We introduce multi-head finite-state dimension, a generalization of finite-state dimension in which a group of finite-state agents (the heads) with oblivious, one-way movement rules, each reporting only one symbol at a time, enable their leader to bet on subsequent symbols in an infinite data stream. In aggregate, such a scheme constitutes an $h$-head finite state gambler whose maximum achievable growth rate of capital in this task, quantified using betting strategies called gales, determines the multi-head finite-state dimension of the sequence. The 1-head case is equivalent to finite-state dimension as defined by Dai, Lathrop, Lutz and Mayordomo (2004). In our main theorem, we prove a strict hierarchy as the number of heads increases, giving an explicit sequence family that separates, for each positive integer $h$, the earning power of $h$-head finite-state gamblers from that of $(h+1)$-head finite-state gamblers. We prove that multi-head finite-state dimension is stable under finite unions but that the corresponding quantity for any fixed number $h>1$ of heads--the $h$-head finite-state predimension--lacks this stability property.

翻译：本文引入了多头有限状态维数，作为有限状态维数的一种推广。在该框架中，一组具有单向无记忆移动规则的有限状态智能体（即"头"）各自每次仅报告一个符号，使其领导者能够对无限数据流中的后续符号进行投注。整体而言，此类方案构成了一个$h$头有限状态赌徒，其在此任务中通过称为"强风"的投注策略所能实现的最大资本增长率决定了序列的多头有限状态维数。单头情形等价于Dai、Lathrop、Lutz和Mayordomo（2004）定义的有限状态维数。在主要定理中，我们证明了随着头数量的增加存在严格层级结构，并构造了明确的序列族：对每个正整数$h$，该序列族能区分$h$头有限状态赌徒与$(h+1)$头有限状态赌徒的盈利能力。我们证明多头有限状态维数在有限并集下保持稳定，但针对任意固定头数$h>1$的对应量——$h$头有限状态预维数——则不具备该稳定性。