Journal of Artificial Intelligence Research

Symbols and Neurons: A Review of Symbolic XAI in Deep Learning

2026-03-10T00:20:31+00:00

Background: Deep neural networks increasingly power language, vision, and decision systems, yet many deployments require explanations that are faithful, compositional, and governance-ready. Symbolic techniques promise these properties, but the literature mixes post-hoc extraction, knowledge injection, and intrinsically hybrid designs without a unifying view.

Objectives: We provide a systematic review and synthesis of symbolic explainable AI (XAI) for deep learning (January 2017– June 2025), organize the field around a three-part taxonomy—Symbolic Knowledge Extraction (SKE), Symbolic Knowledge Injection (SKI), and Hybrid neurosymbolic architectures—and propose a conceptual framework that clarifies training–inference flows, explanation interfaces, human feedback, and governance touchpoints.

Methods: Beginning from ≈50,000 records, we deduplicated and screened full texts, analyzed 393 PDFs, and included 273 primary studies in the synthesis. We coded each paper for model domain, modality, symbolic formalism, explanation scope and stage, evaluation protocol, and governance alignment. Analyses combine descriptive statistics with stratification by domain and formalism; we qualitatively assess evidence for faithfulness, robustness, data efficiency, and constraint satisfaction.

Results: Research activity accelerates after 2020, with a marked turn toward hybrids. Across the corpus, SKE, SKI, and Hybrid account for approximately 29%, 26%, and 45% of studies, respectively. Rule sets/decision trees remain the dominant explanation artifacts, while logic- and program-based formalisms grow in NLP and planning. SKI most often targets constraint satisfaction and robustness improvements; SKE emphasizes global surrogates and faithfulness auditing; hybrids report gains in sample efficiency and traceable reasoning. However, evaluation practices are heterogeneous, human-subject studies are scarce, and explicit links to policy/risk controls appear in a minority of works.

Conclusions: Our framework unifies how data, priors, and symbolic reasoning interact with neural learners, the explanation interface, human stakeholders, and governance. We distill actionable recommendations: (1) report faithfulness and constraintsatisfaction metrics alongside accuracy; (2) specify symbolic assumptions and training-time injections precisely; (3) include user studies or auditor-centric protocols for high-stakes use; and (4) develop benchmarks that couple tasks with machinereadable knowledge bases. We highlight open problems in scalable formal reasoning with foundation models, verifying generated rationales, and measuring causal faithfulness at scale.

An Empirical Comparison of Cost Functions in Inductive Logic Programming

2026-02-17T10:54:14+00:00

Recent inductive logic programming (ILP) approaches learn optimal hypotheses. An optimal hypothesis minimises a given cost function on the training data. There are many cost functions, such as minimising training error, minimising textual complexity, or minimising the description length of hypotheses. However, selecting an appropriate cost function remains a key question. To address this gap, we extend a constraint-based ILP system to learn optimal hypotheses for seven standard cost functions. We then empirically compare the generalisation error of optimal hypotheses induced under these standard cost functions. Our results on over 20 domains and 1,000 tasks, including game playing, program synthesis, and image reasoning, show that, while no cost function consistently outperforms the others, minimising training error or description length has the best overall performance. Notably, our results indicate that minimising the size of hypotheses does not always reduce generalisation error.

Causal Explanations for Image Classifiers

2026-02-13T08:54:00+00:00

Existing algorithms for explaining the output of image classifiers use different definitions of explanations and a variety of techniques to find them. However, none of the existing tools use a principled approach based on formal definitions of cause and explanation.

In this paper we present a novel black-box approach to computing explanations grounded in the theory of actual causality. We prove relevant theoretical results and present an algorithm for computing approximate explanations based on these definitions. We prove termination of our algorithm and discuss its complexity and the amount of approximation compared to the precise definition.

We implemented the framework in a tool, ReX, and we present experimental results and a comparison with state-of-the-art tools. We demonstrate that ReX is the most efficient black-box tool and produces the smallest explanations, in addition to outperforming other black-box tools on standard quality measures.

