---
title: "Physics of Language Models: Part 3.2, Knowledge Manipulation"
description: "The authors test whether language models can do more than store factual… 100% accuracy. 39 references, cited by 64 papers here."
authors: ["Zeyuan Allen-Zhu", "Yuanzhi Li"]
year: 2023
venue: "arXiv:2309.14402"
source: "https://arxiv.org/abs/2309.14402"
doi: "10.48550/arxiv.2309.14402"
arxiv: "2309.14402"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/allen-zhu-2023-physics-language-models-part-3-2/"
---
[Elamur Research](https://md.research.elamur.ai/free/en/) · [this page in HTML](https://research.elamur.ai/free/en/papers/allen-zhu-2023-physics-language-models-part-3-2/)

[home](https://md.research.elamur.ai/free/en/) / [papers](https://md.research.elamur.ai/free/en/papers/) / [2023](https://md.research.elamur.ai/free/en/papers/2023/page/9/) / Physics of Language Models: Part 3.2, Knowledge Manipulation

# Physics of Language Models: Part 3.2, Knowledge Manipulation

**Authors:** [Zeyuan Allen-Zhu](https://md.research.elamur.ai/free/en/authors/zeyuan-allen-zhu/) and [Yuanzhi Li](https://md.research.elamur.ai/free/en/authors/yuanzhi-li/) · **Year:** 2023 · **Venue:** arXiv:2309.14402 · **Licence:** [arXiv — non-exclusive licence to distribute](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2309.14402) · [PDF](https://arxiv.org/pdf/2309.14402) · [DOI 10.48550/arxiv.2309.14402](https://doi.org/10.48550/arxiv.2309.14402) · [arXiv 2309.14402](https://arxiv.org/abs/2309.14402) · [OpenAlex](https://openalex.org/W4410357125)

**Citations:** 8 citations (OpenAlex, 2026-10-14) · [64](https://md.research.elamur.ai/free/en/papers/allen-zhu-2023-physics-language-models-part-3-2/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary

The authors test whether language models can do more than store factual knowledge: can they also use it flexibly at inference time? They consider four basic knowledge operations. The first is attribute retrieval. The second is classification (for example, whether a birth year is even). The third is comparing two people on an attribute. The fourth is inverse search, meaning finding a person from an attribute value. To rule out data contamination, they use a synthetic, augmented corpus of biographies of fictional people. The model sees the biographies of all individuals but sees question–answer pairs for only some of them. Accuracy is measured on the remaining individuals (OOD). Direct retrieval works well. Without Chain of Thought (CoT) at both the training and inference stages, classification and comparison barely beat random guessing, even though the model remembers the facts themselves perfectly. Inverse search gives near-zero accuracy in every setting. The authors found the same weaknesses in GPT-4 and Llama-3. The conclusion for practitioners is that a language model should not be used as a database, and it can manipulate knowledge only when it explicitly spells out the intermediate facts.

## Key points

- The data are synthetic biographies of up to N = 20 million individuals, with birth date and city, university, major, company, and work city. Individuals are split into P_train (with QA) and P_test (biographies only). Order matters: the model can name the company and then the city, but it makes errors in the reverse order. It reproduces the full birth date, but does worse on the year alone.
- Without CoT, the model cannot determine whether the birth month is even. This holds even after 25,000 QA examples and with 100% accuracy at retrieving the month. Comparison among 100 options stays at the level of random guessing even with 2.5 million training examples.
- Adding CoT examples to training does not improve answers without CoT. Better knowledge retrieval does not improve the ability to manipulate that knowledge either.
- Inverse search gives ≈0% regardless of model size, data volume, and prompts. The exception is when the knowledge already appears in the data in the reverse direction. Other researchers have called a similar phenomenon the "reversal curse."
- Tests on real models used 4,779 individuals from WikiBio for GPT-3.5/GPT-4 (forward and inverse search, parity, comparison of birth dates). For inverse search, the authors also used 2,244 four-character Chinese idioms and 233 pairs of lines from Chinese poems.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Meta](https://md.research.elamur.ai/free/en/companies/meta/) — Company, Menlo Park, United States · 214 works · 241,492 citations
- [Mohamed bin Zayed University of Artificial Intelligence](https://md.research.elamur.ai/free/en/universities/mohamed-bin-zayed-university-of-artificial-intelligence/) — University, Abu Dhabi, United Arab Emirates · 138 works · 6,590 citations

