Lecture 11 : Circuit Lower
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چکیده
There are specific kinds of circuits for which lower bounds techniques were successfully developed. One is small-depth circuits, the other is monotone circuits. For constant-depth circuits with AND,OR,NOT gates, people proved that they cannot compute simple functions like PARITY [3, 1] or MAJORITY. For monotone circuits, Alexander A. Razborov proved that CLIQUE, an NP-complete problem, has exponential-sized circuit complexity [5]. We won’t talk about lower bounds for monotone circuits. Definition 1 (NC) NC is the class of languages solved using circuits of depth O(log n), size poly(n), with AND, OR, NOT gates of fan-in ≤ 2. NC stands for “Nick’s Class” named by Steven Cook in honor of Nick Pippenger. NC includes functions f(x1, · · · , xn) which depend on O(1) inputs. Clearly, NC circuits cannot compute all functions. For example, PARITY cannot be computed by NC because the parity value depends on all n inputs, instead of a constant number of inputs. However, we still don’t know the answer yet: Question 2 P = NC? Note: obviously, NC ⊆ P/poly. Definition 3 (AC) AC is the same as NC except that AND, OR gates allow unbounded fan-ins.
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تاریخ انتشار 2013