Hierarchy Theorems

Space Hierarchy Theorem

Relativization

Limits of the Diagonalization Method

Circuit Complexity      x1x1 x2x2 x3x3 output

⊔ 0⊔⊔0 1 q00q00 Cell[1,1] start configuration Cell[t(n),1] Accept position t(n)-th configuration

q 0 … 0q l 1q 0 … 1q l 0 1 …     …

Monotone Circuits

12 3

Richard Rado Paul Erd ö s (3/26, 1913 – 9/20,1996 )

Plucking a sunflower      x 1 x 2 x 1 x 3 x 2 x 3   x 1 x 2 x 1 x 3 x 1 x 3 x 2 x 3  {x 1 x 2, x 1 x 3 }, { x 1 x 3, x 2 x 3 } {x 1 x 2 x 3, x 1 x 3 }

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K-1 …

Z

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