An inequality on guessing and its application to sequential decoding

Let (X,Y) be a pair of discrete random variables with X taking one of M possible values. Suppose the value of X is to be determined, given the value of Y, by asking questions of the form ''Is X equal to x?'' until the answer ''Yes.'' Let G(x / y) denote the number of guesses in any such guessing scheme when X = x, Y = y. We prove that E[G(X \ Y)(rho)] greater than or equal to (1 + ln M)(-rho)Sigma(y)[Sigma(x) P-X,P-Y(x,y)(1/1+rho)](1+rho) for any rho greater than or equal to 0. This provides an operational characterization of Renyi's entropy. Next we apply this inequality to the estimation of the computational complexity of sequential decoding. For this, we regard X as the input, Y as the output of a communication channel. Given Y, the sequential decoding algorithm works essentially by guessing X, one value at a time, until the guess is correct. Thus the computational complexity of sequential decoding, which is a random variable, is given by a guessing function G(X \ Y) that is defined by the order in which nodes in the tree code are hypothesized by the decoder. This observation, combined with the above lower bound on moments of G(X \ Y), yields lower bounds on moments of computation in sequential decoding. The present approach enables the determination of the (previously known) cutoff rate of sequential decoding in a simple manner; it also yields the (previously unknown) cutoff rate region of sequential decoding for multiaccess channels. These results hold for memoryless channels with finite input alphabets.