UNOmaha Problem of the Week (2021-2022 Edition)

The University of Omaha math department\u27s Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester\u27s end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes. Now there are three difficulty tiers to POW problems, roughly corresponding to easy/medium/hard difficulties, with each tier getting twelve problems per semester, and three problems (one of each tier) per week posted online and around campus. The tiers are named after the EPH classification of conic sections (which is connected to many other classifications in math), and in the present compilation they abide by the following color-coding: Cyan, Green, and Magenta. In practice, when creating the problem sets, we begin with a large enough pool of problem drafts and separate out the ones which are most obviously elliptic or hyperbolic, and then the remaining ones fall into parabolic. The tiers don\u27t necessarily reflect workload, though, only prerequisite mathematical background. Ideally, the solutions to elliptic problems, and any parts of solutions to parabolic and hyperbolic problems not covered in standard undergraduate courses, are meant to test participants\u27 creativity. Beware, though, many solutions also include additional commentary which varies wildly in the reader\u27s assumed mathematical maturity

A Verified Software Toolchain for Quantum Programming

Quantum computing is steadily moving from theory into practice, with small-scale quantum computers available for public use. Now quantum programmers are faced with a classical problem: How can they be sure that their code does what they intend it to do? I aim to show that techniques for classical program verification can be adapted to the quantum setting, allowing for the development of high-assurance quantum software, without sacrificing performance or programmability. In support of this thesis, I present several results in the application of formal methods to the domain of quantum programming, aiming to provide a high-assurance software toolchain for quantum programming. I begin by presenting SQIR, a small quantum intermediate representation deeply embedded in the Coq proof assistant, which has been used to implement and prove correct quantum algorithms such as Grover’s search and Shor’s factorization algorithm. Next, I present VOQC, a verified optimizer for quantum circuits that contains state-of-the-art SQIR program optimizations with performance on par with unverified tools. I additionally discuss VQO, a framework for specifying and verifying oracle programs, which can then be optimized with VOQC. Finally, I present exploratory work on providing high assurance for a high-level industry quantum programming language, Q#, in the F* proof assistant

Formal Methods Specification and Analysis Guidebook for the Verification of Software and Computer Systems

This guidebook, the second of a two-volume series, is intended to facilitate the transfer of formal methods to the avionics and aerospace community. The 1st volume concentrates on administrative and planning issues [NASA-95a], and the second volume focuses on the technical issues involved in applying formal methods to avionics and aerospace software systems. Hereafter, the term "guidebook" refers exclusively to the second volume of the series. The title of this second volume, A Practitioner's Companion, conveys its intent. The guidebook is written primarily for the nonexpert and requires little or no prior experience with formal methods techniques and tools. However, it does attempt to distill some of the more subtle ingredients in the productive application of formal methods. To the extent that it succeeds, those conversant with formal methods will also nd the guidebook useful. The discussion is illustrated through the development of a realistic example, relevant fragments of which appear in each chapter. The guidebook focuses primarily on the use of formal methods for analysis of requirements and high-level design, the stages at which formal methods have been most productively applied. Although much of the discussion applies to low-level design and implementation, the guidebook does not discuss issues involved in the later life cycle application of formal methods