Rational Transductors

TL;DR

Rational Transductors extend Transformers with rational state information, enabling them to recognize all regular languages and solve complex problems efficiently, thus overcoming traditional limitations in length generalization and logical reasoning.

Contribution

The paper introduces Rational Transductors, a novel dual-stream architecture that enhances Transformers with WFA-derived rational state information, significantly broadening their expressive power.

Findings

Achieves strict generalization to all regular languages and NC^1 problems.

Maintains O(L + log T) parallel complexity, enabling efficient computation.

Demonstrates improved length generalization on algorithmic tasks.

Abstract

Standard Transformers excel at semantic modeling but struggle with rigid sequential logic and state tracking. Theoretical work establishes that self-attention is limited to $\AC^0$ (under hard attention) or $\TC^0$ (under soft attention), complexity classes that often fail to support robust length generalization on sequential problems without intermediate chain-of-thought. In this work, we introduce \emph{Rational Transductors}, a dual-stream architecture that augments the Transformer with a matrix-valued recurrence derived from Weighted Finite Automata (WFA). By injecting rational state information into the attention mechanism via a \emph{Deep Rational Injection} scheme, our framework strictly generalizes the expressive power of Transformers to capture all Regular Languages, $\NC^1$-complete problems (such as Boolean Formula Evaluation), and fundamental separations like Parity and Modular Counting, while preserving $O(L + \log T)$ parallel time complexity. We ground the architecture in a rigorous learning theory: we prove that \emph{Random Rational Features} act as a universal basis for sequential dependencies, justifying our initialization strategy, while establishing that the \emph{Differentiable Rational Feature} regime is necessary to close the representational compactness gap. Theoretical analysis and empirical results demonstrate that Rational Transductors solve the "Regular Gap," enabling robust length generalization on algorithmic tasks where standard Transformers fail, without the sequential computational bottlenecks of traditional RNNs.