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 <inline-formula> <tex-math notation="LaTeX">$x^{n}=(x_{1},\ldots,x_{n})$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$x^{n}$ </tex-math></inline-formula> 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, <inline-formula> <tex-math notation="LaTeX">$y^{n}=(y_{1},\ldots,y_{n})$ </tex-math></inline-formula>, which is also an individual sequence.