Ebook: Handbook on Neurosymbolic AI and Knowledge Graphs

Neurosymbolic integration has emerged as a very active subfield of artificial intelligence research. Historically, neural approaches have excelled at perceptual tasks—such as pattern recognition in images and natural language—while symbolic frameworks have offered powerful methods for knowledge representation, logical inference, and interpretability. Today’s AI landscape is increasingly defined by hybrid systems that blend these complementary paradigms, enabling both sophisticated pattern recognition and robust reasoning grounded in explicit knowledge. This confluence is particularly relevant in the context of knowledge graphs (KGs), which provide a structured, machine-readable representation of relational data and domain ontologies. KGs serve as a bridge between symbolic logic and the subsymbolic world of deep learning, bringing forward a new era of data-driven discovery and explainable AI.

This book was conceived to capture and organize state-of-the-art neurosymbolic and knowledge graph-based AI. It reflects an ecosystem in which large language models, deep neural networks, and symbolic representations converge. The chapters collected here illustrate the progress that has been made in bridging data-driven machine learning with the semantic richness of KGs, while also uncovering emerging challenges in trustworthiness, interpretability, and scalability. The primary aim is to provide researchers, practitioners, and students with a comprehensive overview of these intersecting fields—unifying theoretical foundations with practical insights and real-world applications.

We have intentionally designed this volume’s structure to move from Foundations and Representation to Construction, Reasoning, Interpretability, and, ultimately, Applications. After setting the conceptual stage in Foundations of Neural and Symbolic AI, the book explores the nuances of Knowledge Graph Representation and Embeddings, where readers gain insight into the cutting-edge methods of turning symbolic knowledge into continuous vector spaces. We then move on to Knowledge Graph Construction, Integration, and Quality, covering key challenges such as entity alignment, canonicalization, fusion, and the critical aspect of uncertainty management. Next, in Neurosymbolic Reasoning and Hybrid Architectures, readers will find an anthology of solutions that seamlessly combine symbolic logic with deep learning pipelines—spanning question answering, program synthesis, dynamic KG methods, and beyond. The subsequent section, Explainable and Interpretable AI, addresses one of the most pressing needs in modern AI: ensuring transparency, accountability, and trust in systems that operate on increasingly complex data. Finally, Interdisciplinary Perspectives and Real-World Applications showcase how these methods tackle domain-specific problems across news analytics, literary studies, life sciences, food computing, social media, and more—demonstrating the far-reaching impact of knowledge-infused neural AI.

Our hope is that this book not only advances the conversation in neurosymbolic integration and knowledge graphs but also serves as a practical guide for building AI systems that are robust, transparent, and ethically grounded. We see this volume as a starting point for researchers and developers who wish to create applications that meaningfully combine data-driven models with explicit symbolic reasoning. By tying together fundamental concepts, advanced methods, and real-world scenarios, we strive to offer a well-rounded perspective suitable for graduate students, academic researchers, and industrial practitioners alike.

We extend our sincere gratitude to the authors and contributors whose expertise, vision, and dedication made this volume possible. Their chapters reflect a remarkable diversity of approaches, each reinforcing the notion that the synergy between symbolic and neural methodologies is indispensable for the future of AI. We also thank our readers for embarking on this journey. May these pages spark your curiosity, inform your practice, and inspire breakthroughs at the vibrant intersection of knowledge graphs, deep learning, and symbolic reasoning.

Pascal Hitzler

Abhilekha Dalal

Mohammad Saeid Mahdavinejad

Sanaz Saki Norouzi

Manhattan, Kansas, December 2024

This chapter offers a detailed overview of key concepts in deep learning. In addition to traditional Neural Networks, several deep learning models such as Convolutional Neural Networks, Recurrent Neural Networks, Transformers, Generative Adversarial Networks, Graph Neural Networks, and Autoencoders are reviewed. Each model is described in terms of its architecture, functionality, and primary applications. The chapter also highlights the significance of popular deep learning frameworks in facilitating model implementation and deployment.

