In this paper we introduce the convex fragment of Lukasiewicz Logic and discuss its possible applications in different learning schemes. Indeed, the provided theoretical results are highly general, because they can be exploited in any learning framework involving logical constraints. The method is of particular interest since the fragment guarantees to deal with convex constraints, which are shown to be equivalent to a set of linear constraints. Within this framework, we are able to formulate learning with kernel machines as well as collective classification as a quadratic programming problem.

Injecting prior knowledge into the learning process of a neural architecture is one of the main challenges currently faced by the artificial intelligence community, which also motivated the emergence of neural-symbolic models. One of the main advantages of these approaches is their capacity to learn competitive solutions with a significant reduction of the amount of supervised data. In this regard, a commonly adopted solution consists of representing the prior knowledge via first-order logic formulas, then relaxing the formulas into a set of differentiable constraints by using a t-norm fuzzy logic. This paper shows that this relaxation, together with the choice of the penalty terms enforcing the constraint satisfaction, can be unambiguously determined by the selection of a t-norm generator, providing numerical simplification properties and a tighter integration between the logic knowledge and the learning objective. When restricted to supervised learning, the presented theoretical framework provides a straight derivation of the popular cross-entropy loss, which has been shown to provide faster convergence and to reduce the vanishing gradient problem in very deep structures. However, the proposed learning formulation extends the advantages of the cross-entropy loss to the general knowledge that can be represented by neural-symbolic methods. In addition, the presented methodology allows the development of novel classes of loss functions, which are shown in the experimental results to lead to faster convergence rates than the approaches previously proposed in the literature.