NDLI: Polynomial Hierarchy, Betti Numbers and a Real Analogue of Toda's Theorem

Content Provider	IEEE Xplore Digital Library
Author	Basu, S. Zell, T.
Copyright Year	2009
Abstract	Toda \cite{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 Turing machines \cite{BSS89}) 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.
Starting Page	73
Ending Page	82
File Size	304863
Page Count	10
File Format	PDF
ISBN	9781424451166
ISSN	02725428
DOI	10.1109/FOCS.2009.70
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2009-10-25
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Polynomials Turing machines Computational complexity Computational geometry Mathematics Analog computers Complexity theory Algorithm design and analysis Computer science History Toda's theorem Polynomial hierarchy Betti numbers Semi-algebraic sets
Content Type	Text
Resource Type	Article

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