In this paper, we address the problem of giving names to predicates in logic rules using Large Language Models (LLMs). In the context of Inductive Logic Programming, various rule generation methods produce rules containing unnamed predicates, with Predicate Invention being a key example. This hinders the readability, interpretability, and reusability of the logic theory. Leveraging recent advancements in LLMs development, we explore their ability to process natural language and code to provide semantically meaningful suggestions for giving a name to unnamed predicates. The evaluation of our approach on some hand-crafted logic rules indicates that LLMs hold potential for this task.

Despite impressive results on curated benchmarks, the practical impact of large language models (LLMs) on research-level neural theorem proving and proof autoformalization is still limited. We introduce RLMEval, an evaluation suite for these tasks, focusing on research-level mathematics from real-world Lean formalization projects. RLMEval targets the evaluation of neural theorem proving and proof autoformalization on challenging research-level theorems by leveraging real Lean Blueprint formalization projects. Our evaluation of state-of-the-art models on RLMEval, comprising 613 theorems from 6 Lean projects, reveals a significant gap: progress on existing benchmarks does not readily translate to these more realistic settings, with the best model achieving only a 10.3 % pass rate. RLMEval provides a new, challenging benchmark designed to guide and accelerate progress in automated reasoning for formal mathematics.

Graph Neural Networks (GNNs) are widely used for node classification, yet their opaque decision-making limits trust and adoption. While local explanations offer insights into individual predictions, global explanation methods, those that characterize an entire class, remain underdeveloped. Existing global explainers rely on motif discovery in small graphs, an approach that breaks down in large, real-world settings where subgraph repetition is rare, node attributes are high-dimensional, and predictions arise from complex structure-attribute interactions. We propose GnnXemplar, a novel global explainer inspired from Exemplar Theory from cognitive science. GnnXemplar identifies representative nodes in the GNN embedding space, exemplars, and explains predictions using natural language rules derived from their neighborhoods. Exemplar selection is framed as a coverage maximization problem over reverse k-nearest neighbors, for which we provide an efficient greedy approximation. To derive interpretable rules, we employ a self-refining prompt strategy using large language models (LLMs). Experiments across diverse benchmarks show that GnnXemplar significantly outperforms existing methods in fidelity, scalability, and human interpretability, as validated by a user study with 60 participants.

In recent years, the expressive power of various neural architectures -- including graph neural networks (GNNs), transformers, and recurrent neural networks -- has been characterised using tools from logic and formal language theory. As the capabilities of basic architectures are becoming well understood, increasing attention is turning to models that combine multiple architectural paradigms. Among them particularly important, and challenging to analyse, are temporal extensions of GNNs, which integrate both spatial (graph-structure) and temporal (evolution over time) dimensions. In this paper, we initiate the study of logical characterisation of temporal GNNs by connecting them to two-dimensional product logics. We show that the expressive power of temporal GNNs depends on how graph and temporal components are combined. In particular, temporal GNNs that apply static GNNs recursively over time can capture all properties definable in the product logic of (past) propositional temporal logic PTL and the modal logic K. In contrast, architectures such as graph-and-time TGNNs and global TGNNs can only express restricted fragments of this logic, where the interaction between temporal and spatial operators is syntactically constrained. These provide us with the first results on the logical expressiveness of temporal GNNs.

Inductive logic programming (ILP) is a form of logical machine learning. Most ILP algorithms learn a single hypothesis from a single training run. Ensemble methods train an ILP algorithm multiple times to learn multiple hypotheses. In this paper, we train an ILP algorithm only once and save intermediate hypotheses. We then combine the hypotheses using a minimum description length weighting scheme. Our experiments on multiple benchmarks, including game playing and visual reasoning, show that our approach improves predictive accuracy by 4% with less than 1% computational overhead.

A large class of Neural-Symbolic (NeSy) methods employs a machine learner to process the input entities, while relying on a reasoner based on First-Order Logic to represent and process more complex relationships among the entities. A fundamental role for these methods is played by the process of logic grounding, which determines the relevant substitutions for the logic rules using a (sub)set of entities. Some NeSy methods use an exhaustive derivation of all possible substitutions, preserving the full expressive power of the logic knowledge. This leads to a combinatorial explosion in the number of ground formulas to consider and, therefore, strongly limits their scalability. Other methods rely on heuristic-based selective derivations, which are generally more computationally efficient, but lack a justification and provide no guarantees of preserving the information provided to and returned by the reasoner. Taking inspiration from multi-hop symbolic reasoning, this paper proposes a parametrized family of grounding methods generalizing classic Backward Chaining. Different selections within this family allow us to obtain commonly employed grounding methods as special cases, and to control the trade-off between expressiveness and scalability of the reasoner. The experimental results show that the selection of the grounding criterion is often as important as the NeSy method itself.

