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Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines Diehl, Scott van Melkebeek, Dieter Time-space lower bounds lower bounds randomness polynomial-time hierarchy 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Scott Diehl and Dieter van Melkebeek 2006 Is Part Of Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.06111.20 urn:nbn:de:0030-drops-6054 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.20 eng https://creativecommons.org/licenses/by/4.0/legalcode