One Adaptive Trailing Head Can Outperform Many Oblivious Trailing Heads

2026-05-28

AI Summary

This study investigates the advantage of adaptive trailing heads over arbitrarily many non-adaptive trailing heads in sequence predictability within the framework of finite-state strong dimension. By integrating multi-head finite-state automata, finite-state dimension theory, and information-theoretic analysis, the authors construct a binary sequence for which the strong dimension under an adaptive two-head model is at least 0.3 lower than that under any non-adaptive multi-head model. This result demonstrates—by a substantial and consistent margin—that even a single adaptive trailing head can outperform any number of non-adaptive heads employing fixed strategies. The finding strengthens and extends existing dimension separation results, highlighting the fundamental superiority of adaptive strategies in finite-state prediction.

Abstract

In the setting of multi-head finite-state dimensions, trailing heads lag behind a leading head, accessing past data to aid a finite-state gambler placing bets on successive bits read by the leading head. Cruz, Glashausser, Li, and Lutz (2026) proved that, for any fixed number of trailing heads, adaptive (data-dependent) movement rules can strictly outperform oblivious (data-independent) movement schedules. In this paper we strengthen that separation by proving that a single trailing head with adaptive movements can outperform, by a large and uniform margin, arbitrarily many trailing heads with oblivious movements. Formally, our main theorem states that there is a binary sequence whose adaptive two-head finite-state strong dimension is less than its oblivious multi-head finite-state dimension, and that the gap is greater than 0.3.

Problem

• adaptive trailing head
• oblivious trailing heads
• finite-state dimension
• multi-head
• strong dimension

Innovation

• adaptive trailing head
• oblivious movement
• finite-state dimension
• multi-head automata
• strong dimension