Operationalizing definitions of fairness is difficult in practice, as multiple definitions can be incompatible while each being arguably desirable. Instead, it may be easier to directly describe algorithmic bias through ad-hoc assumptions specific to a particular real-world task, e.g., based on background information on systemic biases in its context. Such assumptions can, in turn, be used to mitigate this bias during training. Yet, a framework for incorporating such assumptions that is simultaneously principled, flexible, and interpretable is currently lacking. Our approach is to formalize bias assumptions as programs in ProbLog, a probabilistic logic programming language that allows for the description of probabilistic causal relationships through logic. Neurosymbolic extensions of ProbLog then allow for easy integration of these assumptions in a neural network's training process. We propose a set of templates to express different types of bias and show the versatility of our approach on synthetic tabular datasets with known biases. Using estimates of the bias distortions present, we also succeed in mitigating algorithmic bias in real-world tabular and image data. We conclude that ProbLog4Fairness outperforms baselines due to its ability to flexibly model the relevant bias assumptions, where other methods typically uphold a fixed bias type or notion of fairness.

Neurosymbolic AI integrates low-level perception with high-level reasoning, making it well suited for tasks that involve both visual recognition and domain-specific constraints. One such task is the digitization of structured documents like handwritten tax forms, which must satisfy numerous known rules. While neural OCR models are becoming increasingly capable at reading handwritten text, they fail to enforce such constraints on their output, leading to invalid predictions. By combining neural OCR outputs with grammar-based stochastic reasoning over these constraints, neurosymbolic OCR can correct both neural perception errors and user mistakes. This paper demonstrates the application of DeepStochLog, a neurosymbolic AI system, to digitize handwritten IRS 1040 tax forms. Its ability to incorporate background knowledge and ease of use make DeepStochLog an attractive framework for constrained OCR applications.

Neurosymbolic AI (NeSy) aims to integrate the statistical strengths of neural networks with the interpretability and structure of symbolic reasoning. However, current NeSy frameworks like DEEPGRAPHLOG enforce a fixed flow where symbolic reasoning always follows neural processing. This restricts their ability to model complex dependencies, especially in irregular data structures such as graphs. In this work, we introduce DEEPGRAPHLOG, a novel NeSy framework that extends PROBLOG with Graph Neural Predicates. DEEPGRAPHLOG enables multi-layer neural-symbolic reasoning, allowing neural and symbolic components to be layered in arbitrary order. In contrast to DEEPGRAPHLOG, which cannot handle symbolic reasoning via neural methods, DEEPGRAPHLOG treats symbolic representations as graphs, enabling them to be processed by Graph Neural Networks (GNNs). We showcase the capabilities of DEEPGRAPHLOG on tasks in planning, knowledge graph completion with distant supervision, and GNN expressivity. Our results demonstrate that DEEPGRAPHLOG effectively captures complex relational dependencies, overcoming key limitations of existing NeSy systems. By broadening the applicability of neurosymbolic AI to graph-structured domains, DEEPGRAPHLOG offers a more expressive and flexible framework for neural-symbolic integration. Code is available at https://github.com/ML-KULeuven/DeepGraphLog.

Neural-symbolic AI (NeSy) allows neural networks to exploit symbolic background knowledge in the form of logic. It has been shown to aid learning in the limited data regime and to facilitate inference on out-of-distribution data. Probabilistic NeSy focuses on integrating neural networks with both logic and probability theory, which additionally allows learning under uncertainty. A major limitation of current probabilistic NeSy systems, such as DeepProbLog, is their restriction to finite probability distributions, i.e., discrete random variables. In contrast, deep probabilistic programming (DPP) excels in modelling and optimising continuous probability distributions. Hence, we introduce DeepSeaProbLog, a neural probabilistic logic programming language that incorporates DPP techniques into NeSy. Doing so results in the support of inference and learning of both discrete and continuous probability distributions under logical constraints. Our main contributions are 1) the semantics of DeepSeaProbLog and its corresponding inference algorithm, 2) a proven asymptotically unbiased learning algorithm, and 3) a series of experiments that illustrate the versatility of our approach.

Recent advances in neural-symbolic learning, such as DeepProbLog, extend probabilistic logic programs with neural predicates. Like graphical models, these probabilistic logic programs define a probability distribution over possible worlds, for which inference is computationally hard. We propose DeepStochLog, an alternative neural-symbolic framework based on stochastic definite clause grammars, a kind of stochastic logic program. More specifically, we introduce neural grammar rules into stochastic definite clause grammars to create a framework that can be trained end-to-end. We show that inference and learning in neural stochastic logic programming scale much better than for neural probabilistic logic programs. Furthermore, the experimental evaluation shows that DeepStochLog achieves state-of-the-art results on challenging neural-symbolic learning tasks.

DeepProbLog is a neural-symbolic framework that integrates probabilistic logic programming and neural networks. It is realized by providing an interface between the probabilistic logic and the neural networks. Inference in probabilistic neural symbolic methods is hard, since it combines logical theorem proving with probabilistic inference and neural network evaluation. In this work, we make the inference more efficient by extending an approximate inference algorithm from the field of statistical-relational AI. Instead of considering all possible proofs for a certain query, the system searches for the best proof. However, training a DeepProbLog model using approximate inference introduces additional challenges, as the best proof is unknown at the start of training which can lead to convergence towards a local optimum. To be able to apply DeepProbLog on larger tasks, we propose: 1) a method for approximate inference using an A*-like search, called DPLA* 2) an exploration strategy for proving in a neural-symbolic setting, and 3) a parametric heuristic to guide the proof search. We empirically evaluate the performance and scalability of the new approach, and also compare the resulting approach to other neural-symbolic systems. The experiments show that DPLA* achieves a speed up of up to 2-3 orders of magnitude in some cases.

Neuro-symbolic and statistical relational artificial intelligence both integrate frameworks for learning with logical reasoning. This survey identifies several parallels across seven different dimensions between these two fields. These cannot only be used to characterize and position neuro-symbolic artificial intelligence approaches but also to identify a number of directions for further research.

We introduce DeepProbLog, a probabilistic logic programming language that incorporates deep learning by means of neural predicates. We show how existing inference and learning techniques can be adapted for the new language. Our experiments demonstrate that DeepProbLog supports (i) both symbolic and subsymbolic representations and inference, (ii) program induction, (iii) probabilistic (logic) programming, and (iv) (deep) learning from examples. To the best of our knowledge, this work is the first to propose a framework where general-purpose neural networks and expressive probabilistic-logical modeling and reasoning are integrated in a way that exploits the full expressiveness and strengths of both worlds and can be trained end-to-end based on examples.