Time-space lower bounds for SAT on uniform and non-uniform machines

The arguments used by R. Kannan (1984), L. Fortnow (1997), and Lipton-Viglas (1999) are generalized and combined with a new argument for diagonalizing over machines taking n bits of advice on inputs of length n to obtain the first nontrivial time-space lower bounds for SAT on non-uniform machines. In particular we show that for any a <√2 and any b<1, SAT cannot be computed by a random-access deterministic Turing machine using n^a time, n^o(1) space and n^o(1) advice, nor by a random-access deterministic Turing machine using n^1+o(1) time, n^b space and n ^o(1) advice. More generally, we show that for all δ>0 and any ε<1, SAT cannot be solved by a random-access deterministic Turing machine using n^½((√ε²+8-ε-)-δ) time, n² space and n^o(1) advice. Similar lower bounds for computing SAT on random-access nondeterministic Turing machines taking n^o(1) advice are also obtained. Moreover we show that SAT does not have NC¹ circuits of size n^1+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 ε>0 and all rationals r⩾1, DTISP(n^r, n^l-ε)⊂≠NTIM E(n ^r). We show how extending our uniform separations can lead to a separation of SC and NP