Honey, I Shrunk the Hypothesis Space (Through Logical Preprocessing)

2026-01-29T00:07:47+00:00

Inductive logic programming (ILP) is a form of logical machine learning. The goal is to search a hypothesis space for a hypothesis that generalises training examples and background knowledge. We introduce an approach that shrinks the hypothesis space before an ILP system searches it. Our approach uses background knowledge to find rules that cannot be in an optimal hypothesis regardless of the training examples. For instance, our approach discovers relationships such as even numbers cannot be odd and prime numbers greater than 2 are odd. It then removes violating rules from the hypothesis space. We implement our approach using answer set programming and use it to shrink the hypothesis space of a constraint-based ILP system. Our experiments on multiple domains, including visual reasoning and game playing, show that our approach can substantially reduce learning times whilst maintaining predictive accuracies. For instance, given just 10 seconds of preprocessing time, our approach can reduce learning times from over 10 hours to only 2 seconds.

Kernel-Based Learning of Safety Barriers

2026-02-27T01:50:07+00:00

The rapid integration of AI algorithms in safety-critical applications such as autonomous driving and healthcare is raising significant concerns about the ability to meet stringent safety standards. Traditional tools for formal safety verification struggle with the black-box nature of AI-driven systems and lack the flexibility needed to scale to the complexity of real-world applications. In this paper, we present a data-driven approach for safety verification and synthesis of black-box systems with discrete-time stochastic dynamics. We employ the concept of control barrier certificates, which can guarantee safety of the system, and learn the certificate directly from a set of system trajectories. We use conditional mean embeddings to embed data from the system into a reproducing kernel Hilbert space (RKHS) and construct an RKHS ambiguity set that can be inflated to robustify the result to out-of-distribution behavior. We provide the theoretical results on how to apply the approach to general classes of temporal logic specifications beyond safety. For the data-driven computation of safety barriers, we leverage a finite Fourier expansion to cast a typically intractable semi-infinite optimization problem as a linear program. The resulting spectral barrier allows us to leverage the fast Fourier transform to generate the relaxed problem efficiently, offering a scalable yet distributionally robust framework for verifying safety. Our work moves beyond restrictive assumptions on system dynamics and uncertainty, as demonstrated on two case studies including a black-box system with a neural network controller.

R-Mod: Minimal Structural Revision of S5 Epistemic Models

2026-01-15T05:11:35+00:00

Revising what an agent knows in response to new information is a central problem in formal epistemology. In doxastic logics such as KD45, belief revision proceeds by reordering plausibility: the agent simply re-ranks which worlds it considers most credible. This strategy fails for S5 knowledge. Because knowledge is factive (Kφ → φ), an agent cannot come to know phi merely by finding φ-worlds more plausible; if the actual world falsifies φ, then Kφ remains unsatisfiable regardless of any reordering. Accommodating new modal information in S5 therefore requires genuine model transformation: adjusting the equivalence-based accessibility structure, the valuation, or both.

We develop R-Mod, a selection-based revision operator that realizes this transformation as minimal structural repair. Given an S5 model and a target formula, R-Mod searches for a closest model, measured by a bisimulation-aware distance on quotient structures, that satisfies the formula while preserving S5 constraints. At the skeptical level, R-Mod satisfies success, consistency preservation, and deductive closure; classical AGM postulates such as Inclusion and Superexpansion fail due to permissible structural amplification, though we identify conditions under which they re-emerge. Computationally, the decision problem is NP-complete, and we provide tractable fragments exploiting structural locality.

While recent work has advanced AGM-style postulate analysis for S5 and topological semantics via simplicial complexes, these approaches do not provide goal-driven optimization with algorithmic guarantees. R-Mod fills this gap by combining modal invariance, explicit distance minimization, and fine-grained complexity analysis. Our results reframe revision in S5 as knowledge-model revision rather than belief revision, offering a foundation for algorithmic implementations and extensions to richer epistemic semantics.

Gradual Semantics for Weighted Higher-Order Argumentation Frameworks

2026-02-17T02:08:25+00:00

This paper investigates complex argumentation settings, called weighted higher-order argumentation frameworks (wHO-AFs), where both arguments and attacks carry initial weights and may be subject to attacks from arguments. It focuses on developing gradual semantics capable of rationally evaluating these elements by assigning each argument and attack a numerical value, representing their respective degrees of acceptance and seriousness.

