Chain-of-Thought Shows the Path to a Tree: Realizing Branching Complexity

The increasing popularity of \emph{reasoning} models -- language models that output a series of reasoning or thought tokens before producing an answer -- is justified, in part, by theoretical results showing that chain-of-thought (CoT) transformers can simulate Turing machines, and thus perform arbitrary computation. However, the Turing machine, while suitable for complexity-theoretic analysis, is not convenient, intuitive, or efficient for discussing algorithms. Algorithms are typically designed and analyzed at a higher level of abstraction, captured by the \emph{Word RAM} model with random-access memory and unit-cost operations on $\bigO(\log n)$-bit words. As a result, Word RAM algorithms can be substantially more efficient than their Turing machine counterparts, raising the question: \emph{Can CoT transformers efficiently simulate Word RAM algorithms?} For instance, can they sort $n$ items in $\bigO(n \log n)$ steps or run Dijkstra's algorithm in $\bigO(E + V \log V)$ steps? We answer affirmatively, up to poly-logarithmic overhead. We first establish this for finite-precision transformers with poly-logarithmic width and rightmost unique hard attention, then strengthen the result to two more practical settings with finite width and log-precision: \emph{continuous} CoT, where reasoning takes the form of vectors rather than tokens, and a \emph{hybrid} architecture in which transformer layers sit atop a recurrent (linear RNN) layer. In all three cases, we find that CoT \emph{can} efficiently simulate any Word RAM algorithm with only a poly-logarithmic overhead in $n$. This overhead reduces to log-square when the Word RAM has a ``flat'' instruction set, and only logarithmic for multiplication-free flat instructions -- in stark contrast to known CoT simulations of Turing machines, which require quadratic overhead over Word RAM.

We develop a learning-theoretic framework for understanding Chain of Thought (CoT). We model CoT as the interaction between an answer map and a chain rule that generates intermediate questions autoregressively, and define the reasoning risk of a hypothesis under this interaction. Our first result is a tight canonical decomposition of this risk into two terms with opposing roles: an oracle-trajectory risk (OTR), which captures the benefit of CoT and reduces to a target-domain risk in a domain adaptation problem, and a trajectory-mismatch risk (TMR), which captures the cost of CoT through error accumulation along mismatched reasoning trajectories. We then show that this cost is unavoidable without structure: if any one of the loss, the hypothesis answer map, or the chain rule lacks stability, the TMR can be arbitrarily large even when the OTR is zero and the hypothesis is uniformly close to the ground truth. Conversely, under stability, we prove a tight upper bound on the TMR governed by an exact amplification factor that identifies bounded, linear, and exponential error-growth regimes. Together, these results give a precise theory of when CoT helps, when it hurts, and what controls the transition between the two.

We tightly characterize the VC dimension of depth-$L$ Transformers with a total of $W$ parameters, mapping an input sequence of length $T$ to a single output, establishing an upper bound of $O(L W \log (T W))$ and a nearly matching lower bound of $Ω(L W \log (T W / L))$. We further tightly characterize the sample complexity of chain-of-thought learning using such a Transformer, showing teacher forcing (i.e. selecting a predictor consistent with the entire chain-of-thought on training data) learns with sample complexity $O\left(L W \log \left(\left(T+T^{\prime}\right) W\right)\right)$ and that any learning rule that uses chain-of-thought data requires at least $Ω\left(L W \log \left(\left(T+T^{\prime}\right) W / L\right)\right)$ examples, where $T$ is the input length and $T^{\prime}$ is the number of autoregressive steps.