Interleaving and lock-step semantics for analysis and verification of GPU kernels

Graphics Processing Units (GPUs) from leading vendors employ predicated (or guarded) execution to eliminate branching and increase performance. Similarly, a recent GPU verification technique uses predication to reduce verification of GPU kernels (the massively parallel programs that run on GPUs) to verification of a sequential program. Prior work on the formal semantics of lock-step predicated execution for kernels focused on structured programs, where control is organised using if- and while-statements. We provide lock-step execution semantics for GPU kernels that are represented by arbitrary reducible control flow graphs. We present a traditional interleaving semantics and a novel lock-step semantics based on predication, and show that for terminating kernels either both semantics compute identical results or both behave erroneously. The method allows reducing GPU kernel verification to the verification of a sequential, lock-step program to be applied to GPU kernels with arbitrary reducible control flow. We have implemented the method in the GPUVerify tool, and present an evaluation using a set of 163 open source and commercial GPU kernels. Among these kernels, 42 exhibit unstructured control flow which our novel lock-step predication technique can handle fully automatically. This generality comes at a modest price: verification across our benchmark set was on average 2.25 times slower than using an existing approach that specifically targets structured kernels

The inverse moment problem for convex polytopes

The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.Comment: LaTeX2e, 24 pages including 1 appendi

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Six-minute stepper test: a valid clinical exercise tolerance test for COPD patients

JM Grosbois,1,2 C Riquier,3 B Chehere,4 J Coquart,5 H Béhal,6 F Bart,2 B Wallaert,2,3 C Chenivesse3 1FormAction Santé, Pérenchies, France; 2Department of Respiratory Medicine, Centre Hospitalier Germon et Gauthier, Béthune, France; 3Department of Respiratory Medicine Immunology and Allergy, Centre Hospitalier Universitaire de Lille, Competence Center for rare lung diseases, University Lille 2, Lille, France; 4EA 7369, URePSSS, Multidisciplinary Research Unit in Sport Health Society, University Lille 2, Lille, France; 5Faculty of Sport Sciences, Sports and Physical Activity, Center for Ecology and Transformation, University of Rouen, Mont Saint Aignan, France; 6Department of Statistical Methods and Biostatistics, Centre Hospitalier Universitaire de Lille, University of Lille Nord, Lille, France Introduction: Exercise tolerance testing is an integral part of the pulmonary rehabilitation (PR) management of patients with chronic obstructive pulmonary disease (COPD). The 6-minute stepper test (6MST) is a new, well-tolerated, reproducible exercise test, which can be performed without any spatial constraints.Objective: The aim of this study was to compare the results of the 6MST to those obtained during a 6-minute walk test (6MWT) and cardiopulmonary exercise testing (CPET) in a cohort of COPD patients.Methods: Ninety-one COPD patients managed by outpatient PR and assessed by 6MST, 6MWT, and CPET were retrospectively included in this study. Correlations between the number of steps on the 6MST, the distance covered on the 6MWT, oxygen consumption, and power at the ventilatory threshold and at maximum effort during CPET were analyzed before starting PR, and the improvement on the 6MST and 6MWT was compared after PR.Results: The number of steps on the 6MST was significantly correlated with the distance covered on the 6MWT (r=0.56; P<0.0001), the power at maximum effort (r=0.46; P<0.0001), and oxygen consumption at maximum effort (r=0.39; P<0.005). Performances on the 6MST and 6MWT were significantly improved after PR (570 vs 488 steps, P=0.001 and 448 vs 406 m, respectively; P<0.0001). Improvements of the 6MST and 6MWT after PR were significantly correlated (r=0.34; P=0.03).Conclusion: The results of this study show that the 6MST is a valid test to evaluate exercise tolerance in COPD patients. The use of this test in clinical practice appears to be particularly relevant for the assessment of patients managed by home PR. Keywords: 6-minute stepper test, 6-minute walk test, exercise tolerance, pulmonary rehabilitation, cardiopulmonary exercise testing, validit