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Multi-Head Finite-State Dimension

Authors Xiang Huang , Xiaoyuan Li , Jack H. Lutz , Neil Lutz

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LIPIcs.MFCS.2026.59.pdf
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ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Finite-state dimension
  • effective dimension
  • algorithmic randomness

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Abstract

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.

Cite As Get BibTex

Xiang Huang, Xiaoyuan Li, Jack H. Lutz, and Neil Lutz. Multi-Head Finite-State Dimension. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.59

Author Details

Xiang Huang
  • University of Illinois Springfield, IL, USA
Xiaoyuan Li
  • Iowa State University, Ames, IA, USA
Jack H. Lutz
  • Iowa State University, Ames, IA, USA
Neil Lutz
  • University of Notre Dame, IN, USA

Funding

  • Huang, Xiang: This research was supported in part by U.S. Department of Energy Grant DE-SC0024278.
  • Li, Xiaoyuan: This research was supported in part by National Science Foundation Grants 1545028 and 1900716.
  • Lutz, Jack H.: This research was supported in part by National Science Foundation Grants 1545028 and 1900716.
  • Lutz, Neil: This research was conducted while Neil Lutz was at Swarthmore College.

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