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

Abstract

Abstract. Toda [35] 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 Turing machines [9]) 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 semialgebraic 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.

A Computational Framework for the Study of Partition Functions and Graph Polynomials

“…However, although there are various analogues for P in BSS, cf. [7], [8], [9], [10] there seemed to be no counterpart for P-completeness in the BSS model suitable for graph polynomials. In [7], [8] what is proposed as P for BSS counts the zeros of a polynomial, in case this is finite.…”

A Complexity Theory of Constructible Functions and Sheaves

“…As mentioned before, in order to get around certain difficulties caused by nonlocally-closed sets and non-proper maps, a restricted polynomial hierarchy was defined in [17]. We now recall the definition of this compact analog, PH c R , of PH R .…”

“…It is important to note that the the B-S-S complexity class P R (as well as P c R ) is stable under certain operations. In fact, many results (such as the analog of Toda's theorem in the B-S-S model proved in [17] as an illustrative example) depend only on these stability properties of the class P R and not on its actual definition involving B-S-S machines. We will formulate similar stability properties for the sheaf-theoretic generalization of the class P c R .…”

“…In the first case, we use a construction used in [17]. Namely, given a sequence (Y n ⊂ S m1(n) × S (m(n) ) n>0 ∈ P R , the sequence…”

“…We will see later that many discrete valued functions that appear in complexity theory including functions such as the characteristic functions of constructible as well as semi-algebraic sets, ranks of matrices and higher dimensional tensors, topological invariants such as the Betti numbers or Euler-Poincaré characteristics, local dimensions of semi-algebraic sets are all examples of such functions. Constructible functions in the place of so called "counting" functions have already appeared in B-S-S style complexity theory over R and C. For example, in [17] (respectively, [9]) a real (respectively, complex) analog of Toda's theorem of discrete complexity theory was obtained. The notion of counting the number of satisfying assignments of a Boolean equation was replaced in these papers by the problem of computing the Poincaré polynomial of a semi-algebraic/constructible set (see also [27,26]).…”