Towards justifying computer algebra algorithms in Isabelle/HOL

As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results. This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis. In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains. The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework

Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora

Intuition in formal proof : a novel framework for combining mathematical tools

This doctoral thesis addresses one major difficulty in formal proof: removing obstructions to intuition which hamper the proof endeavour. We investigate this in the context of formally verifying geometric algorithms using the theorem prover Isabelle, by first proving the Graham’s Scan algorithm for finding convex hulls, then using the challenges we encountered as motivations for the design of a general, modular framework for combining mathematical tools. We introduce our integration framework — the Prover’s Palette, describing in detail the guiding principles from software engineering and the key differentiator of our approach — emphasising the role of the user. Two integrations are described, using the framework to extend Eclipse Proof General so that the computer algebra systems QEPCAD and Maple are directly available in an Isabelle proof context, capable of running either fully automated or with user customisation. The versatility of the approach is illustrated by showing a variety of ways that these tools can be used to streamline the theorem proving process, enriching the user’s intuition rather than disrupting it. The usefulness of our approach is then demonstrated through the formal verification of an algorithm for computing Delaunay triangulations in the Prover’s Palette

Discrete Event Simulations

Considered by many authors as a technique for modelling stochastic, dynamic and discretely evolving systems, this technique has gained widespread acceptance among the practitioners who want to represent and improve complex systems. Since DES is a technique applied in incredibly different areas, this book reflects many different points of view about DES, thus, all authors describe how it is understood and applied within their context of work, providing an extensive understanding of what DES is. It can be said that the name of the book itself reflects the plurality that these points of view represent. The book embraces a number of topics covering theory, methods and applications to a wide range of sectors and problem areas that have been categorised into five groups. As well as the previously explained variety of points of view concerning DES, there is one additional thing to remark about this book: its richness when talking about actual data or actual data based analysis. When most academic areas are lacking application cases, roughly the half part of the chapters included in this book deal with actual problems or at least are based on actual data. Thus, the editor firmly believes that this book will be interesting for both beginners and practitioners in the area of DES

Measurements, Models, Systems and Design

531 s.

Algebraic verification of hybrid systems in Isabelle/HOL

The thesis describes an open modular semantic framework for the verification of hybrid systems in a general-purpose proof assistant. We follow this approach to create the first algebraic based verification components for hybrid systems in Isabelle/HOL. The framework benefits from various design choices. Firstly, an algebra for programs such as Kleene algebras with tests or modal Kleene algebras captures the verification condition generation by providing rules for each programming construct. Intermediate relational or state transformer semantics instantiated to a concrete model of the program store allow the framework to handle assignments and ordinary differential equations (ODEs). The verification rules for ODEs require user-provided solutions, differential invariants or analytical descriptions of the continuous dynamics of the system. The construction is a shallow embedding which makes the approach quickly extensible and modular. Taking advantage of these features, we derive differential Hoare logic (dH), a minimalistic logic for the verification of hybrid systems, and the differential refinement calculus (dR) for their stepwise construction. Yet the approach is not limited to these formalisms. We also present a hybrid weakest liberal precondition calculus based on predicate transformers which subsumes powerful deductive verification approaches like differential dynamic logic. The framework is also compositional: we combine it with lenses to vary the model of the program store. We also support it with a formalisation of affine and linear systems of ordinary differential equations in Isabelle/HOL. This integration simplifies various certifications that the proof assistant requires such as guarantees of existence and uniqueness of the corresponding solutions. Verification examples illustrate the approach at work. Formalisations of our solutions to problems of the international friendly competition ARCH2020, where our components participated, further evidence their effectiveness. Finally, a larger case study certifying an invariant for a PID controller of the roll angle in a quadcopter’s flight complements these verifications

Data Mining

Data mining is a branch of computer science that is used to automatically extract meaningful, useful knowledge and previously unknown, hidden, interesting patterns from a large amount of data to support the decision-making process. This book presents recent theoretical and practical advances in the field of data mining. It discusses a number of data mining methods, including classification, clustering, and association rule mining. This book brings together many different successful data mining studies in various areas such as health, banking, education, software engineering, animal science, and the environment

The art and architecture of mathematics education: a study in metaphors

This chapter presents the summary of a talk given at the Eighth European Summer University, held in Oslo in 2018. It attempts to show how art, literature, and history, can paint images of mathematics that are not only useful but relevant to learners as they can support their personal development as well as their appreciation of mathematics as a discipline. To achieve this goal, several metaphors about and of mathematics are explored