  Is intelligence realized by connectionist or classicist? While connectionist
approaches have achieved superhuman performance, there has been growing
evidence that such task-specific superiority is particularly fragile in
systematic generalization. This observation lies in the central debate between
connectionist and classicist, wherein the latter continually advocates an
algebraic treatment in cognitive architectures. In this work, we follow the
classicist's call and propose a hybrid approach to improve systematic
generalization in reasoning. Specifically, we showcase a prototype with
algebraic representation for the abstract spatial-temporal reasoning task of
Raven's Progressive Matrices (RPM) and present the ALgebra-Aware
Neuro-Semi-Symbolic (ALANS) learner. The ALANS learner is motivated by abstract
algebra and the representation theory. It consists of a neural visual
perception frontend and an algebraic abstract reasoning backend: the frontend
summarizes the visual information from object-based representation, while the
backend transforms it into an algebraic structure and induces the hidden
operator on the fly. The induced operator is later executed to predict the
answer's representation, and the choice most similar to the prediction is
selected as the solution. Extensive experiments show that by incorporating an
algebraic treatment, the ALANS learner outperforms various pure connectionist
models in domains requiring systematic generalization. We further show the
generative nature of the learned algebraic representation; it can be decoded by
isomorphism to generate an answer.
