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Multi-Head Finite-State Dimension

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

TL;DR

It is proved 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.

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.

Multi-Head Finite-State Dimension

TL;DR

It is proved that multi-head finite-state dimension is stable under finite unions but that the corresponding quantity for any fixed number of heads--the -head finite-state predimension--lacks this stability property.

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 -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 , the earning power of -head finite-state gamblers from that of -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 of heads--the -head finite-state predimension--lacks this stability property.

Paper Structure

This paper contains 14 sections, 14 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. String and Sequence Notation
  4. s-Gales and Martingales
  5. Kolmogorov Complexity
  6. Multi-Head Finite-State Gamblers
  7. Multi-Head Finite-State Dimensions
  8. A Hierarchy of Predimensions
  9. Instability of h-Head Finite-State Predimensions
  10. Stability of Multi-Head Finite-State Dimension
  11. Conclusion
  12. Appendix: Proofs from Section \ref{['sec:hierarchy']}
  13. Appendix: Lemmas and Proofs from Section \ref{['sec:instability']}
  14. Appendix: Proof of Theorem \ref{['thm:appd']}

Key Result

Theorem 5.1

For each $h\in\mathbb{Z}^+$, there is a sequence $Y$ such that

Figures (1)

  • Figure 1: $F_3(S)[150]=S[20]\oplus S[29] \oplus S[29]\oplus S[44]=S[20]\oplus S[44]$.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • proof
  • ...and 33 more