布尔和的代数代价
The Algebraic Cost of a Boolean Sum
AI总结：
本文研究了永久多项式VNP完备性证明无法适用于行列式的原因，分离出三个证明VNP困难性的基本性质，通过“布尔和的代价”揭示了两者在代数复杂度上的根本差异。
摘要：
众所周知，永久多项式对于复杂度类 VNP 是完备的，且人们普遍怀疑行列式尽管具有类似的表达式，却不具备此性质。我们研究了为何永久多项式的 VNP 完备性证明在行列式上会失败。我们分离出了三个足以证明多项式序列是 VNP 困难的基本性质，其中两个为永久多项式和行列式所共有。我们接着证明，永久多项式满足第三个性质，我们将其称为“布尔和的代价”，而行列式不满足，从而展示了这两个多项式族之间的根本差异。我们进一步指出，这种区分也适用于边界复杂度设定，并且我们的结果适用于计数复杂度。
英文摘要：
It is a well-known fact that the permanent polynomial is complete for the complexity class VNP, and it is largely suspected that the determinant does not share this property, despite its similar expression. We study the question of why the VNP-completeness proof of the permanent fails for the determinant. We isolate three fundamental properties that are sufficient to prove a polynomial sequence is VNP-hard, of which two are shared by both the permanent and the determinant. We proceed to show that the permanent satisfies the third property, which we refer to as the ``cost of a boolean sum," while the determinant does not, showcasing the fundamental difference between the polynomial families. We further note that this differentiation also applies in the border complexity setting and that our results apply for counting complexity.
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