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

2026-03-24

AI Summary

This study addresses the problem of modeling recursive mastery propagation under deep prerequisite structures in knowledge tracing, introducing circuit complexity theory—specifically the classes NC¹ and TC⁰—into this domain for the first time. By analyzing the recursive majority task, the authors theoretically establish that it resides in NC¹ yet resists uniform TC⁰ simulation, while further revealing a depth hierarchy of monotone threshold circuits within alternating ALL/ANY prerequisite trees. Empirically, they demonstrate that standard log-precision Transformers tend to exploit permutation-invariant shortcuts; however, incorporating structure-aware subtree auxiliary supervision substantially enhances their ability to capture hierarchical dependencies, achieving near-perfect accuracy on tasks with depth 3–4.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• knowledge tracing
• hierarchical prerequisites
• circuit complexity
• recursive-majority
• transformer limitations

Innovation

• circuit complexity
• hierarchical knowledge tracing
• log-precision transformers
• recursive-majority propagation
• monotone threshold circuits