📚 Shelves (6): [🖥️ Computer Science](https://md.research.elamur.ai/free/en/shelves/computer-science/) · 18,614 works · [🤖 Artificial Intelligence](https://md.research.elamur.ai/free/en/shelves/computer-science/artificial-intelligence/) · 17,965 works · [🎭 Arts and Humanities](https://md.research.elamur.ai/free/en/shelves/arts-and-humanities/) · 17,450 works · [🗣️ Language and Linguistics](https://md.research.elamur.ai/free/en/shelves/arts-and-humanities/language-and-linguistics/) · 17,296 works · [🧠 Neuroscience](https://md.research.elamur.ai/free/en/shelves/neuroscience/) · 186 works · [🧠 Cognitive Neuroscience](https://md.research.elamur.ai/free/en/shelves/neuroscience/cognitive-neuroscience/) · 185 works

🧭 Topics (1): [🧠 Language models & pretraining](https://md.research.elamur.ai/free/en/topics/language-models/) · 3,894 works

## Cite

Cite as: Zeyuan Allen-Zhu and Yuanzhi Li. 2023. Physics of Language Models: Part 3.2, Knowledge Manipulation. arXiv:2309.14402.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{allenzhu2023physics,
  title = {{Physics of Language Models: Part 3.2, Knowledge Manipulation}},
  author = {Zeyuan Allen-Zhu and Yuanzhi Li},
  year = {2023},
  doi = {10.48550/arxiv.2309.14402},
  eprint = {2309.14402},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2309.14402}
}
```

- [← previous Chain-of-Thought Hub: A Continuous Effort to Measure Large Language Models' Reasoning Performance](https://md.research.elamur.ai/free/en/papers/fu-2023-chain-thought-hub-continuous-effort/)
- [next → Skeleton-of-Thought: Prompting LLMs for Efficient Parallel Generation](https://md.research.elamur.ai/free/en/papers/ning-2023-skeleton-thought-prompting-llms-efficient/)
- [↑ back to papers of 2023](https://md.research.elamur.ai/free/en/papers/2023/page/9/)

## References

39 references, 21 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

1. Zeyuan Allen-Zhu and Yuanzhi Li. Physics of Language Models: Part 1, Learning Hierarchical Language Structures. ArXiv e-prints, abs/2305.13673, May 2023. Full version available at http://arxiv.org/abs/2305.13673. [↗](http://arxiv.org/abs/2305.13673)
2. Zeyuan Allen-Zhu and Yuanzhi Li. [Physics of Language Models: Part 3.1, Knowledge Storage and Extraction](https://md.research.elamur.ai/free/en/papers/allen-zhu-2023-physics-language-models-part-3/). In ICML, 2024. Full version available at http://arxiv.org/abs/2309.14316. [↗](http://arxiv.org/abs/2309.14316)
3. Zeyuan Allen-Zhu and Yuanzhi Li. Physics of Language Models: Part 3.3, Knowledge Capacity Scaling Laws. ArXiv e-prints, abs/2404.05405, April 2024. Full version available at http://arxiv.org/abs/2404.05405. [↗](http://arxiv.org/abs/2404.05405)
4. Lukas Berglund, Meg Tong, Max Kaufmann, Mikita Balesni, Asa Cooper Stickland, Tomasz Korbak, and Owain Evans. The Reversal Curse: LLMs trained on ”A is B” fail to learn ”B is A”. arXiv preprint arXiv:2309.12288, September 2023. [find](https://scholar.google.com/scholar?q=Lukas+Berglund%2C+Meg+Tong%2C+Max+Kaufmann%2C+Mikita+Balesni%2C+Asa+Cooper+Stickland%2C+Tomasz+Korbak%2C+and+Owain+Evans%2E+The+Reversal+Curse%3A+LLMs+trained+on+%E2%80%9DA+is+B%E2%80%9D+fail+to+learn+%E2%80%9DB+is+A%E2%80%9D%2E+arXiv+preprint+arXiv%3A)
