arXiv subjects

Simon Foster

Publications and source records attributed to Simon Foster.

At least 19 recordsLinked to original sources

arXiv2026

Formal-Method-Guided Vibe Coding: Closing the Verification Loop on AI-Generated Safety-Critical Software Through Model-Driven Engineering

Vibe coding -- accepting LLM-generated source from natural-language intent with minimal review -- is fast and may be adequate for low-criticality consumer software. But for safety-critical systems governed by DO-178C, IEC 61508, or ISO 26262, it offers no path to certification: large language models (LLMs) provide no formal correctness guarantees, and existing remedies target verification-aware languages (Dafny, Verus, Lean) that are scarce in pretraining data and absent from industrial toolchains. This paper closes the gap. We present Forge (Formal method Oriented Refinement loop for GEnerated code): a closed-loop pipeline that guides vibe coding through formal verification using established Model-Driven Engineering (MDE) infrastructure. Through vibe coding, we generate Java source code; our pipeline then extracts -- via model transformations -- formal artefacts in three different formalisms, each checked by a complementary verifier: deductive verification (Dafny), Communicating Sequential Processes (CSP) refinement via the Failures-Divergences Refinement checker (FDR4), and theorem proving using Z-Machines in Isabelle; every verification failure becomes a structured correction prompt that drives the next code-generation iteration. The LLM is the draft generator, the MDE chain is the discriminator, and the developer never has to read the formal models. Empirically, we find that the pipeline produces standards-relevant verification evidence for LLM-generated Java -- a step toward certification.

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arXiv2026

Formal Evidence Generation for Assurance Cases for Robotic Software Models

Robotics and Autonomous Systems are increasingly deployed in safety-critical domains, so that demonstrating their safety is essential. Assurance Cases (ACs) provide structured arguments supported by evidence, but generating and maintaining this evidence is labour-intensive, error-prone, and difficult to keep consistent as systems evolve. We present a model-based approach to systematically generating AC evidence by embedding formal verification into the assurance workflow. The approach addresses three challenges: systematically deriving formal assertions from natural language requirements using templates, orchestrating multiple formal verification tools to handle diverse property types, and integrating formal evidence production into the workflow. Leveraging RoboChart, a domain-specific modelling language with formal semantics, we combine model checking and theorem proving in our approach. Structured requirements are automatically transformed into formal assertions using predefined templates, and verification results are automatically integrated as evidence. Case studies demonstrate the effectiveness of our approach.

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arXiv2025

Formal Analysis of the Sigmoid Function and Formal Proof of the Universal Approximation Theorem

This paper presents a formalized analysis of the sigmoid function and a fully mechanized proof of the Universal Approximation Theorem (UAT) in Isabelle/HOL, a higher-order logic theorem prover. The sigmoid function plays a fundamental role in neural networks; yet, its formal properties, such as differentiability, higher-order derivatives, and limit behavior, have not previously been comprehensively mechanized in a proof assistant. We present a rigorous formalization of the sigmoid function, proving its monotonicity, smoothness, and higher-order derivatives. We provide a constructive proof of the UAT, demonstrating that neural networks with sigmoidal activation functions can approximate any continuous function on a compact interval. Our work identifies and addresses gaps in Isabelle/HOL's formal proof libraries and introduces simpler methods for reasoning about the limits of real functions. By exploiting theorem proving for AI verification, our work enhances trust in neural networks and contributes to the broader goal of verified and trustworthy machine learning.

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arXiv2025

Verifying Numerical Methods with Isabelle/HOL

Modern machine learning pipelines are built on numerical algorithms. Reliable numerical methods are thus a prerequisite for trustworthy machine learning and cyber-physical systems. Therefore, we contribute a framework for verified numerical methods in Isabelle/HOL based on ITrees. Our user-friendly specification language enables the direct declaration of numerical programs that can be annotated with variants and invariants for reasoning about correctness specifications. The generated verification conditions can be discharged via automated proof methods and lemmas from the HOL-Analysis library. The ITrees foundation interacts with Isabelle's code generator to export source code. This provides an end-to-end path from formal specifications with machine-checked guarantees to executable sources. We illustrate the process of modelling numerical methods and demonstrate the effectiveness of the verification by focusing on two well-known methods, the bisection method and the fixed-point iteration method. We also contribute crucial extensions to the libraries of formalised mathematics required for this objective: higher-order derivatives and Taylor's theorem in Peano form. Finally, we qualitatively evaluate the use of the framework for verifying numerical methods.

