Time-space tradeoffs for nondeterministic computation

We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose random-access Turing machines in rime n(1.618) and space n(o(1)). This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less than the golden ratio, we prove that satisfiability cannot be salved in time n(a) and space n(b) for some positive constant b. Our techniques allow us to establish this result for b < 1/2(a+2/a(2) - a). We can do better for a close to the golden ratio, for example, satisfiability cannot be solved by a random-access Turing machine using n(1.46) time and n(.11) space. We also shore tradeoffs for nondeterministic linear time computations using sublinear space. For example, there exists a language computable in nondeterministic linear rime and n(.619) space that cannot be computed in deterministic n(1.618) time and n(o(1)) space. Higher up the polynomial-time hierarchy we can get better bounds. We show that linear-time Sigma(l)-computations require essentially n(l) time on deterministic machines that use only n(o(1)) space. We also show new lower bounds on conondeterministic versus nondeterministic computation.