5. Sid Black, Stella Biderman, Eric Hallahan, Quentin Anthony, Leo Gao, Laurence Golding, Horace He, Connor Leahy, Kyle McDonell, Jason Phang, Michael Pieler, USVSN Sai Prashanth, Shivanshu Purohit, Laria Reynolds, Jonathan Tow, Ben Wang, and Samuel Weinbach. GPT-NeoX-20B: An open-source autoregressive… — [GPT-NeoX-20B: An Open-Source Autoregressive Language Model](https://md.research.elamur.ai/free/en/papers/black-2022-gpt-neox-20b-open-source/)
6. Deng Cai, Yan Wang, Lemao Liu, and Shuming Shi. Recent advances in retrieval-augmented text generation. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 3417–3419, 2022. [find](https://scholar.google.com/scholar?q=Deng+Cai%2C+Yan+Wang%2C+Lemao+Liu%2C+and+Shuming+Shi%2E+Recent+advances+in+retrieval%2Daugmented+text+generation%2E+In+Proceedings+of+the+45th+International+ACM+SIGIR+Conference+on+Research+and+Development+in+Inf)
7. Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, Dan Roth, and Jonathan Berant. Did aristotle use a laptop? a question answering benchmark with implicit reasoning strategies. Transactions of the Association for Computational Linguistics, 9:346–361, 2021. [find](https://scholar.google.com/scholar?q=Mor+Geva%2C+Daniel+Khashabi%2C+Elad+Segal%2C+Tushar+Khot%2C+Dan+Roth%2C+and+Jonathan+Berant%2E+Did+aristotle+use+a+laptop%3F+a+question+answering+benchmark+with+implicit+reasoning+strategies%2E+Transactions+of+the+As)
8. Fabian Gloeckle, Badr Youbi Idrissi, Baptiste Rozière, David Lopez-Paz, and Gabriel Synnaeve. [Better & faster large language models via multi-token prediction](https://md.research.elamur.ai/free/en/papers/gloeckle-2024-better-faster-large-language-models/). arXiv preprint arXiv:2404.19737, 2024.
9. Olga Golovneva, Zeyuan Allen-Zhu, Jason Weston, and Sainbayar Sukhbaatar. Reverse training to nurse the reversal curse. arXiv preprint arXiv:2403.13799, 2024. — [arXiv:2403.13799](https://arxiv.org/abs/2403.13799) [find](https://scholar.google.com/scholar?q=Olga+Golovneva%2C+Zeyuan+Allen%2DZhu%2C+Jason+Weston%2C+and+Sainbayar+Sukhbaatar%2E+Reverse+training+to+nurse+the+reversal+curse%2E+arXiv+preprint+arXiv%3A2403%2E13799%2C+2024%2E+%E2%80%94+%5BarXiv%3A2403%2E13799%5D%28https%3A%2F%2Farxiv%2Eorg%2Fab)
10. Qingyan Guo, Rui Wang, Junliang Guo, Xu Tan, Jiang Bian, and Yujiu Yang. Mitigating reversal curse via semantic-aware permutation training. arXiv preprint arXiv:2403.00758, 2024. — [arXiv:2403.00758](https://arxiv.org/abs/2403.00758) [find](https://scholar.google.com/scholar?q=Qingyan+Guo%2C+Rui+Wang%2C+Junliang+Guo%2C+Xu+Tan%2C+Jiang+Bian%2C+and+Yujiu+Yang%2E+Mitigating+reversal+curse+via+semantic%2Daware+permutation+training%2E+arXiv+preprint+arXiv%3A2403%2E00758%2C+2024%2E+%E2%80%94+%5BarXiv%3A2403%2E00758%5D%28)
11. Evan Hernandez, Belinda Z Li, and Jacob Andreas. Measuring and manipulating knowledge representations in language models. arXiv preprint arXiv:2304.[00740, 2023](https://md.research.elamur.ai/free/en/papers/hernandez-2023-inspecting-editing-knowledge-representations-language/).
12. Edward J Hu, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, Weizhu Chen, et al. [LoRA: Low-Rank Adaptation of Large Language Models](https://md.research.elamur.ai/free/en/papers/hu-2021-lora-low-rank-adaptation-large/). In ICLR, 2021.
