The Expressive Power of Low Precision Softmax Transformers with (Summarized) Chain-of-Thought

• URL: http://arxiv.org/abs/2605.18079v1
• Date: Mon, 18 May 2026 08:57:53 GMT
• Title: The Expressive Power of Low Precision Softmax Transformers with (Summarized) Chain-of-Thought
• Abstract summary: Existing expressivity results for transformers typically rely on hardmax attention, high precision, and other architectural modifications that disconnect them from the models used in practice.<n>We bridge this gap by analyzing transformer decoders with softmax attention and rounding of activations and attention weights, while allowing depth and width to grow logarithmically with the context length.<n>As an intermediate step, we construct hardmax transformers with ternary activations and well-separated attention scores that simulate Turing machines using Chain-of-Thought (CoT)
• Score: 1.9193579706947885
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Existing expressivity results for transformers typically rely on hardmax attention, high precision, and other architectural modifications that disconnect them from the models used in practice. We bridge this gap by analyzing standard transformer decoders with softmax attention and rounding of activations and attention weights, while allowing depth and width to grow logarithmically with the context length. As an intermediate step, we construct hardmax transformers with ternary activations and well-separated attention scores that simulate Turing machines using Chain-of-Thought (CoT). This lets us convert the constructions to equivalent softmax transformers without the unrealistic parameter magnitudes or activation precision that prior approaches would require. Using the same technique, we analyze a recently proposed summarized CoT paradigm and show that it simulates Turing machines more efficiently, with model size scaling logarithmically in a space bound rather than a time bound. We empirically test predictions made by our results on a Sudoku reasoning task and find better alignment with learnability than for prior high-precision results. Our code is available at https://github.com/moritzbroe/transformer-expressivity.

Related papers

• Constant Bit-size Transformers Are Turing Complete [6.70318953627082]
  We prove that any Turing machine running on inputs of arbitrary length can be simulated by a constant bit-size transformer.<n>Our approach relies on simulating Post machines, a Turing-complete computational model.
  arXiv  Detail & Related papers  (2025-05-22T02:45:38Z)
• On the Convergence of Encoder-only Shallow Transformers [62.639819460956176]
  We build the global convergence theory of encoder-only shallow Transformers under a realistic setting. Our results can pave the way for a better understanding of modern Transformers, particularly on training dynamics.
  arXiv  Detail & Related papers  (2023-11-02T20:03:05Z)
• Softmax-free Linear Transformers [90.83157268265654]
  Vision transformers (ViTs) have pushed the state-of-the-art for visual perception tasks. Existing methods are either theoretically flawed or empirically ineffective for visual recognition. We propose a family of Softmax-Free Transformers (SOFT)
  arXiv  Detail & Related papers  (2022-07-05T03:08:27Z)
• cosFormer: Rethinking Softmax in Attention [60.557869510885205]
  kernel methods are often adopted to reduce the complexity by approximating the softmax operator. Due to the approximation errors, their performances vary in different tasks/corpus and suffer crucial performance drops. We propose a linear transformer called cosFormer that can achieve comparable or better accuracy to the vanilla transformer.
  arXiv  Detail & Related papers  (2022-02-17T17:53:48Z)
• SOFT: Softmax-free Transformer with Linear Complexity [112.9754491864247]
  Vision transformers (ViTs) have pushed the state-of-the-art for various visual recognition tasks by patch-wise image tokenization followed by self-attention. Various attempts on approximating the self-attention with linear complexity have been made in Natural Language Processing. We identify that their limitations are rooted in keeping the softmax self-attention during approximations. For the first time, a softmax-free transformer or SOFT is proposed.
  arXiv  Detail & Related papers  (2021-10-22T17:57:29Z)
• Predicting Attention Sparsity in Transformers [0.9786690381850356]
  We propose Sparsefinder, a model trained to identify the sparsity pattern of entmax attention before computing it. Our work provides a new angle to study model efficiency by doing extensive analysis of the tradeoff between the sparsity and recall of the predicted attention graph.
  arXiv  Detail & Related papers  (2021-09-24T20:51:21Z)
• Stable, Fast and Accurate: Kernelized Attention with Relative Positional Encoding [63.539333383965726]
  We propose a novel way to accelerate attention calculation for Transformers with relative positional encoding (RPE) Based upon the observation that relative positional encoding forms a Toeplitz matrix, we mathematically show that kernelized attention with RPE can be calculated efficiently using Fast Fourier Transform (FFT)
  arXiv  Detail & Related papers  (2021-06-23T17:51:26Z)
• FNet: Mixing Tokens with Fourier Transforms [0.578717214982749]
  We show that Transformer encoder architectures can be massively sped up with limited accuracy costs. We replace the self-attention sublayers with simple linear transformations that "mix" input tokens. The resulting model, which we name FNet, scales very efficiently to long inputs.
  arXiv  Detail & Related papers  (2021-05-09T03:32:48Z)
• Finetuning Pretrained Transformers into RNNs [81.72974646901136]
  Transformers have outperformed recurrent neural networks (RNNs) in natural language generation. A linear-complexity recurrent variant has proven well suited for autoregressive generation. This work aims to convert a pretrained transformer into its efficient recurrent counterpart.
  arXiv  Detail & Related papers  (2021-03-24T10:50:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.