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arXiv2024

Brownian Motion in Isabelle/HOL

In order to formally verify robotic controllers, we must tackle the inherent uncertainty of sensing and actuation in a physical environment. We can model uncertainty using stochastic hybrid systems, which combine discrete jumps with continuous, stochastic behaviour. The verification of these systems is intractable for state-exploration based approaches, so we instead propose a deductive verification approach. As a first step towards a deductive verification tool, we present a mechanisation of Brownian motion within Isabelle/HOL. For this, we mechanise stochastic kernels and Markov semigroups, which allow us to specify a range of processes with stationary, independent increments. Further, we prove the Kolmogorov-Chentsov theorem, which allows us to construct H\"older continuous modifications of processes that satisfy certain bounds on their expectation. This paves the way for modelling and verifying stochastic hybrid systems in Isabelle/HOL.

cs.LO↗

arXiv2024

Unifying Model Execution and Deductive Verification with Interaction Trees in Isabelle/HOL

Model execution allows us to prototype and analyse software engineering models by stepping through their possible behaviours, using techniques like animation and simulation. On the other hand, deductive verification allows us to construct formal proofs demonstrating satisfaction of certain critical properties in support of high-assurance software engineering. To ensure coherent results between execution and proof, we need unifying semantics and automation. In this paper, we mechanise Interaction Trees (ITrees) in Isabelle/HOL to produce an execution and verification framework. ITrees are coinductive structures that allow us to encode infinite labelled transition systems, yet they are inherently executable. We use ITrees to create verification tools for stateful imperative programs, concurrent programs with message passing in the form of the CSP and \Circus languages, and abstract system models in the style of the Z and B methods. We demonstrate how ITrees can account for diverse semantic presentations, such as structural operational semantics, a relational program model, and CSP's failures-divergences trace model. Finally, we demonstrate how ITrees can be executed using the Isabelle code generator to support the animation of models.

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arXiv2024

ACCESS: Assurance Case Centric Engineering of Safety-critical Systems

Assurance cases are used to communicate and assess confidence in critical system properties such as safety and security. Historically, assurance cases have been manually created documents, which are evaluated by system stakeholders through lengthy and complicated processes. In recent years, model-based system assurance approaches have gained popularity to improve the efficiency and quality of system assurance activities. This becomes increasingly important, as systems becomes more complex, it is a challenge to manage their development life-cycles, including coordination of development, verification and validation activities, and change impact analysis in inter-connected system assurance artifacts. Moreover, there is a need for assurance cases that support evolution during the operational life of the system, to enable continuous assurance in the face of an uncertain environment, as Robotics and Autonomous Systems (RAS) are adopted into society. In this paper, we contribute ACCESS - Assurance Case Centric Engineering of Safety-critical Systems, an engineering methodology, together with its tool support, for the development of safety critical systems around evolving model-based assurance cases. We show how model-based system assurance cases can trace to heterogeneous engineering artifacts (e.g. system architectural models, system safety analysis, system behaviour models, etc.), and how formal methods can be integrated during the development process. We demonstrate how assurance cases can be automatically evaluated both at development and runtime. We apply our approach to a case study based on an Autonomous Underwater Vehicle (AUV).

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arXiv2024

Scalable Automated Verification for Cyber-Physical Systems in Isabelle/HOL

We formally introduce IsaVODEs (Isabelle verification with Ordinary Differential Equations), a framework for the verification of cyber-physical systems. We describe the semantic foundations of the framework's formalisation in the Isabelle/HOL proof assistant. A user-friendly language specification based on a robust state model makes our framework flexible and adaptable to various engineering workflows. New additions to the framework increase both its expressivity and proof automation. Specifically, formalisations related to forward diamond correctness specifications, certification of unique solutions to ordinary differential equations (ODEs) as flows, and invariant reasoning for systems of ODEs contribute to the framework's scalability and usability. Various examples and an evaluation validate the effectiveness of our framework.

