### Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines [chapter]

Scott Diehl, Dieter van Melkebeek
2005 Lecture Notes in Computer Science
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 QSAT ℓ cannot be computed by such machines in time n c and space n d , where QSAT ℓ 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
more » ... 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 n a and ℓ − 1 alternations cannot be simulated by randomized machines with two-sided error running in time n c and space n d , where d approaches a/2 from below as c approaches 1 from above for ℓ = 2 and d approaches a from below as c approaches 1 from above for ℓ ≥ 3. Corresponding to ℓ = 1, we prove that 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 n 1.759 and space n d . As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result. * A preliminary version of this work appeared as an extended abstract in the Proceedings of the 32nd International Colloquium on Automata, Languages, and Programming . † Lemma 5 (Folklore). Let ℓ be a positive integer and T a time function. Then Σ ℓ TIME[T ] Π ℓ TIME[o(T )]. Our lower bounds for space-bounded alternating linear time require a stronger diagonalization result which is both time-and space-sensitive. Lemma 6 (). Let T be a time function and S a space function. Then for any integer ℓ > 0, Σ ℓ TISP[T, S] Π ℓ TISP[o(T ), o(S)]. Earlier Techniques We now outline some of the techniques common to many time-space lower bound arguments. We also use them for our results. 7