Designing models that can learn to reason in a systematic way is an important and long-standing challenge. In recent years, a wide range of solutions have been proposed for the specific case of systematic relational reasoning, including Neuro-Symbolic approaches, variants of the Transformer architecture, and specialised Graph Neural Networks. However, existing benchmarks for systematic relational reasoning focus on an overly simplified setting, based on the assumption that reasoning can be reduced to composing relational paths. In fact, this assumption is hard-baked into the architecture of several recent models, leading to approaches that can perform well on existing benchmarks but are difficult to generalise to other settings. To support further progress in the field of systematic relational reasoning with neural networks, we introduce NoRA, a new benchmark which adds several levels of difficulty and requires models to go beyond path-based reasoning.

We investigate a relatively underexplored class of hybrid neurosymbolic models integrating symbolic learning with neural reasoning to construct data generators meeting formal correctness criteria. In \textit{Symbolic Neural Generators} (SNGs), symbolic learners examine logical specifications of feasible data from a small set of instances -- sometimes just one. Each specification in turn constrains the conditional information supplied to a neural-based generator, which rejects any instance violating the symbolic specification. Like other neurosymbolic approaches, SNG exploits the complementary strengths of symbolic and neural methods. The outcome of an SNG is a triple $(H, X, W)$, where $H$ is a symbolic description of feasible instances constructed from data, $X$ a set of generated new instances that satisfy the description, and $W$ an associated weight. We introduce a semantics for such systems, based on the construction of appropriate \textit{base} and \textit{fibre} partially-ordered sets combined into an overall partial order, and outline a probabilistic extension relevant to practical applications. In this extension, SNGs result from searching over a weighted partial ordering. We implement an SNG combining a restricted form of Inductive Logic Programming (ILP) with a large language model (LLM) and evaluate it on early-stage drug design. Our main interest is the description and the set of potential inhibitor molecules generated by the SNG. On benchmark problems -- where drug targets are well understood -- SNG performance is statistically comparable to state-of-the-art methods. On exploratory problems with poorly understood targets, generated molecules exhibit binding affinities on par with leading clinical candidates. Experts further find the symbolic specifications useful as preliminary filters, with several generated molecules identified as viable for synthesis and wet-lab testing.

Cognitive studies and artificial intelligence have developed distinct models for various inferential mechanisms (categorization, induction, abduction, causal inference, contrast, merge, ...). Yet, both natural and artificial views on cognition lack apparently a unifying framework. This paper formulates a speculative answer attempting to respond to this gap. To postulate on higher-level activation processes from a material perspective, we consider inferential mechanisms informed by symbolic AI modelling techniques, through the simplistic lenses of electronic circuits based on logic gates. We observe that a logic gate view entails a different treatment of implication and negation compared to standard logic and logic programming. Then, by combinatorial exploration, we identify four main forms of dependencies that can be realized by these inferential circuits. Looking at how these forms are generally used in the context of logic programs, we identify eight common inferential patterns, exposing traditionally distinct inferential mechanisms in an unifying framework. Finally, following a probabilistic interpretation of logic programs, we unveil inner functional dependencies. The paper concludes elaborating in what sense, even if our arguments are mostly informed by symbolic means and digital systems infrastructures, our observations may pinpoint to more generally applicable structures.

Modern generative models excel at producing realistic samples--images, text, molecules-- but often lack guarantees about structural properties. A protein generator may produce plausible sequences that violate stability constraints; a network topology generator may create graphs that fail connectivity requirements; a molecule generator may output structures violating chemical valence rules. Standard GAN discriminators provide a global "real vs. fake" signal, but they cannot pinpoint which specific structural constraint failed or guarantee that generated samples satisfy formal specifications. Meanwhile, mathematical logic has developed precise tools for reasoning about structural properties. Ehrenfeucht-Fraïssé (EF) games [1, 2] characterize when two structures are indistinguishable by logical formulas up to a given complexity (quantifier depth k). First-order (FO) and monadic second-order (MSO) logics can express rich structural properties--connectivity, bipartiteness, planarity, acyclicity--that are crucial in applications but invisible to standard discriminators.