# Rethinking Circuit Completeness in Language Models: AND, OR, and ADDER Gates

arXiv: https://arxiv.org/abs/2505.10039 · HTML: https://mlsanity.com/paper/2505.10039/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 23 of its 26 references that are in the map.
- First posted on arXiv 15 May 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)
- Mechanistic interpretability / [Circuits and mechanistic analysis](https://mlsanity.com/mechinterp/circuits-and-mechanistic-analysis/)

## Builds on (most cited first, 23 of 23)

- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [BeaverTails: Towards Improved Safety Alignment of LLM via a Human-Preference Dataset](https://mlsanity.com/paper/2307.04657/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models](https://mlsanity.com/paper/2403.19647/) (2024)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [How to use and interpret activation patching](https://mlsanity.com/paper/2404.15255/) (2024)
- [Attribution Patching Outperforms Automated Circuit Discovery](https://mlsanity.com/paper/2310.10348/) (2023)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [Localizing Model Behavior with Path Patching](https://mlsanity.com/paper/2304.05969/) (2023)
- [Does Circuit Analysis Interpretability Scale? Evidence from Multiple Choice Capabilities in Chinchilla](https://mlsanity.com/paper/2307.09458/) (2023)
- [Comparing Kullback-Leibler Divergence and Mean Squared Error Loss in Knowledge Distillation](https://mlsanity.com/paper/2105.08919/) (2021)
- [The Evolution of Statistical Induction Heads: In-Context Learning Markov Chains](https://mlsanity.com/paper/2402.11004/) (2024)
- [A Practical Review of Mechanistic Interpretability for Transformer-Based Language Models](https://mlsanity.com/paper/2407.02646/) (2024)
- [Knowledge Circuits in Pretrained Transformers](https://mlsanity.com/paper/2405.17969/) (2024)
- [Finding Transformer Circuits with Edge Pruning](https://mlsanity.com/paper/2406.16778/) (2024)
- [Identifying Semantic Induction Heads to Understand In-Context Learning](https://mlsanity.com/paper/2402.13055/) (2024)
- [Induction Heads as an Essential Mechanism for Pattern Matching in In-context Learning](https://mlsanity.com/paper/2407.07011/) (2024)
- [Sparse Interventions in Language Models with Differentiable Masking](https://mlsanity.com/paper/2112.06837/) (2021)
- [Efficient Automated Circuit Discovery in Transformers using Contextual Decomposition](https://mlsanity.com/paper/2407.00886/) (2024)
- [Missed Causes and Ambiguous Effects: Counterfactuals Pose Challenges for Interpreting Neural Networks](https://mlsanity.com/paper/2407.04690/) (2024)
- [Discovering Knowledge-Critical Subnetworks in Pretrained Language Models](https://mlsanity.com/paper/2310.03084/) (2023)
- [Position-aware Automatic Circuit Discovery](https://mlsanity.com/paper/2502.04577/) (2025)

## Built on by (most cited first, 10 of 10)

- [Conditional Co-Ablation: Recovering Self-Repair Backups in Transformer Circuits](https://mlsanity.com/paper/2607.01940/) (2026)
- [Addressing divergent representations from causal interventions on neural networks](https://mlsanity.com/paper/2511.04638/) (2025)
- [Discovering Transformer Circuits via a Hybrid Attribution and Pruning Framework](https://mlsanity.com/paper/2510.03282/) (2025)
- [Navigating by Old Maps: The Pitfalls of Static Mechanistic Localization in LLM Post-Training](https://mlsanity.com/paper/2605.06076/) (2026)
- [Beyond Transfer Accuracy: Mechanism-Guided Controlled Adaptation for Low-Resource Languages](https://mlsanity.com/paper/2601.08146/) (2026)
- [Can Graph Learning Learn Circuits?](https://mlsanity.com/paper/2608.08536/) (2026)
- [Beyond Static Interpretability: Anticipating Post-SFT Mechanisms from Pre-SFT Parameters for Better Tuning](https://mlsanity.com/paper/2608.24482/) (2026)
- [RECLAIM: Can Agents Reproduce the Claims of Machine Learning Papers?](https://mlsanity.com/paper/2609.28850/) (2026)
- [Every Ablation Is a Dose: Counterweights and the Semblance of Self-Repair](https://mlsanity.com/paper/2610.02173/) (2026)
- [Slaying the Hydra: Interaction-Aware Circuit Discovery in Language Models](https://mlsanity.com/paper/2610.04017/) (2026)

## Abstract (arXiv)

Circuit discovery has gradually become one of the prominent methods for mechanistic interpretability, and research on circuit completeness has also garnered increasing attention. Methods of circuit discovery that do not guarantee completeness not only result in circuits that are not fixed across different runs but also cause key mechanisms to be omitted. The nature of incompleteness arises from the presence of OR gates within the circuit, which are often only partially detected in standard circuit discovery methods. To this end, we systematically introduce three types of logic gates: AND, OR, and ADDER gates, and decompose the circuit into combinations of these logical gates. Through the concept of these gates, we derive the minimum requirements necessary to achieve faithfulness and completeness. Furthermore, we propose a framework that combines noising-based and denoising-based interventions, which can be easily integrated into existing circuit discovery methods without significantly increasing computational complexity. This framework is capable of fully identifying the logic gates and distinguishing them within the circuit. In addition to the extensive experimental validation of the framework's ability to restore the faithfulness, completeness, and sparsity of circuits, using this framework, we uncover fundamental properties of the three logic gates, such as their proportions and contributions to the output, and explore how they behave among the functionalities of language models.