The Potential and Challenges of CAD with Equational Constraints for SC-Square

Cylindrical algebraic decomposition (CAD) is a core algorithm within Symbolic Computation, particularly for quantifier elimination over the reals and polynomial systems solving more generally. It is now finding increased application as a decision procedure for Satisfiability Modulo Theories (SMT) solvers when working with non-linear real arithmetic. We discuss the potentials from increased focus on the logical structure of the input brought by the SMT applications and SC-Square project, particularly the presence of equational constraints. We also highlight the challenges for exploiting these: primitivity restrictions, well-orientedness questions, and the prospect of incrementality.Comment: Accepted into proceedings of MACIS 201

Improved cross-validation for classifiers that make algorithmic choices to minimise runtime without compromising output correctness

Our topic is the use of machine learning to improve software by making choices which do not compromise the correctness of the output, but do affect the time taken to produce such output. We are particularly concerned with computer algebra systems (CASs), and in particular, our experiments are for selecting the variable ordering to use when performing a cylindrical algebraic decomposition of $n$-dimensional real space with respect to the signs of a set of polynomials. In our prior work we explored the different ML models that could be used, and how to identify suitable features of the input polynomials. In the present paper we both repeat our prior experiments on problems which have more variables (and thus exponentially more possible orderings), and examine the metric which our ML classifiers targets. The natural metric is computational runtime, with classifiers trained to pick the ordering which minimises this. However, this leads to the situation were models do not distinguish between any of the non-optimal orderings, whose runtimes may still vary dramatically. In this paper we investigate a modification to the cross-validation algorithms of the classifiers so that they do distinguish these cases, leading to improved results.Comment: 16 pages. Accepted into the Proceedings of MACIS 2019. arXiv admin note: text overlap with arXiv:1906.0145

Recent advances in real geometric reasoning

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather than "solving all problems to do with these polynomials"

Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition

There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. The present work extends to have ML select the variable ordering directly, and to try a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human constructed heuristics for the problem: Brown's heuristic and sotd. On this dataset all of the ML approaches outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201

An ongoing case-control study to evaluate the NHS Bowel Cancer Screening Programme

© 2014 Massat et al.; licensee BioMed Central. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated

Machine Learning for Mathematical Software

While there has been some discussion on how Symbolic Computation could be used for AI there is little literature on applications in the other direction. However, recent results for quantifier elimination suggest that, given enough example problems, there is scope for machine learning tools like Support Vector Machines to improve the performance of Computer Algebra Systems. We survey the authors own work and similar applications for other mathematical software. It may seem that the inherently probabilistic nature of machine learning tools would invalidate the exact results prized by mathematical software. However, algorithms and implementations often come with a range of choices which have no effect on the mathematical correctness of the end result but a great effect on the resources required to find it, and thus here, machine learning can have a significant impact.Comment: To appear in Proc. ICMS 201

Sodium channel Nav1.6 accumulates at the site of infraorbital nerve injury

<p>Abstract</p> <p>Background</p> <p>Sodium channel (NaCh) expressions change following nerve and inflammatory lesions and this change may contribute to the activation of pain pathways. In a previous study we found a dramatic increase in the size and density of NaCh accumulations, and a remodeling of NaChs at intact and altered myelinated sites at a location just proximal to a combined partial axotomy and chromic suture lesion of the rat infraorbital nerve (ION) with the use of an antibody that identifies all NaCh isoforms. Here we evaluate the contribution of the major nodal NaCh isoform, Na_v1.6, to this remodeling of NaChs following the same lesion. Sections of the ION from normal and ION lesioned subjects were double-stained with antibodies against Na_v1.6 and caspr (contactin-associated protein; a paranodal protein to identify nodes of Ranvier) and then z-series of optically sectioned images were captured with a confocal microscope. ImageJ (NIH) software was used to quantify the average size and density of Na_v1.6 accumulations, while additional single fiber analyses measured the axial length of the nodal gap, and the immunofluorescence intensity of Na_v1.6 in nodes and of caspr in the paranodal region.</p> <p>Results</p> <p>The findings showed a significant increase in the average size and density of Na_v1.6 accumulations in lesioned IONs when compared to normal IONs. The results of the single fiber analyses in caspr-identified typical nodes showed an increased axial length of the nodal gap, an increased immunofluorescence intensity of nodal Na_v1.6 and a decreased immunofluorescence intensity of paranodal caspr in lesioned IONs when compared to normal IONs. In the lesioned IONs, Na_v1.6 accumulations were also seen in association with altered caspr-relationships, such as heminodes.</p> <p>Conclusion</p> <p>The results of the present study identify Na_v1.6 as one isoform involved in the augmentation and remodeling of NaChs at nodal sites following a combined partial axotomy and chromic suture ION lesion. The augmentation of Na_v1.6 may result from an alteration in axon-Schwann cell signaling mechanisms as suggested by changes in caspr expression. The changes identified in this study suggest that the participation of Na_v1.6 should be considered when examining changes in the excitability of myelinated axons in neuropathic pain models.</p

Need Polynomial Systems Be Doubly-Exponential?

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of zeros of a zero-dimensional system is singly-exponential in the number of variables. How should this contradiction be reconciled? We first note that [Mayr and Ritscher, 2013] shows that the doubly exponential nature of Gr\"{o}bner bases is with respect to the dimension of the ideal, not the number of variables. This inspires us to consider what can be done for Cylindrical Algebraic Decomposition which produces a doubly-exponential number of polynomials of doubly-exponential degree. We review work from ISSAC 2015 which showed the number of polynomials could be restricted to doubly-exponential in the (complex) dimension using McCallum's theory of reduced projection in the presence of equational constraints. We then discuss preliminary results showing the same for the degree of those polynomials. The results are under primitivity assumptions whose importance we illustrate.Comment: Extended Abstract for ICMS 2016 Presentation. arXiv admin note: text overlap with arXiv:1605.0249

Giant Solitary Fibrous Tumor of the Pleura: An Analysis of Five Patients

Ó The Author(s) 2010. This article is published with open access at Springerlink.com Background Solitary fibrous tumor of the pleura (SFTP) represents a clinical entity rarely encountered, especially in giant forms. Complete surgical resection for giant tumor of pleura is a challenge. The aim of this article is to present five new cases of giant SFTP, and to discuss their clinical characteristics and the treatment strategy of such neoplasms. Methods We performed a retrospective review of the clinical records of five patients who underwent surgery for a huge SFTP ([18 cm in diameter) between 2007 and 2009. Results Four patients were symptomatic. All five patients underwent angiography and embolization of the tumorsupplying vessels within 24 h of surgery. All giant tumors were removed completely by extended postlateral thoracotomy with moderate intraoperative bleeding. Two wedge resections and one lobectomy were performed in three cases where the parenchyma had been encroached. Tumors in three patients were pathologically benign; those in the other two were malignant. The symptoms disappeared in all cases after surgery. Conclusions Complete resection remains the mainstay of cure for giant SFTP. We recommend preoperative angiography and embolization for giant SFTP which can reduce the risk of hemorrhage and can contribute to piecemeal removal for radical excision