  Large language models exhibit surprising emergent generalization properties,
yet also struggle on many simple reasoning tasks such as arithmetic and parity.
This raises the question of if and when Transformer models can learn the true
algorithm for solving a task. We study the scope of Transformers' abilities in
the specific setting of length generalization on algorithmic tasks. Here, we
propose a unifying framework to understand when and how Transformers can
exhibit strong length generalization on a given task. Specifically, we leverage
RASP (Weiss et al., 2021) -- a programming language designed for the
computational model of a Transformer -- and introduce the RASP-Generalization
Conjecture: Transformers tend to length generalize on a task if the task can be
solved by a short RASP program which works for all input lengths. This simple
conjecture remarkably captures most known instances of length generalization on
algorithmic tasks. Moreover, we leverage our insights to drastically improve
generalization performance on traditionally hard tasks (such as parity and
addition). On the theoretical side, we give a simple example where the
"min-degree-interpolator" model of learning from Abbe et al. (2023) does not
correctly predict Transformers' out-of-distribution behavior, but our
conjecture does. Overall, our work provides a novel perspective on the
mechanisms of compositional generalization and the algorithmic capabilities of
Transformers.
