Should AI continue to be driven by a single paradigm, or does real progress lie in combining the strengths and weaknesses of many? Professor Luc De Raedt of KU Leuven has spent much of his career persistently addressing this question. Through pioneering work that bridges logic, probability, and machine learning, he has helped shape the field of neurosymbolic AI. In our conversation at IJCAI 2025 in Montreal, he spoke about what continues to fascinate him in this line of research, how he responds to criticisms of neurosymbolic AI, and why reconciling multiple paradigms is such an exciting challenge. Hello Professor De Raedt, thank you very much for joining me.

We contribute a theoretical and operational framework for neurosymbolic AI called DeepLog. DeepLog introduces building blocks and primitives for neurosymbolic AI that make abstraction of commonly used representations and computational mechanisms used in neurosymbolic AI. DeepLog can represent and emulate a wide range of neurosymbolic systems. It consists of two key components. The first is the DeepLog language for specifying neurosymbolic models and inference tasks. This language consists of an annotated neural extension of grounded first-order logic, and makes abstraction of the type of logic, e.g. boolean, fuzzy or probabilistic, and whether logic is used in the architecture or in the loss function. The second DeepLog component is situated at the computational level and uses extended algebraic circuits as computational graphs. Together these two components are to be considered as a neurosymbolic abstract machine, with the DeepLog language as the intermediate level of abstraction and the circuits level as the computational one. DeepLog is implemented in software, relies on the latest insights in implementing algebraic circuits on GPUs, and is declarative in that it is easy to obtain different neurosymbolic models by making different choices for the underlying algebraic structures and logics. The generality and efficiency of the DeepLog neurosymbolic machine is demonstrated through an experimental comparison between 1) different fuzzy and probabilistic logics, 2) between using logic in the architecture or in the loss function, and 3) between a standalone CPU-based implementation of a neurosymbolic AI system and a DeepLog GPU-based one.

However, to deal with ambiguity and partial information, new approache s have emerged - examples of which are fuzzy logic, probabilistic modal logic, Bayesian networks and belief-based systems. Even though progress has been made, these approaches genera lly have a limitation: the probability or degree of belief, in general, being kept out of the l ogical semantics, remaining at another level of interpretation on a deterministic model. In other w ords, maintaining the binary characteristic of truth - true or false, with uncertainty being treate d as associated with models, rather than a property of logical language in itself. The proposed logic will be used to solve the problem of tackling certain types of uncertainty and imprecision with Bayesian Networks. The aim is to take advantage of the conceptual and practical benefits of this sy stem in practical situations that have not yet been adequately explored.

Brains perform decision-making by Bayes theorem. The theorem quantifies events as probabilities and, based on probability rules, renders the decisions. Learning from this, Bayes theorem can be applied to enable efficient user-scene interactions. However, given the probabilistic nature, implementing Bayes theorem in hardware using conventional deterministic computing can incur excessive computational cost and decision latency. Though challenging, here we present a probabilistic computing approach based on memristors to implement the Bayes theorem. We integrate memristors with Boolean logics and, by exploiting the volatile stochastic switching of the memristors, realise probabilistic logic operations, key for hardware Bayes theorem implementation. To empirically validate the efficacy of the hardware Bayes theorem in user-scene interactions, we develop lightweight Bayesian inference and fusion hardware operators using the probabilistic logics and apply the operators in road scene parsing for self-driving, including route planning and obstacle detection. The results show our operators can achieve reliable decisions in less than 0.4 ms (or equivalently 2,500 fps), outperforming human decision-making and the existing driving assistance systems.

This paper mainly focuses on (1) a generalized treatment of fuzzy sets of type $n$, where $n$ is an integer larger than or equal to $1$, with an example, mathematical discussions, and real-life interpretation of the given mathematical concepts; (2) the potentials and links between fuzzy logic and probability logic that have not been discussed in one document in literature; (3) representation of real-life random and fuzzy uncertainties and ambiguities that arise in data-driven real-life problems, due to uncertain mathematical and vague verbal terms in datasets.

