Probabilistic Polynomial Time

A language $L \in BPP$ (Bounded from half Probabilistic Polytime)

If there exists a machine M in PPT such that for all x in L the probability over M’s coin flips that M(x) = 1 > 2/3.

For x not in L, the probability that M(x) = 1 < 1/3

Proof using Chernoff Bound

Let $x_1, x_2, …, x_n$ be IID.

Define $Y = \sum_{i=1}^n x_i$

$P[Y \geq (1 + \beta) \cdot \mathbb{E}(Y)] < e^{-\beta^2\dfrac{\mathbb{E}(Y)}{2}}$

where $\beta < 1$ and $\mathbb{E}(Y) = p\cdot n$.

Define $x_1 = 1$ if an experiment $i$ makes a mistake.

Let $p=\dfrac{1}{3}$ and set $\beta = \dfrac{1}{2}$.

Plugging into Chernoff bound:

$P(Y\geq (1 + \dfrac{1}{2})\dfrac{n}{3} = \dfrac{n}{2}) < e^{-\dfrac{1}{4}\dfrac{n}{3}\dfrac{1}{2}}$

$P(Y \geq \dfrac{n}{2}) < e^{-\dfrac{n}{24}}$

Chebyshev Bound Proof

For pairwise independent

$P[|(Y-E(Y))|> \delta\cdot n] = 2e^{-\dfrac{\delta^2 n}{2}}$

Puzzle: array of 3 entries with random bits. Observer promises to only look at 2 coins but at random. What’s the minimum number of bits for the randomness to be the same across all bits?

Answer: 2 flips, last one XOR.