Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines

In this talk, we establish lower bounds for the running time of randomized machines with two-sided error which use a small amount of workspace to solve complete problems in the polynomial-time hierarchy. In particular, we show that for integers $l > 1$, a randomized machine with two-sided error using subpolynomial space requires time $n^{l - o(1)}$ to solve QSATl, where QSATl denotes the problem of deciding the validity of a Boolean first-order formula with at most $l-1$ quantifier alternations. This represents the first time-space lower bounds for complete problems in the polynomial-time hierarchy on randomized machines with two-sided error. Corresponding to $l = 1$, we show that a randomized machine with one-sided error using subpolynomial space requires time $n^{1.759}$ to decide the set of Boolean tautologies. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result.