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of Language Models: Part 1, Learning Hierarchical Language Structures Thanks: The title Physics of Language Models was jointly conceived and designed by ZA and Xiaoli Xu. V1 appeared on this date; V2 polishes writing and adds Appendix G ; V3 polishes writing and changes the title; V4 improves writing and adds Appendix H (more uniform attention results, May 18, 2025). The first six papers in the Physics of Language Models series were presented as a two-hour tutorial at ICML 2024 in Austria ( youtu.be/yBL7J0kgldU ). A 100-min deep dive into Part 1 is available at youtu.be/kf_eGgVtOcs . Future updates and code release can be found on SSRN and the project page physics.allen-zhu.com . We would like to thank Lin Xiao, Sida Wang and Hu Xu for many helpful conversations. We would like to extend special thanks to Ian Clark, Gourab De, Anmol Mann, and Max Pfeifer from W&B , as well as Nabib Ahmed, Giri Anantharaman, Lucca Bertoncini, Henry Estela, Liao Hu, Caleb Ho, Will Johnson, Apostolos Kokolis, and Shubho Sengupta from Meta FAIR NextSys; without their invaluable support, the experiments in this paper would not have been possible.","offset":0,"level":1,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-1.html#text"},{"title":"Abstract","offset":1153,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-1.html#text"},{"title":"1 Introduction","offset":2673,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-1.html#text"},{"title":"2 Our Synthetic Context-Free Grammars","offset":11025,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-1.html#text"},{"title":"2.1 Definition and Notations","offset":11582,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-1.html#text"},{"title":"Definition 2.1 .","offset":15914,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-1.html#text"},{"title":"2.2 Why Such CFGs","offset":17939,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-2.html#text"},{"title":"3 Results 1-3: Transformer Can Learn Such 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(Generation)","offset":71460,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-5.html#text"},{"title":"B.1 Generation Diversity via Birthday Paradox","offset":73531,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-5.html#text"},{"title":"Definition B.1 .","offset":74186,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-5.html#text"},{"title":"Definition B.2 .","offset":74857,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a72219d4bd65d1e84ce31079b6c788d7e6c297e71d4b083af5d4cffccfdcff72/part-5.html#text"},{"title":"B.2 Marginal Distribution 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