# K-MSHC: Unmasking Minimally Sufficient Head Circuits in Large Language Models with Experiments on Syntactic Classification Tasks

arXiv: https://arxiv.org/abs/2505.12268 · HTML: https://mlsanity.com/paper/2505.12268/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Circuits and mechanistic analysis, one of 13 directions in Mechanistic interpretability.
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 18 May 2025.

## Placed in

- Mechanistic interpretability / [Circuits and mechanistic analysis](https://mlsanity.com/mechinterp/circuits-and-mechanistic-analysis/)

## Builds on (most cited first, 22 of 22)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Analyzing Multi-Head Self-Attention: Specialized Heads Do the Heavy Lifting, the Rest Can Be Pruned](https://mlsanity.com/paper/1905.09418/) (2019)
- [Are Sixteen Heads Really Better than One?](https://mlsanity.com/paper/1905.10650/) (2019)
- [BLiMP: The Benchmark of Linguistic Minimal Pairs for English](https://mlsanity.com/paper/1912.00582/) (2019)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models](https://mlsanity.com/paper/2403.19647/) (2024)
- [Transcoders Find Interpretable LLM Feature Circuits](https://mlsanity.com/paper/2406.11944/) (2024)
- [Do Large Language Models Latently Perform Multi-Hop Reasoning?](https://mlsanity.com/paper/2402.16837/) (2024)
- [Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics](https://mlsanity.com/paper/2410.21272/) (2024)
- [A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis](https://mlsanity.com/paper/2305.15054/) (2023)
- [Language Models Use Trigonometry to Do Addition](https://mlsanity.com/paper/2502.00873/) (2025)
- [Evidence of Learned Look-Ahead in a Chess-Playing Neural Network](https://mlsanity.com/paper/2406.00877/) (2024)
- [The Same But Different: Structural Similarities and Differences in Multilingual Language Modeling](https://mlsanity.com/paper/2410.09223/) (2024)
- [Automatically Identifying Local and Global Circuits with Linear Computation Graphs](https://mlsanity.com/paper/2405.13868/) (2024)
- [Planning in a recurrent neural network that plays Sokoban](https://mlsanity.com/paper/2407.15421/) (2024)
- [The Computational Complexity of Circuit Discovery for Inner Interpretability](https://mlsanity.com/paper/2410.08025/) (2024)
- [Interpreting Emergent Planning in Model-Free Reinforcement Learning](https://mlsanity.com/paper/2504.01871/) (2025)

## Built on by (most cited first, 2 of 2)

- [Formal Mechanistic Interpretability: Automated Circuit Discovery with Provable Guarantees](https://mlsanity.com/paper/2602.16823/) (2026)
- [CircuitTree: Composing Circuits into A Syllable Transcriber](https://mlsanity.com/paper/2610.12520/) (2026)

## Abstract (arXiv)

Understanding which neural components drive specific capabilities in mid-sized language models (≤10B parameters) remains a key challenge. We introduce the (bmK, ε)-Minimum Sufficient Head Circuit (K-MSHC), a methodology to identify minimal sets of attention heads crucial for classification tasks as well as Search-K-MSHC, an efficient algorithm for discovering these circuits. Applying our Search-K-MSHC algorithm to Gemma-9B, we analyze three syntactic task families: grammar acceptability, arithmetic verification, and arithmetic word problems. Our findings reveal distinct task-specific head circuits, with grammar tasks predominantly utilizing early layers, word problems showing pronounced activity in both shallow and deep regions, and arithmetic verification demonstrating a more distributed pattern across the network. We discover non-linear circuit overlap patterns, where different task pairs share computational components at varying levels of importance. While grammar and arithmetic share many "weak" heads, arithmetic and word problems share more consistently critical "strong" heads. Importantly, we find that each task maintains dedicated "super-heads" with minimal cross-task overlap, suggesting that syntactic and numerical competencies emerge from specialized yet partially reusable head circuits.