13. Albert Q Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, et al. Mistral 7b. arXiv preprint arXiv:2310.06825, 2023. [find](https://scholar.google.com/scholar?q=Albert+Q+Jiang%2C+Alexandre+Sablayrolles%2C+Arthur+Mensch%2C+Chris+Bamford%2C+Devendra+Singh+Chaplot%2C+Diego+de+las+Casas%2C+Florian+Bressand%2C+Gianna+Lengyel%2C+Guillaume+Lample%2C+Lucile+Saulnier%2C+et+al%2E+Mistral+7b)
14. Zhengbao Jiang, Frank F Xu, Luyu Gao, Zhiqing Sun, Qian Liu, Jane Dwivedi-Yu, Yiming Yang, Jamie Callan, and Graham Neubig. [Active retrieval augmented generation](https://md.research.elamur.ai/free/en/papers/jiang-2023-active-retrieval-augmented-generation/). arXiv preprint arXiv:2305.06983, 2023.
15. Mojtaba Komeili, Kurt Shuster, and Jason Weston. [Internet-augmented dialogue generation](https://md.research.elamur.ai/free/en/papers/komeili-2022-internet-augmented-dialogue-generation/). arXiv preprint arXiv:2107.07566, 2021.
16. Rémi Lebret, David Grangier, and Michael Auli. Generating text from structured data with application to the biography domain. CoRR, abs/1603.[07771, 2016](https://md.research.elamur.ai/free/en/papers/lebret-2016-neural-text-generation-structured-data/). URL http://arxiv.org/abs/1603.07771. [↗](http://arxiv.org/abs/1603.07771)
17. Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, Sebastian Riedel, and Douwe Kiela. [Retrieval-augmented generation for knowledge-intensive nlp tasks](https://md.research.elamur.ai/free/en/papers/lewis-2020-retrieval-augmented-generation-knowledge-intensive/). In H. Larochelle, M. Ranzato, R. Hadsell,…
18. Shangqing Liu, Yu Chen, Xiaofei Xie, Jingkai Siow, and Yang Liu. Retrieval-augmented generation for code summarization via hybrid gnn. arXiv preprint arXiv:2006.05405, 2020. — [arXiv:2006.05405](https://arxiv.org/abs/2006.05405) [find](https://scholar.google.com/scholar?q=Shangqing+Liu%2C+Yu+Chen%2C+Xiaofei+Xie%2C+Jingkai+Siow%2C+and+Yang+Liu%2E+Retrieval%2Daugmented+generation+for+code+summarization+via+hybrid+gnn%2E+arXiv+preprint+arXiv%3A2006%2E05405%2C+2020%2E+%E2%80%94+%5BarXiv%3A2006%2E05405%5D%28https)
19. Yuning Mao, Pengcheng He, Xiaodong Liu, Yelong Shen, Jianfeng Gao, Jiawei Han, and Weizhu Chen. [Generation-augmented retrieval for open-domain question answering](https://md.research.elamur.ai/free/en/papers/mao-2020-generation-augmented-retrieval-open-domain/). arXiv preprint arXiv:2009.08553, 2020.
20. Tahira Naseem, Srinivas Ravishankar, Nandana Mihindukulasooriya, Ibrahim Abdelaziz, Young-Suk Lee, Pavan Kapanipathi, Salim Roukos, Alfio Gliozzo, and Alexander Gray. A semantics-aware transformer model of relation linking for knowledge base question answering. In Proceedings of the 59th Annual Meet… [find](https://scholar.google.com/scholar?q=Tahira+Naseem%2C+Srinivas+Ravishankar%2C+Nandana+Mihindukulasooriya%2C+Ibrahim+Abdelaziz%2C+Young%2DSuk+Lee%2C+Pavan+Kapanipathi%2C+Salim+Roukos%2C+Alfio+Gliozzo%2C+and+Alexander+Gray%2E+A+semantics%2Daware+transformer+mod)
21. Anh Nguyen, Nikos Karampatziakis, and Weizhu Chen. Meet in the middle: A new pre-training paradigm. Advances in Neural Information Processing Systems, 36, 2024. [find](https://scholar.google.com/scholar?q=Anh+Nguyen%2C+Nikos+Karampatziakis%2C+and+Weizhu+Chen%2E+Meet+in+the+middle%3A+A+new+pre%2Dtraining+paradigm%2E+Advances+in+Neural+Information+Processing+Systems%2C+36%2C+2024%2E)
