https://scribbletogether.com/whiteboard/FCEE879C-5C12-4584-9F83-739BE48698DF

Overview

• Computing all functions
• Circuits as strings
• Shannon’s lower bound

Today:

• More on Shannon’s lower bound
• Universal circuit
• PECTT
• Computing with growing inputs

Theorem: There is an encoding from some $(n,m,s)$ circuit to a bit string such that the length of the encoding $\leq 12(n+s)\log_2(n+s)$

$(n,m,s,\{m\},\{s\cdot 2\})$

• of variables

• of outputs

• of gates

• $m$ output indices
• 2 incoming gates for each $s$ gate

$3 + m + 2\cdot s \leq 3 + n + s + 2s = n+3s + 3$, each of magnitude $n+s$.

Thus, length $\leq (n+3s+3) \cdot \log_2 (n+s)$

Theorem: There exists a function $f:\{0,1\}^n \to \{0,1\}$ that has no NAND-CIRCUIT of size $\leq c\cdot \dfrac{2^n}{n}$ for some fixed constant $c>0$.