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 </spl radic/2 and any b<1, SAT cannot be computed by a random-access deterministic Turing machine using n/sup a/ time, n/sup o(1)/ space and n/sup o(1)/ advice, nor by a random-access deterministic Turing machine using n/sup 1+o(1)/ time, n/sup b/ space and n/sup o(1)/ advice. More generally, we show that for all /spl delta/>0 and any /spl epsiv/<1, SAT cannot be solved by a random-access deterministic Turing machine using n/sup 1/2 /((/spl radic//spl epsiv//sup 2/+8-/spl epsiv/-)-/spl delta/) time, n/sup 2/ space and n/sup o(1)/ advice. Similar lower bounds for computing SAT on random-access nondeterministic Turing machines taking n/sup o(1)/ advice are also obtained. Moreover we show that SAT does not have NC/sup 1/ circuits of size n/sup 1+o(1)/ generated by a nondeterministic log-space machine taking n/sup o(1)/ advice. Additionally new separations of uniform classes are obtained. We show that for all /spl epsiv/>0 and all rationals r/spl ges/1, DTISP(n/sup r/, n/sup l-/spl epsiv//)/spl sub//spl ne/NTIM E(n/sup r/). We show how extending our uniform separations can lead to a separation of SC and NP.

[1]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[2]  V. Rich.  Personal communication , 1989, Nature.

[3]  Lance Fortnow,et al.  Time-space tradeoffs for nondeterministic computation , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[4]  Richard J. Lipton,et al.  On the complexity of SAT , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[5]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[6]  Richard Edwin Stearns,et al.  Two-Tape Simulation of Multitape Turing Machines , 1966, JACM.

[7]  Stephen A. Cook,et al.  Short Propositional Formulas Represent Nondeterministic Computations , 1988, Inf. Process. Lett..

[8]  Michael J. Fischer,et al.  Relations Among Complexity Measures , 1979, JACM.

[9]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[10]  Sheila A. Greibach,et al.  Time- and Tape-Bounded Turing Acceptors and AFLs , 1970, J. Comput. Syst. Sci..

[11]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[12]  Endre Szemerédi,et al.  On determinism versus non-determinism and related problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  John Michael Robson,et al.  An O (T log T) Reduction from RAM Computations to Satisfiability , 1991, Theor. Comput. Sci..

[14]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[15]  Lance Fortnow.  Nondeterministic polynomial time versus nondeterministic logarithmic space: time-space tradeoffs for satisfiability , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.