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.
@InProceedings{diehl_et_al:DagSemProc.06111.20, author = {Diehl, Scott and van Melkebeek, Dieter}, title = {{Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines}}, booktitle = {Complexity of Boolean Functions}, pages = {1--33}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.20}, URN = {urn:nbn:de:0030-drops-6054}, doi = {10.4230/DagSemProc.06111.20}, annote = {Keywords: Time-space lower bounds, lower bounds, randomness, polynomial-time hierarchy} }
Feedback for Dagstuhl Publishing