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

Structural Test Coverage Criteria for Integration Testing of LUSTRE/SCADE Programs

International audienceLustre is a formal synchronous declarative language widely used for modeling and specifying safety-critical applications in the fi elds of avionics, transportation, and energy production. In such applications, the testing activity to ensure correctness of the system plays a crucial role in the development process. To enable adequacy measurement of test cases over applications speci ed in Lustre (or SCADE), a hierarchy of structural coverage criteria for Lustre programs has been recently de ned. A drawback with the current de nition of the criteria is that they can only be applied for unit testing, i.e., to single modules without calls to other modules. The criteria experiences scalability issues when used over large systems with several modules and calls between modules. We propose an extension to the criteria de nition to address this scalability issue. We formally de ne the extension by introducing an operator to abstract calls to other modules. This extension allows coverage metrics to be applied to industrial-sized software without an exponential blowup in the number of activation conditions.We conduct a preliminary evaluation of the extended criteria using an Alarm Management System