Bella Subbotovskaya

Bella Abramovna Subbotovskaya (17 December 1937 – 23 September 1982) was a Soviet mathematician who founded the short-lived Jewish People's University (1978–1983) in Moscow. The school's purpose was to offer free education to those affected by structured anti-Semitism within the Soviet educational system. Its existence was outside Soviet authority and it was investigated by the KGB. Subbotovskaya herself was interrogated a number of times by the KGB and shortly thereafter was hit by a truck and died, in what has been speculated was an assassination.

Academic work

Prior to founding the Jewish People's University, Subbotovskaya published papers in mathematical logic. Her results on Boolean formulas written in terms of $\wedge$, $\vee$, and $\mathrm{\neg }$ were influential in the then nascent field of computational complexity theory.

Random restrictions

Subbotovskaya invented the method of random restrictions to Boolean functions. Starting with a function $f$, a restriction $\rho$ of $f$ is a partial assignment to $n-k$ of the $n$ variables, giving a function $f_{\rho }$ of fewer variables. Take the following function:

$f(x_1,x_2,x_3)=(x_1\vee x_2\vee x_3)\wedge (\mathrm{\neg }{x}_1\vee x_2)\wedge (x_1\vee \mathrm{\neg }{x}_3)$.

The following is a restriction of one variable

$f_{\rho }(y_1,y_2)=f(1,y_1,y_2)=(1\vee y_1\vee y_2)\wedge (\mathrm{\neg }1\vee y_1)\wedge (1\vee \mathrm{\neg }{y}_2)$.

Under the usual identities of Boolean algebra this simplifies to $f_{\rho }(y_1,y_2)=y_1$.

To sample a random restriction, retain $k$ variables uniformly at random. For each remaining variable, assign it 0 or 1 with equal probability.

Formula Size and Restrictions

As demonstrated in the above example, applying a restriction to a function can massively reduce the size of its formula. Though $f$ is written with 7 variables, by only restricting one variable, we found that $f_{\rho }$ uses only 1.

Subbotovskaya proved something much stronger: if $\rho$ is a random restriction of $n-k$ variables, then the expected shrinkage between $f$ and $f_{\rho }$ is large, specifically

$𝔼[L(f_{\rho })]\le {\left(\frac{k}{n}\right)}^{3/2}L(f)$,

where $L$ is the minimum number of variables in the formula. Applying Markov's inequality we see

$\mathrm{Pr}[L(f_{\rho })\le 4{\left(\frac{k}{n}\right)}^{3/2}L(f)]\ge \frac{3}{4}$.

Example application

Take $f$ to be the parity function over $n$ variables. After applying a random restriction of $n-1$ variables, we know that $f_{\rho }$ is either $x_i$ or $\mathrm{\neg }{x}_i$ depending the parity of the assignments to the remaining variables. Thus clearly the size of the circuit that computes $f_{\rho }$ is exactly 1. Then applying the probabilistic method, for sufficiently large $n$, we know there is some $\rho$ for which

$L(f_{\rho })\le 4{\left(\frac{1}{n}\right)}^{3/2}L(f)$.

Plugging in $L(f_{\rho })=1$, we see that $L(f)\ge n^{3/2}/4$. Thus we have proven that the smallest circuit to compute the parity of $n$ variables using only $\wedge ,\vee ,\mathrm{\neg }$ must use at least this many variables.

Influence

Although this is not an exceptionally strong lower bound, random restrictions have become an essential tool in complexity. In a similar vein to this proof, the exponent $3/2$ in the main lemma has been increased through careful analysis to $1.63$ by Paterson and Zwick (1993) and then to $2$ by Håstad (1998). Additionally, Håstad's Switching lemma (1987) applied the same technique to the much richer model of constant depth Boolean circuits.