Collaborative Research: FMitF: Track I: Formally Verified Numerical Methods

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When computer algorithms calculate the behavior of physical systems -- planets, self-driving cars, rockets, wi-fi antennas, medicines – some of the mathematical equations can't be solved exactly. Thus, the algorithms approximate the answers as precisely as needed. Numerical analysis is the science of calculating how accurate these algorithms are. The more precision is needed in a certain type of computation, the more time it takes to compute the answer. When one absolutely needs to know in advance how accurate the answers will be, traditional numerical methods are hard to combine into a complete and trustworthy solution. This project's novelties are (1) to integrate all the reasoning about numerical software into a single system, (2) to connect all the levels of reasoning, from mathematics down to software code, producing an end-to-end guarantee of correctness and accuracy, and (3) to produce that guarantee in the form of a mathematical theorem whose proof can be checked fully automatically. The project's broader significance and importance are both scientific and economic: the novel methods and tools developed can (1) prevent software bugs that can be extremely expensive to diagnose by conventional means; (2) allow faster algorithms to be used (when one can now prove that they are accurate); and (3) assure the safety of control software in safety-critical applications such as aircraft and vehicle control.The project takes a layered approach to foundational verification of correctness and accuracy of numerical software---that is, formal machine-checked proofs about programs (not just algorithms), with no assumptions except specifications of instruction-set architectures. The researchers build, improve, and use appropriate tools at each layer: proving in the real numbers about discrete algorithms; proving how floating-point algorithms approximate real-number algorithms; reasoning about C program implementation of floating-point algorithms; and connecting all proofs end-to-end in the Coq proof assistant.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.