The Tensor Brain (TB) has been introduced as a computational model for perception and memory. This paper provides an overview of the TB model, incorporating recent developments and insights into its functionality. The TB is composed of two primary layers: the representation layer and the index layer. The representation layer serves as a model for the subsymbolic global workspace, a concept derived from consciousness research. Its state represents the cognitive brain state, capturing the dynamic interplay of sensory and cognitive processes. The index layer, in contrast, contains symbolic representations for concepts, time instances, and predicates. In a bottom-up operation, sensory input activates the representation layer, which then triggers associated symbolic labels in the index layer. Conversely, in a top-down operation, symbols in the index layer activate the representation layer, which in turn influences earlier processing layers through embodiment. This top-down mechanism underpins semantic memory, enabling the integration of abstract knowledge into perceptual and cognitive processes. A key feature of the TB is its use of concept embeddings, which function as connection weights linking the index layer to the representation layer. As a concept’s “DNA,” these embeddings consolidate knowledge from diverse experiences, sensory modalities, and symbolic representations, providing a unified framework for learning and memory. Although the TB is primarily a computational model, it has been hypothesized to reflect certain aspects of actual brain function. Notably, the sequential generation of symbols in the TB may represent a precursor to the development of natural language. The model incorporates an attention mechanism and supports multitasking through multiplexing, simulating the brain’s ability to rapidly switch between mental states. Additionally, the TB emphasizes multimodality, with the representation layer integrating inputs across multiple sensory and cognitive dimensions.

Graph neural networks are prominent models for representation learning over graph-structured data. While the capabilities and limitations of these models are well-understood for simple graphs, our understanding remains incomplete in the context of knowledge graphs. Our goal is to provide a systematic understanding of the landscape of graph neural networks for knowledge graphs pertaining to the prominent task of link prediction. Our analysis entails a unifying perspective on seemingly unrelated models and unlocks a series of other models. The expressive power of various models is characterized via a precise logical characterization of the class of functions captured by a class of graph neural networks. The theoretical findings presented in this paper explain the benefits of some widely employed practical design choices, which are validated empirically.

Graph representation learning has garnered significant attention due to its outstanding performance across numerous real-world applications, such as social network analysis, bioinformatics, and recommendation systems. However, supervised graph representation learning models often struggle with label sparsity, as data labeling is time-consuming and resource-intensive. To address this, few-shot learning on graphs (FSLG) has been proposed, combining the strengths of graph representation learning and few-shot learning to mitigate performance issues caused by limited annotated data. This chapter comprehensively presents the body of work in FSLG. It begins by introducing the challenges and foundational concepts of FSLG. The chapter categorizes and summarizes existing FSLG research into three major graph mining tasks at different granularity levels: node, edge, and graph. Node-level tasks involve predicting labels for individual nodes, edge-level tasks focus on predicting relationships between nodes, and graph-level tasks involve predicting properties of entire graphs. This organization provides a clear overview of various methodologies and applications in FSLG. Finally, the chapter discusses potential future research directions in FSLG, which aims to inspire further investigation and innovation in FSLG, advancing its development and application to create more effective AI solutions.

Transformer-based language models are known to encode factual information in their hidden representations, however, an open question is how to explicate this implicitly encoded knowledge. Approaches to unveil encoded factual associations relate to prompt-based methods, techniques to localize or edit specific model weights, and to trace the information flow during inference. Inspired by evidence of entire dependency graphs in embedding (sub-)spaces of these models, this chapter presents an approach to utilize linear transformation of a model’s vector representation space to uncover the properties of knowledge graphs. Subjects and objects of factual knowledge graph triples, i.e., from Wikidata, are expected to be near neighbors, a proximity of knowledge entity embeddings that can be measured with simple cosine similarity and approximated by training a transformation matrix in a supervised manner on minimizing the distance between the actual and predicted similarity scores. The obtained results are promising in terms of Pearson and Spearman correlations as well as mean squared error rates, indicating that there is a specific vector sub-space that encodes factual associations in the last layer of neural language models.