Deep learning has proven effective for various application tasks, but its applicability is limited by the reliance on annotated examples. Self-supervised learning has emerged as a promising direction to alleviate the supervision bottleneck, but existing work focuses on leveraging co-occurrences in unlabeled data for task-agnostic representation learning, as exemplified by masked language model pretraining. In this chapter, we explore task-specific self-supervision, which leverages domain knowledge to automatically annotate noisy training examples for end applications, either by introducing labeling functions for annotating individual instances, or by imposing constraints over interdependent label decisions. We first present deep probabilistic logic(DPL), which offers a unifying framework for task-specific self-supervision by composing probabilistic logic with deep learning. DPL represents unknown labels as latent variables and incorporates diverse self-supervision using probabilistic logic to train a deep neural network end-to-end using variational EM. Next, we present self-supervised self-supervision(S4), which adds to DPL the capability to learn new self-supervision automatically. Starting from an initial seed self-supervision, S4 iteratively uses the deep neural network to propose new self supervision. These are either added directly (a form of structured self-training) or verified by a human expert (as in feature-based active learning). Experiments on real-world applications such as biomedical machine reading and various text classification tasks show that task-specific self-supervision can effectively leverage domain expertise and often match the accuracy of supervised methods with a tiny fraction of human effort.

Labeling training examples at scale is a perennial challenge in machine learning. Self-supervision methods compensate for the lack of direct supervision by leveraging prior knowledge to automatically generate noisy labeled examples. Deep probabilistic logic (DPL) is a unifying framework for self-supervised learning that represents unknown labels as latent variables and incorporates diverse self-supervision using probabilistic logic to train a deep neural network end-to-end using variational EM. While DPL is successful at combining pre-specified self-supervision, manually crafting self-supervision to attain high accuracy may still be tedious and challenging. In this paper, we propose Self-Supervised Self-Supervision (S4), which adds to DPL the capability to learn new self-supervision automatically. Starting from an initial "seed," S4 iteratively uses the deep neural network to propose new self supervision. These are either added directly (a form of structured self-training) or verified by a human expert (as in feature-based active learning). Experiments show that S4 is able to automatically propose accurate self-supervision and can often nearly match the accuracy of supervised methods with a tiny fraction of the human effort.

We present an implementation of a probabilistic first-order logic called TensorLog, in which classes of logical queries are compiled into differentiable functions in a neural-network infrastructure such as Tensorflow or Theano. This leads to a close integration of probabilistic logical reasoning with deep-learning infrastructure: in particular, it enables high-performance deep learning frameworks to be used for tuning the parameters of a probabilistic logic. The integration with these frameworks enables use of GPU-based parallel processors for inference and learning, making TensorLog the first highly parallellizable probabilistic logic. Experimental results show that TensorLog scales to problems involving hundreds of thousands of knowledge-base triples and tens of thousands of examples.

The exclusive or (xor) function is one of the simplest examples that illustrate why nonlinear feedforward networks are superior to linear regression for machine learning applications. We review the xor representation and approximation problems and discuss their solutions in terms of probabilistic logic and associative copula functions. After briefly reviewing the specification of feedforward networks, we compare the dynamics of learned error surfaces with different activation functions such as RELU and tanh through a set of colorful three-dimensional charts. The copula representations extend xor from Boolean to real values, thereby providing a convenient way to demonstrate the concept of cross-validation on in-sample and out-sample data sets. Our approach is pedagogical and is meant to be a machine learning prolegomenon. Keywords: machine learning; neural networks; probabilistic logic; copulas; error surfaces; xor.

In order to reason about probabilistic knowledge, we must reason about time and actions as well. When we say, for example, that "the probability of'heads' after a coin toss is 50% and that of'tails' is 50%", we implicitly assume that there is an action (in this example, tossing a coin) which can bring the world to different states in the next moment in time. The uncertainty lies in the state transition: the world may end up in a state where the coin shows heads or in a state where it shows tails. Despite the evident dependence of our informal notion of probability on the notions of action and time, the formal mathematical languages that we use to talk about probabilities rarely support mentioning action and time explicitly. Kolmogorov's probability theory, for example, merely defines probability as the measure function in a measure space with total measure 1 [9]. The task of modeling time-dependent actions and their possible outcomes in terms of events in a probabilistic space remains informal. While this informality is not problematic in the simplest situations (e.g. when we are interested in the possible outcomes of a single action, or when multiple actions are independent of each other), slightly more complex situations may already lead to confusion and difficulty. A famous example is the Monty Hall problem [10]. Another inconvenience of dealing with probabilities just in terms of a measure space is that its set-theoretic language (where events are represented as subsets of the sample space) is rather limited.