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

2022/01/26 by Diehl, Scott, van Melkebeek, Dieter
#Time-space lower bounds #lower bounds #polynomial-time hierarchy #randomness

paper · doi:10.4230/dagsemproc.06111.20

Abstract

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.

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