The rise of multi-paradigm languages challenges traditional classification methods, leading to practical software engineering issues like interoperability defects. This systematic literature review (SLR) maps the formal foundations of programming paradigms. Our objective is twofold: (1) to assess the state of the art of classification formalisms and their limitations, and (2) to identify the conceptual primitives and mathematical frameworks for a more powerful, reconstructive approach. Based on a synthesis of 74 primary studies, we find that existing taxonomies lack conceptual granularity, a unified formal basis, and struggle with hybrid languages. In response, our analysis reveals a strong convergence toward a compositional reconstruction of paradigms. This approach identifies a minimal set of orthogonal, atomic primitives and leverages mathematical frameworks, predominantly Type theory, Category theory and Unifying Theories of Programming (UTP), to formally guarantee their compositional properties. We conclude that the literature reflects a significant intellectual shift away from classification towards these promising formal, reconstructive frameworks. This review provides a map of this evolution and proposes a research agenda for their unification.

Recent advances in visual-language machine learning models have demonstrated exceptional ability to use natural language and understand visual scenes by training on large, unstructured datasets. However, this training paradigm cannot produce interpretable explanations for its outputs, requires retraining to integrate new information, is highly resource-intensive, and struggles with certain forms of logical reasoning. One promising solution involves integrating neural networks with external symbolic information systems, forming neural symbolic systems that can enhance reasoning and memory abilities. These neural symbolic systems provide more interpretable explanations to their outputs and the capacity to assimilate new information without extensive retraining. Utilizing powerful pre-trained Vision-Language Models (VLMs) as the core neural component, augmented by external systems, offers a pragmatic approach to realizing the benefits of neural-symbolic integration. This systematic literature review aims to categorize techniques through which visual-language understanding can be improved by interacting with external symbolic information systems.

As neural networks (NNs) become increasingly prevalent in safety-critical neural network-controlled cyber-physical systems (NNCSs), formally guaranteeing their safety becomes crucial. For these systems, safety must be ensured throughout their entire operation, necessitating infinite-time horizon verification. To verify the infinite-time horizon safety of NNCSs, recent approaches leverage Differential Dynamic Logic (dL). However, these dL-based guarantees rely on idealized, real-valued NN semantics and fail to account for roundoff errors introduced by finite-precision implementations. This paper bridges the gap between theoretical guarantees and real-world implementations by incorporating robustness under finite-precision perturbations -- in sensing, actuation, and computation -- into the safety verification. We model the problem as a hybrid game between a good Demon, responsible for control actions, and a bad Angel, introducing perturbations. This formulation enables formal proofs of robustness w.r.t. a given (bounded) perturbation. Leveraging this bound, we employ state-of-the-art mixed-precision fixed-point tuners to synthesize sound and efficient implementations, thus providing a complete end-to-end solution. We evaluate our approach on case studies from the automotive and aeronautics domains, producing efficient NN implementations with rigorous infinite-time horizon safety guarantees.

This paper presents a comprehensive refutation of the so-called "blockchain trilemma," a widely cited but formally ungrounded claim asserting an inherent trade-off between decentralisation, security, and scalability in blockchain protocols. Through formal analysis, empirical evidence, and detailed critique of both methodology and terminology, we demonstrate that the trilemma rests on semantic equivocation, misuse of distributed systems theory, and a failure to define operational metrics. Particular focus is placed on the conflation of topological network analogies with protocol-level architecture, the mischaracterisation of Bitcoin's design--including the role of miners, SPV clients, and header-based verification--and the failure to ground claims in complexity-theoretic or adversarial models. By reconstructing Bitcoin as a deterministic, stateless distribution protocol governed by evidentiary trust, we show that scalability is not a trade-off but an engineering outcome. The paper concludes by identifying systemic issues in academic discourse and peer review that have allowed such fallacies to persist, and offers formal criteria for evaluating future claims in blockchain research.

This paper contributes to the Alpay Algebra by demonstrating that the stable outcome of a self referential process, obtained by iterating a transformation through all ordinal stages, is identical to the unique equilibrium of an unbounded revision dialogue between a system and its environment. The analysis initially elucidates how classical fixed point theorems guarantee such convergence in finite settings and subsequently extends the argument to the transfinite domain, relying upon well founded induction and principles of order theoretic continuity. Furthermore, the resulting transordinal fixed point operator is embedded into dependent type theory, a formalization which permits every step of the transfinite iteration and its limit to be verified within a modern proof assistant. This procedure yields a machine checked proof that the iterative dialogue necessarily stabilizes and that its limit is unique. The result provides a foundation for Alpay's philosophical claim of semantic convergence within the framework of constructive logic. By unifying concepts from fixed point theory, game semantics, ordinal analysis, and type theory, this research establishes a broadly accessible yet formally rigorous foundation for reasoning about infinite self referential systems and offers practical tools for certifying their convergence within computational environments.

