Efficient Turing Machine Simulation with Transformers

• URL: http://arxiv.org/abs/2512.00003v2
• Date: Tue, 02 Dec 2025 03:55:12 GMT
• Title: Efficient Turing Machine Simulation with Transformers
• Authors: Qian Li, Yuyi Wang,
• Abstract summary: Constant bit-size Transformers are known to be Turing complete.<n>Existing constructions require $(s(n))$ chain-of-thought steps per simulated Turing machine step.<n>We prove that any $(t(n),s(n)$-bounded multi-tape TM can be simulated by a constant bit-size Transformer.
• Score: 6.70318953627082
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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