Resumen de alto nivel del método de aproximaciones de Razborov

What is Razborov's method of approximations? Can someone give a high-level overview and the intuition behind it?

Sea f una función booleana en n bits. Sea Z = f - 1 ( 0 ) ⊆ 2 n . Sea C un circuito en n bits y de tamaño m y puertas g 1 , ... , g m . g i también denota la función en n bits calculada por subcircuito con g i como la última puerta. Las primeras n puertas son para la entrada x 1 , ... , x $f$$n$$Z = f^{-1}(0) \subseteq 2^n$$C$$m$$g_1, \ldots, g_m$$g_i$$n$$g_i$$n$$x_1, \ldots, x_n$ . El objetivo es mostrar que C de tamaño $C$$m$ cannot compute $f$. Consider all computations of $C$ on inputs from $Z$. A computation assigns values to outputs of gates. Let $B$ be the Boolean algebra of $P(Z)$.

The idea is to consider for any function $g$ on $n$-bits how well it approximates $f$ on $Z$. Let $||g|| = \{w \in Z \mid g(w) \neq 0 \}$.

For an ultrafilter $F \subseteq B$ we can define a new computation by ultraproduct from it: $c(g_i) = 0$ iff $||g_i|| \not\in F$. Because an ultrafilter is essentially a set of consistent computations for 0 values the resulting $c$ is a valid computation. It would follow that $f(c_1, \ldots, c_n) = 0$. We created a new computation from existing ones. Since all ultrafilters on finite sets are principal $c_1,\ldots,c_n \in Z$. This works for any circuit, we have not exploited the fact that the circuit is of size $m$.

The next idea is now to exploit the finiteness of the circuit to construct a new input that is outside $Z$ and $f(w) \neq 0$ but the circuit does not notice because of its limited size and therefore still outputs 0. So it does not compute $f$.

We need to relax the definition of ultrafilter so that we can get an input outside $Z$. In place of ultrafilters we use upward-closed subsets of $B$ ($a \in F$ and $a \subseteq b$ implies $b \in F$) that preserve meets($a,b \in F$ implies $a \cap b \in F$).

Let $W_F = \{w \in 2^n \mid w_i = 0 \to ||\lnot x_i|| \in F, w_i \neq 0 \to ||x_i|| \in F\}$. $W_F$ is the set of inputs consistent with $F$. If $F$ is prime ($a \cup b \in F$ implies $a \in F$ or $b \in F$) and nonfull ($\emptyset \not\in F$) then for each $i$, $F$ contains either $||x_i||$ or $||\lnot x_i||$ and $W_F$ contains only a single input.

We are going to relax preservation of meets. In place of all meets in the Boolean algebra we will preserve a small number of them. Let $|f|$ be the smallest number $k$ of meets $M = (a_1 \cap b_1, \ldots, a_k \cap b_k)$ such that for all upward-closed, nonfull, $M$-preserving $F$, $W_F \subseteq Z$.

Let $m$ be the circuit complexity of $f$. Razborov proved that $\frac{1}{2}|f| \leq m \leq O(|f|^3 + n^3)$.

Note that this inequality holds for all functions. To prove a circuit size lower bound $m$ show that for all $m$-meets $M$, there is a $F$ that satisfies the conditions but its $W_F$ is not contained in $Z$. Moreover any strong circuit lower bound can be proven by this method because of the second inequality.

The actual part of a circuit lower bound proof is to show that for given $m$, for any $m$-meets there is such an $F$. In the case of monotone circuits the condition about $W_F$ simplifies to $w_i \neq 0 \to ||x_i|| \in F$ so coming up with $F$ is easier.

Alexander Razborov, On the Method of Approximation, 1989. pdf

Mauricio Karchmer, On Proving Lower Bounds for Circuit Size, 1995.

Tim Gowers, Razborov's method of approximation, 2009. pdf

Disclaimer: This is only a high-level overview intended to give some intuition to the methods used in Blum's recent paper.

I will attempt to use notation that is closer to what is used in the aforementioned paper.

Let $f$ be a Boolean function on $n$ variables $x_1,\dots,x_n$. Suppose we want to prove that any Boolean network computing $f$ has large size.

Given some Boolean network $\beta$ computing $f$ at its output node, consider the following process.

1. Order the gates in $\beta$ according to some topological order $g_1,g_2,\dots,g_m$ where the last node is the output node.
2. For each time step $t=1,\dots,m$ we will approximate the function computed at gate $g_t$ by a “simple” function $f_{g_t}$. This approximation may change the functions computed at nodes downstream of $g_t$ (in particular, the function at the output node $g_m$ may have changed).

At the end of this process, we will have approximated the function computed at $g_m$ by a simple function $f_{g_m}$.

Next construct a group of test inputs $T\subseteq \{0,1\}^n$.

Suppose we can prove the following statements:

• The approximation of each individual node is good (i.e., at most $e$-many mistakes are introduced on inputs from $T$ at each approximation step).
• No simple function approximates $f$ well (i.e., for any simple function $f_{g_m}$, we have $f_{g_m}\neq f$ on more than a $d$-fraction of $T$).

Then by simply counting the number of errors we get that $\beta$ must have at least $\tfrac{d\lvert T\rvert}{e}$-many gates.

If this approximation scheme can be shown to work for any network $\beta$ computing the function $f$, then we arrive at a lower bound for the circuit complexity of $f$.