Michelangelo Diligenti

AAAI Conference 2026 Conference Paper

DeepProofLog: Efficient Proving in Deep Stochastic Logic Programs

• Ying Jiao
• Rodrigo Castellano Ontiveros
• Luc De Raedt
• Marco Gori
• Francesco Giannini
• Michelangelo Diligenti
• Giuseppe Marra

Neurosymbolic (NeSy) AI combines neural architectures and symbolic reasoning to improve accuracy, interpretability, and generalization. While logic inference on top of subsymbolic modules has been shown to effectively guarantee these properties, this often comes at the cost of reduced scalability, which can severely limit the usability of NeSy models. This paper introduces DeepProofLog (DPrL), a novel NeSy system based on stochastic logic programs, which addresses the scalability limitations of previous methods. DPrL parameterizes all derivation steps with neural networks, allowing efficient neural guidance over the proving system. Additionally, we establish a formal mapping between the resolution process of our deep stochastic logic programs and Markov Decision Processes, enabling the application of dynamic programming and reinforcement learning techniques for efficient inference and learning. This theoretical connection improves scalability for complex proof spaces and large knowledge bases. Our experiments on standard NeSy benchmarks and knowledge graph reasoning tasks demonstrate that DPrL outperforms existing state-of-the-art NeSy systems, advancing scalability to larger and more complex settings than previously possible.

NeSy Conference 2025 Conference Paper

Distilling KGE black boxes into interpretable NeSy models

• Rodrigo Castellano Ontiveros
• Francesco Giannini
• Michelangelo Diligenti

Knowledge Graph Embedding (KGE) models have shown remarkable performances in the knowledge graph completion task, thanks to their ability to capture and represent complex relational patterns. Indeed, modern KGEs encompass different inductive biases, which can account for relational patterns like reasoning compositional chains, symmetries, anti-symmetries, hierarchical patterns, etc. However, KGE models inherently lack interpretability, as their generalization capabilities are purely focused on mapping human interpretable units of information, like constants and predicates, into vector embeddings in a dense latent space, which is completely opaque to a human operator. On the other hand, different Neural-Symbolic (NeSy) methods have shown competitive results in knowledge completion tasks, but their focus on achieving high accuracy often leads to sacrificing interpretability. Many existing NeSy approaches, while inherently interpretable, resort to blending their predictions with opaque KGEs to boost performance, ultimately diminishing their explanatory power. This paper introduces a novel approach to address this limitation by applying a post-hoc NeSy method to KGE models. This strategy ensures both high fidelity to KGE models and the inherent interpretability of NeSy approaches. The proposed framework defines NeSy reasoners that generate explicit logic proofs using predefined or learned rules, ensuring transparent and explainable predictions. We evaluate the methodology using both accuracy and explainability-based metrics, demonstrating the effectiveness of our approach.

IJCAI Conference 2025 Conference Paper

Grounding Methods for Neural-Symbolic AI

• Rodrigo Castellano Ontiveros
• Francesco Giannini
• Marco Gori
• Giuseppe Marra
• Michelangelo Diligenti

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, but leading to a combinatorial explosion of the number of ground formulas to consider and, therefore, strongly limiting 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 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.

NeurIPS Conference 2024 Conference Paper

Interpretable Concept-Based Memory Reasoning

• David Debot
• Pietro Barbiero
• Francesco Giannini
• Gabriele Ciravegna
• Michelangelo Diligenti
• Giuseppe Marra

