Prediction with Restricted Resources and Finite Automata

TL;DR

This paper introduces a method to measure the complexity of random sequences generated by finite automata and develops optimal algorithms for predicting such sequences, linking automata size to sequence complexity.

Contribution

It proposes a new complexity index based on finite automata and provides optimal prediction algorithms for sequences generated by automata with a fixed number of states.

Findings

Complexity can be indexed by automata states

Optimal prediction algorithms are developed for automata-generated sequences

Sequence complexity correlates with automata size

Abstract

We obtain an index of the complexity of a random sequence by allowing the role of the measure in classical probability theory to be played by a function we call the generating mechanism. Typically, this generating mechanism will be a finite automata. We generate a set of biased sequences by applying a finite state automata with a specified number, $m$, of states to the set of all binary sequences. Thus we can index the complexity of our random sequence by the number of states of the automata. We detail optimal algorithms to predict sequences generated in this way.