{"id":"arxiv:2509.25239","title":"A Formal Comparison Between Chain of Thought and Latent Thought","date":"2025-09-25","section":"Reasoning and the \"physics\" of language models","quality":0.129176,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/","source_url":"https://arxiv.org/abs/2509.25239","text_available":true,"authors":["Kevin Xu","Issei Sato"],"outgoing":[{"source":"arxiv:2509.25239","target":"arxiv:1807.03819","evidence":"title","id":"arxiv:1807.03819","title":"Universal Transformers","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/1d843516d980c3404cb622a2a490edb535a3afe27a85a980335a54b5fcd2220a/"},{"source":"arxiv:2509.25239","target":"arxiv:2405.14838","evidence":"id","id":"arxiv:2405.14838","title":"From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/a242b6148f6f84466ce4315ac4937278586a242dd3f847146e142f3c7e27c39c/"},{"source":"arxiv:2509.25239","target":"arxiv:2412.06769","evidence":"title","id":"arxiv:2412.06769","title":"Training Large Language Models to Reason in a Continuous Latent Space","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/68c71dc1a042dedd5e3c4d9c1ad32c9d046fd31b8dc79080bd6dff1c9800abc7/"},{"source":"arxiv:2509.25239","target":"arxiv:2507.10524","evidence":"id","id":"arxiv:2507.10524","title":"Mixture-of-Recursions: Learning Dynamic Recursive Depths for Adaptive Token-Level Computation","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/7b25702c35f07446cea97d6a98afc489c8fdda54560c05fbeb1b3b139a9da622/"}],"incoming":[{"source":"arxiv:2605.12466","target":"arxiv:2509.25239","evidence":"id","id":"arxiv:2605.12466","title":"Solve the Loop: Attractor Models for Language and Reasoning","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/eb46f81cd5718eb6b9256c583cf9dd2e97b793b4d14337e487c16285f96e2b9a/"},{"source":"arxiv:2606.31779","target":"arxiv:2509.25239","evidence":"title","id":"arxiv:2606.31779","title":"Bridging the Gap Between Latent and Explicit Reasoning with Looped Transformers","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/bf07d5c19e507d9ec91a48f9067e6c29fbe8965d001923e2de0e96da143c72e1/"},{"source":"arxiv:2608.09888","target":"arxiv:2509.25239","evidence":"title","id":"arxiv:2608.09888","title":"BDH-CQ: In-Context Learning with Recurrent Latent Reasoning","url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/f0c77a8a0f50254e9efdf9a511e08fc2a40677bfa029864bdd0a9cbce561c666/"}],"text_parts":[{"start":0,"end":15974,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html"},{"start":15974,"end":31710,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html"},{"start":31710,"end":47035,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html"},{"start":47035,"end":62116,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html"},{"start":62116,"end":77818,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html"},{"start":77818,"end":93498,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-6.html"},{"start":93498,"end":109327,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html"},{"start":109327,"end":125071,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html"},{"start":125071,"end":137425,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html"}],"contents":[{"title":"A Formal Comparison Between Chain of Thought and Latent Thought","offset":0,"level":1,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Abstract","offset":67,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Keywords:","offset":918,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"1 Introduction","offset":936,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"1.1 Our Contributions","offset":3271,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Latent thought enables parallel reasoning.","offset":3757,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"CoT enables approximate counting.","offset":5210,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"2 Background","offset":6336,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"2.1 Models of Computation","offset":6353,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Definition 2.1 (CoT, following Merrill and Sabharwal (2024) ) .","offset":6672,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Definition 2.2 (Coconut) .","offset":7903,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Definition 2.3 (Looped TF) .","offset":8961,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"2.2 Related Work","offset":10023,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Computational power of chain of thought.","offset":10376,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Computational power of latent thought.","offset":11151,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"3 Latent Thought Enables Parallel Reasoning","offset":12227,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"3.1 Problem Setting","offset":12618,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Definition 3.1 (Computation graph) .","offset":12821,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Assumptions on models.","offset":13827,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"On the fairness of comparing steps and loops.","offset":14644,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"3.2 CoT Suffices with Size-scaled Steps and Latent Thought Suffices with Depth-scaled Iterations","offset":15334,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Definition 3.2 ( Merrill and Sabharwal, 2023 ) .","offset":15719,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Assumption 3.3 (Poly-size graph) .","offset":15931,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-1.html#text"},{"title":"Assumption 3.4 (Poly-efficient approximation, cf. ( Feng et al., 2023 ) ) .","offset":16047,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.5 (CoT for DAGs) .","offset":16768,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Proof sketch.","offset":17293,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.6 (Latent thought for DAGs) .","offset":17710,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Proof sketch.","offset":18308,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Remark.","offset":18679,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"3.3 