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 time n/sup 1.618/ and space n/sup 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 solved in time n/sup a/ and space n/sup /spl delta// for some positive constant b. Our techniques allow us to establish this result for b< 1/2 (/spl alpha/+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/sup 1.46/ time and n/sup .11/ space. We also show tradeoffs for nondeterministic linear time computations using sublinear space. For example, there exists a language computable in nondeterministic linear time and n/sup 619/ space that cannot be computed in deterministic n/sup 1.618/ time and n/sup o(1)/ space. Higher up the polynomial-time hierarchy we can get better bounds. We show that linear-time /spl Sigma//sub l/-computations require essentially n/sup l/ time on deterministic machines that use only n/sup o(1)/ space. We also show new lower bounds on conondeterministic versus nondeterministic computation.