Guessing Individual Sequences: Generating Randomized Guesses Using Finite-State Machines
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Motivated by earlier results on universal randomized guessing, we consider an individual-sequence approach to the guessing problem: in this setting, the goal is to guess a secret, individual (deterministic) vector x^n=(x_1,...,x_n), by using a finite-state machine that sequentially generates randomized guesses from a stream of purely random bits. We define the finite-state guessing exponent as the asymptotic normalized logarithm of the minimum achievable moment of the number of randomized guesses, generated by any finite-state machine, until x^n is guessed successfully. We show that the finite-state guessing exponent of any sequence is intimately related to its finite-state compressibility (due to Lempel and Ziv), and it is asymptotically achieved by the decoder of (a certain modified version of) the 1978 Lempel-Ziv data compression algorithm (a.k.a. the LZ78 algorithm), fed by purely random bits. The results are also extended to the case where the guessing machine has access to a side information sequence, y^n=(y_1,...,y_n), which is also an individual sequence.
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