The contributions of the paper are five-fold: i) It identifies key technical challenges for semantics in wHO-AFs, such as handling the distinction between arguments and attacks, and determining how their values should be combined. To address these challenges, the paper introduces a set of guiding principles that capture various semantic strategies. ii) It introduces a broad family of semantics that treat arguments and attacks uniformly, evaluating them in the same way. This family encompasses both semantics inspired by extension-based approaches and those that are purely gradual. iii) It defines a second extensive family of semantics that differentiate between arguments and attacks. This includes instances where a single function evaluates both types of elements, as well as hybrid variants that apply distinct evaluation functions to arguments and attacks. iv) It offers a formal analysis and comprehensive comparison of the resulting diverse semantics. v) It presents an empirical evaluation of these semantics. The results are promising as they show that the values of all elements can be computed in fewer than 20 iterations, even for large frameworks, while maintaining very low runtime. The paper ultimately delivers a catalogue of semantics, each offering distinct formal properties and behaviors, contributing a flexible toolkit for reasoning in rich argumentative contexts.

Jump Over Dynamic Block (JODB): An Efficient Line-of-Sight Checker for Dynamic Grid/Voxel Maps with Sparse Obstacles

2026-02-19T07:20:15+00:00

Background: Line-of-sight (LOS) checking plays a critical role in the computational cost of collision avoidance, particularly in large-scale maps with sparse obstacles, since it requires a cell-by-cell state traversal.
Objectives: To provide a general method for reducing the time cost of LOS checking and thus accelerating path and motion planning, we present an efficient LOS checker, JODB.
Methods: JODB partitions the traversable space into blocks. When our algorithm encounters the surface of a block, its LOS check skips cell-by-cell traversal inside the block and directly jumps to the opposite surface. To handle dynamic maps, we propose a novel data structure, the Space Dyadic Tree (SDT), inspired by quadtree and OctoMap, which recursively partitions the traversable space into blocks. In addition, SDT employs a decision-tree strategy to maximize block size by merging neighboring blocks whenever possible.
Results: Experimental results demonstrate that SDT can be updated in near real-time on large-scale 2D/3D maps with sparse obstacles, and that using SDT significantly reduces the time cost of LOS checking in such maps. Therefore, the time cost of path planning can be significantly reduced by applying SDT-accelerated LOS checking. Specifically, the time cost of LOS checking accelerated by SDT is approximately 40%–60% of that using a quadtree/OctoMap and 22%–70% of that using traditional LOS checking.
Conclusions: Compared with traditional quadtree/OctoMap, SDT is more efficient to initialize and update in dynamic environments, and it makes LOS checking more efficient. To facilitate further research within the community, we have made the source code of the proposed algorithm publicly available. Links to the source code can be found in the Results section.

Formal Logic Inference Guided Uncertainty Quantification for Personalized Federated Learning

2026-01-16T05:23:35+00:00

Federated Learning (FL) enables privacy-preserving model training across heterogeneous distributed systems, such as smartgrid forecasting or traffic-flow prediction from geographically dispersed sensors and devices. A key challenge in such settings is capturing client-specific patterns while addressing data heterogeneity and uncertainty at scale. Existing approaches, including Bayesian Neural Networks (BNNs) and clustering-based methods, struggle with scalability and consistent personalization. We propose LogiCP, a novel FL framework that integrates formal logic reasoning with uncertainty quantification (UQ) to support scalable and personalized learning with theoretical guarantees. LogiCP uses Signal Temporal Logic (STL) to extract temporal patterns and form semantically coherent client clusters, controlling intra-cluster heterogeneity. Within each cluster, LogiCP applies decentralized Conformal Prediction (CP) to produce distribution-free prediction intervals with mathematical guarantees that encompass the real value. LogiCP dynamically assigns clients to clusters at runtime without retraining, improving practicality. Evaluations on three real-world datasets—traffic, temperature, and electricity—show that LogiCP consistently outperforms BNN-, clustering-, and CP-based baselines, achieving up to a 95% improvement in client-level MSE while maintaining strong scalability.

Explaining Multivariate Decision Trees: Characterising Tractable Languages

2025-12-20T06:48:44+00:00

We study multivariate decision trees (MDTs), in particular, classes of MDTs determined by the language of relations that can be used to split feature space. An abductive explanation (AXp) of the classification of a particular instance, viewed as a set of feature-value assignments, is a minimal subset of the instance which is sufficient to lead to the same decision. We investigate when finding a single AXp is tractable. We identify tractable languages for real, integer and boolean features. Indeed, in the case of boolean languages, we provide a P/NP-hard dichotomy. We extend this dichotomy to languages defined by formulas whose literals correspond to splits of ordered domains of arbitrary finite size. Experiments indicate that MDTs can provide more compact models than classical decision trees while conserving accuracy and explainability.