We give the first nontrivial model-independent time-space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for log-space uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...

We show that non-deterministic time NT IME(n) is not contained in deterministic time n # 2-# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2-# ) and poly-logarithmic space. A similar result is presented for uniform circuits.

. A serious limitation of the theory of P-completeness is that it fails to distinguish between those P-complete problems that do have polynomial speedup on parallel machines from those that don't. We introduce the notion of strict P-completeness and develop tools to prove precise limits on the possible speedups obtainable for a number of Pcomplete problems. Key words. Parallel computation; P-completeness. Subject classifications. 68Q15, 68Q22. 1. Introduction A major goal of the theory of parallel computation is to understand how much speedup is obtainable in solving a problem on parallel machines over sequential machines. The theory of P-completeness has successfully classified many problems as unlikely to have polylog time algorithms on a parallel machine with a polynomial number of processors. However, the theory fails to distinguish between those P-complete problems that do have significant, polynomial speedup on parallel machines from those that don't. Yet this distinction is e...