Ehrenfeucht-Haussler Rank and Chain of Thought

The notion of rank of a Boolean function has been a cornerstone in PAC learning theory, 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 f corresponds to the minimum number of Chain of Thought (CoT) steps required by a single-layer Transformer with hard attention to compute f . Based on this characterization we establish tight bounds on the number of CoT steps required for specific problems, showing that ℓ-fold function composition necessitates exactly ℓ CoT steps. Furthermore, we analyze the problem of identifying the position of the k-th occurrence of 1 in a Boolean sequence, proving that it requires k CoT steps. Finally, we introduce the notion of the multi-head rank that captures multi-head single-layer transformers, and perform the analysis of PAC-learnability of the classes of functions with bounded multi-head rank.