Adaptive Multi-Head Finite-State Gamblers

TL;DR

This paper introduces an adaptive model for multi-head finite-state gamblers, demonstrating that adaptivity increases predictive power and establishing a hierarchy in predimensions as the number of heads varies.

Contribution

The paper extends the finite-state gambler model by allowing adaptive head movements, showing this improves predictability and creates a strict hierarchy in predimensions.

Findings

Adaptive heads outperform oblivious heads in predictability.

A strict hierarchy exists in predimensions as the number of heads increases.

Sequences can be constructed where adding more heads reduces predimension.

Abstract

Multi-head finite-state dimensions and predimensions quantify the predictability of a sequence by a gambler with trailing heads acting as "probes to the past." These additional heads allow the gambler to exploit patterns that are simple but non-local, such as in a sequence $S$ with $S[n]=S[2n]$ for all $n$. In the original definitions of Huang, Li, Lutz, and Lutz (2025), the head movements were required to be oblivious (i.e., data-independent). Here, we introduce a model in which head movements are adaptive (i.e., data-dependent) and compare it to the oblivious model. We establish that for each $h\geq 2$, adaptivity enhances the predictive power of $h$-head finite-state gamblers, in the sense that there are sequences whose oblivious $h$-head finite-state predimensions strictly exceed their adaptive $h$-head finite-state predimensions. We further prove that adaptive finite-state predimensions admit a strict hierarchy as the number of heads increases, and in fact that for all $h\geq 1$ there is a sequence whose adaptive $(h+1)$-head finite-state predimension is strictly less than its adaptive $h$-head predimension.