We define the notion of diversity for families of finite functions and express the limitations of a simple class of holographic algorithms, called elementary algorithms, in terms of limitations on diversity. We show that this class of elementary algorithms is too weak to solve the Boolean circuit value problem, or Boolean satisfiability, or the permanent. The lower bound argument is a natural but apparently novel combination of counting and algebraic dependence arguments that is viable in the holographic framework. We go on to describe polynomial time holographic algorithms that go beyond the elementarity restriction in the two respects that they use exponential size fields, and multiple oracle calls in the form of polynomial interpolation. These new algorithms, which use bases of three components, compute the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case, the parity can also be computed for any one slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes.

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关于全息算法的一些观察

我们为有限函数族定义了多样性的概念，并根据多样性的限制表达了一类简单的全息算法（称为基本算法）的局限性。我们表明，这类基本算法太弱，无法解决布尔电路值问题，或布尔可满足性，或永久问题。下界论证是计数和代数依赖论证的自然但显然新颖的组合，在全息框架中是可行的。我们继续描述多项式时间全息算法，这些算法在使用指数大小字段和多项式插值形式的多个预言机调用这两个方面超越了基本性限制。这些新算法使用三个组件的基础，计算三阶平面无向图的以下数量的奇偶性：直到颜色排列的 3-colorings 的数量、连接的顶点覆盖的数量以及诱导森林或反馈顶点集的数量。在每种情况下，也可以为问题的任何一个切片计算奇偶校验，特别是对于第一种颜色使用特定次数的着色，或者连接的顶点覆盖、反馈集或诱导森林具有特定数量的着色的节点。