Adaptive Multi-Head Finite-State Gamblers

2026-03-16

AI Summary

This study addresses the challenge of more effectively quantifying the predictability of sequences exhibiting simple yet non-local patterns. To this end, it proposes an adaptive multi-head finite-state gambler model, wherein the movement of read-write heads is dynamically guided by the input data, thereby enhancing the model’s capacity to capture non-local regularities. The core innovation lies in the first introduction of a data-dependent adaptive head movement mechanism. Theoretical analysis establishes that: (1) the proposed model strictly outperforms conventional oblivious models in predictive power; (2) for any $h \geq 2$, there exist sequences whose $h$-head predictive dimension under the adaptive model is strictly lower than under the oblivious counterpart; and (3) for any $h \geq 1$, there are sequences for which the $(h+1)$-head adaptive predictive dimension is strictly less than the $h$-head case, revealing a strict hierarchy in predictive dimension as the number of heads increases in the adaptive setting.

Technology Category

• Cognitive Modeling & Cognitive Systems: Adaptive Behavior
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• finite-state gamblers
• adaptive head movement
• sequence predictability
• multi-head model
• predimension

Innovation

• adaptive head movement
• finite-state dimension
• multi-head gambler
• sequence predictability
• predimension hierarchy