A large class of Neural-Symbolic (NeSy) methods employs a machine learner to process the input entities, while relying on a reasoner based on First-Order Logic to represent and process more complex relationships among the entities. A fundamental role for these methods is played by the process of logic grounding, which determines the relevant substitutions for the logic rules using a (sub)set of entities. Some NeSy methods use an exhaustive derivation of all possible substitutions, preserving the full expressive power of the logic knowledge. This leads to a combinatorial explosion in the number of ground formulas to consider and, therefore, strongly limits their scalability. Other methods rely on heuristic-based selective derivations, which are generally more computationally efficient, but lack a justification and provide no guarantees of preserving the information provided to and returned by the reasoner. Taking inspiration from multi-hop symbolic reasoning, this paper proposes a parametrized family of grounding methods generalizing classic Backward Chaining. Different selections within this family allow us to obtain commonly employed grounding methods as special cases, and to control the trade-off between expressiveness and scalability of the reasoner. The experimental results show that the selection of the grounding criterion is often as important as the NeSy method itself.

Inductive logic programming (ILP) (Muggleton 1996) has We propose first-order extensions of LNNs that can been of long-standing interest where the goal is to learn tackle ILP. Since vanilla backpropagation is insufficient for logical rules from labeled data. Since rules are explicitly constraint optimization, we propose flexible learning algorithms symbolic, they provide certain advantages over black box capable of handling a variety of (linear) inequality and models. For instance, learned rules can be inspected, understood equality constraints. We experiment with diverse benchmarks and verified forming a convenient means of storing for ILP including gridworld and knowledge base completion learned knowledge. Consequently, a number of approaches (KBC) that call for learning of different kinds of rules have been proposed to address ILP including, but not limited and show how our approach can tackle both effectively. In to, statistical relational learning (Getoor and Taskar 2007) fact, our KBC results represents a 4-16% relative improvement and more recently, neuro-symbolic methods.