Simulating two pushdown stores by one tape in O(n1.5v) time

Based on two graph separator theorems, we present two unexpected upper bounds and resolve several open problems for on-line computations. (1) 1 tape nondeterministic machines can simulate 2 pushdown stores in time O(n1.5√logn) (true for both on-line and off-line machines). Together with the Ω(n1.5/√logn) lower bound, this solves the open problem 1 in [DGPR] for the 1 tape vs. 2 pushdown case. It also disproves the commonly conjectured Ω(n2) lower bound. (2) The languages defined by Maass and Freivalds, aimed to obtain optimal lower bound for 1 tape nondeterministic machines, can be accepted in O(n2loglogn√logn) and O(n1.5√logn) time by a 1 tape TM, respectively. (3) 3 pushdown stores are better than 2 pushdown stores. This answers a rather old open problem by Book and Greibach, and Duris and Galil. An Ω(n4/3/loge n) lower bound is also obtained. (4) 1 tape can nondeterministically simulate 1 queue in O(n1.5/√logn) time. This disproves the conjectured Ω(n2) lower bound. Also 1 queue can simulate 2 pushdowns in time O(n1.5√logn).