Classic machine learning tasks, such as clustering and link prediction, can be applied to Knowledge Graphs making used of the so-called graph embeddings, mathematical vector representations of the nodes present within the graph structure. Often, the data structure of Knowledge Graphs in Digital Humanities is at the same time versatile and complex, challenging the machine learning tasks. In this work, we compare algorithms on two different subgraphs extracted from a large knowledge graph developed in the cultural heritage domain: one is randomly selected while the other is built to maximise the triple density. Using the European Olfactory Knowledge Graph (EOKG) as use-case, we show that embedding dense subgraph can improve the performances of state-of-art algorithms.

Knowledge Graph Embeddings (KGEs) are widely used in many tasks beyond link prediction, such as node classification, analogical reasoning, or as background knowledge, e.g., for recommender systems. While they exhibit good results, little work has been done in understanding what different knowledge graph embedding methods actually learn. In this chapter, we present two studies: the first one analyzes which ontological patterns KGEs are capable of capturing and which they are not. The second one takes a closer look at the similarity relation in different embedding spaces, analyzing how reliable the common KGE entity similarity assumption – stating that similar entities are projected close to one another in the embedding space – actually is.

Over 100 billions RDF assertions are available on the Web. With this high availability of knowledge expressed in RDF knowledge bases comes the need to make them amenable to Web-scale machine learning. Knowledge graph embeddings cater for enabling the use of knowledge bases in neural settings. However, they do so by discarding the explicit semantics that underpin knowledge bases. In contrast, class expression learning makes explicit use of the semantics of RDF knowledge bases expressed in description logics. In contrast to neural approaches, this form of machine learning generates models that can be translated into natural language and can thus be understood by domain experts. However, most implementations of this paradigm fail to scale to the large knowledge bases found on the Web and in real-life applications. The corresponding literature suggests that one common bottleneck of these approaches is the instance retrieval function. We address this drawback by introducing an approach based on worst-case optimal multi-way joins for the evaluation of SPARQL queries that correspond to ALC class expressions. We implement our algorithm into a tensor-based triple store and use this triple store as backend to efficiently answer retrieval queries in ALC under the closed-world assumption. We evaluate the implementation of our approach on five benchmark datasets against four state-of-the-art graph storage solutions for RDF knowledge graphs. The results of our extensive evaluation show that our approach outperforms its competition across all datasets and that it is the only one able to scale to large datasets. With our approach, class expression learning can now be used on Web-scale knowledge bases.

Geometric relational embeddings map relational data as geometric objects that combine vector information suitable for machine learning and structured/symbolic information suitable for symbolic reasoning, typically in low dimensions. Their preservation of relational structures and their appealing properties and interpretability have led to their uptake for multiple neuro-symbolic reasoning tasks such as knowledge graph completion, ontology and hierarchy reasoning, logical query answering, and hierarchical multi-label classification. We survey methods that underly geometric relational embeddings and categorize them based on (i) the embedding geometries that are used to represent the data; and (ii) the relational reasoning tasks that they aim to improve. We identify the desired properties (i.e., inductive biases) of each kind of embedding and discuss some potential future work.

LLMs, trained on a diverse range of resources, internalize a substantial amount of knowledge. Understanding this internalized knowledge is crucial for assessing the accuracy and reliability of their outputs, guiding improvements in model design, and addressing potential biases. In this chapter, we explore two primary forms of knowledge within LLMs: taxonomy and factual knowledge. We delve into the nature of internalized knowledge, examining how these models store and utilize information. Additionally, we introduce CRAG, a comprehensive benchmark for evaluating both the internalized knowledge of LLMs and their performance in retrieved-augmented generation tasks.

In recent years, Knowledge Graphs (KGs) have sprung up in various fields to support a variety of applications, like question answering, recommendations, etc. When integrating knowledge from different KGs, a common task is to identify some equivalent entities from multiple KGs describing real-world objects, which is called the entity Alignment (EA) task. In recent years, unsupervised EA methods have attracted the interest of many researchers and achieved many research results. However, there are few literatures that comprehensively compare and analyze the modeling frameworks, research ideas and techniques of these methods. This chapter focuses on the research ideas and status of this field in order to provide practical insights. We summarize and analyze 22 unsupervised EA methods that have emerged in recent years, and propose a general framework for unsupervised EA methods. Then the related methods are divided into two categories: EA methods with training and EA methods without training, and the characteristics and implementation techniques of the related methods are analyzed. In addition, the current research challenges in this field are summarized and future research directions are prospected.