We investigate lifted inference on ordered domains with predecessor relations, where the elements of the domain respect a total (cyclic) order, and every element has a distinct (clockwise) predecessor. Previous work has explored this problem through weighted first-order model counting (WFOMC), which computes the weighted sum of models for a given first-order logic sentence over a finite domain. In WFOMC, the order constraint is typically encoded by the linear order axiom introducing a binary predicate in the sentence to impose a linear ordering on the domain elements. The immediate and second predecessor relations are then encoded by the linear order predicate. Although WFOMC with the linear order axiom is theoretically tractable, existing algorithms struggle with practical applications, particularly when the predecessor relations are involved. In this paper, we treat predecessor relations as a native part of the axiom and devise a novel algorithm that inherently supports these relations. The proposed algorithm not only provides an exponential speedup for the immediate and second predecessor relations, which are known to be tractable, but also handles the general k -th predecessor relations. The extensive experiments on lifted inference tasks and combinatorics math problems demonstrate the efficiency of our algorithm, achieving speedups of a full order of magnitude.

Satisfiability Modulo Theory (SMT) solvers have advanced automated reasoning, solving complex formulas across discrete and continuous domains. Recent progress in propositional model counting motivates extending SMT capabilities toward model counting, especially for hybrid SMT formulas. Existing approaches, like bit-blasting, are limited to discrete variables, highlighting the challenge of counting solutions projected onto the discrete domain in hybrid formulas. We introduce pact, an SMT model counter for hybrid formulas that uses hashing-based approximate model counting to estimate solutions with theoretical guarantees. pact makes a logarithmic number of SMT solver calls relative to the projection variables, leveraging optimized hash functions. pact achieves significant performance improvements over baselines on a large suite of benchmarks. In particular, out of 14,202 instances, pact successfully finished on 603 instances, while Baseline could only finish on 13 instances.

Neuro-symbolic artificial intelligence aims to combine neural architectures with symbolic approaches that can represent knowledge in a human-interpretable formalism. Continual learning concerns with agents that expand their knowledge over time, improving their skills while avoiding to forget previously learned concepts. Most of the existing approaches for neuro-symbolic artificial intelligence are applied to static scenarios only, and the challenging setting where reasoning along the temporal dimension is necessary has been seldom explored. In this work we introduce LTLZinc, a benchmarking framework that can be used to generate datasets covering a variety of different problems, against which neuro-symbolic and continual learning methods can be evaluated along the temporal and constraint-driven dimensions. Our framework generates expressive temporal reasoning and continual learning tasks from a linear temporal logic specification over MiniZinc constraints, and arbitrary image classification datasets. Fine-grained annotations allow multiple neural and neuro-symbolic training settings on the same generated datasets. Experiments on six neuro-symbolic sequence classification and four class-continual learning tasks generated by LTLZinc, demonstrate the challenging nature of temporal learning and reasoning, and highlight limitations of current state-of-the-art methods. We release the LTLZinc generator and ten ready-to-use tasks to the neuro-symbolic and continual learning communities, in the hope of fostering research towards unified temporal learning and reasoning frameworks.

Network protocols are fundamental to the functioning of modern society, as more and more services are provided online, including critical infrastructure. Historically, these services have mostly been implemented with wired networks that connect conventional computers. But with the emergence of wireless and cellular technologies, their reach has extended to Internet of Things (IoT) devices and sensors, robotic systems, and autonomous vehicles. The next generation of cellular technologies, often referred to as NextG, will further expand the set of services that can be delivered over a network, including real-time remote healthcare, industrial automation, precision agriculture, smart infrastructure, holographic communication, space-based communication, and more. Experience shows that formal specification can provide significant practical benefits for network protocol development by enhancing clarity, trustworthiness, and productivity. Adoption of formal methods has been hindered not by lack of scalability but by cultural gaps that result in tools and methods that do not align with engineering practice.

Language models for program synthesis are usually trained and evaluated on programming competition datasets (MBPP, APPS). However, these datasets are limited in size and quality, while these language models are extremely data hungry. Additionally, the language models have a misaligned program synthesis process compared to humans. While humans iteratively develop code with the help of a compiler, most program synthesis models currently produce code in one go. To solve these issues, we introduce a bootstrapping algorithm for program synthesis, that supports teaching models how to repair. We show that bootstrapping consistently outperforms regular fine-tuning. Compared to other work, our bootstrapped model performs on par with fine-tuned models that are 68\% larger. Notably, bootstrapping with repairing also improves non-repairing performance compared to regular bootstrapping during inference. However, on our models, repairing during inference is likely inferior to simply sampling the same number of solutions. Furthermore, we find that there are issues with the example test cases in the training portion of the APPS dataset that are valuable to the community, as many repairing and reinforcement learning methods rely on them.