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Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

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Abstract

Toda (in SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #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 #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 in Bull. Am. Math. Soc. (NS) 21(1): 1–46, 1989) 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: 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 may be of independent interest to researchers in algorithmic semi-algebraic geometry.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA

    Saugata Basu

  2. School of Mathematics and Computing Sciences, Lenoir-Rhyne University, Hickory, NC, 28603, USA

    Thierry Zell

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  1. Saugata Basu
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  2. Thierry Zell
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Correspondence to Saugata Basu.

Additional information

Communicated By Peter Bürgisser.

S. Basu was supported in part by an NSF grant CCF-0634907.

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Basu, S., Zell, T. Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem. Found Comput Math 10, 429–454 (2010). https://doi.org/10.1007/s10208-010-9062-4

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