Neural-symbolic models bridge the gap between sub-symbolic and symbolic approaches, both of which have significant limitations. Sub-symbolic approaches, like neural networks, require a large amount of labeled data to be successful, whereas symbolic approaches, like logic reasoners, require a small amount of prior domain knowledge but do not easily scale to large collections of data. This chapter presents a general approach to integrate learning and reasoning that is based on the translation of the available prior knowledge into an undirected graphical model. Potentials on the graphical model are designed to accommodate dependencies among random variables by means of a set of trainable functions, like those computed by neural networks. The resulting neural-symbolic framework can effectively leverage the training data, when available, while exploiting high-level logic reasoning in a certain domain of discourse. Although exact inference is intractable within this model, different tractable models can be derived by making different assumptions. In particular, three models are presented in this chapter: Semantic-Based Regularization, Deep Logic Models and Relational Neural Machines. Semantic-Based Regularization is a scalable neural-symbolic model, that does not adapt the parameters of the reasoner, under the assumption that the provided prior knowledge is correct and must be exactly satisfied. Deep Logic Models preserve the scalability of Semantic-Based Regularization, while providing a flexible exploitation of logic knowledge by co-training the parameters of the reasoner during the learning procedure. Finally, Relational Neural Machines provide the fundamental advantages of perfectly replicating the effectiveness of training from supervised data of standard deep architectures, and of preserving the same generality and expressive power of Markov Logic Networks, when considering pure reasoning on symbolic data. The bonding between learning and reasoning is very general as any (deep) learner can be adopted, and any output structure expressed via First-Order Logic can be integrated. However, exact inference within a Relational Neural Machine is still intractable, and different factorizations are discussed to increase the scalability of the approach.
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