  Neural network language models (NNLMs) have achieved ever-improving accuracy
due to more sophisticated architectures and increasing amounts of training
data. However, the inductive bias of these models (formed by the distributional
hypothesis of language), while ideally suited to modeling most running text,
results in key limitations for today's models. In particular, the models often
struggle to learn certain spatial, temporal, or quantitative relationships,
which are commonplace in text and are second-nature for human readers. Yet, in
many cases, these relationships can be encoded with simple mathematical or
logical expressions. How can we augment today's neural models with such
encodings?
  In this paper, we propose a general methodology to enhance the inductive bias
of NNLMs by incorporating simple functions into a neural architecture to form a
hierarchical neural-symbolic language model (NSLM). These functions explicitly
encode symbolic deterministic relationships to form probability distributions
over words. We explore the effectiveness of this approach on numbers and
geographic locations, and show that NSLMs significantly reduce perplexity in
small-corpus language modeling, and that the performance improvement persists
for rare tokens even on much larger corpora. The approach is simple and
general, and we discuss how it can be applied to other word classes beyond
numbers and geography.