cs.LO↗

arXiv2023

Checking and Automating Confidentiality Theory in Isabelle/UTP

The severity of recent vulnerabilities discovered on modern CPUs, e.g., Spectre [1], highlights how information leakage can have devas-tating effects to the security of computer systems. At the same time, it suggests that confidentiality should be promoted as a normal part of program verification, to discover and mitigate such vulnerabili-ties early in development. The theory we propose is primarily based on Bank's theory [2], a framework for reasoning about confidentiali-ty properties formalised in the Unifying Theories of Programming (UTP) [3]. We mechanised our encoding in the current implementa-tion of UTP in the Isabelle theorem prover, Isabelle/UTP [4]. We have identified some theoretical issues in Bank's original framework. Finally, we demonstrate how our mechanisation can be used to for-mally verify of some of the examples from Bank's work.

cs.CR↗

arXiv2023

Formally Verified Animation for RoboChart using Interaction Trees

RoboChart is a core notation in the RoboStar framework. It is a timed and probabilistic domain-specific and state machine-based language for robotics. RoboChart supports shared variables and communication across entities in its component model. It has formal denotational semantics given in CSP. The semantic technique of Interaction Trees (ITrees) represents behaviours of reactive and concurrent programs interacting with their environments. Recent mechanisation of ITrees, along with ITree-based CSP semantics and a Z mathematical toolkit in Isabelle/HOL, bring new applications of verification and animation for state-rich process languages, such as RoboChart. In this paper, we use ITrees to give RoboChart novel operational semantics, implement it in Isabelle, and use Isabelle's code generator to generate verified and executable animations. We illustrate our approach using an autonomous chemical detector model and a patrol robot model additionally exhibiting nondeterminism and using shared variables. With animation, we show two concrete scenarios for the chemical detector when the robot encounters different environmental inputs and three concrete scenarios for the patrol robot when its calibrated position is in different sections of a corridor. We also verify that the animated scenarios are truly trace refinements of the CSP denotational semantics of the RoboChart models using FDR, a refinement model checker for CSP. This ensures that our approach to resolving nondeterminism using CSP operators with priority is sound and correct.

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arXiv2023

Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: semantics and automated reasoning with theorem proving

Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic unifying relations (ProbURel), takes a step towards our vision to tackle these challenges. Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic; (2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers; (3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem; (4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics; (5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops; and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving. We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.

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arXiv2023

Automated Reasoning for Physical Quantities, Units, and Measurements in Isabelle/HOL

Formal verification of cyber-physical and robotic systems requires that we can accurately model physical quantities that exist in the real-world. The use of explicit units in such quantities can allow a higher degree of rigour, since we can ensure compatibility of quantities in calculations. At the same time, improper use of units can be a barrier to safety and therefore it is highly desirable to have automated sanity checking in physical calculations. In this paper, we contribute a mechanisation of the International System of Quantities (ISQ) and the associated SI unit system in Isabelle/HOL. We show how Isabelle can be used to provide a type system for physical quantities, and automated proof support. Quantities are parameterised by dimension types, which correspond to base vectors, and thus only quantities of the same dimension can be equated. Since the underlying "algebra of quantities" induces congruences on quantity and SI types, specific tactic support is developed to capture these. Our construction is validated by a test-set of known equivalences between both quantities and SI units. Moreover, the presented theory can be used for type-safe conversions between the SI system and others, like the British Imperial System (BIS).

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arXiv2022

GC Insights: Space sector careers resources in the UK need a greater diversity of roles

Educational research highlights that improved careers education is needed to increase participation in science, technology, engineering, and mathematics (STEM). Current UK careers resources concerning the space sector, however, are found to perhaps not best reflect the diversity of roles present and may in fact perpetuate misconceptions about the usefulness of science. We, therefore, compile a more diverse set of space-related jobs, which will be used in the development of a new space careers resource.

physics.ed-ph↗

arXiv2021

Hybrid Systems Verification with Isabelle/HOL: Simpler Syntax, Better Models, Faster Proofs

