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.

翻译：多头有限状态维度与预维度通过一种带有“过去探针”作用的尾随头（heads）来量化赌徒对序列的可预测性。这些额外头允许赌徒利用简单但非局部的模式，例如序列$S$满足对所有$n$有$S[n]=S[2n]$的情况。在黄、李、卢茨与卢茨（2025）的原始定义中，头的移动被要求是 oblivious（即与数据无关）。本文引入一种头的移动具有自适应性的模型（即与数据相关），并将其与oblivious模型进行比较。我们证明：对于每个$h \geq 2$，自适应性增强了$h$头有限状态赌徒的预测能力——具体地，存在一些序列，其oblivious $h$头有限状态预维度严格大于自适应$h$头有限状态预维度。我们进一步证明，随着头数增加，自适应有限状态预维度呈现严格层次结构；实际上，对所有$h \geq 1$，存在一个序列，其自适应$(h+1)$头有限状态预维度严格小于其自适应$h$头预维度。