Research and practice in Intelligent Design (ID) have significantly enhanced engineering innovation, efficiency, quality, and productivity over recent decades, fundamentally reshaping how engineering designers think, behave, and interact with design processes. The recent emergence of Foundation Models (FMs), particularly Large Language Models (LLMs), has demonstrated general knowledge-based reasoning capabilities, and open new avenues for further transformation in engineering design. In this context, this paper introduces Intelligent Design 4.0 (ID 4.0) as an emerging paradigm empowered by foundation model-based agentic AI systems. We review the historical evolution of ID across four distinct stages: rule-based expert systems, task-specific machine learning models, large-scale foundation AI models, and the recent emerging paradigm of foundation model-based multi-agent collaboration. We propose an ontological framework for ID 4.0 and discuss its potential to support end-to-end automation of engineering design processes through coordinated, autonomous multi-agent-based systems. Furthermore, we discuss challenges and opportunities of ID 4.0, including perspectives on data foundations, agent collaboration mechanisms, and the formulation of design problems and objectives. In sum, these insights provide a foundation for advancing Intelligent Design toward greater adaptivity, autonomy, and effectiveness in addressing the growing complexity of engineering design.

Artificial intelligence deployed in risk-sensitive domains such as healthcare, finance, and security must not only achieve predictive accuracy but also ensure transparency, ethical alignment, and compliance with regulatory expectations. Hybrid neuro symbolic models combine the pattern-recognition strengths of neural networks with the interpretability and logical rigor of symbolic reasoning, making them well-suited for these contexts. This paper surveys hybrid architectures, ethical design considerations, and deployment patterns that balance accuracy with accountability. We highlight techniques for integrating knowledge graphs with deep inference, embedding fairness-aware rules, and generating human-readable explanations. Through case studies in healthcare decision support, financial risk management, and autonomous infrastructure, we show how hybrid systems can deliver reliable and auditable AI. Finally, we outline evaluation protocols and future directions for scaling neuro symbolic frameworks in complex, high stakes environments.

We introduce deep neural networks for end-to-end differentiable theorem proving that operate on dense vector representations of symbols. These neural networks are recursively constructed by following the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. The resulting neural network can be trained to infer facts from a given incomplete knowledge base using gradient descent. By doing so, it learns to (i) place representations of similar symbols in close proximity in a vector space, (ii) make use of such similarities to prove facts, (iii) induce logical rules, and (iv) it can use provided and induced logical rules for complex multi-hop reasoning. On four benchmark knowledge bases we demonstrate that this architecture outperforms ComplEx, a state-of-the-art neural link prediction model, while at the same time inducing interpretable function-free first-order logic rules.

Current state-of-the-art methods for automated Knowledge Base (KB) completion use neural link prediction models to learn distributed vector representations of symbols ( i.e. subsymbolic representations)

We study the problem of learning probabilistic first-order logical rules for knowledge base reasoning.

Large Language Models (LLMs) excel in many natural language processing tasks but often exhibit factual inconsistencies in knowledge-intensive settings. Integrating external knowledge resources, particularly knowledge graphs (KGs), provides a transparent and updatable foundation for more reliable reasoning. Knowledge Base Question Answering (KBQA), which queries and reasons over KGs, is central to this effort, especially for complex, multi-hop queries. However, multi-hop reasoning poses two key challenges: (1)~maintaining coherent reasoning paths, and (2)~avoiding prematurely discarding critical multi-hop connections. To tackle these challenges, we introduce iQUEST, a question-guided KBQA framework that iteratively decomposes complex queries into simpler sub-questions, ensuring a structured and focused reasoning trajectory. Additionally, we integrate a Graph Neural Network (GNN) to look ahead and incorporate 2-hop neighbor information at each reasoning step. This dual approach strengthens the reasoning process, enabling the model to explore viable paths more effectively. Detailed experiments demonstrate the consistent improvement delivered by iQUEST across four benchmark datasets and four LLMs.

This paper considers the learning of Boolean rules in either disjunctive normal form (DNF, OR-of-ANDs, equivalent to decision rule sets) or conjunctive normal form (CNF, AND-of-ORs) as an interpretable model for classification. An integer program is formulated to optimally trade classification accuracy for rule simplicity. Column generation (CG) is used to efficiently search over an exponential number of candidate clauses (conjunctions or disjunctions) without the need for heuristic rule mining. This approach also bounds the gap between the selected rule set and the best possible rule set on the training data. To handle large datasets, we propose an approximate CG algorithm using randomization. Compared to three recently proposed alternatives, the CG algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets. When maximized for accuracy, CG is competitive with rule learners designed for this purpose, sometimes finding significantly simpler solutions that are no less accurate.

We present the Multi-value Rule Set (MRS) for interpretable classification with feature efficient presentations. Compared to rule sets built from single-value rules, MRS adopts a more generalized form of association rules that allows multiple values in a condition. Rules of this form are more concise than classical single-value rules in capturing and describing patterns in data. Our formulation also pursues a higher efficiency of feature utilization, which reduces possible cost in data collection and storage. We propose a Bayesian framework for formulating an MRS model and develop an efficient inference method for learning a maximum a posteriori, incorporating theoretically grounded bounds to iteratively reduce the search space and improve the search efficiency. Experiments on synthetic and real-world data demonstrate that MRS models have significantly smaller complexity and fewer features than baseline models while being competitive in predictive accuracy.

Neural Information Processing Systems http://nips.cc/

Rule-based systems must solve complex matching problems within tight time constraints to be effective in real-time applications, such as planning and reactive control for AI agents, as well as low-latency relational database querying. Pattern-matching systems can encounter issues where exponential time and space are required to find matches for rules with many underconstrained variables, or which produce combinatorial intermediate partial matches (but are otherwise well-constrained). When online AI systems automatically generate rules from example-driven induction or code synthesis, they can easily produce worst-case matching patterns that slow or halt program execution by exceeding available memory. In our own work with cognitive systems that learn from example, we've found that aggressive forms of anti-unification-based generalization can easily produce these circumstances. To make these systems practical without hand-engineering constraints or succumbing to unpredictable failure modes, we introduce a new matching algorithm called CORGI (Collection-Oriented Relational Graph Iteration). Unlike RETE-based approaches, CORGI offers quadratic time and space guarantees for finding single satisficing matches, and the ability to iteratively stream subsequent matches without committing entire conflict sets to memory. CORGI differs from RETE in that it does not have a traditional $β$-memory for collecting partial matches. Instead, CORGI takes a two-step approach: a graph of grounded relations is built/maintained in a forward pass, and an iterator generates matches as needed by working backward through the graph. This approach eliminates the high-latency delays and memory overflows that can result from populating full conflict sets. In a performance evaluation, we demonstrate that CORGI significantly outperforms RETE implementations from SOAR and OPS5 on a simple combinatorial matching task.