Separation in Polylogarithmic Iterations","offset":19036,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Definition 3.7 (Complexity Classes 𝖢𝗈𝖳 \\mathsf{CoT} , 𝖢𝖳 \\mathsf{CT} and 𝖫𝗈𝗈𝗉 \\mathsf{Loop} ) .","offset":19385,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Definition 3.8 (Informal) .","offset":20356,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.9 ( Li et al., 2024 ) .","offset":20986,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.10 .","offset":21208,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Definition 3.11 .","offset":21874,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.12 .","offset":22393,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Proof sketch.","offset":22829,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Lemma 3.13 .","offset":23559,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Proof.","offset":23764,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.14 .","offset":24406,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 3.15 .","offset":24965,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Remark.","offset":25544,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"4 CoT Enables Approximate Counting","offset":25834,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"4.1 Preliminaries","offset":26525,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Approximate counting.","offset":26548,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Definition 4.1 (Informal: Self-reducibility ( Schnorr, 1976 ) ) .","offset":27023,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Definition 4.2 (FPRAS) .","offset":27741,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Probabilistic models of computation.","offset":28902,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"4.2 Separation in Approximate Counting","offset":30107,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Lemma 4.3 (Informal) .","offset":30715,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Proof sketch.","offset":31276,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"4.3 Separation in Approximate Sampling","offset":31666,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-2.html#text"},{"title":"Theorem 4.4 .","offset":32103,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Proof sketch.","offset":32832,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Theorem 4.5 .","offset":34133,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"5 Experiments","offset":34467,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"5.1 Experimental Setting","offset":34841,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Fundamental algorithmic reasoning tasks.","offset":34871,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Approximate counting tasks.","offset":36000,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Training strategy.","offset":36495,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"5.2 Results","offset":37158,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"6 Conclusion","offset":38656,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Impact Statement","offset":39624,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Appendix A Formal Definitions","offset":39845,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"A.1 Notation","offset":39879,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"A.2 Transformer Block","offset":41786,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Definition A.1 (Transformer) .","offset":42052,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"A.3 Chain of Thought","offset":45089,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Definition A.2 (CoT) .","offset":45115,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"A.4 Continuous Thought","offset":46972,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"Definition A.3 (Coconut) .","offset":47000,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-3.html#text"},{"title":"A.5 Looped Transformer","offset":48965,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Definition A.4 (Looped TF) .","offset":48993,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Appendix B Deferred Proofs for Section 3","offset":50146,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.1 Precision Modeling","offset":50191,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Definition B.1 (Floating-point Representation, cf. ( Li et al., 2024 ) ) .","offset":50330,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Definition B.2 (Correct Rounding, cf. ( Li et al., 2024 ) ) .","offset":50668,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Definition B.3 (Summation with Iterative Rounding ( Li et al., 2024 ) ) .","offset":51437,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.2 Definition of Assumption 3.4","offset":52650,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Definition B.4 (Polynomially-efficient approximation) .","offset":52892,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.3 Technical lemmas","offset":54052,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.3.1 Orthogonal Vectors","offset":54155,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Lemma B.5 ( Li et al., 2024 ) .","offset":55294,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.3.2 Position Selector","offset":56138,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Lemma B.6 .","offset":56168,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Proof.","offset":56615,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.3.3 Feedforward layers","offset":58142,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Lemma B.7 .","offset":58173,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Proof.","offset":58987,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Lemma B.8 .","offset":60239,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"Proof.","offset":61204,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-4.html#text"},{"title":"B.4 Proof for Theorem 3.5","offset":66616,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Proof.","offset":66647,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Embedding","offset":69082,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Attention