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

13 years 10 months ago

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We establish the ﬁrst polynomial-strength time-space lower bounds for problems in the lineartime hierarchy on randomized machines with 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 quantiﬁed Boolean formula with at most ℓ − 1 quantiﬁer alternations. Moreover, d approaches 1/2 from below as c approaches 1 from above for ℓ = 2, and d approaches 1 from below as c approaches 1 from above for ℓ ≥ 3. In fact, we establish the stronger result that for any constants a ≤ 1 and c < 1+(ℓ−1)a, there exists a positive constant d such that linear-time alternating machines using space na and ℓ − 1 alternations cannot be simulated by randomized machines with two-sided error running in time nc and space nd , where d approaches a/2 from below as c approaches 1 fr...

Scott Diehl, Dieter van Melkebeek

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