  Logic-based machine learning aims to learn general, interpretable knowledge
in a data-efficient manner. However, labelled data must be specified in a
structured logical form. To address this limitation, we propose a
neural-symbolic learning framework, called Feed-Forward Neural-Symbolic Learner
(FFNSL), that integrates a logic-based machine learning system capable of
learning from noisy examples, with neural networks, in order to learn
interpretable knowledge from labelled unstructured data. We demonstrate the
generality of FFNSL on four neural-symbolic classification problems, where
different pre-trained neural network models and logic-based machine learning
systems are integrated to learn interpretable knowledge from sequences of
images. We evaluate the robustness of our framework by using images subject to
distributional shifts, for which the pre-trained neural networks may predict
incorrectly and with high confidence. We analyse the impact that these shifts
have on the accuracy of the learned knowledge and run-time performance,
comparing FFNSL to tree-based and pure neural approaches. Our experimental
results show that FFNSL outperforms the baselines by learning more accurate and
interpretable knowledge with fewer examples.
