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- W2950038686 abstract "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." @default.
- W2950038686 created "2019-06-27" @default.
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- W2950038686 date "2008-12-05" @default.
- W2950038686 modified "2023-09-27" @default.
- W2950038686 title "Polynomial hierarchy, Betti numbers and a real analogue of Toda's theorem" @default.
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