Algebraic complexity theory via the algebraic geometry and representation theory of generalised continued fractions

通过代数几何和广义连分数表示论的代数复杂性理论

• 批准号: EP/W014882/2
• 负责人: Christian Ikenmeyer
• 金额: $ 37.03万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW014882%2F2
• 关键词: Algebraic; complexity; theory; via; algebraic

High-end computers can solve extremely difficult tasks, but even the best computers have limits in what they can do. These limits are studied in the field of computational complexity theory. The most famous question in computational complexity theory is the P vs NP conjecture, where a price of $1,000,000 will be awarded to the person or group who resolves it. The question can be phrased in terms of Boolean circuits and algorithms as follows. A Boolean circuit is a description of a finite sequence of logical operations that is performed a list of input values, much like a computer program, but with no loops. Given a Boolean circuit we want to know if there exists an input that makes it output TRUE. The P vs NP conjecture states that there is no efficient algorithm for this task. More precisely, the computation time required to answer this question for a circuit that is described using n symbols grows faster than any polynomial in n: Superpolynomially fast. This is independent of the algorithm that is used to answer the question. It is generally assumed among researchers that we are very far from resolving the P vs NP conjecture though. Instead of studying Boolean circuits many algorithmic breakthroughs are based on algebraic insights, for example the Euclidean algorithm, the effective Chinese Remainder Theorem, Gaussian elimination, Newton iteration, fast matrix multiplication, fast polynomial multiplication, fast Fourier transform, Elliptic Curve Cryptography, Reed-Solomon codes, or solving systems of polynomial equations via Gröbner bases. One more important example is Horner's method, a way of evaluating polynomials in a single variable efficiently. In 1966 Victor Pan proved that Horner's method is optimal if we count the number of arithmetic operations instead of Boolean operations. Using this way of measuring the complexity of a problem, in 1979 Leslie Valiant FSR defined algebraic analogues of the P vs NP problem. In this proposal we focus on Valiant's VF vs VNP conjecture. This conjecture says that the required size of arithmetic formulas counting the number of Hamiltonian cycles in directed graphs, a problem from combinatorics, grows superpolynomially. This is closely related to the P vs NP conjecture: If P equals NP, then the so-called polynomial hierarchy collapses, which most complexity theorists think does not happen. If one proves that this collapse does not happen, then this also implies that VF is not equal to VNP by a result of Karp and Lipton in 1980, so in a sense the VF vs VNP conjecture has to be resolved in order to make progress.We hope to make progress on the VF vs VNP conjecture, because a recent paper (Bringmann, Ikenmeyer, Zuiddam 2017) shows a deep connection between arithmetic formula size and a multivariate (i.e., several variables) version of Horner's method. This means that separating VF and VNP boils down to understanding the multivariate Horner's method on a deep level. It has connections to continued fractions and the continuant polynomial, which was recently studied by Charles Conley and Valentin Ovsienko (2018). Other than their work, only preliminary algebraic, geometric, and algorithmic results are known so far (very recently by Shpilka and Medini) and we aim to provide the missing foundations. We aim to connect our new insights in several ways with Geometric Complexity Theory, a mathematical approach for complexity lower bounds that uses algebraic geometry and representation theory and that was founded in 2001 by Ketan Mulmuley and Milind Sohoni. This will lead to new ways of proving lower bounds on arithmetic formula size, which then ultimately can help to separate VF and VNP.

高端计算机可以解决极其困难的任务，但即使是最好的计算机也有其局限性。这些限制在计算复杂性理论领域进行了研究。计算复杂性理论中最著名的问题是P vs NP猜想，解决这个问题的人或团体将获得100万美元的奖励。这个问题可以用布尔电路和算法来表述，如下所示。布尔电路是对有限的逻辑运算序列的描述，该逻辑运算是对输入值列表执行的，很像计算机程序，但没有循环。给定一个布尔电路，我们想知道是否存在使其输出为TRUE的输入。P vs NP猜想指出，对于这个任务没有有效的算法。更准确地说，对于使用n个符号描述的电路，回答这个问题所需的计算时间比n中的任何多项式都要快：超多项式快。这与用于回答问题的算法无关。研究人员普遍认为，我们离解决P vs NP猜想还很远。许多算法的突破不是基于布尔电路的研究，而是基于代数的见解，例如欧几里得算法、有效的中国剩余定理、高斯消去法、牛顿迭代法、快速矩阵乘法、快速多项式乘法、快速傅立叶变换、椭圆曲线密码学、里德-所罗门码或通过Gröbner基求解多项式方程组。一个更重要的例子是霍纳方法，一种有效地计算单变量多项式的方法。在1966年维克托潘证明，霍纳的方法是最佳的，如果我们计数的算术运算，而不是布尔运算。1979年，Leslie Valiant FSR使用这种方法来测量问题的复杂性，定义了P vs NP问题的代数类似物。在这个建议中，我们专注于Valiant的VF与VNP猜想。这个猜想说，计算有向图中哈密尔顿圈数的算术公式所需的大小，是一个组合学问题，以超多项式方式增长。这与P vs NP猜想密切相关：如果P等于NP，那么所谓的多项式层次崩溃，大多数复杂性理论家认为这不会发生。如果证明了这种坍缩不发生，那么这也意味着VF不等于1980年Karp和Lipton的结果，所以在某种意义上，VF vs VNP猜想必须得到解决才能取得进展。（Bringmann，Ikenmeyer，Zuiddam 2017）显示了算术公式大小和多变量（即，几个变量）版本的霍纳的方法。这意味着分离VF和VNP归结为在深层次上理解多变量Horner方法。它与连续分数和连续多项式有关，最近由Charles Conley和Valentin Ovsienko（2018）研究。除了他们的工作，只有初步的代数，几何和算法的结果是已知的，到目前为止（最近由Shpilka和Medini），我们的目标是提供失踪的基础。我们的目标是以多种方式将我们的新见解与几何复杂性理论联系起来，这是一种使用代数几何和表示理论的复杂性下界的数学方法，由Ketan Mulmuley和Milind Sohoni于2001年创立。这将导致新的方法来证明算术公式大小的下限，然后最终可以帮助分离VF和VNP。