Arithmetic in Transformers Explained

While recent work has shown transformers can learn addition, previous models exhibit poor prediction accuracy and are limited to small numbers. Furthermore, the relationship between single-task and multitask arithmetic capabilities remains unexplored. In this work, we analyze 44 autoregressive transformer models trained on addition, subtraction, or both. These include 16 addition-only models, 2 subtraction-only models, 8 "mixed" models trained to perform addition and subtraction, and 14 mixed models initialized with parameters from an addition-only model. The models span 5- to 15-digit questions, 2 to 4 attention heads, and 2 to 3 layers. We show that the addition models converge on a common logical algorithm, with most models achieving >99.999% prediction accuracy. We provide a detailed mechanistic explanation of how this algorithm is implemented within the network architecture. Subtraction-only models have lower accuracy. With the initialized mixed models, through parameter transfer experiments, we explore how multitask learning dynamics evolve, revealing that some features originally specialized for addition become polysemantic, serving both operations, and boosting subtraction accuracy. We explain the mixed algorithm mechanically. Finally, we introduce a reusable library of mechanistic interpretability tools to define, locate, and visualize these algorithmic circuits across multiple models.