Efficient Turing Machine Simulation with Transformers
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917117468606464 |
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| author | Li, Qian Wang, Yuyi |
| author_facet | Li, Qian Wang, Yuyi |
| contents | Constant bit-size Transformers are known to be Turing complete, but existing constructions require $Ω(s(n))$ chain-of-thought (CoT) steps per simulated Turing machine (TM) step, leading to impractical reasoning lengths. In this paper, we significantly reduce this efficiency gap by proving that any $(t(n),s(n))$-bounded multi-tape TM can be simulated by a constant bit-size Transformer with an optimal $O(s(n))$-long context window and only $O(s(n)^c)$ CoT steps per TM step, where $c>0$ can be made arbitrarily small by letting the Transformers' head-layer product sufficiently large. In addition, our construction shows that sparse attention with fixed geometric offsets suffices for efficient universal computation. Our proof leverages multi-queue TMs as a bridge. The main technical novelty is a more efficient simulation of multi-tape TMs by synchronous multi-queue TMs, improving both time and space complexity under stricter model assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient Turing Machine Simulation with Transformers Li, Qian Wang, Yuyi Computational Complexity Data Structures and Algorithms Machine Learning Constant bit-size Transformers are known to be Turing complete, but existing constructions require $Ω(s(n))$ chain-of-thought (CoT) steps per simulated Turing machine (TM) step, leading to impractical reasoning lengths. In this paper, we significantly reduce this efficiency gap by proving that any $(t(n),s(n))$-bounded multi-tape TM can be simulated by a constant bit-size Transformer with an optimal $O(s(n))$-long context window and only $O(s(n)^c)$ CoT steps per TM step, where $c>0$ can be made arbitrarily small by letting the Transformers' head-layer product sufficiently large. In addition, our construction shows that sparse attention with fixed geometric offsets suffices for efficient universal computation. Our proof leverages multi-queue TMs as a bridge. The main technical novelty is a more efficient simulation of multi-tape TMs by synchronous multi-queue TMs, improving both time and space complexity under stricter model assumptions. |
| title | Efficient Turing Machine Simulation with Transformers |
| topic | Computational Complexity Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2512.00003 |