layer","offset":70781,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Feed-forward layer","offset":73162,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Output layer","offset":76685,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"B.5 Proof of Theorem 3.6 for Looped Transformer","offset":77750,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Proof.","offset":77803,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-5.html#text"},{"title":"Summary","offset":83784,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-6.html#text"},{"title":"B.6 Proof of Theorem 3.6 for Continuous Thought","offset":88949,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-6.html#text"},{"title":"Proof.","offset":89002,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-6.html#text"},{"title":"B.7 Proof for Theorem 3.12","offset":93466,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-6.html#text"},{"title":"Theorem B.9 ( Li et al., 2024 ) .","offset":94259,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Lemma B.10 ( Li et al., 2024 ) .","offset":94674,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Proof for Theorem 3.12 .","offset":95038,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Appendix C Deferred Proofs for Section 4","offset":95904,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"C.1 Definition for Models of Computation","offset":95949,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.1 (Language model with CoT) .","offset":96367,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.2 (Language model with Coconut) .","offset":97505,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.3 (Language model with looped TF) .","offset":98597,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.4 (Complexity Classes 𝗉𝖢𝗈𝖳 \\mathsf{pCoT} , 𝗉𝖢𝖳 \\mathsf{pCT} , and 𝗉𝖫𝖮𝖮𝖯 \\mathsf{pLOOP} ) .","offset":100177,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"C.2 Lemma: Universality of Chain of Thought","offset":100923,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.5 .","offset":100972,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Lemma C.6 ( Nowak et al., 2024 ) .","offset":102306,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"C.3 Self-Reducibility and Complexity of Approximate Counting","offset":103018,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.7 (Relation) .","offset":103164,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.8 (Counting) .","offset":103436,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.9 ( p p -relation) .","offset":103645,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.10 .","offset":103999,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.11 (FPTAS) .","offset":104861,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.12 (FPRAS) .","offset":105196,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Proposition C.13 ( Jerrum et al., 1986 ) .","offset":105751,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Proposition C.14 .","offset":106353,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Proof.","offset":106555,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Proof for Proposition C.13 .","offset":107560,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-7.html#text"},{"title":"Definition C.15 ( Schnorr, 1976 ) .","offset":110328,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Proposition C.16 ( Jerrum et al., 1986 ) .","offset":112847,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Proof.","offset":113023,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"C.4 Proof for Theorem 4.3","offset":114156,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Lemma C.17 (Formal Statement of Lemma 4.3 ) .","offset":114187,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Proof.","offset":115133,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"C.5 Proof for Theorem 4.4","offset":116274,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Definition C.18 (FPAUS) .","offset":116305,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Theorem C.19 ( Jerrum et al., 1986 ) .","offset":117080,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Proof of Theorem 4.4 .","offset":117309,"level":6,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Appendix D Experimental Details","offset":122084,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"D.1 Fundamental Algorithmic Reasoning Tasks","offset":122120,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"D.1.1 Task Settings","offset":122169,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Word Problem","offset":122195,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Connectivity","offset":122773,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Arithmetic Expression Evaluation","offset":124338,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"Edit Distance","offset":125050,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-8.html#text"},{"title":"D.1.2 Training Configuration","offset":126380,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"Configuration of chain of thought","offset":126415,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"Optimization and model details.","offset":128197,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"Uniform selection.","offset":129207,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"Stepwise internalization.","offset":129854,"level":5,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"D.2 Approximate Counting and Approximate Sampling","offset":130668,"level":3,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"D.2.1 Approximate Counting of DNF Formulas","offset":130723,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"D.2.2 Approximate Sampling of Graph Colorings","offset":132296,"level":4,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"},{"title":"Instructions for reporting errors","offset":136461,"level":2,"url":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/part-9.html#text"}],"text_code_points":137425,"source_markdown":"https://vinnik-dmitry07.github.io/favorite-papers/agents/papers/8399816be05fff6a7002f5c3439ec43ffe7b468424759b4cc70f5efc7739ea9e/source.md"}