Polynomial hierarchy, Betti numbers and a real analogue of Toda's theorem

TL;DR

This paper establishes a real analogue of Toda's theorem, linking the polynomial hierarchy over the reals with Betti number computations using topological methods, advancing the understanding of complexity in real algebraic geometry.

Contribution

It formulates and proves a topological version of Toda's theorem for real computation, connecting the polynomial hierarchy to Betti number calculations in semi-algebraic geometry.

Findings

Proves a topological analogue of Toda's theorem over the reals.

Relates the complexity of deciding first-order sentences to computing Betti numbers.

Provides a polynomial time reduction from decision problems to Betti number computation.

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.