This survey explores the integration of learning and reasoning in two different fields of artificial intelligence: neurosymbolic and statistical relational artificial intelligence. Neurosymbolic artificial intelligence (NeSy) studies the integration of symbolic reasoning and neural networks, while statistical relational artificial intelligence (StarAI) focuses on integrating logic with probabilistic graphical models. This survey identifies seven shared dimensions between these two subfields of AI. These dimensions can be used to characterize different NeSy and StarAI systems. They are concerned with (1) the approach to logical inference, whether model or proof-based; (2) the syntax of the used logical theories; (3) the logical semantics of the systems and their extensions to facilitate learning; (4) the scope of learning, encompassing either parameter or structure learning; (5) the presence of symbolic and subsymbolic representations; (6) the degree to which systems capture the original logic, probabilistic, and neural paradigms; and (7) the classes of learning tasks the systems are applied to. By positioning various NeSy and StarAI systems along these dimensions and pointing out similarities and differences between them, this survey contributes fundamental concepts for understanding the integration of learning and reasoning.

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As machine learning models, specifically neural networks, are becoming increasingly popular, there are concerns regarding their trustworthiness, especially in safety-critical applications, e.g., actions of an autonomous vehicle must be safe. There are approaches that can train neural networks where such domain requirements are enforced as constraints, but they either cannot guarantee that the constraint will be satisfied by all possible predictions (even on unseen data) or they are limited in the type of constraints that can be enforced. In this work, we present an approach to train neural networks which can enforce a wide variety of constraints and guarantee that the constraint is satisfied by all possible predictions. The approach builds on earlier work where learning linear models is formulated as a constraint satisfaction problem (CSP). To make this idea applicable to neural networks, two crucial new elements are added: constraint propagation over the network layers, and weight updates based on a mix of gradient descent and CSP solving. Evaluation on various machine learning tasks demonstrates that our approach is flexible enough to enforce a wide variety of domain constraints and is able to guarantee them in neural networks.

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Increasing global food demand, accompanied by the limited number of expert growers, brings the need for more sustainable and efficient horticulture. The controlled environment of greenhouses enable data collection and precise control. For optimally controlling the greenhouse climate, a grower not only looks at crop production, but rather aims at maximising the profit. However this is a complex, long term optimisation task. In this paper, Constraint Programming (CP) is applied to task of optimal greenhouse economic control, by leveraging a learned greenhouse climate model through a CP embedding. In collaboration with an industrial partner, we demonstrate how to model the greenhouse climate with an LSTM model, embed this LSTM into a CP optimisation framework, and optimise the expected profit of the grower. This data-to-decision pipeline is being integrated into a decision support system for multiple greenhouses in the Netherlands.

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In many real world applications of machine learning, models have to meet certain domain-based requirements that can be expressed as constraints (for example, safety-critical constraints in autonomous driving systems). Such constraints are often handled by including them in a regularization term, while learning a model. This approach, however, does not guarantee 100% satisfaction of the constraints: it only reduces violations of the constraints on the training set rather than ensuring that the predictions by the model will always adhere to them. In this paper, we present a framework for learning models that provably fulfill the constraints under all circumstances (i.e., also on unseen data). To achieve this, we cast learning as a maximum satisfiability problem, and solve it using a novel SaDe algorithm that combines constraint satisfaction with gradient descent. We compare our method against regularization based baselines on linear models and show that our method is capable of enforcing different types of domain constraints effectively on unseen data, without sacrificing predictive performance.

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Inductive logic programming (ILP) is a form of machine learning. The goal of ILP is to induce a hypothesis (a set of logical rules) that generalises training examples. As ILP turns 30, we provide a new introduction to the field. We introduce the necessary logical notation and the main learning settings; describe the building blocks of an ILP system; compare several systems on several dimensions; describe four systems (Aleph, TILDE, ASPAL, and Metagol); highlight key application areas; and, finally, summarise current limitations and directions for future research.

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Inductive logic programming (ILP) is a form of logic-based machine learning. The goal is to induce a hypothesis (a logic program) that generalises given training examples and background knowledge. As ILP turns 30, we review the last decade of research. We focus on (i) new meta-level search methods, (ii) techniques for learning recursive programs, (iii) new approaches for predicate invention, and (iv) the use of different technologies. We conclude by discussing current limitations of ILP and directions for future research.@en