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

## TL;DR

This paper studies a finite-state machine approach to guessing individual sequences, linking the guessing difficulty to sequence compressibility and demonstrating the effectiveness of a modified Lempel-Ziv algorithm in this context.

## Contribution

It introduces the finite-state guessing exponent and connects it to sequence compressibility, showing that a modified Lempel-Ziv algorithm asymptotically achieves this bound.

## Key findings

- Finite-state guessing exponent relates to sequence compressibility.
- Modified Lempel-Ziv algorithm asymptotically achieves the guessing exponent.
- Results extend to scenarios with side information sequences.

## Abstract

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.