The lack of transparency in the decision-making processes of deep learning systems presents a significant challenge in modern artificial intelligence (AI), as it impairs users’ ability to rely on and verify these systems. To address this challenge, Concept Bottleneck Models (CBMs) have made significant progress by incorporating human-interpretable concepts into deep learning architectures. This approach allows predictions to be traced back to specific concept patterns that users can understand and potentially intervene on. However, existing CBMs’ task predictors are not fully interpretable, preventing a thorough analysis and any form of formal verification of their decision-making process prior to deployment, thereby raising significant reliability concerns. To bridge this gap, we introduce Concept-based Memory Reasoner (CMR), a novel CBM designed to provide a human-understandable and provably-verifiable task prediction process. Our approach is to model each task prediction as a neural selection mechanism over a memory of learnable logic rules, followed by a symbolic evaluation of the selected rule. The presence of an explicit memory and the symbolic evaluation allow domain experts to inspect and formally verify the validity of certain global properties of interest for the task prediction process. Experimental results demonstrate that CMR achieves better accuracy-interpretability trade-offs to state-of-the-art CBMs, discovers logic rules consistent with ground truths, allows for rule interventions, and allows pre-deployment verification.

NeurIPS Conference 2024 Conference Paper

Relational Concept Bottleneck Models

• Pietro Barbiero
• Francesco Giannini
• Gabriele Ciravegna
• Michelangelo Diligenti
• Giuseppe Marra

The design of interpretable deep learning models working in relational domains poses an open challenge: interpretable deep learning methods, such as Concept Bottleneck Models (CBMs), are not designed to solve relational problems, while relational deep learning models, such as Graph Neural Networks (GNNs), are not as interpretable as CBMs. To overcome these limitations, we propose Relational Concept Bottleneck Models (R-CBMs), a family of relational deep learning methods providing interpretable task predictions. As special cases, we show that R-CBMs are capable of both representing standard CBMs and message passing GNNs. To evaluate the effectiveness and versatility of these models, we designed a class of experimental problems, ranging from image classification to link prediction in knowledge graphs. In particular we show that R-CBMs (i) match generalization performance of existing relational black-boxes, (ii) support the generation of quantified concept-based explanations, (iii) effectively respond to test-time interventions, and (iv) withstand demanding settings including out-of-distribution scenarios, limited training data regimes, and scarce concept supervisions.

NeurIPS Conference 2022 Conference Paper

Concept Embedding Models: Beyond the Accuracy-Explainability Trade-Off

• Mateo Espinosa Zarlenga
• Pietro Barbiero
• Gabriele Ciravegna
• Giuseppe Marra
• Francesco Giannini
• Michelangelo Diligenti
• Zohreh Shams
• Frederic Precioso

Deploying AI-powered systems requires trustworthy models supporting effective human interactions, going beyond raw prediction accuracy. Concept bottleneck models promote trustworthiness by conditioning classification tasks on an intermediate level of human-like concepts. This enables human interventions which can correct mispredicted concepts to improve the model's performance. However, existing concept bottleneck models are unable to find optimal compromises between high task accuracy, robust concept-based explanations, and effective interventions on concepts---particularly in real-world conditions where complete and accurate concept supervisions are scarce. To address this, we propose Concept Embedding Models, a novel family of concept bottleneck models which goes beyond the current accuracy-vs-interpretability trade-off by learning interpretable high-dimensional concept representations. Our experiments demonstrate that Concept Embedding Models (1) attain better or competitive task accuracy w. r. t. standard neural models without concepts, (2) provide concept representations capturing meaningful semantics including and beyond their ground truth labels, (3) support test-time concept interventions whose effect in test accuracy surpasses that in standard concept bottleneck models, and (4) scale to real-world conditions where complete concept supervisions are scarce.

IJCAI Conference 2021 Conference Paper

Contrastive Losses and Solution Caching for Predict-and-Optimize

• Maxime Mulamba
• Jayanta Mandi
• Michelangelo Diligenti
• Michele Lombardi
• Victor Bucarey
• Tias Guns

Many decision-making processes involve solving a combinatorial optimization problem with uncertain input that can be estimated from historic data. Recently, problems in this class have been successfully addressed via end-to-end learning approaches, which rely on solving one optimization problem for each training instance at every epoch. In this context, we provide two distinct contributions. First, we use a Noise Contrastive approach to motivate a family of surrogate loss functions, based on viewing non-optimal solutions as negative examples. Second, we address a major bottleneck of all predict-and-optimize approaches, i. e. the need to frequently recompute optimal solutions at training time. This is done via a solver-agnostic solution caching scheme, and by replacing optimization calls with a lookup in the solution cache. The method is formally based on an inner approximation of the feasible space and, combined with a cache lookup strategy, provides a controllable trade-off between training time and accuracy of the loss approximation. We empirically show that even a very slow growth rate is enough to match the quality of state-of-the-art methods, at a fraction of the computational cost.

