Multihead Finite-State Dimension

2025-09-26

AI Summary

This paper introduces and studies multihead finite-state dimension—a generalized predictive betting model for infinite sequences, where multiple one-way finite-state agents collaboratively place bets. Methodologically, it formalizes a Gale-based multihead finite-state gambler framework, precisely capturing the multi-agent coordination mechanism and information aggregation rules. The main contributions are threefold: (1) It establishes a strict hierarchy of computational power with respect to head count—namely, $(k+1)$-head finite-state gamblers are strictly more powerful than $k$-head ones; (2) it proves that the $1$-head case coincides exactly with the classical finite-state dimension; and (3) it demonstrates that multihead finite-state dimension is not preserved under finite union, providing an explicit family of separating examples. These results advance the understanding of the interplay between finite-state computational resources and information compression capabilities.

Technology Category

• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce multihead 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 multihead 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 multihead 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.

Problem

• Generalizes finite-state dimension using multihead finite-state agents
• Establishes strict hierarchy in earning power with increasing heads
• Analyzes stability properties of multihead finite-state dimension

Innovation

• Multihead finite-state dimension generalizes single-head version
• Hierarchy theorem separates earning power by head count
• Multihead dimension stable under finite unions unlike predimension