Abstract This paper describes a formal theory of smooth vector fields, Lie groups and the Lie algebra of a Lie group in the theorem prover Isabelle. Lie groups are abstract structures that are composable, invertible and differentiable; they are useful in the study of continuous transformations in fields such as particle physics and robotics. The formalisation of this theory in an interactive theorem prover poses challenges beyond those encountered in textbook developments. We comment on representational choices we made to integrate involved concepts, such as smoothness of vector fields, with the simple type theory of higher-order logic (HOL) and existing material in Isabelle/HOL.

Neurosymbolic artificial intelligence (AI) is an increasingly active area of research that combines symbolic reasoning methods with deep learning to leverage their complementary benefits. As knowledge graphs (KGs) are becoming a popular way to represent heterogeneous and multirelational data, methods for reasoning on graph structures have attempted to follow this neurosymbolic paradigm. Traditionally, such approaches have utilized either rule-based inference or generated representative numerical embeddings from which patterns could be extracted. However, several recent studies have attempted to bridge this dichotomy to generate models that facilitate interpretability, maintain competitive performance, and integrate expert knowledge. Therefore, we survey methods that perform neurosymbolic reasoning tasks on KGs and propose a novel taxonomy by which we can classify them. Specifically, we propose three major categories: 1) logically informed embedding approaches; 2) embedding approaches with logical constraints; and 3) rule-learning approaches. Alongside the taxonomy, we provide a tabular overview of the approaches and links to their source code, if available, for more direct comparison. Finally, we discuss the unique characteristics and limitations of these methods and then propose several prospective directions toward which this field of research could evolve.

Abstract We present a formal framework for process composition based on actions that are specified by their input and output resources. The correctness of these compositions is verified by translating them into deductions in intuitionistic linear logic. As part of the verification we derive simple conditions on the compositions which ensure well-formedness of the corresponding deduction when satisfied. We mechanise the whole framework, including a deep embedding of ILL, in the proof assistant Isabelle/HOL. Beyond the increased confidence in our proofs, this allows us to automatically generate executable code for our verified definitions. We demonstrate our approach by formalising part of the simulation game Factorio and modelling a manufacturing process in it. Our framework guarantees that this model is free of bottlenecks.

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Identifying groups of patients with similar morbid-ity profiles can help us understand the relationships between their pre-existing conditions and the risks of adverse events in the ICU. To find such groups, common approaches apply clustering algorithms such as k-means and latent class analysis. However, these techniques present drawbacks such as the lack of principled methods for choosing the number of clusters, the need for assumptions about the relationships between variables, and outputs which are hard to explain. To overcome these limitations, we map the problem of patient clustering to that of community detection in complex networks. We construct a bipartite network in which nodes represent patients and their features, including morbidities and demographics. Then, we find homogeneous groups of patients using stochastic block modeling (SBM), an unsupervised probabilistic approach to find structure in networks. We show that this approach has several advantages over traditional clustering methods, and enables us to retrieve more fine-grained clusters that are commonly missed by existing approaches. We also show that these clusters have a stronger relationship with mortality and sepsis rates of patients in the ICU.

Abstract Special relativity is a cornerstone of modern physical theory. While a standard coordinate model is well known and widely taught today, multiple axiomatic systems for SR have been constructed over the past century. This paper reports on the formalisation of one such system, which is closer in spirit to Hilbert’s axiomatic approach to Euclidean geometry than to the vector space approach employed by Minkowski. We present a mechanisation in Isabelle/HOL of the system of axioms as well as theorems relating to temporal order. Some proofs are discussed, particularly where the formal work required additional steps, alternative approaches or corrections to Schutz’ prose.

Collaboration across institutional boundaries is widespread and increasing today. It depends on federations sharing data that often have governance rules or external regulations restricting their use. However, the handling of data governance rules (aka. data-use policies) remains manual, time-consuming and error-prone, limiting the rate at which collaborations can form and respond to challenges and opportunities, inhibiting citizen science and reducing data providers' trust in compliance. Using an automated system to facilitate compliance handling reduces substantially the time needed for such non-mission work, thereby accelerating collaboration and improving productivity. We present a framework, Dr.Aid, that helps individuals, organisations and federations comply with data rules, using automation to track which rules are applicable as data is passed between processes and as derived data is generated. It encodes data-governance rules using a formal language and performs reasoning on multi-input-multi-output data-flow graphs in decentralised contexts. We test its power and utility by working with users performing cyclone tracking and earthquake modelling to support mitigation and emergency response. We query standard provenance traces to detach Dr.Aid from details of the tools and systems they are using, as these inevitably vary across members of a federation and through time. We evaluate the model in three aspects by encoding real-life data-use policies from diverse fields, showing its capability for real-world usage and its advantages compared with traditional frameworks. We argue that this approach will lead to more agile, more productive and more trustworthy collaborations and show that the approach can be adopted incrementally. This, in-turn, will allow more appropriate data policies to emerge opening up new forms of collaboration.

Intrahospital transfers are a common but hazardous aspect of hospital care, with a large number of incidents posing a threat to patient safety. A growing body of work advocates the use of checklists for minimizing intrahospital transfer risk, but the majority of existing checklists are not guaranteed to be error-free and are difficult to adapt to different clinical settings or changing hospital policies. This paper details an approach that addresses these challenges through the employment of workflow technologies and formal methods for generating structured checklists. A three-phased methodology is proposed, where intrahospital transfer processes are first conceptualized, then rigorously composed into workflows that are mechanically verified, and finally, translated into a set of checklists that support hospital staff while maintaining the dependencies between different transfer tasks. A case study is presented, highlighting the feasibility of this approach, and the correctness and maintainability benefits brought by the logical underpinning of this methodology. A checklist evaluation is discussed, with promising results regarding their usefulness.

We review a generic framework for rigorous workflow modelling and verification that was recently applied to healthcare collaboration patterns, and we show how it can be utilised to help both medical staff and health informaticians build a systematic understanding of informal practices followed during intra-hospital patient transfers. A case study is discussed, demonstrating how the logical foundations of our approach help capture and enforce significant aspects of intra-hospital transfers that are pertinent to their improvement.

We propose a computer-based framework for the formal verification of collaboration patterns in healthcare teams. In this, the patterns are constructed diagrammatically as compositions of keystones that are viewed as abstract processes. The approach provides mechanisms for ensuring that safety properties are enforced and exceptional events are handled systematically. Additionally, a fully verified, executable model is obtained as an end product, enabling a simulation of its associated collaboration scenarios.

We review recently proposed notions of healthcare patterns for collaborative work and show how these can be cast in terms of composition of processes. The approach uses a purely diagrammatic language to drive a logic-based verification engine, resulting in fully-verified workflows that capture the information flow in these patterns.

Writing dependently typed functional programs that capture non-trivial program properties is difficult in current systems due to lack of proof automation. We identify proof patterns that occur when programming with dependent types and detail how automating such patterns allow us to work more comfortably with types that capture, for example, membership, ordering and non-linear arithmetic properties. We describe the role of the rippling heuristic, both for inductive and non-inductive proofs, and generalisation in providing such automation. We then discuss an implementation of our ideas in Coq with practical examples of dependently typed programs, that capture useful program properties, which can be verified automatically. We demonstrate that our proof automation is generic in that it can provide support for working with theorems involving user-defined functions and inductive data types.