Bella Subbotovskaya facts for kids
Quick facts for kids
Bella Abramovna Subbotovskaya
|
|
|---|---|
| Белла Абрамовна Субботовская | |
Subbotovskaya in 1961
|
|
| Born | December 17, 1937 |
| Died | November 23, 1982 (aged 44) |
| Cause of death | Car crash (suspected assassination) |
| Resting place | Vostryakovo Jewish Cemetery, Moscow |
| Nationality | Russian |
| Alma mater | Faculty of Mechanics and Mathematics, Moscow State University |
| Known for | Boolean formula complexity Jewish People's University |
| Spouse(s) |
Ilya Muchnik
(m. 1961–1971) |
| Scientific career | |
| Fields | Mathematical logic Mathematics education |
| Thesis | "A criterion for the comparability of bases for the realisation of Boolean functions by formulas" (1963) |
| Academic advisors | Oleg Lupanov |
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 orchestrated accident.
Contents
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 
, 
, and 
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 
, a restriction 
of is a partial assignment to 
of the 
variables, giving a function 
of fewer variables. Take the following function:
.
The following is a restriction of one variable
.
Under the usual identities of Boolean algebra this simplifies to 
.
To sample a random restriction, retain 
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 is written with 7 variables, by only restricting one variable, we found that uses only 1.
Subbotovskaya proved something much stronger: if is a random restriction of variables, then the expected shrinkage between and is large, specifically
![\mathbb{E} \left [ L(f_\rho) \right ] \le \left ( \frac k n \right )^{3/2} L(f)](/images/math/d/e/4/de4f6436db1ca46c67fde201cef3c3f0.png)
,
where 
is the minimum number of variables in the formula. Applying Markov's inequality we see
![\Pr \left [ L(f_\rho) \le 4 \left ( \frac k n \right )^{3/2} L(f) \right ] \ge \frac 3 4](/images/math/2/8/0/280f4e2a55f96e82bc8110e50b74408f.png)
.
Example application
Take to be the parity function over variables. After applying a random restriction of 
variables, we know that is either 
or 
depending the parity of the assignments to the remaining variables. Thus clearly the size of the circuit that computes is exactly 1. Then applying the probabilistic method, for sufficiently large , we know there is some for which
.
Plugging in 
, we see that 
. Thus we have proven that the smallest circuit to compute the parity of variables using only 
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 
in the main lemma has been increased through careful analysis to 
by Paterson and Zwick (1993) and then to 
by Håstad (1998). Additionally, Håstad's Switching lemma (1987) applied the same technique to the much richer model of constant depth Boolean circuits.
See also
In Spanish: Bela Subbotovskaya para niños
