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

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 430 Found Comput Math (2010) 10: 429-454 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.

On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials

“…We prove in this paper (see Theorems 2.1 and 1.12 below) that the number of homotopy types of such limits can indeed be bounded single exponentially in terms of the format of the formulas defining the one-parameter family. The techniques introduced by Zell in [19] (as well certain semi-algebraic constructions described in [6]) play a crucial role in the proof of our bound. These intermediate results may be of independent interest.…”

“…Definition 2.10 differs slightly from the original definition in [6] in that we have replaced the predicate ∧ 0≤i≤p T i ≥ 0 in the original definition by the predicate 0≤i≤p T 2 i ≤ 4. Also, because of the application in [6] the semi-algebraic fibered join was defined as a semi-algebraic subset of a sphere of large enough radius. This is not essential in this paper and this leads to a further simplification of the formula.…”

“…The next proposition proved in [6] is important in the proof of Proposition 2.3; it relates up to p-equivalence the semi-algebraic set J Proposition 2.12.…”

“…Using a semi-algebraic realization of the fibered join described in [6], we then consider for any fixed p ≥ 0, a semi-algebraic set J…”

“…Definition 2.10 (The semi-algebraic fibered join [6] ). For a semi-algebraic subset S ⊂ R k contained in B k (0, R), defined by a P-formula Φ and f : S → T a semialgebraic map, we define…”