Guessing Individual Sequences: Generating Randomized Guesses Using Finite-State Machines

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 $x{n}=(x_{1},\ldots,x_{n})$ , 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, $y{n}=(y_{1},\ldots,y_{n})$ , which is also an individual sequence.