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Ehrenfeucht-Haussler Rank and Chain of Thought

22 January 2025
Pablo Barceló
Alexander Kozachinskiy
Tomasz Steifer
    LRM
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Abstract

The notion of rank of a Boolean function has been a cornerstone in the theory of PAC learning, enabling quasipolynomial-time learning algorithms for polynomial-size decision trees. We present a novel characterization of rank, grounded in the well-known Transformer architecture. We show that the rank of a function fff corresponds to the minimum number of Chain of Thought (CoT) steps required by a single-layer transformer decoder with hard attention to compute fff. Based on this characterization we establish tight bounds on the number of CoT steps required for specific problems, showing that ℓ\ellℓ-fold function composition necessitates exactly ℓ\ellℓ CoT steps. Furthermore, we analyze the problem of identifying the position of the kkk-th occurrence of 1 in a Boolean sequence, proving that it requires kkk CoT steps.

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