Transformers as Formal Transducers: Expressivity and Limits

Core Concepts

Transformers can express surprisingly large classes of string-to-string transductions, including first-order rational, regular, and polyregular functions, which can be simulated using variants of the RASP programming language.

Abstract

The paper studies the sequence-to-sequence mapping capacity of transformers by relating them to finite transducers. It introduces three variants of the RASP programming language as an intermediate representation:

B-RASP, which operates on Boolean values, can express exactly the first-order rational transductions.

B-RASP[pos], which adds positional information, can express all first-order regular transductions and more.

S-RASP, which adds a prefix-sum operation, can express all first-order polyregular transductions and more.

The paper shows that average-hard attention transformers can simulate S-RASP, and as a corollary, that transformer decoders are Turing-complete.
The paper also provides examples of transductions expressible in these RASP variants, including rotate-right, map-reverse, map-duplicate, copy-first-half, marked-square, and majority-rules. It demonstrates that some of these transductions, like majority-rules, are not polyregular.

Stats

The paper does not contain any key metrics or important figures to support the author's logic.

Quotes

"We study the sequence-to-sequence mapping capacity of transformers by relating them to finite transducers, and find that they can express surprisingly large classes of transductions."
"Finally, we show that masked average-hard attention transformers can simulate S-RASP."
"A corollary of our results is a new proof that transformer decoders are Turing-complete."

Key Insights Distilled From

Transformers as Transducers

by Lena Strobl,... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.02040.pdf

Deeper Inquiries

What other formal language classes or computational models could be used to further analyze the expressivity of transformers

In addition to the formal language classes and computational models mentioned in the paper, other models that could be used to further analyze the expressivity of transformers include finite-state transducers, pushdown automata, and Turing machines. Finite-state transducers are particularly relevant as they can capture more complex mappings between input and output sequences. Pushdown automata, with their ability to handle context-free languages, could provide insights into the hierarchical structure of sequences processed by transformers. Turing machines, being the most powerful computational model, could help explore the full extent of the capabilities of transformers in terms of computability and complexity.

How might the results in this paper be extended to analyze the capabilities of transformers in practical natural language processing tasks

The results presented in this paper could be extended to analyze the capabilities of transformers in practical natural language processing tasks by investigating their performance on specific tasks such as machine translation, text summarization, and question answering. By mapping the functions performed by transformers to the transduction classes identified in the paper, researchers can gain a deeper understanding of the types of transformations and manipulations that transformers are capable of. This analysis could lead to improvements in model design, training strategies, and task-specific fine-tuning, ultimately enhancing the performance of transformers in real-world applications.

Are there any real-world applications or implications of the transductions studied in this paper, beyond the theoretical analysis

The transductions studied in this paper have several real-world applications and implications beyond theoretical analysis. For example, the ability to compute string homomorphisms, map-reverse, map-duplicate, and other transductions can be directly applied in data preprocessing tasks in natural language processing pipelines. These transductions can help in data cleaning, normalization, and transformation, leading to improved model performance and accuracy. Additionally, understanding the expressivity of transformers in terms of transductions can guide the development of more efficient and effective transformer architectures for various NLP tasks, contributing to advancements in the field of artificial intelligence and language processing.