22. Reham Omar, Omij Mangukiya, Panos Kalnis, and Essam Mansour. Chatgpt versus traditional question answering for knowledge graphs: Current status and future directions towards knowledge graph chatbots. arXiv preprint arXiv:2302.06466, 2023. — [arXiv:2302.06466](https://arxiv.org/abs/2302.06466) [find](https://scholar.google.com/scholar?q=Reham+Omar%2C+Omij+Mangukiya%2C+Panos+Kalnis%2C+and+Essam+Mansour%2E+Chatgpt+versus+traditional+question+answering+for+knowledge+graphs%3A+Current+status+and+future+directions+towards+knowledge+graph+chatbots%2E+)
23. OpenAI. [Gpt-4 technical report](https://md.research.elamur.ai/free/en/papers/openai-2023-gpt-4-technical-report/), 2023.
24. Md Rizwan Parvez, Wasi Uddin Ahmad, Saikat Chakraborty, Baishakhi Ray, and Kai-Wei Chang. [Retrieval augmented code generation and summarization](https://md.research.elamur.ai/free/en/papers/parvez-2021-retrieval-augmented-code-generation-summarization/). arXiv preprint arXiv:2108.11601, 2021.
25. Hao Peng, Xiaozhi Wang, Shengding Hu, Hailong Jin, Lei Hou, Juanzi Li, Zhiyuan Liu, and Qun Liu. Copen: Probing conceptual knowledge in pre-trained language models. arXiv preprint arXiv:2211.04079, 2022. [find](https://scholar.google.com/scholar?q=Hao+Peng%2C+Xiaozhi+Wang%2C+Shengding+Hu%2C+Hailong+Jin%2C+Lei+Hou%2C+Juanzi+Li%2C+Zhiyuan+Liu%2C+and+Qun+Liu%2E+Copen%3A+Probing+conceptual+knowledge+in+pre%2Dtrained+language+models%2E+arXiv+preprint+arXiv%3A2211%2E04079%2C+20)
26. Fabio Petroni, Tim Rocktäschel, Patrick Lewis, Anton Bakhtin, Yuxiang Wu, Alexander H Miller, and Sebastian Riedel. [Language models as knowledge bases?](https://md.research.elamur.ai/free/en/papers/petroni-2019-language-models-as-knowledge-bases/) arXiv preprint arXiv:1909.01066, 2019.
27. Jacob Pfau, William Merrill, and Samuel R Bowman. Let’s think dot by dot: Hidden computation in transformer language models. arXiv preprint arXiv:2404.15758, 2024. [find](https://scholar.google.com/scholar?q=Jacob+Pfau%2C+William+Merrill%2C+and+Samuel+R+Bowman%2E+Let%E2%80%99s+think+dot+by+dot%3A+Hidden+computation+in+transformer+language+models%2E+arXiv+preprint+arXiv%3A2404%2E15758%2C+2024%2E)
28. Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019. [find](https://scholar.google.com/scholar?q=Alec+Radford%2C+Jeffrey+Wu%2C+Rewon+Child%2C+David+Luan%2C+Dario+Amodei%2C+Ilya+Sutskever%2C+et+al%2E+Language+models+are+unsupervised+multitask+learners%2E+OpenAI+blog%2C+1%288%29%3A9%2C+2019%2E)
29. Ori Ram, Yoav Levine, Itay Dalmedigos, Dor Muhlgay, Amnon Shashua, Kevin Leyton-Brown, and Yoav Shoham. [In-context retrieval-augmented language models](https://md.research.elamur.ai/free/en/papers/ram-2023-context-retrieval-augmented-language-models/). arXiv preprint arXiv:2302.00083, 2023.
30. Kyle Richardson and Ashish Sabharwal. [What does my QA model know? devising controlled probes using expert knowledge](https://md.research.elamur.ai/free/en/papers/richardson-2020-what-does-my-qa-model/). Transactions of the Association for Computational Linguistics, 8:572–588, 2020. doi: 10.1162/tacl a 00331. URL https://aclanthology.org/2020.tacl-1.37. [↗](https://aclanthology.org/2020.tacl-1.37)