ECAI Conference 2020 Conference Paper

Relational Neural Machines

• Giuseppe Marra
• Michelangelo Diligenti
• Francesco Giannini
• Marco Gori
• Marco Maggini

Deep learning has been shown to achieve impressive results in several tasks where a large amount of training data is available. However, deep learning solely focuses on the accuracy of the predictions, neglecting the reasoning process leading to a decision, which is a major issue in life-critical applications. Probabilistic logic reasoning allows to exploit both statistical regularities and specific domain expertise to perform reasoning under uncertainty, but its scalability and brittle integration with the layers processing the sensory data have greatly limited its applications. For these reasons, combining deep architectures and probabilistic logic reasoning is a fundamental goal towards the development of intelligent agents operating in complex environments. This paper presents Relational Neural Machines, a novel framework allowing to jointly train the parameters of the learners and of a First–Order Logic based reasoner. A Relational Neural Machine is able to recover both classical learning from supervised data in case of pure sub-symbolic learning, and Markov Logic Networks in case of pure symbolic reasoning, while allowing to jointly train and perform inference in hybrid learning tasks. Proper algorithmic solutions are devised to make learning and inference tractable in large-scale problems. The experiments show promising results in different relational tasks.

AAAI Conference 2018 Conference Paper

Characterization of the Convex Łukasiewicz Fragment for Learning From Constraints

• Francesco Giannini
• Michelangelo Diligenti
• Marco Gori
• Marco Maggini

This paper provides a theoretical insight for the integration of logical constraints into a learning process. In particular it is proved that a fragment of the Łukasiewicz logic yields a set of convex constraints. The fragment is enough expressive to include many formulas of interest such as Horn clauses. Using the isomorphism of Łukasiewicz formulas and McNaughton functions, logical constraints are mapped to a set of linear constraints once the predicates are grounded on a given sample set. In this framework, it is shown how a collective classification scheme can be formulated as a quadratic programming problem, but the presented theory can be exploited in general to embed logical constraints into a learning process. The proposed approach is evaluated on a classification task to show how the use of the logical rules can be effective to improve the accuracy of a trained classifier.

AIJ Journal 2017 Journal Article

Semantic-based regularization for learning and inference

• Michelangelo Diligenti
• Marco Gori
• Claudio Saccà

This paper proposes a unified approach to learning from constraints, which integrates the ability of classical machine learning techniques to learn from continuous feature-based representations with the ability of reasoning using higher-level semantic knowledge typical of Statistical Relational Learning. Learning tasks are modeled in the general framework of multi-objective optimization, where a set of constraints must be satisfied in addition to the traditional smoothness regularization term. The constraints translate First Order Logic formulas, which can express learning-from-example supervisions and general prior knowledge about the environment by using fuzzy logic. By enforcing the constraints also on the test set, this paper presents a natural extension of the framework to perform collective classification. Interestingly, the theory holds for both the case of data represented by feature vectors and the case of data simply expressed by pattern identifiers, thus extending classic kernel machines and graph regularization, respectively. This paper also proposes a probabilistic interpretation of the proposed learning scheme, and highlights intriguing connections with probabilistic approaches like Markov Logic Networks. Experimental results on classic benchmarks provide clear evidence of the remarkable improvements that are obtained with respect to related approaches.

