable, and is incorrect with probability $\epsilon $. We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities - this question simultaneously minimizes all the guessing moments for any value of $\epsilon $. In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$ -ary question, and then observe Carole's answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob's question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$ ) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$ -ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$.

KW - Information theory

KW - combinatorics

KW - computational complexity

UR - http://www.scopus.com/inward/record.url?scp=85097343969&partnerID=8YFLogxK

U2 - 10.1109/TIT.2020.3033305

DO - 10.1109/TIT.2020.3033305

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AN - SCOPUS:85097343969

SN - 0018-9448

VL - 66

SP - 7528

EP - 7538

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 12

M1 - 9237980

ER -