Neurosymbolic integration (NeSy) blends neural‐network learning with symbolic reasoning. The field can be split between methods injecting hand-crafted rules into neural models, and methods inducing symbolic rules from data. We introduce Logic of Hypotheses (LoH), a novel language that unifies these strands, enabling the flexible integration of data-driven rule learning with symbolic priors and expert knowledge. LoH extends propositional logic syntax with a choice operator, which has learnable parameters and selects a subformula from a pool of options. Using fuzzy logic, formulas in LoH can be directly compiled into a differentiable computational graph, so the optimal choices can be learned via backpropagation. This framework subsumes some existing NeSy models, while adding the possibility of arbitrary degrees of knowledge specification. Moreover, the use of Gödel fuzzy logic and the recently developed Gödel trick yields models that can be discretized to hard Boolean-valued functions without any loss in performance. We provide experimental analysis on such models, showing strong results on tabular data and on two NeSy tasks with a perceptual component.

A fundamental challenge in neurosymbolic systems is applying continuous gradient-based optimization to discrete logical domains. While fuzzy relaxations provide differentiability, they often lack a formal structural alignment with classical logic. In this work, we show that Gödel semantics addresses this limitation through a homomorphism that maps its continuous interpretations to Boolean ones, allowing discrete variables to be encoded while maintaining full differentiability. Building on this foundation, we show that gradient-based optimization on Gödel logic instantiates a discrete local search for Boolean satisfiability. Our formal analysis proves that each optimization step identifies and modifies a single variable within an unsatisfied clause, precisely mimicking the steps of a discrete solver. We identify local optima as the primary limitation of such dynamics and introduce the Gödel Trick, a stochastic reparameterization technique designed to improve the exploration of the solution space. We further show a formal connection between this approach, probabilistic inference, and the Gumbel-Max trick. Experimental results on SAT benchmarks and the Visual Sudoku task validate our theoretical findings, demonstrating that our approach effectively navigates complex combinatorial landscapes and provides a solid foundation for differentiable discrete search.

Abstract Despite significant progress in Embodied AI, current agents largely operate under generic task specifications and struggle to reason about user-specific semantics that naturally arise in human-centered environments. In domestic settings, objects are often associated with particular individuals, requiring agents to interpret personalized instructions such as ownership and preference when navigating and acting in 3D spaces. We introduce PersONAL-3D ( PERS onalized O bject N avigation A nd L ocalization), a benchmark designed to study personalized spatial reasoning in embodied environments. PersONAL-3D focuses on domestic scenarios in which an agent must navigate to target objects associated with specific individuals, given natural-language instructions such as “find Lily’s backpack” . The benchmark includes 2,000+ curated evaluation episodes across 30+ photorealistic HM3D homes. Each episode pairs a natural-language scene description that specifies object ownership with a user-specific query, requiring models to ground personalized semantics in 3D space. PersONAL-3D supports two evaluation settings: (1) Personalized Active Navigation in previously unseen environments, and (2) Personalized Object Grounding in pre-explored scenes or directly on 3D point clouds. Experiments with state-of-the-art baselines reveal a substantial gap to human performance, with the best navigation model underperforming humans by about 45 percentage points in Success Rate and 25 percentage points in Path Efficiency, indicating that current methods struggle to perceive, act, and reason over personalized information in embodied contexts. This work highlights personalization as a critical and largely unsolved challenge for embodied AI systems operating in real-world assistive scenarios.

State-of-the-art approaches for integrating symbolic knowledge with deep learning architectures have demonstrated promising results in static domains. However, methods to handle temporal logic specifications remain underexplored. The only existing approach relies on an explicit representation of a finite-state automaton corresponding to the temporal specification. Instead, we aim at directly injecting the temporal knowledge into the neural model without having to rely on a separate symbolic structure. Specifically, we propose a neurosymbolic framework designed to incorporate temporal logic specifications, expressed in Linear Temporal Logic over finite traces (LTLf), directly into deep learning architectures for sequence-based tasks. We extend the Iterative Local Refinement (ILR) neurosymbolic algorithm, leveraging the recent introduction of fuzzy LTLf interpretations. We name this proposed method Temporal Iterative Local Refinement (T-ILR). We assess T-ILR on an existing benchmark for temporal neurosymbolic architectures, consisting of the classification of image sequences in the presence of temporal knowledge. The results demonstrate improved accuracy and computational efficiency compared to the state-of-the-art method.

Neuro-Symbolic (NeSy) integration combines symbolic reasoning with Neural Networks (NNs) for tasks requiring perception and reasoning. Most NeSy systems rely on continuous relaxation of logical knowledge, and no discrete decisions are made within the model pipeline. Furthermore, these methods assume that the symbolic rules are given. In this paper, we propose Deep Symboilic Learning (DSL), a NeSy system that learns NeSy-functions, i.e., the composition of a (set of) perception functions which map continuous data to discrete symbols, and a symbolic function over the set of symbols. DSL simultaneously learns the perception and symbolic functions while being trained only on their composition (NeSy-function). The key novelty of DSL is that it can create internal (interpretable) symbolic representations and map them to perception inputs within a differentiable NN learning pipeline. The created symbols are automatically selected to generate symbolic functions that best explain the data. We provide experimental analysis to substantiate the efficacy of DSL in simultaneously learning perception and symbolic functions.

Abstract Recent work has shown learning systems can use logical background knowledge to compensate for a lack of labeled training data. Many methods work by creating a loss function that encodes this knowledge. However, often the logic is discarded after training, even if it is still helpful at test time. Instead, we ensure neural network predictions satisfy the knowledge by refining the predictions with an extra computation step. We introduce differentiable refinement functions that find a corrected prediction close to the original prediction. We study how to effectively and efficiently compute these refinement functions. Using a new algorithm called iterative local refinement (ILR), we combine refinement functions to find refined predictions for logical formulas of any complexity. ILR finds refinements on complex SAT formulas in significantly fewer iterations and frequently finds solutions where gradient descent can not. Finally, ILR produces competitive results in the MNIST addition task.