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PENCIL: Long Thoughts with Short Memory

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arxiv 2503.14337 v2 pith:FUC5QTYO submitted 2025-03-18 cs.LG cs.CL

PENCIL: Long Thoughts with Short Memory

classification cs.LG cs.CL
keywords pencilthoughtscontextefficientintermediatelongmemoryproblems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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While state-of-the-art LLMs have demonstrated great promise of using long Chains-of-Thought (CoT) to boost reasoning, scaling it up to more challenging problems at test-time is fundamentally limited by suboptimal memory usage -- intermediate computations accumulate indefinitely in context even when no longer needed for future thoughts. We introduce PENCIL, which incorporates a novel reduction mechanism into the autoregressive generation process that recursively cleans up intermediate thoughts based on patterns learned from training. By iteratively generating and erasing thoughts, PENCIL can think deeper to solve harder problems using shorter context and less compute. Empirically, we observe PENCIL is significantly more effective and efficient than CoT. For example, we demonstrate PENCIL with a small 25M-parameter transformer and 2048 context length solves Einstein's puzzle -- a task that challenges much larger models like GPT-4. Theoretically, we prove PENCIL can perform universal efficient computation by simulating any Turing machines with optimal time and space complexity, and thus can solve arbitrary computable tasks that are otherwise intractable for vanilla CoT.

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Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    The sample complexity of exact-trace learning for autoregressive Chain-of-Thought is O((DSdim(H) + log(1/δ))/ε), matching the local next-token class with no dependence on rollout length.

  2. Tight Sample Complexity of Transformers

    cs.LG 2026-06 unverdicted novelty 8.0

    Depth-L transformers with W parameters have VC dimension Theta(L W log(T W)), yielding matching O(L W log((T+T')W)) upper and Omega(L W log((T+T')W/L)) lower bounds on sample complexity for chain-of-thought learning.

  3. Rethinking the Role of Positional Encoding: Sliding-Window Transformers without PE Remain Turing Complete

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    Sliding-window transformers without positional encodings are Turing complete because the sliding window breaks permutation symmetry and suffices to simulate Post machines via a constant-size histogram state.

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  8. Null Space Constrained Contrastive Visual Forgetting for MLLM Unlearning

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  9. MEMENTO: Teaching LLMs to Manage Their Own Context

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  11. Pseudo-Formalization for Automatic Proof Verification

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