Time-space tradeoffs for SAT on nonuniform machines

The arguments used by R. Karman (1984, Math. Systems Theory, 17, 29-45), L. Fortnow (1997, in "Proceedings, Twelfth Annual IEEE Conference on Computational Complexity, Ulm, Germany, 24-27 June, 1997," pp. 52-60), and R. J. Lipton and A. Viglas (1999, in "40th Annual Symposium on Foundations of Computer Science, New York, 17-19 Oct. 1999," pp. 459-469) are generalized and combined with an argument for diagonalizing over machines taking n bits of advice on inputs of length it to obtain the first nontrivial time-space lower bounds for SAT on nonuniform machines. In particular, we show that for any a < root2 and any epsilon > 0, SAT cannot be computed by a random access deterministic Turing machine using n(a) time, n(0(1)) space, and o(n (root2/2-epsilon)) advice nor by a random access deterministic Turing machine using n(1+sigma (1)) time, n(1-epsilon) space, and n(1-epsilon) advice. More generally, we show that if for some epsilon > 0 there exists a random access deterministic Turing machine solving SAT using n(a) time, n(b) space, and o(n((a+b)/2-epsilon)) advice, then a greater than or equal to 1/2 (root b2 + 8 - b). Lower bounds for computing R-AT on random access nondeterministic Turing machines taking sublinear advice are also obtained. Moreover, we show that SAT does not have NC1 circuits of size n(l+o(1)) generated by a nondeterministic log-space machine taking n(o(1)) advice. Additionally, new separations of uniform classes are obtained. We show that for all a > 0 and all rational numbers r greater than or equal to 1, DTISP(n(r), n(1-epsilon)) is properly contained in NTIME(n(r)). (C) 2001 Academic Press.