# Knee-Deep in C-RASP: A Transformer Depth Hierarchy

arXiv: https://arxiv.org/abs/2506.16055 · HTML: https://mlsanity.com/paper/2506.16055/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 37 references that are in the map.
- First posted on arXiv 19 Jun 2025.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Chain of Thought Empowers Transformers to Solve Inherently Serial Problems](https://mlsanity.com/paper/2402.12875/) (2024)
- [Thinking Like Transformers](https://mlsanity.com/paper/2106.06981/) (2021)
- [Transformers, parallel computation, and logarithmic depth](https://mlsanity.com/paper/2402.09268/) (2024)
- [Exposing Attention Glitches with Flip-Flop Language Modeling](https://mlsanity.com/paper/2306.00946/) (2023)
- [A Logic for Expressing Log-Precision Transformers](https://mlsanity.com/paper/2210.02671/) (2022)
- [Masked Hard-Attention Transformers Recognize Exactly the Star-Free Languages](https://mlsanity.com/paper/2310.13897/) (2023)
- [A Formal Framework for Understanding Length Generalization in Transformers](https://mlsanity.com/paper/2410.02140/) (2024)
- [Theoretical limitations of multi-layer Transformer](https://mlsanity.com/paper/2412.02975/) (2024)
- [Counting Like Transformers: Compiling Temporal Counting Logic Into Softmax Transformers](https://mlsanity.com/paper/2404.04393/) (2024)
- [Logical Languages Accepted by Transformer Encoders with Hard Attention](https://mlsanity.com/paper/2310.03817/) (2023)

## Built on by (most cited first, 12 of 12)

- [Quantitative Bounds for Length Generalization in Transformers](https://mlsanity.com/paper/2510.27015/) (2025)
- [Length Generalization Bounds for Transformers](https://mlsanity.com/paper/2603.02238/) (2026)
- [Characterizing the Expressivity of Local Attention in Transformers](https://mlsanity.com/paper/2605.00768/) (2026)
- [Probability Distributions Computed by Autoregressive Transformers](https://mlsanity.com/paper/2510.27118/) (2025)
- [On the Ability of Transformers to Verify Plans](https://mlsanity.com/paper/2603.19954/) (2026)
- [Rational Transductors](https://mlsanity.com/paper/2602.07599/) (2026)
- [Barriers to Universal Reasoning With Transformers (And How to Overcome Them)](https://mlsanity.com/paper/2604.25800/) (2026)
- [Revisiting Padded Transformer Expressivity: Which Architectural Choices Matter and Which Don't](https://mlsanity.com/paper/2605.30523/) (2026)
- [Disentangling the Expressivity of RoPE](https://mlsanity.com/paper/2608.11909/) (2026)
- [Synthesis and Verification of Transformer Programs (Technical Report)](https://mlsanity.com/paper/2602.16473/) (2026)
- [Encoder-Decoder Transformers: Logical Characterizations and Periodicity](https://mlsanity.com/paper/2605.07705/) (2026)
- [Length Generalization for Transformers via Compression](https://mlsanity.com/paper/2609.08851/) (2026)

## Abstract (arXiv)

It has been observed that transformers with greater depth (that is, more layers) have more capabilities, but can we establish formally which capabilities are gained? We answer this question with a theoretical proof followed by an empirical study. First, we consider transformers that round to fixed precision except inside attention. We show that this subclass of transformers is expressively equivalent to the programming language C-RASP and this equivalence preserves depth. Second, we prove that deeper C-RASP programs are more expressive than shallower C-RASP programs, implying that deeper transformers are more expressive than shallower transformers (within the subclass mentioned above). The same is also proven for transformers with positional encodings (like RoPE and ALiBi). These results are established by studying a temporal logic with counting operators equivalent to C-RASP. Finally, we provide empirical evidence that our theory predicts the depth required for transformers without positional encodings to length-generalize on a family of sequential dependency tasks.