Lecture 8: BPP, P/Poly

Karp-Lipton

Th (Karp-Lipton):

If $NP ⊆ P/poly$, then $PH ⊆ P/poly$

Proof:

We show that $Σ_i^p ⊆ P/poly$ for every $i≥1$ by induction on $i$.

Inductive case: Let $L∈ Σ_{i+1}^p$

\[L = \lbrace x \mid ∃y; \; (x, y) ∈ L' \rbrace, \; L'∈ℳ_i^p ⊆ P/poly \text{ (by i.h. + neg.)}\]

There is a family $(w_n)_{n∈ℕ}$ of words of poly size (lists for the family of poly circuits $(𝒞_n)_n$ decide $L’$)

\[L = \lbrace \underbrace{x}_{\text{ size } n} \mid ∃ \underbrace{y}_{\text{ size } p(n)}; \; eval((x,y), w_{n+p(n)+cst}) \; true\rbrace\]

AM-Games

AM (Arthur & Merlin) Games:

fixed number of rounds

A word protocol on $\lbrace A, M \rbrace$:

a word $∈ \lbrace A, M \rbrace^\ast$

An AM-game (with a protocol prot):

: is a triple $(𝒜, M, D)$ where

- $𝒜$ is a randomized TM
- $M$ is a function
- $D ∈ P$

$L ∈ BP \cdot 𝒞 ⟺ ∃ L’ ∈ 𝒞$

  • if $x∈L, P_r((x,r) ∈ L’) \text{ large } (≥ 1- \frac{1}{2^{q(n)}})$
  • if $x∉L, P_r((x,r) ∈ L’) \text{ small } (≤ \frac{1}{2^{q(n)}})$
  • $BP \cdot \textbf{P} = \textbf{BPP}$
  • $BP \cdot \textbf{NP} = \textbf{AM} = \textbf{AM_{lazy}}$
  • $co BP \cdot 𝒞 = BP \cdot (co 𝒞)$
  • if $𝒞 ⊆ 𝒞’$, then $BP \cdot 𝒞 = BP \cdot 𝒞’$

Quantifiers

$∃, E$: for $f: \underbrace{X}_{\text{finite}} ⟶ [0, 1]$

  • $∃x. f(x) ≝ \max_{x∈X} f(x)$
  • $Ex. f(x) ≝ \frac{1}{\vert X \vert} \sum_{x∈X} f(x)$

$prot ∈ \lbrace A, M \rbrace^\ast$

⟶ $\textbf{prot} \text{ i.e. } ∃ L; ∃(A, M, \underbrace{D}_{∈ \textbf{P}})$

  • if $x∈L, P_{r_1, r_2}(x # r_1 # \underbrace{y_1}_{ = M(x # r_1)} # r_2 # y_2 ∈ D) \text{ large } (≥ 1- \frac{1}{2^{q(n)}})$

  • if $x∉L, ∀M’, P_{r_1, r_2}(x # r_1 # \underbrace{y_1}_{ = M’(x # r_1)} # r_2 # y_2 ∈ D) \text{ small } (≤ \frac{1}{2^{q(n)}})$

AMAM ⟶ \(\underbrace{E}_{A} r_1, ∃ y_1, E r_2, ∃ y_2, ⋯\)

Prop: \(L∈ \textbf{AMAM} ⟺ ∀q, ∃D ∈ P;\)

  • if $x∈L, P(E r_1, ∃ y_1, E r_2, ∃ y_2. x # r_1 # y_1 # r_2 # y_2 ∈ D) \text{ large } (≥ 1- \frac{1}{2^{q(n)}})$
  • if $x∉L, P(E r_1, ∃ y_1, E r_2, ∃ y_2. x # r_1 # y_1 # r_2 # y_2 ∈ D) \text{ small } (≤ \frac{1}{2^{q(n)}})$

Proof: Skolemization trick:

\[E \underbrace{r}_{∈R}. ∃ y. F(r, y) = ∃ f: R ⟶ Y. E r. F(r, f(r))\] 
  digraph {
    rankdir=BT;
    "ε = P" -> "M = NP", "A = BPP";
    "M = NP" -> MA;
    "A = BPP" -> "MA" -> "AM"
  }

And $\textbf{AM}$ is equal to all the other classes.

