CS 710 : Complexity Theory 1 / 28 / 2010 Lecture 4 : Time - Bounded Nondeterminism
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چکیده
In the previous lecture, we discussed NP-completeness and gave some strong results pertaining to the complexity of SAT, viz. that it is complete for NP under a reduction computable in logarithmic space and polylogarithmic time, and that it is complete for NQLIN (the set of NP problems solvable in quasilinear time) under quasi-linear time mapping reductions. The conclusion that all naturally occurring NP-complete problems are also NQLIN-complete raised two questions: are there any complete problems in NP that are not also in NQLIN, and if P 6= NP, then are there NP-intermediate problems, i.e., problems that are in NP \ P but that are not NP-complete? In this lecture, we investigate time-bounded nondeterminism more closely to address these questions. First, we present a time hierarchy theorem for nondeterministic computation analogous to that for deterministic computation; given more time, one can perform strictly more computation. Second, we show the existence of NP-intermediate problems by constructing a specific (artificial) example of one. Finally, we discuss the impact of relativization on the question of whether P = NP and how it can be answered.
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