多项式族、Betti数和Toda定理的一个实类比
Polynomial hierarchy, Betti numbers and a real analogue of Toda's theorem
摘要：
户田在1989年证明了（离散的）多项式时间层次结构$\mathbf{PH}$包含在类$\mathbf{P}^{#\mathbf{P}}$中，即可以由图灵机在多项式时间内决定的语言类，该图灵机可以访问具有计算计数复杂性类$#\mathbf{P}$中的函数的能力的预言。这个结果说明了计数的力量，被认为是计算复杂性理论中的一个开创性的结果。到目前为止，在实数上的复杂性理论（在Blum-Shub-Smale真实机器的意义上）中还没有类似的结果。在本文中，我们制定和证明一个真正的模拟户田定理。不像户田的证明在离散的情况下，这依赖于复杂的组合参数，我们的证明是拓扑的性质。由于我们的技术，我们也能够涉及到计算硬度的两个非常好的研究问题，在算法半代数几何-即问题的决定句子在一阶理论的实数与常数数量的量词交替，并计算贝蒂数的半代数集。我们得到了一个多项式时间减少的第一个问题的紧凑版本的第二个。后者的结果可能是独立的兴趣，研究人员在算法半代数几何。
英文摘要：
Toda proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class $\mathbf{P}^{#\mathbf{P}}$, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class $#\mathbf{P}$. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines) has been missing so far. In this paper we formulate and prove a real analogue of Toda's theorem. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry -- namely the problem of deciding sentences in the first order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result might be of independent interest to researchers in algorithmic semi-algebraic geometry.
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