On Approximate Majority and Probabilistic Time

Abstract

We prove new results on the circuit complexity of approximate majority, which is the problem of computing the majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, ∑O(1)Time (t). Our main results are the following: We prove that depth-3 circuits with bottom fan-in (log n)/2 that compute approximate majority on n bits must have size at least $$2^{n^{0.1}}$$. As a corollary we obtain that there is no black-box proof that BPTime (t) $$\subseteq \sum_2$$Time (o(t2)). This complements the (black-box) result that BPTime (t) $$\subseteq \sum_2$$Time (t2 · poly log t) (Sipser and Gacs, STOC ’83; Lautemann, IPL ’83). We prove that approximate majority is computable by uniform polynomial-size circuits of depth 3. Prior to our work, the only known polynomial-size depth-3 circuits for approximate majority were non-uniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) $$\subseteq \sum_3$$Time (t · poly log t). This complements our results in (1). We prove new lower bounds for solving QSAT3 $$\in \sum_3$$Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT3 requires time n1+Ω(1) on Turing machines with a random-access input tape and a sequential-access work tape that is initialized with random bits. No nontrivial lower bound was previously known on this model (for a function computable in linear space).