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Adaptive Multi-Head Finite-State Gamblers

Julianne Cruz, Sho Glashausser, Xiaoyuan Li, Neil Lutz

Abstract

Multi-head finite-state dimensions and predimensions quantify the predictability of a sequence by a gambler with trailing heads acting as "probes to the past." These additional heads allow the gambler to exploit patterns that are simple but non-local, such as in a sequence $S$ with $S[n]=S[2n]$ for all $n$. In the original definitions of Huang, Li, Lutz, and Lutz (2025), the head movements were required to be oblivious (i.e., data-independent). Here, we introduce a model in which head movements are adaptive (i.e., data-dependent) and compare it to the oblivious model. We establish that for each $h\geq 2$, adaptivity enhances the predictive power of $h$-head finite-state gamblers, in the sense that there are sequences whose oblivious $h$-head finite-state predimensions strictly exceed their adaptive $h$-head finite-state predimensions. We further prove that adaptive finite-state predimensions admit a strict hierarchy as the number of heads increases, and in fact that for all $h\geq 1$ there is a sequence whose adaptive $(h+1)$-head finite-state predimension is strictly less than its adaptive $h$-head predimension.

Adaptive Multi-Head Finite-State Gamblers

Abstract

Multi-head finite-state dimensions and predimensions quantify the predictability of a sequence by a gambler with trailing heads acting as "probes to the past." These additional heads allow the gambler to exploit patterns that are simple but non-local, such as in a sequence with for all . In the original definitions of Huang, Li, Lutz, and Lutz (2025), the head movements were required to be oblivious (i.e., data-independent). Here, we introduce a model in which head movements are adaptive (i.e., data-dependent) and compare it to the oblivious model. We establish that for each , adaptivity enhances the predictive power of -head finite-state gamblers, in the sense that there are sequences whose oblivious -head finite-state predimensions strictly exceed their adaptive -head finite-state predimensions. We further prove that adaptive finite-state predimensions admit a strict hierarchy as the number of heads increases, and in fact that for all there is a sequence whose adaptive -head finite-state predimension is strictly less than its adaptive -head predimension.
Paper Structure (15 sections, 11 theorems, 107 equations)

This paper contains 15 sections, 11 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Adaptive Multi-Head Finite-State Gamblers
  3. String and Sequence Notation
  4. Adaptive Multi-Head Finite-State Gamblers
  5. Oblivious Multi-Head Finite-State Gamblers
  6. Algorithmic Information Preliminaries
  7. $s$-Gales and Martingales
  8. Kolmogorov Complexity and Martin-Löf Randomness
  9. Using Kolmogorov Complexity to Bound Martingales
  10. Adaptive Multi-Head Finite-State Dimensions
  11. Defining our Family of Sequences
  12. Upper Bound for Adaptive Multi-Head Finite-State Predimensions
  13. Lower Bound for Oblivious Multi-Head Finite-State Predimensions
  14. A Hierarchy of Adaptive Predimensions
  15. Conclusion

Key Result

Lemma 3.1

Let $\Sigma$ be a finite, non-unary alphabet, $h\geq 1$, $G$ be any adaptive or oblivious $h$-head finite-state gambler over $\Sigma$, $\alpha,\delta\in(0,1)\cap\mathbb{Q}$, $S\in\Sigma^\omega$, $m,n\in\mathbb{N}$, and $U$ any superset of the positions of the trailing heads while the leading head re If $n-m$ is sufficiently large and $K(S[[m,n]]\mid S[U])\ge (1-\alpha+\delta)(n-m)\log|\Sigma|$, th

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:dtok']}
  • Definition 5: mhfsd
  • Theorem 6.1
  • Lemma 6.2
  • proof
  • ...and 14 more