Curated knowledge graphs (CKGs) play a fundamental role in both academia and industry. They require significant human involvement to pre-define the ontology and cannot quickly adapt to new domains and new data. To solve this problem, open information extraction (OIE) methods are leveraged to automatically extract structure information in the form of non-canonicalized triples <noun phrase, relation phrase, noun phrase> from unstructured text. OIE can be used to create large open knowledge graphs (OKGs). However, noun phrases and relation phrases in such OKGs are not canonicalized, which results in scattered and redundant facts. In order to disambiguate and eliminate redundancy in such OKGs, the task of OKG canonicalization is proposed to cluster synonymous noun phrases and relation phrases into the same group and assign them unique identifiers. Nevertheless, this task is challenging due to the high sparsity and limited information of OKGs. This chapter provides an overview and analysis of the neuro-symbolic techniques used in this task.

Knowledge fusion aims to integrate different knowledge graphs into a unified, consistent, and concise format, thereby establishing interoperability for applications that leverage multi-source knowledge. In this chapter, we introduce the research process of knowledge fusion. We begin with an exploration of the fundamental principles of multi-source knowledge fusion, followed by a survey of the technical advancements in knowledge fusion. Subsequently, we outline potential future research directions and conclude with a summary. We believe that the combination and mutual promotion of knowledge fusion and other knowledge graph technologies will make a big difference.

This chapter traverses the evolution of Artificial Intelligence (AI) from structured, deterministic systems to those proficient in managing ambiguity and uncertainty. It begins by detailing methods of knowledge modeling using ontologies, rules, and texts to establish a foundation of certain knowledge. The discourse progresses to explain how knowledge is symbolically and parametrically represented, notably through advanced language models, enhancing the depth and utility of AI systems. Further, the chapter explores the dynamic processes of knowledge acquisition, emphasizing the extraction of intricate data patterns and probing the embedded knowledge within pre-trained language models. It highlights techniques like Retrieval-Augmented Generation (RAG) and symbol-guided parametric knowledge editing to enhance AI capabilities. Concluding with the handling of uncertain knowledge, the chapter underscores strategies enabling AI to operate effectively amidst incomplete and ambiguous information, illustrating the shift towards systems that can robustly navigate the complexities of real-world applications. This concise overview encapsulates the core advancements in AI knowledge processes, marking a critical evolution from certainty to the proficient management of uncertainty.

Knowledge graphs (KGs) are essential in human-centered AI by reducing the need for extensive labeled machine-learning datasets, enhancing retrieval-augmented generation, and facilitating explanations. However, modern KG construction has evolved into a complex, semi-automated process, increasingly reliant on opaque deep-learning models and a multitude of heterogeneous data sources to achieve the scale and meet the requirements of downstream applications. As such, the KG lifecycle poses the same risks as any other AI context; designing AI assistants for KG construction requires a deeper understanding of emerging knowledge engineering practices, with a greater focus on automation with human oversight, as well as bespoke solutions to ensure transparency, accountability, fairness, and other trustworthy AI concerns. This chapter undertakes a comprehensive exploration of trustworthy KG construction through a systematic literature review combined with a series of use cases and user studies. It highlights tasks in the KG lifecycle where automation is underexplored, introduces new AI assistants for such KG tasks, and discusses gaps in research, datasets, and capabilities to use AI responsibly.

