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\notag
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$
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$
$
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$
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$
Back to notes
Boolean Circuits
My notes on Boolean Circuits
| Topics |
Progress |
HTML |
PDF |
| Kannan's theorems |
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| $\MA/1\notin\SIZE[n^k]$ |
|
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| $\MA_{\EXP}\notin\Ppoly$ |
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| Collapse Theorems |
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| The Easy Witness Lemma |
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| 3(n-o(1)) circuit lower bound from gate elimination |
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| Natural Proof |
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| Open Problems in Circuit Complexity Theory |
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| Parity is not in $\AC{0}$ |
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|
Definitions
Basic Results
Most boolean functions are hard
For a random boolean function $f:\bit^n\rightarrow\bit$, we have $c(f)\geq(1-o(1))2^n/n$ with probability at least $1-o(1)$.
For any boolean function $f:\bit^n\rightarrow\bit$, we have $c(f)\leq(1+o(1))2^n/n$.
Let $H(n):=\max_{f:\bit^n\rightarrow\bit}c(f)$, we have $\class{E}s^{\Sigma_2^p}\not\subseteq\SIZE[H(n)-1]$.