To show that, $prot ⊆ AM$ for every prot:

  • $prot ≝ Aw$ ⟶ $AAM = AM$
  • $prot ≝ Mw$ ⟶ $MAM ⊆ AM$

Goldwasser and Sipser theorem

$Σ = \lbrace 0, 1 \rbrace = \underbrace{𝔽_2}_{\text{field}}$

  • $Σ^m$ is a $m$-dimensional vector space over $𝔽_2$
  • $h: Σ^m ⟶ Σ^{m’}$ hash function
A collision for $h$ in $X⊆ Σ^m$:

is an $x∈X$ s.t. $∃x’ ∈ X; \; x’≠x ∧ h(x’) = h(x)$

Let \(H ≝ \lbrace h_1, ⋯, h_l: Σ^m ⟶ Σ^{m'} \rbrace\)

A collision for $H$ in $X⊆ Σ^m$:

is an $x∈X$ s.t. $∀j ∈ ⟦1, l⟧, ∃x’_j ∈ X; \; x’_j≠x ∧ h_j(x’_j) = h_j(x)$

$X$ has a collision for $H$:

iff there exists a collision $x∈X$ for $H$ in $X$

Lemma 1 (Sipser): If $\vert X \vert ≤ 2^{m’-1}$, $l ≥ m’$:

\[P_H(X \text{ has a collision in } H) ≤ \frac{1}{2^{l-m'+1}}\]

Lemma 2 (Sipser): If $\vert X \vert > l × 2^{m’}$, then $H$ has a collision in $X$.

\[\underbrace{AM[k]}_{AM ⋯ AM} ⊆ IP[k]\]

Th: (Goldwasser-Sipser): $∀k ≥ 1$,

\[IP[k] ⊆ AM[k+1]\]

Corollary:

\(∀k ≥ 1, IP[k] = AM[k] = AM\) because \(IP[k] ⊆ AM[k+1] = AM = AM[k] ⊆ IP[k]\)

$\overline{ISO}$:

  • Input: $G_0 ≝ (V_0, E_0), G_1 ≝ (V_1, E_1)$

  • Question: are $G_0$ and $G_1$ non-isomoprhic?

In $IP[1]$:

  • $A$ draws $b ∈ Σ ≝ \lbrace 0, 1 \rbrace$ at random, $π ∈ 𝔖_n$ at random, then sends $π[G_b]$

  • $M$ answers $b’∈Σ$

  • accept if $b = b’$

Prop: \(\overline{ISO} ∈ AM\)

Prop: \(AM = AM_{tot}\)

\[L∈ AM_{tot} ⟺ ∀q \text{ poly}, ∀x, ∃(M, \underbrace{D}_{∈ P})\\ \begin{cases} ℙ_r(x \# r \# M(x \# r) ∈ D) = 1 \text{ if } x∈L \\ ∀ M', ℙ_r(x \# r \# M'(x \# r) ∈ D) ≤ \frac 1 {2^{q(n)}} \text{ else} \end{cases}\]

Let $L ∈ AM = BP \cdot NP$, there is a $L’∈NP$:

  • if $x∈L$, $ℙ_r((x,r) ∈ L’) ≥ 1 - \frac 1 n$
  • if $x∉L$, $ℙ_r((x,r) ∈ L’) ≤ \frac 1 n$

NB: $L’ = \lbrace (x, r) \mid ∃y, (x, r, y)∈ D’ \rbrace, D’∈P$

Let $R = \lbrace r \mid (x, r)∈L’ \rbrace$ whise gap is $\sim n$

New protocol:

  • $A$ draws $H = \lbrace h_1, ⋯, h_l \rbrace$
  • $M$ returns $r, r_1, ⋯, r_l, y, y_1, ⋯, y_l$

  • $A$ checks:

    • $r∈R: (x,r, y)∈D’$
    • $∀j, (x, r_j, y_j)∈D’$
    • $∀j, r_j ≠ r, h_j(r_j) = h_j(r)$

So:

  • if $x∈L$, Sipser II ⟶ there is a collision ⟹ accept always
  • if $x∉L$, $ℙ_r(\text{collision in } R \text{ for } H) ≤ \frac 1 {2^{q(n)}}$

Corollary: \(AM ⊆ Π_2^P\)

Thm: If $ISO$ is $NP$-complete, then $PH$ collapse at level 2.

NB: If $ISO$ is $NP$-complete, then $\overline{ISO}$ is $coNP$-complete, and in $AM$.

Hence $coNP ⊆ AM$, because is closed under polytime reduction.

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