Circuit Complexity of Hierarchical Knowledge Tracing and Implications for Log-Precision Transformers

• URL: http://arxiv.org/abs/2603.23823v1
• Date: Wed, 25 Mar 2026 01:10:42 GMT
• Title: Circuit Complexity of Hierarchical Knowledge Tracing and Implications for Log-Precision Transformers
• Abstract summary: Empirically, transformer encoders trained on recursion-invariant trees converge to permutation-invariant shortcuts.<n>These findings motivate structure-aware objectives and iterative mechanisms for prerequisite-sensitive knowledge tracing on deep hierarchies.
• Score: 2.688207424884465
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Knowledge tracing models mastery over interconnected concepts, often organized by prerequisites. We analyze hierarchical prerequisite propagation through a circuit-complexity lens to clarify what is provable about transformer-style computation on deep concept hierarchies. Using recent results that log-precision transformers lie in logspace-uniform $\mathsf{TC}^0$, we formalize prerequisite-tree tasks including recursive-majority mastery propagation. Unconditionally, recursive-majority propagation lies in $\mathsf{NC}^1$ via $O(\log n)$-depth bounded-fanin circuits, while separating it from uniform $\mathsf{TC}^0$ would require major progress on open lower bounds. Under a monotonicity restriction, we obtain an unconditional barrier: alternating ALL/ANY prerequisite trees yield a strict depth hierarchy for \emph{monotone} threshold circuits. Empirically, transformer encoders trained on recursive-majority trees converge to permutation-invariant shortcuts; explicit structure alone does not prevent this, but auxiliary supervision on intermediate subtrees elicits structure-dependent computation and achieves near-perfect accuracy at depths 3--4. These findings motivate structure-aware objectives and iterative mechanisms for prerequisite-sensitive knowledge tracing on deep hierarchies.

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