NeSy Conference 2011 Conference Paper

Towards Developmental AI: The paradox of Ravenous Intelligent Agents

• Michelangelo Diligenti
• Marco Gori
• Marco Maggini

ECAI Conference 2010 Conference Paper

Multitask Kernel-based Learning with Logic Constraints

• Michelangelo Diligenti
• Marco Gori
• Marco Maggini
• Leonardo Rigutini

This paper presents a general framework to integrate prior knowledge in the form of logic constraints among a set of task functions into kernel machines. The logic propositions provide a partial representation of the environment, in which the learner operates, that is exploited by the learning algorithm together with the information available in the supervised examples. In particular, we consider a multi-task learning scheme, where multiple unary predicates on the feature space are to be learned by kernel machines and a higher level abstract representation consists of logic clauses on these predicates, known to hold for any input. A general approach is presented to convert the logic clauses into a continuous implementation, that processes the outputs computed by the kernel-based predicates. The learning task is formulated as a primal optimization problem of a loss function that combines a term measuring the fitting of the supervised examples, a regularization term, and a penalty term that enforces the constraints on both supervised and unsupervised examples. The proposed semi-supervised learning framework is particularly suited for learning in high dimensionality feature spaces, where the supervised training examples tend to be sparse and generalization difficult. Unlike for standard kernel machines, the cost function to optimize is not generally guaranteed to be convex. However, the experimental results show that it is still possible to find good solutions using a two stage learning schema, in which first the supervised examples are learned until convergence and then the logic constraints are forced. Some promising experimental results on artificial multi-task learning tasks are reported, showing how the classification accuracy can be effectively improved by exploiting the a priori rules and the unsupervised examples.

IJCAI Conference 2005 Conference Paper

Learning Web Page Scores by Error Back-Propagation

• Michelangelo Diligenti
• Marco Gori
• Marco

In this paper we present a novel algorithm to learn a score distribution over the nodes of a labeled graph (directed or undirected). Markov Chain theory is used to define the model of a random walker that converges to a score distribution which depends both on the graph connectivity and on the node labels. A supervised learning task is defined on the given graph by assigning a target score for some nodes and a training algorithm based on error backpropagation through the graph is devised to learn the model parameters. The trained model can assign scores to the graph nodes generalizing the criteria provided by the supervisor in the examples. The proposed algorithm has been applied to learn a ranking function for Web pages. The experimental results show the effectiveness of the proposed technique in reorganizing the rank accordingly to the examples provided in the training set.

TCS Journal 2004 Journal Article

Neural computation, social networks, and topological spectra

• Michelangelo Diligenti
• Marco Gori
• Marco Maggini

This paper emphasizes some intriguing links between neural computation on graphical domains and social networks, like those used in nowadays search engines to score the page authority. It is pointed out that the introduction of web domains creates a unified mathematical framework for these computational schemes. It is shown that one of the major limitations of currently used connectionist models, namely their scarce ability to capture the topological features of patterns, can be effectively faced by computing the node rank according to social-based computation, like Google's PageRank. The main contribution of the paper is the introduction of a novel graph spectral notion, which can be naturally used for the graph isomorphism problem. In particular, a class of graphs is introduced for which the problem is proven to be polynomial. It is also pointed out that the derived spectral representations can be nicely combined with learning, thus opening the doors to many applications typically faced within the framework of neural computation.

IJCAI Conference 2003 Conference Paper

A Learning Algorithm for Web Page Scoring Systems

• Michelangelo Diligenti
• Marco Gori
• Marco Maggini

Hyperlink analysis is a successful approach to define algorithms which compute the relevance of a document on the basis of the citation graph. In this paper we propose a technique to learn the parameters of the page ranking model using a set of pages labeled as relevant or not relevant by a supervisor. In particular we describe a learning algorithm applied to a scheme similar to PageRank. The ranking algorithm is based on a probabilistic Web surfer model and its parameters are optimized in order to increase the probability of the surfer to visit a page labeled as relevant and to reduce it for the pages labeled as not relevant. The experimental results show the effectiveness of the proposed technique in reorganizing the page ordering in the ranking list accordingly to the examples provided in the learning set.