We extend a semantic verification framework for hybrid systems with the Isabelle/HOL proof assistant by an algebraic model for hybrid program stores, a shallow expression model for hybrid programs and their correctness specifications, and domain-specific deductive and calculational support. The new store model yields clean separations and dynamic local views of variables, e.g. discrete/continuous, mutable/immutable, program/logical, and enhanced ways of manipulating them using combinators, projections and framing. This leads to more local inference rules, procedures and tactics for reasoning with invariant sets, certifying solutions of hybrid specifications or calculating derivatives with increased proof automation and scalability. The new expression model provides more user-friendly syntax, better control of name spaces and interfaces connecting the framework with real-world modelling languages.

cs.LO↗

arXiv2021

Formally Verified Simulations of State-Rich Processes using Interaction Trees in Isabelle/HOL

Simulation and formal verification are important complementary techniques necessary in high assurance model-based systems development. In order to support coherent results, it is necessary to provide unifying semantics and automation for both activities. In this paper we apply Interaction Trees in Isabelle/HOL to produce a verification and simulation framework for state-rich process languages. We develop the core theory and verification techniques for Interaction Trees, use them to give a semantics to the CSP and Circus languages, and formally link our new semantics with the failures-divergences semantic model. We also show how the Isabelle code generator can be used to generate verified executable simulations for reactive and concurrent programs.

cs.LO↗

arXiv2021

Certifying Differential Equation Solutions from Computer Algebra Systems in Isabelle/HOL

The Isabelle/HOL proof assistant has a powerful library for continuous analysis, which provides the foundation for verification of hybrid systems. However, Isabelle lacks automated proof support for continuous artifacts, which means that verification is often manual. In contrast, Computer Algebra Systems (CAS), such as Mathematica and SageMath, contain a wealth of efficient algorithms for matrices, differential equations, and other related artifacts. Nevertheless, these algorithms are not verified, and thus their outputs cannot, of themselves, be trusted for use in a safety critical system. In this paper we integrate two CAS systems into Isabelle, with the aim of certifying symbolic solutions to ordinary differential equations. This supports a verification technique that is both automated and trustworthy.

cs.LO↗

arXiv2020

Integration of Formal Proof into Unified Assurance Cases with Isabelle/SACM

Assurance cases are often required to certify critical systems. The use of formal methods in assurance can improve automation, increase confidence, and overcome errant reasoning. However, assurance cases can never be fully formalised, as the use of formal methods is contingent on models that are validated by informal processes. Consequently, assurance techniques should support both formal and informal artifacts, with explicated inferential links between them. In this paper, we contribute a formal machine-checked interactive language, called Isabelle/SACM, supporting the computer-assisted construction of assurance cases compliant with the OMG Structured Assurance Case Meta-Model. The use of Isabelle/SACM guarantees well-formedness, consistency, and traceability of assurance cases, and allows a tight integration of formal and informal evidence of various provenance. In particular, Isabelle brings a diverse range of automated verification techniques that can provide evidence. To validate our approach, we present a substantial case study based on the Tokeneer secure entry system benchmark. We embed its functional specification into Isabelle, verify its security requirements, and form a modular security case in Isabelle/SACM that combines the heterogeneous artifacts. We thus show that Isabelle is a suitable platform for critical systems assurance.

cs.SE↗

arXiv2020

Automated Verification of Reactive and Concurrent Programs by Calculation

Reactive programs combine traditional sequential programming constructs with primitives to allow communication with other concurrent agents. They are ubiquitous in modern applications, ranging from components systems and web services, to cyber-physical systems and autonomous robots. In this paper, we present an algebraic verification strategy for concurrent reactive programs, with a large or infinite state space. We define novel operators to characterise interactions and state updates, and an associated equational theory. With this we can calculate a reactive program's denotational semantics, and thereby facilitate automated proof. Of note is our reasoning support for iterative programs with reactive invariants, based on Kleene algebra, and for parallel composition. We illustrate our strategy by verifying a reactive buffer. Our laws and strategy are mechanised in Isabelle/UTP, our implementation of Hoare and He's Unifying Theories of Programming (UTP) framework, to provide soundness guarantees and practical verification support.

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