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

We establish the first polynomial-strength time-space lower bounds for problems in the linear-time hierarchy on randomized machines with bounded two-sided error. We show that for any integer l > 1 and constant c < l, there exists a positive constant d such that QSATl cannot be computed by such machines in time nc and space nd, where QSATl denotes the problem of deciding the validity of a Boolean first-order formula with at most l–1 quantifier alternations. Corresponding to l = 1, we prove that for any constant c < φ ≈ 1.618, there exists a positive constant d such that the set of Boolean tautologies cannot be decided by a randomized machine with one-sided error in time nc and space nd.

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