This article presents the implementation of a neuro-symbolic system within the Open Research Knowledge Graph (ORKG), a platform designed to collect and organize scientific knowledge in a structured, machine-readable format. Our approach leverages the strengths of symbolic knowledge representation to encode complex relationships and domain-specific rules, combined with the pattern recognition capabilities of neural networks to process large volumes of unstructured data, in particular scientific articles in the form of narrative text documents. With the ORKG, we developed a hybrid system that integrates a knowledge graph (symbolic system) with neural networks (subsymbolic system), enabling the machine-assisted extraction of scientific knowledge from research papers and symbolic knowledge representation, curation, organisation, and reuse. The implementation involves constructing a symbolic knowledge representation of the ORKG, training neural models on annotated datasets, and designing algorithms to synergize the two systems. Our findings suggest that neuro-symbolic integration enhances the ORKG’s ability to support complex queries, infer new relationships, and provide more robust and explainable AI-driven insights. This research contributes to the broader field of AI by showcasing the practical benefits of combining neural and symbolic methodologies in knowledge management systems.

Knowledge graphs (KGs) are increasingly utilized for data integration, representation, and visualization. While KG population is critical, it is often costly, especially when data must be extracted from unstructured text in natural language, which presents challenges, such as ambiguity and complex interpretations. Large Language Models (LLMs) offer promising capabilities for such tasks, excelling in natural language understanding and content generation. However, their tendency to “hallucinate” can produce inaccurate outputs. Despite these limitations, LLMs offer rapid and scalable processing of natural language data, and with prompt engineering and fine-tuning, they can approximate human-level performance in extracting and structuring data for KGs. This study investigates LLM effectiveness for the KG population, focusing on the Enslaved.org Hub Ontology. In this paper, we report that compared to the ground truth, LLM’s can extract ≈90% of triples, when provided a modular ontology as guidance in the prompts.

The integration of Large Language Models (LLMs) with logic-based Knowledge Graphs (KGs) and more generally with Knowledge Representation and Reasoning (KRR) methodologies has rapidly emerged as a pivotal area of research. Such a synergy is aimed at enhancing transparency and accountability in AI-driven applications, which is paramount for big data processing and robust decision-making over high-stakes domains such as finance and biomedicine. Indeed, despite the adaptability and human-centric understanding that LLMs bring, they inherently lack systematic reasoning capabilities, often operating opaquely with limited factuality and common sense. On the other hand, ontological reasoning with knowledge graphs offers robust and scalable reasoning, enriched with the step-by-step explainability of the inferred insights, but is often restricted by the rigidity of its structured rule-based formalism and falls short in providing the semantic understanding required in today’s human-data interaction. In this chapter, we address the intrinsic limitations affecting the above paradigms individually and introduce KGLM, a novel neurosymbolic framework that synergistically combines state-of-the-art LLMs with powerful KRR approaches to perform complex reasoning tasks over large knowledge graphs. Through KGLM, language models such as Llama 3 are enhanced with domain awareness and transparency, enabling them to act as natural language interfaces to KGs. Conversely, ontological reasoning systems such as our Vadalog engine are augmented with human-like flexibility to capture semantic nuances in the data. The framework can be seamlessly integrated into existing data processing pipelines and tools to power data-intensive decision-making processes in complex real-world domains.

Knowledge Graph Question Answering (KGQA) is an evolving field that aims to leverage structured knowledge graphs to provide precise answers to user queries. As Knowledge Graphs continue to expand in complexity and size, efficiently navigating and extracting relevant information from these vast datasets has become increasingly challenging. Recent advancements in Large Language Models (LLMs), offer promising capabilities in understanding and processing natural language. By integrating LLMs with KGQA systems, it is possible to enhance the accuracy and contextual relevance of answers generated. In this chapter, we explore the intersection of KGQA and LLMs, evaluating their combined potential to fetch information from knowledge graphs.

We survey neurosymbolic program synthesis, an emerging research area at the interface of deep learning and symbolic artificial intelligence. As in classical machine learning, the goal in neurosymbolic program synthesis is to learn functions from data. However, these functions are represented as programs that use symbolic primitives, often in conjunction with neural network components, and must, in some cases, satisfy certain additional behavioral constraints. The programs are induced using a combination of symbolic search and gradient-based optimization. In this survey, we categorize the main ways in which symbolic and neural learning techniques come together in this area. We also showcase the key advantages of the approach — specifically, greater reliability, interpretability, verifiability, and compositionality — over end-to-end deep learning.