Multi-Head Finite-State Dimension
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908827153072128 |
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| author | Huang, Xiang Li, Xiaoyuan Lutz, Jack H. Lutz, Neil |
| author_facet | Huang, Xiang Li, Xiaoyuan Lutz, Jack H. Lutz, Neil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22912 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-Head Finite-State Dimension Huang, Xiang Li, Xiaoyuan Lutz, Jack H. Lutz, Neil Information Theory Formal Languages and Automata Theory Optimization and Control 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. |
| title | Multi-Head Finite-State Dimension |
| topic | Information Theory Formal Languages and Automata Theory Optimization and Control |
| url | https://arxiv.org/abs/2509.22912 |