Journal of Artificial Intelligence Research

Optimal Clustering with Dependent Costs in Bayesian Networks

2026-08-04T12:53:55+00:00

Background: Clustering of nodes in Bayesian Networks (BNs) and related graphical models such as Dynamic BNs (DBNs) has been demonstrated to enhance computational efficiency and improve model learning. It typically involves partitioning the underlying Directed Acyclic Graph (DAG) into cliques or optimising for some cost or criteria.

Objectives: Given a DAG, we focus on a critical but understudied aspect of optimal clustering involving cost dependency. This is where inference outcomes and hence clustering costs depend on both nodes within a cluster and the mapping of clusters that are connected by at least one arc.

Methods: We propose a novel algorithm called Dependent Cluster MAPping (DCMAP) which can, given an arbitrary, positive cost function, iteratively and rapidly find near-optimal, then optimal cluster mappings.

Results: DCMAP is shown analytically to be optimal in terms of finding all of the least cost cluster mapping solutions and with no more iterations than an equally informed algorithm. Demonstrated on a complex systems seagrass DBN with 9.91 × 10⁹ and 1.51 × 10²¹ possible cluster mappings for 25 and 50 node configurations, it took 856 and 1569 iterations on average to find the first optimal solution, respectively.

Conclusions: The effectiveness of DCMAP enables future research in BN learning using optimisation, such as through enhancing computational efficiency or minimising entropy for learning. This is critically important as computation of marginal distributions or updating model parameters is NP-hard.

Hierarchy Generation and Exploitation for Enhanced Multi-Class Classification

2026-06-01T09:17:41+00:00

Background: Hierarchical Classification (HC) has long been recognized for improving predictive performance by exploiting relationships between classes. However, most tabular multi-class datasets lack predefined class hierarchies, limiting the broader applicability of hierarchy-aware learning methods.

Objectives: This study introduces HiGEC (Hierarchy Generation and Exploitation for Classification), a unified framework that enables hierarchy-aware learning for standard flat-label tabular classification problems through automatically generated class hierarchies. The objectives are to determine whether data-driven hierarchies can consistently improve performance over Flat Classification (FC), whether hierarchy-aware evaluation metrics improve under enhanced exploitation strategies, and whether probabilistic aggregation mitigates hierarchical error propagation.

Methods: HiGEC systematically integrates Hierarchy Generation (HG) and Hierarchy Exploitation (HE) within a unified framework. Two enhanced HE schemes are proposed: HE+, which mitigates error propagation through probabilistic path aggregation, and HE+F, which combines hierarchical and flat predictions through calibrated convex fusion. A large-scale benchmark involving 100 tabular datasets and ten classifiers—including gradient boosting, ensemble, generative, instancebased, and transformer-based methods—was conducted. Evaluation included both conventional classification metrics and hierarchy-aware metrics, together with statistical significance testing and runtime analysis.

Results: HiGEC consistently improved predictive performance over FC baselines, particularly in F1-score, with statistically significant gains across diverse classifiers. Hierarchy-aware evaluation further demonstrated improvements in structural consistency, with enhanced HE schemes achieving higher hierarchical F-measure (hF) and lower path-based loss than baseline hierarchical methods. Runtime analysis showed that optimized HE+F configurations typically incurred moderate computational overhead (approximately 2–3× slower than FC), while certain lightweight HE+ configurations achieved both higher predictive performance and lower runtime than their FC counterparts.

Conclusions: The empirical findings support the three formal hypotheses of this study: (1) HiGEC improves flat predictive performance over FC, (2) enhanced HE strategies improve hierarchy-aware evaluation metrics, and (3) probabilistic aggregation mitigates hierarchical error propagation. Factor-wise ranking analysis further demonstrated that HE strategies contribute more strongly to performance variation than HG methods under aggregated evaluation. Overall, HiGEC provides a unified and reproducible framework for hierarchy-aware learning on tabular multi-class benchmarks.

Certified Neural Networks: from Verification to Synthesis

2026-07-25T08:25:05+00:00

Neural networks find applications in many safety-critical systems that raise concerns about their deployment: Are we sure they never advise doing anything catastrophic? Formal verification has been recently applied to prove whether an existing neural network is certified for some property; i.e., if it satisfies the property for all possible inputs or not. Formal verification can prove that a network satisfies a property but cannot fix the network in case it doesn’t. In this paper we focus on the automated synthesis of certified neural networks, that is, on how to automatically build a network that is guaranteed to respect some required properties expressed as logical constraints. We exploit a Counter Example Guided Inductive Synthesis (CEGIS) loop that alternates Deep Learning, Formal Verification, and a novel data generation technique that augments the training data to synthesize certified networks in a fully automated way. We identify a few conditions guaranteeing termination of the approach. We also investigate a soft constraint acceleration technique to reduce the number of CEGIS iterations and we test our approach to synthesize safety neural controllers for four case studies: a social robot application scenario, an expense prediction scenario, and two versions of the Airborne Collision Avoidance System X for unmanned aircraft benchmark (HCAS and ACASXu).

