Universal Transformers for Circuit Computations: Perfect Length Generalization in Tiny Transformers

2026-08-31

AI Summary

研究通过一种仅含280个参数的变压器模型解决了神经网络在算法计算中的泛化问题，采用电路嵌入和深度追踪编码方法实现了任意长度布尔表达式的精确评估。

Abstract

Learning generalizable algorithmic computations remains a challenge for neural networks, as reflected in persistent failures on compositional and length generalization benchmarks. We present a provably correct, transformer parameterization (with only 280 learnable parameters for Boolean algebra tasks) capable of learning and evaluating problems of any depth or length. We assume inputs are fully parenthesized, well-formed expressions. Our approach conceptualizes algorithmic tasks as circuit models embedded in transformers, enabling depth-1 circuit reduction in a single forward pass. To achieve depth generalization, we introduce a positional encoding that tracks each gate's depth within the circuit, enabling the model to identify evaluable subexpressions at each iteration via masked hard attention, with $O(n)$ per-iteration complexity via linear attention. Combined with an autonomous halting criterion, the model terminates after $d$ iterations for problems of depth $d$, yielding $O(n \cdot d)$ total complexity. We show that training on shallow problem instances (depth 1 and depth 2) effectively recovers interpretable parameters that {\em snap} into place, resulting in exact length generalization. Though we establish that our construction provably evaluates Boolean expressions -- a universal symbolic computation -- of arbitrary length perfectly, in other experiments we also demonstrate that our transformer variant can learn and generalize perfectly (100% accuracy) on other common length generalization benchmarks, including modular arithmetic and ListOps.

Problem

• neural networks
• generalization
• algorithmic computations
• length generalization
• Boolean algebra

Innovation

• Universal Transformers
• Circuit Models
• Positional Encoding
• Masked Hard Attention
• Length Generalization