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

Toda 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 BlumShub-Smale real Turing 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.