AF: Small: The Power of Randomness in Decision and Verification

做出随机选择的能力似乎简化了几项计算任务。例如，为了确定像x*x-y*y和(x+y)*(x-y)这样的两个公式是否相等，只需插入x和y的随机值，并验证这两个公式是否产生相同的值。即使对于复杂的公式，这也是一个可靠的策略，比操作抽象的公式更容易。随机性的一种更复杂的使用包括抽查计算的方法，这种方法需要的工作量比执行计算本身少得多-在将计算委托给云的时代，这是一个至关重要的特征。这个项目调查了在没有随机性的情况下在决策和核查中实现类似效率的可能性。它包括研究生和本科生的培训和教育，并包括开发一本教科书，使计算思维和量子算法领域更容易为更广泛的公众所接受。该项目探索了去随机化领域的两个新方向。在具有有界误差的多项式时间决策过程(即，有界误差概率多项式时间类，BPP)的背景下，研究者提出了一种基于消失理想的去随机化多项式恒等式检验(PIT)的原则性方法。PIT抓住了上述公式等价性问题，并在去随机化领域发挥了核心作用，因为已知的结果表明，去随机化PIT可能使所有BPP去随机化。研究人员已经描述了广泛使用的伪随机发生器用于坑道的消失理想，并计划为其他发电机开发这种方法，并研究其后果。在被称为Arthur-Merlin(AM)协议的多项式时间交互验证过程的设置中，调查人员希望进一步发展下限和去随机化之间的联系，特别是通过适应AM的设置最近为BPP的设置建立的实例连接。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The ability to make random choices seems to simplify several computational tasks. For example, in order to decide whether two formulas like x*x-y*y and (x+y)*(x-y) are equivalent, one can merely plug in random values for x and y and verify whether the two formulas yield the same value. Even for complicated formulas, this is a reliable strategy, easier than manipulating the abstract formulas. A more involved use of randomness consists of a way to spot-check computations that requires significantly less effort than performing the computations themselves---a feature of paramount importance in the day and age of delegating computations to the cloud. This project investigates the potential of achieving similar efficiency in decision and verification without randomness. It incorporates graduate and undergraduate training and education, and includes the development of a textbook that makes computational thinking and the area of quantum algorithms more accessible to a broader public.The project explores two new directions in the area of derandomization. In the setting of polynomial-time decision procedures with bounded error (i.e., bounded-error probabilistic polynomial time class, BPP), the investigators propose a principled approach towards derandomizing Polynomial Identity Testing (PIT) based on the notion of a vanishing ideal. PIT captures the above formula equivalence problem and plays a central role in the area of derandomization as known results suggest that derandomizing PIT may enable derandomizing all of BPP. The investigators already characterized the vanishing ideal of a widely used pseudorandom generator for PIT, plan to develop the approach for other generators, and to research the ramifications. In the setting of polynomial-time interactive verification processes known as Arthur-Merlin (AM) protocols, the investigators want to further develop the connections between lower bounds and derandomization, in particular by adapting to the setting of AM the instance-wise connection that was recently established for the setting of BPP.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.