31. Karan Singhal, Shekoofeh Azizi, Tao Tu, S Sara Mahdavi, Jason Wei, Hyung Won Chung, Nathan Scales, Ajay Tanwani, Heather Cole-Lewis, Stephen Pfohl, et al. [Large language models encode clinical knowledge](https://md.research.elamur.ai/free/en/papers/singhal-2022-large-language-models-encode-clinical/). arXiv preprint arXiv:2212.13138, 2022.
32. Shamane Siriwardhana, Rivindu Weerasekera, Elliott Wen, Tharindu Kaluarachchi, Rajib Rana, and Suranga Nanayakkara. [Improving the domain adaptation of retrieval augmented generation (rag) models for open domain question answering](https://md.research.elamur.ai/free/en/papers/siriwardhana-2023-improving-domain-adaptation-retrieval-augmented/). Transactions of the Association for Computational Linguistics, 11:1–1…
33. Jianlin Su, Yu Lu, Shengfeng Pan, Bo Wen, and Yunfeng Liu. [Roformer: Enhanced transformer with rotary position embedding](https://md.research.elamur.ai/free/en/papers/su-2021-roformer-enhanced-transformer-rotary-position/), 2021.
34. Kai Sun, Yifan Ethan Xu, Hanwen Zha, Yue Liu, and Xin Luna Dong. Head-to-tail: How knowledgeable are large language models (llm)? aka will llms replace knowledge graphs? arXiv preprint arXiv:2308.10168, 2023. [find](https://scholar.google.com/scholar?q=Kai+Sun%2C+Yifan+Ethan+Xu%2C+Hanwen+Zha%2C+Yue+Liu%2C+and+Xin+Luna+Dong%2E+Head%2Dto%2Dtail%3A+How+knowledgeable+are+large+language+models+%28llm%29%3F+aka+will+llms+replace+knowledge+graphs%3F+arXiv+preprint+arXiv%3A2308%2E1016)
35. Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. [Llama: Open and efficient foundation language models](https://md.research.elamur.ai/free/en/papers/touvron-2023-llama-open-efficient-foundation-language/). arXiv preprint arXiv:2302.13971, 2023.
36. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. [Chain-of-thought prompting elicits reasoning in large language models](https://md.research.elamur.ai/free/en/papers/wei-2022-chain-thought-prompting-elicits-reasoning/). Advances in neural information processing systems, 35:24824–24837, 2022.
37. Tian Ye, Zicheng Xu, Yuanzhi Li, and Zeyuan Allen-Zhu. [Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process](https://md.research.elamur.ai/free/en/papers/ye-2024-physics-language-models-part-2/). arXiv preprint arXiv:xxxx.xxxxx, 2024. to appear.
38. Tian Ye, Zicheng Xu, Yuanzhi Li, and Zeyuan Allen-Zhu. Physics of Language Models: Part 2.2, How to Learn From Mistakes on Grade-School Math Problems. arXiv preprint arXiv:xxxx.xxxxx, 2024. to appear. [find](https://scholar.google.com/scholar?q=Tian+Ye%2C+Zicheng+Xu%2C+Yuanzhi+Li%2C+and+Zeyuan+Allen%2DZhu%2E+Physics+of+Language+Models%3A+Part+2%2E2%2C+How+to+Learn+From+Mistakes+on+Grade%2DSchool+Math+Problems%2E+arXiv+preprint+arXiv%3Axxxx%2Exxxxx%2C+2024%2E+to+appear%2E)
39. Chunting Zhou, Pengfei Liu, Puxin Xu, Srini Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, et al. Lima: Less is more for alignment. arXiv preprint arXiv:2305.11206, 2023. [find](https://scholar.google.com/scholar?q=Chunting+Zhou%2C+Pengfei+Liu%2C+Puxin+Xu%2C+Srini+Iyer%2C+Jiao+Sun%2C+Yuning+Mao%2C+Xuezhe+Ma%2C+Avia+Efrat%2C+Ping+Yu%2C+Lili+Yu%2C+et+al%2E+Lima%3A+Less+is+more+for+alignment%2E+arXiv+preprint+arXiv%3A2305%2E11206%2C+2023%2E)

*Updated 2 November 2026*