Independence Is Not an Issue in Neurosymbolic AI

A popular approach to neurosymbolic AI is to take the output of the last layer of a neural network, e.g. a softmax activation, and pass it through a sparse computation graph encoding certain logical constraints one wishes to enforce. This induces a probability distribution over a set of random variables, which happen to be conditionally independent of each other in many commonly used neurosymbolic AI models. Such conditionally independent random variables have been deemed harmful as their presence has been observed to co-occur with a phenomenon dubbed deterministic bias, where systems learn to deterministically prefer one of the valid solutions from the solution space over the others. We provide evidence contesting this conclusion and show that the phenomenon of deterministic bias is an artifact of improperly applying neurosymbolic AI.

Have Large Language Models Learned to Reason? A Characterization via 3-SAT Phase Transition

Large Language Models (LLMs) have been touted as AI models possessing advanced reasoning abilities. In theory, autoregressive LLMs with Chain-of-Thought (CoT) can perform more serial computations to solve complex reasoning tasks. However, recent studies suggest that, despite this capacity, LLMs do not truly learn to reason but instead fit on statistical features. To study the reasoning capabilities in a principled fashion, we adopt a computational theory perspective and propose an experimental protocol centered on 3-SAT -- the prototypical NP-complete problem lying at the core of logical reasoning and constraint satisfaction tasks. Specifically, we examine the phase transitions in random 3-SAT and characterize the reasoning abilities of state-of-the-art LLMs by varying the inherent hardness of the problem instances. By comparing DeepSeek R1 with other LLMs, our findings reveal two key insights (1) LLM accuracy drops significantly on harder instances, suggesting all current models struggle when statistical shortcuts are unavailable (2) Unlike other LLMs, R1 shows signs of having learned the underlying reasoning. Following a principled experimental protocol, our study moves beyond the benchmark-driven evidence often found in LLM reasoning research. Our findings highlight important gaps and suggest clear directions for future research.

Neurosymbolic Decision Trees

Neurosymbolic (NeSy) AI studies the integration of neural networks (NNs) and symbolic reasoning based on logic. Usually, NeSy techniques focus on learning the neural, probabilistic and/or fuzzy parameters of NeSy models. Learning the symbolic or logical structure of such models has, so far, received less attention. We introduce neurosymbolic decision trees (NDTs), as an extension of decision trees together with a novel NeSy structure learning algorithm, which we dub NeuID3. NeuID3 adapts the standard top-down induction of decision tree algorithms and combines it with a neural probabilistic logic representation, inherited from the DeepProbLog family of models. The key advantage of learning NDTs with NeuID3 is the support of both symbolic and subsymbolic data (such as images), and that they can exploit background knowledge during the induction of the tree structure, In our experimental evaluation we demonstrate the benefits of NeSys structure learning over more traditonal approaches such as purely data-driven learning with neural networks.

A Fast Convoluted Story: Scaling Probabilistic Inference for Integer Arithmetic

As illustrated by the success of integer linear programming, linear integer arithmetic is a powerful tool for modelling combinatorial problems. Furthermore, the probabilistic extension of linear programming has been used to formulate problems in neurosymbolic AI. However, two key problems persist that prevent the adoption of neurosymbolic techniques beyond toy problems. First, probabilistic inference is inherently hard, #P-hard to be precise. Second, the discrete nature of integers renders the construction of meaningful gradients challenging, which is problematic for learning. In order to mitigate these issues, we formulate linear arithmetic over integer-valued random variables as tensor manipulations that can be implemented in a straightforward fashion using modern deep learning libraries. At the core of our formulation lies the observation that the addition of two integer-valued random variables can be performed by adapting the fast Fourier transform to probabilities in the log-domain. By relying on tensor operations we obtain a differentiable data structure, which unlocks, virtually for free, gradient-based learning. In our experimental validation we show that tensorising probabilistic linear integer arithmetic and leveraging the fast Fourier transform allows us to push the state of the art by several orders of magnitude in terms of inference and learning times.

KLay: Accelerating Neurosymbolic AI

A popular approach to neurosymbolic AI involves mapping logic formulas to arithmetic circuits (computation graphs consisting of sums and products) and passing the outputs of a neural network through these circuits. This approach enforces symbolic constraints onto a neural network in a principled and end-to-end differentiable way. Unfortunately, arithmetic circuits are challenging to run on modern AI accelerators as they exhibit a high degree of irregular sparsity. To address this limitation, we introduce knowledge layers (KLay), a new data structure to represent arithmetic circuits that can be efficiently parallelized on GPUs. Moreover, we contribute two algorithms used in the translation of traditional circuit representations to KLay and a further algorithm that exploits parallelization opportunities during circuit evaluations. We empirically show that KLay achieves speedups of multiple orders of magnitude over the state of the art, thereby paving the way towards scaling neurosymbolic AI to larger real-world applications.

