On Probabilistic Time versus Alternating Time

We prove several new results regarding the relationship between probabilistic time, BPTime (t), and alternating time, ΣO(1)Time (t). Our main results are the following: 1. We prove that BPTime (t) ⊆ Σ3Time (t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time ( t2 · log t ) (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime (t) ⊆ ΣcTime (t) for a large constant c > 3 (Ajtai, Adv. in Comp. Complexity Theory ’93). 2. We prove that BPTime (t) 6⊆ Σ2Time ( o(t2) ) with respect to some oracle. This complements our result (1), and shows that the running time of the Sipser-GácsLautemann simulation is optimal, up to a log t factor, for relativizing techniques. (All the results in (1) relativize.) This result is obtained as a corollary from a new circuit lower bound for approximate majority : poly(n)-size depth-3 circuits for approximate majority have bottom fan-in Ω(log n). 3. We prove that solving QSAT3 ∈ Σ3Time (n · poly log n) requires time n1+Ω(1) on probabilistic Turing machines using space n.9, with random access to input and work tapes, and two-way sequential access to the random-bit tape. This is the first lower bound of the form t = n1+Ω(1) on a model with random access to the input and two-way access to the random bits. 4. We prove that solving QSAT3 ∈ Σ3Time (n · poly log n) requires time n1+Ω(1) on Turing machines with an input tape and a sequential work tape that is initialized with random bits. This is the first lower bound on a probabilistic extension of the off-line Turing machine model with one work tape. ∗Research supported by NSF grant CCR-0133096, US-Israel BSF grant 2002246, ONR grant N-0001404-1-0478. 0 Electronic Colloquium on Computational Complexity, Report No. 137 (2005)