  Tractable Boolean and arithmetic circuits have been studied extensively in AI
for over two decades now. These circuits were initially proposed as "compiled
objects," meant to facilitate logical and probabilistic reasoning, as they
permit various types of inference to be performed in linear-time and a
feed-forward fashion like neural networks. In more recent years, the role of
tractable circuits has significantly expanded as they became a computational
and semantical backbone for some approaches that aim to integrate knowledge,
reasoning and learning. In this article, we review the foundations of tractable
circuits and some associated milestones, while focusing on their core
properties and techniques that make them particularly useful for the broad aims
of neuro-symbolic AI.