Symbol Grounding in Neuro-Symbolic AI: A Gentle Introduction to Reasoning Shortcuts

2026-04-17T07:44:38+00:00

Neuro-symbolic (NeSy) AI aims to develop deep neural networks whose predictions comply with prior knowledge encoding, e.g., safety or structural constraints. As such, it represents one of the most promising avenues for reliable and trustworthy AI. The core idea behind NeSy AI is to combine neural and symbolic steps: neural networks are typically responsible for mapping low-level inputs into high-level symbolic concepts, while symbolic reasoning infers predictions compatible with the extracted concepts and the prior knowledge. Despite their promise, it was recently shown that – whenever the concepts are not supervised directly – NeSy models can be affected by Reasoning Shortcuts (RSs). That is, they can achieve high label accuracy by grounding the concepts incorrectly. RSs can compromise the interpretability of the model’s explanations, performance in out-of-distribution scenarios, and therefore reliability. At the same time, RSs are difficult to detect and prevent unless concept supervision is available, which is typically not the case. However, the literature on RSs is scattered, making it difficult for researchers and practitioners to understand and tackle this challenging problem. This overview addresses this issue by providing a gentle introduction to RSs, discussing their causes and consequences in intuitive terms. It also reviews and elucidates existing theoretical characterizations of this phenomenon. Finally, it details methods for dealing with RSs, including mitigation and awareness strategies, and maps their benefits and limitations. By reformulating advanced material in a digestible form, this overview aims to provide a unifying perspective on RSs to lower the bar to entry for tackling them. Ultimately, we hope this overview contributes to the development of reliable NeSy and trustworthy AI models.

Text Generation: A Systematic Literature Review of Tasks, Evaluation, and Challenges

2026-05-18T08:11:16+00:00

Text generation has become more accessible than ever, and the growing interest in these systems, especially those using large language models, has spurred a surge in related publications. We provide a systematic literature review comprising 257 papers, covering the period from January 2017 to December 2025. This review categorizes text generation contributions into five main tasks: open-ended text generation, summarization, translation, paraphrasing, and question answering. For each task in our taxonomy, we review relevant characteristics and key subtasks. We assess current approaches for evaluating text generation systems, covering model-free, model-based, and human evaluation. Our investigation shows several task-specific challenges (e.g., missing datasets for multi-document summarization, lack of coherence in story generation, and difficulties in complex reasoning for question answering). We further discuss nine challenges common to all tasks and sub-tasks in recent text generation papers: bias, reasoning, hallucinations, misuse, privacy, interpretability, transparency, datasets, and computing. This systematic literature review targets two main audiences: early-career researchers in natural language processing seeking an overview of the field and promising research directions, and senior researchers who need a recent overview of the main tasks, evaluation, challenges, and mitigation strategies.

How to Discover Short, Shorter, and the Shortest Proofs of Unsatisfiability: A Branch-and-Bound Approach for Resolution Proof Length Minimization

2026-03-09T04:05:51+00:00

Background: Modern SAT solvers are indispensable reasoning tools that are widely applied in verification, planning, and combinatorial design. Beyond answering satisfiability, they also produce unsatisfiability proofs, which are valuable for certification and analysis. Since there are no guarantees that their length is close to optimal, however, those proofs are insufficient to make claims about the gap between the solver proofs and the shortest possible proof. Understanding this gap is important for analyzing solver behavior, as it can be seen as a proxy for evaluating the room for improvement of the solver for the input formula in question.

Objectives: We study the problem of finding the shortest resolution proofs for unsatisfiable formulas. Our goal is to understand how far solver-generated proofs deviate from the optimum, and to develop algorithms that can discover proofs substantially shorter than the solver-produced proofs.

Methods: We propose a novel branch-and-bound algorithm for resolution proof length minimization. To facilitate it, we introduce a layer list representation that eliminates all symmetries from clause permutations, thereby improving upon an earlier SAT-encoding approach to proof length minimization. Further, we accelerate the search by integrating pruning techniques based on proof length lower bounds and clause subsumption.

Results: Our algorithm consistently reduces solver proof lengths by 25–50% for synthetic instances and by 15—50% for formulas from the SAT Competition editions from 2002 to 2025, as well as halves the proof lengths for half of the available formulas from planning problems. As an exact method, it solves twice as many minimally unsatisfiable instances as the state-of-the-art approach and is faster by orders of magnitude on the overlapping instances.

Conclusions: These results show that solver-generated proofs can often be substantially shortened—not by removing redundant claims, but by an entirely different chain of derivations—and that resolution proof length minimization is more tractable in practice than earlier approaches suggested. Our work highlights proof length minimization as a useful perspective for analyzing solver behavior and opens avenues for applying similar techniques to stronger proof systems.

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.

Abduction in Łukasiewicz Logic With Rational Interval Terms

2026-08-13T14:37:30+00:00

We explore abductive reasoning in contexts involving fuzzy statements (statements with truth degrees) such as ‘the lift is heavily loaded’ (meaning, e.g., that the lift is carrying more than 70% of its maximal load), ‘the symptoms are severe’, ‘it is cold outside’, etc. Such statements are both used to express observations and to explain observed events. That is, both the observed event and our explanation of it may have truth degrees.

To formalise these contexts, we use infinitely-valued Łukasiewicz fuzzy logic Ł with semantics defined over the real-valued interval [0,1]. Here, 0 and 1 are interpreted as ‘absolutely false’ and ‘absolutely true’, respectively, and the remaining values as degrees of truth. We consider two standard entailment relations for Ł: the ‘truth-preserving’ one (if the premise is absolutely true, then the conclusion is absolutely true) and the ‘degree-preserving’ (the truth degree of the premise should be at most as high as the truth degree of the conclusion).

For each of these entailment relations, we define and motivate the notions of abduction problems and explanations in the language of Ł expanded with ‘interval literals’ of the form p≥c, p≤c, and their negations that express the set of values a~variable can have. We analyse the complexity of standard abductive reasoning tasks (solution recognition, solution existence, and relevance / necessity of hypotheses) in Ł for the case of the full language and for the case of theories containing only disjunctive clauses and show that, in contrast to classical propositional logic, the abduction in the clausal fragment has lower complexity than in the general case.

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.