Lower Bounds for Computation with Limited Nondeterminism

We investigate the e ect of limiting the number of available nondeterministic bits in di erent computational models. First we relate formula size to one-way communication complexity and derive lower bounds of (n = log n) on the size of formulae with n = log n nondeterministic bits for 0 < 1=2. Next we prove a rounds-communication hierarchy for communication complexity with limited nondeterminism solving an open problem of [HrS96]. Given a bound s on the number of nondeterministic bits and a number k of rounds we construct a function which can be computed deterministically in k rounds with O(sk logn) bits communication, but requires (n=(sk logn)) in k 1 rounds though s nondeterministic bits are available. We apply this result to show a reversal hierarchy for 2-way automata with limited nondeterminism exhibiting exponential gaps. Furthermore we investigate the e ect of limited nondeterminism on monotone circuit depth. All results presented in the paper have the common core that limited nondeterministic communication has high round dependence.