Can Large Language Models Reason? A Characterization via 3-SAT

Large Language Models (LLMs) are said to possess advanced reasoning abilities. However, some skepticism exists as recent works show how LLMs often bypass true reasoning using shortcuts. Current methods for assessing the reasoning abilities of LLMs typically rely on open-source benchmarks that may be overrepresented in LLM training data, potentially skewing performance. We instead provide a computational theory perspective of reasoning, using 3-SAT -- the prototypical NP-complete problem that lies at the core of logical reasoning and constraint satisfaction tasks. By examining the phase transitions in 3-SAT, we empirically characterize the reasoning abilities of LLMs and show how they vary with the inherent hardness of the problems. Our experimental evidence shows that LLMs cannot perform true reasoning, as is required for solving 3-SAT problems.

REvolve: Reward Evolution with Large Language Models for Autonomous Driving

Designing effective reward functions is crucial to training reinforcement learning (RL) algorithms. However, this design is non-trivial, even for domain experts, due to the subjective nature of certain tasks that are hard to quantify explicitly. In recent works, large language models (LLMs) have been used for reward generation from natural language task descriptions, leveraging their extensive instruction tuning and commonsense understanding of human behavior. In this work, we hypothesize that LLMs, guided by human feedback, can be used to formulate human-aligned reward functions. Specifically, we study this in the challenging setting of autonomous driving (AD), wherein notions of "good" driving are tacit and hard to quantify. To this end, we introduce REvolve, an evolutionary framework that uses LLMs for reward design in AD. REvolve creates and refines reward functions by utilizing human feedback to guide the evolution process, effectively translating implicit human knowledge into explicit reward functions for training (deep) RL agents. We demonstrate that agents trained on REvolve-designed rewards align closely with human driving standards, thereby outperforming other state-of-the-art baselines.

Probabilistic Neural Circuits

Probabilistic circuits (PCs) have gained prominence in recent years as a versatile framework for discussing probabilistic models that support tractable queries and are yet expressive enough to model complex probability distributions. Nevertheless, tractability comes at a cost: PCs are less expressive than neural networks. In this paper we introduce probabilistic neural circuits (PNCs), which strike a balance between PCs and neural nets in terms of tractability and expressive power. Theoretically, we show that PNCs can be interpreted as deep mixtures of Bayesian networks. Experimentally, we demonstrate that PNCs constitute powerful function approximators.

Semirings for Probabilistic and Neuro-Symbolic Logic Programming

The field of probabilistic logic programming (PLP) focuses on integrating probabilistic models into programming languages based on logic. Over the past 30 years, numerous languages and frameworks have been developed for modeling, inference and learning in probabilistic logic programs. While originally PLP focused on discrete probability, more recent approaches have incorporated continuous distributions as well as neural networks, effectively yielding neural-symbolic methods. We provide a unified algebraic perspective on PLP, showing that many if not most of the extensions of PLP can be cast within a common algebraic logic programming framework, in which facts are labeled with elements of a semiring and disjunction and conjunction are replaced by addition and multiplication. This does not only hold for the PLP variations itself but also for the underlying execution mechanism that is based on (algebraic) model counting.

Differentiable Sampling of Categorical Distributions Using the CatLog-Derivative Trick

Categorical random variables can faithfully represent the discrete and uncertain aspects of data as part of a discrete latent variable model. Learning in such models necessitates taking gradients with respect to the parameters of the categorical probability distributions, which is often intractable due to their combinatorial nature. A popular technique to estimate these otherwise intractable gradients is the Log-Derivative trick. This trick forms the basis of the well-known REINFORCE gradient estimator and its many extensions. While the Log-Derivative trick allows us to differentiate through samples drawn from categorical distributions, it does not take into account the discrete nature of the distribution itself. Our first contribution addresses this shortcoming by introducing the CatLog-Derivative trick - a variation of the Log-Derivative trick tailored towards categorical distributions. Secondly, we use the CatLog-Derivative trick to introduce IndeCateR, a novel and unbiased gradient estimator for the important case of products of independent categorical distributions with provably lower variance than REINFORCE. Thirdly, we empirically show that IndeCateR can be efficiently implemented and that its gradient estimates have significantly lower bias and variance for the same number of samples compared to the state of the art.