Neuro-symbolic learning generally consists of two separated worlds, i.e.,
neural network training and symbolic constraint solving, whose success hinges
on symbol grounding, a fundamental problem in AI. This paper presents a novel,
softened symbol grounding process, bridging the gap between the two worlds, and
resulting in an effective and efficient neuro-symbolic learning framework.
Technically, the framework features (1) modeling of symbol solution states as a
Boltzmann distribution, which avoids expensive state searching and facilitates
mutually beneficial interactions between network training and symbolic
reasoning;(2) a new MCMC technique leveraging projection and SMT solvers, which
efficiently samples from disconnected symbol solution spaces; (3) an annealing
mechanism that can escape from %being trapped into sub-optimal symbol
groundings. Experiments with three representative neuro symbolic learning tasks
demonstrate that, owining to its superior symbol grounding capability, our
framework successfully solves problems well beyond the frontier of the existing
proposals.

神经符号学习中的符号基础问题是人工智能中的一个基本问题。本文提出了一种新的、软化的符号基础过程，能够桥接神经网络训练和符号约束求解两个分离的领域，从而构建一个高效且有效的神经符号学习框架，并通过实验证明该框架具有优越的符号基础能力。