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：多头有限状态维数与预维数通过具有追踪头（作为“对过去的探测”）的赌徒来量化序列的可预测性。这些额外的头使赌徒能够利用简单但非局部的模式，例如在所有$n$上满足$S[n]=S[2n]$的序列$S$。在Huang、Li、Lutz和Lutz（2025）的原始定义中，头的移动被要求是遗忘的（即数据无关的）。本文引入了一种头移动为自适应（即数据相关）的模型，并将其与遗忘模型进行比较。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，即存在一些序列，其遗忘$h$头有限状态预维数严格大于其自适应$h$头有限状态预维数。我们进一步证明，自适应有限状态预维数随着头数量的增加呈现严格层级关系，并且事实上对于所有$h\geq 1$，存在一个序列，其自适应$(h+1)$头有限状态预维数严格小于其自适应$h$头预维数。