Probabilistische Logica om te Leren en Redeneren

From erratic weather to unpredictable traffic jams, the world is filled with uncertainty. Reasoning with such uncertainties allows artificial intelligence (AI) systems to make decisions that are robust to unpredictable events or outcomes. This thesis focuses on methods for AI systems to handle uncertainty, in particular methods for modeling uncertainty as well as methods for reasoning and learning in such models. More specifically, this research is situated in the field of probabilistic logic programming (PLP), a programming paradigm that combines logic and probability theory with the aim of automating inference and learning. Probability naturally expresses uncertainty, whereas logic formalizes reasoning. PLP implements these ideas with two primitives: probabilistic facts and rules. Probabilistic facts are logic statements whose truthfulness is only known with a probability. Rules allow new statements to be derived from previous statements. Because probabilistic facts are only probabilistically true, all derived statements are probabilistically true as well. The main task of PLP is to find the probability that a statement is true. The contributions of this thesis are in three distinct domains: positive unlabeled (PU) learning, neuro-symbolic (NeSy) learning and epistemic uncertainty. The common elements over all three domains are synthesized in a general architecture for AI systems that can reason and learn with uncertainty. PU learning is the problem of training a classifier using data with only positively labeled or unlabeled instances. Negative examples are never labeled. A method has been developped to model the uncertainty of unlabeled instances in PLP. This allows PLP methods to be used to solve PU learning problems. NeSy systems consist of a neural and a symbolic component. The neural component usually handles perception whereas the symbolic component reasons over what is perceived. NeSy data often only contains raw input for the neural component (e.g. pixel values of an image) and a symbolic output (e.g. there are more women than men in the image), but no intermediate representation (e.g. which parts of the image depict a man). We developped a sampling based method to deal with the unknown intermediate representation when training a NeSy system. Uncertainty can be aleatoric or epistemic in nature. Aleatoric uncertainty is fundamental uncertainty that cannot be reduced with more information. Epistemic uncertainty in contrast is uncertainty due to the lack of information. Differentiating between these two types is important to gauge the limits and confidence of AI systems. We developped new semantics that allow both aleatoric and epistemic uncertainty to be modeled in PLP. Furthemore, methods are provided for reasoning and learning with both types of uncertainty.