The recent availability of large-scale data combining multiple data modalities has opened various research and commercial opportunities in Artificial Intelligence (AI). Machine Learning (ML) has achieved important results in this area mostly by adopting a sub-symbolic distributed representation. It is generally accepted now that such purely sub-symbolic approaches can be data inefficient and struggle at extrapolation and reasoning. By contrast, symbolic AI is based on rich, high-level representations ideally based on human-readable symbols. Despite being more explainable and having success at reasoning, symbolic AI usually struggles when faced with incomplete knowledge or inaccurate, large data sets and combinatorial knowledge.

Neurosymbolic AI attempts to benefit from the strengths of both approaches combining reasoning with complex representation of knowledge and efficient learning from multiple data modalities. Hence, neurosymbolic AI seeks to ground rich knowledge into efficient sub-symbolic representations and to explain sub-symbolic representations and deep learning by offering high-level symbolic descriptions for such learning systems. Logic Tensor Networks (LTN) are a neurosymbolic AI system for querying, learning and reasoning with rich data and abstract knowledge. LTN introduces Real Logic, a fully differentiable first-order language with concrete semantics such that every symbolic expression has an interpretation that is grounded onto real numbers in the domain. In particular, LTN converts Real Logic formulas into computational graphs that enable gradient-based optimization. This chapter presents the LTN framework and illustrates its use on knowledge completion tasks to ground the relational predicates (symbols) into a concrete interpretation (vectors and tensors). It then investigates the use of LTN on semi-supervised learning, learning of embeddings and reasoning. LTN has been applied recently to many important AI tasks, including semantic image interpretation, ontology learning and reasoning, and reinforcement learning, which use LTN for supervised classification, data clustering, semi-supervised learning, embedding learning, reasoning and query answering. The chapter presents some of the main recent applications of LTN before analyzing results in the context of related work and discussing the next steps for neurosymbolic AI and LTN-based AI models.