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Resolve the TC^0 versus NC^1 circuit complexity conjecture

Establish whether the constant-depth threshold circuit class TC^0 is strictly different from the log-depth circuit class NC^1; equivalently, determine whether TC^0 ≠ NC^1 holds in circuit complexity theory.

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Background

The paper’s main optimization results rely on the widely held belief that TC0 is strictly weaker than NC1. Under this conjecture, showing that constant-depth transformers can learn NC1-complete tasks via chain-of-thought reasoning demonstrates capability beyond TC0.

TC0 consists of constant-depth, polynomial-size circuits with threshold gates, while NC1 consists of log-depth, polynomial-size circuits with bounded fan-in. Proving or refuting the strict separation TC0 ≠ NC1 is a central open problem in circuit complexity with broad implications for parallel versus inherently sequential computation.

References

The first two results further establish that the model can learn the solution of state-tracking problems for non-solvable groups, which is \mathsf{NC}1-complete and lies outside \mathsf{TC}0 unless the widely held conjecture \mathsf{TC}0 \neq \mathsf{NC}1 in circuit complexity theory fails.

— Transformers Provably Learn Chain-of-Thought Reasoning with Length Generalization (2511.07378 - Huang et al., 10 Nov 2025) in Introduction, Item 3 in Contributions