James H. Davenport

James H. Davenport, Matthew England, Scott McCallum

This paper presents two enhancements to cylindrical algebraic decomposition (CAD) based quantifier elimination (QE) for cases in which multiple equational constraints are present in the given input formula $ϕ^*$. The first enhancement provides more detail in the output when there is a conceptual partition of the set of variables of $ϕ^*$ into parameters and unknowns. In such cases, we describe how to partition the parameter space so that: (1) in each open set of the partition the number $ν$ of associated unknowns is a finite constant or is infinite; and (2) for each such open set for which $ν$ is finite, an expression for the unknowns in terms of the parameters is provided. The second enhancement is an efficiency gain achievable in certain situations. Indeed, when certain conditions are met, the second CAD equational projection step can be reduced more significantly than is supported by the prior existing theory. Relevant theorems and worked examples for both enhancements are provided. Application areas include approximation theory, cuspidal manipulator classification, and biological/chemical systems.

James H. Davenport

Formal methods have been largely thought of in the context of safety-critical systems, where they have achieved major acceptance. Tens of millions of people trust their lives every day to such systems, based on formal proofs rather than ``we haven't found a bug'' (yet!). Why is ``we haven't found a bug'' an acceptable basis for systems trusted with hundreds of millions of people's personal data? This paper looks at some of the issues in CyberSecurity, and the extent to which formal methods, ranging from ``fully verified'' to better tool support, could help. Alas The Royal Society (2016) only recommended formal methods in the limited context of ``safety critical applications'': we suggest this is too limited.

Tom Crick, James H. Davenport, Alastair Irons et al.

This paper presents a national case study-based analysis of the numerous dimensions to cybersecurity education and how they are prioritised, implemented and accredited; from understanding the interaction of hardware and software, moving from theory to practice (and vice versa), to human factors, policy and politics (as well as various other important facets). A multitude of model curricula and recommendations have been presented and discussed in international fora in recent years, with varying levels of impact on education, policy and practice. This paper address three key questions: i) what is taught and what should be taught for cybersecurity to general computer science students; ii) should cybersecurity be taught stand-alone or in an integrated manner to general computer science students; and iii) can accreditation by national professional, statutory and regulatory bodies enhance the provision of cybersecurity within a body's jurisdiction? Evaluating how cybersecurity is taught in all aspects of computer science is clearly a task of considerable size, one that is beyond the scope of this paper. Instead a case study-based research approach -- primarily focusing on the UK -- has been adopted to evaluate the evidence of the teaching of cybersecurity within general computer science to university-level students. Thus, in the context of widespread international computer science/engineering curriculum reform, what does this need to embed cybersecurity knowledge and skills mean more generally for institutions and educators, and how can we teach this subject more effectively? Through this UK case study, and by contrasting with related initiatives in the US, we demonstrate the positive effect that national accreditation requirements can have, and offer some recommendations both for future research and curriculum developments.

Zongyan Huang, Matthew England, David Wilson et al.

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in the size of the input, which is often encountered in practice. It has been observed that for many problems a change in algorithm settings or problem formulation can cause huge differences in runtime costs, changing problem instances from intractable to easy. A number of heuristics have been developed to help with such choices, but the complicated nature of the geometric relationships involved means these are imperfect and can sometimes make poor choices. We investigate the use of machine learning (specifically support vector machines) to make such choices instead. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we apply it in two case studies: the first to select between heuristics for choosing a CAD variable ordering; the second to identify when a CAD problem instance would benefit from Groebner Basis preconditioning. These appear to be the first such applications of machine learning to Symbolic Computation. We demonstrate in both cases that the machine learned choice outperforms human developed heuristics.

Zongyan Huang, Matthew England, James H. Davenport et al.

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.

Zongyan Huang, Matthew England, David Wilson et al.

Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.