1. On Derandomizing Algorithms that Err Extremely Rarely
Oded Goldreich
Weizmann Institute of Science
Note difference in titles.
Based on Joint work with Avi Wigderson

2. Standard Derandomization Challenges
Given a circuit C (from a certain class) such that Prob[C(x)=1] > ½, find an input x such that C(x)=1.
Famous frontier: Solve it in poly-time for the class AC0.
A two-sided error version: Given a circuit C (from a certain class) such that Prob[C(x)=] > ⅔, for some , find an input x such that C(x)= .
Ignore the other versions
A black-box version: Given circuit parameters, find a set of inputs S such that every circuit C that satisfies Prob[C(x)=1] > ½, there exists xS such that C(x)=1 .

3. Quantified Derandomization Challenges (new)
For a class C and a bound B, given an n-bit input circuit C from C such that |x:C(x)=0}| < B(n), find an input x such that C(x)=1. The above is called the (C,B)-search problem. Focus: Small B; e.g., quasi-polynomial, sub-exponential.
Q: Is the (P/poly,nlog n)-search problem solvable in (deter.) poly-time?
Ignore the other versions
THM1: The (AC0,exp(n0.999))-search problem is solvable in (deter.) poly-time.
THM2: Standard derandomization of AC0 is reducible to the (AC0,exp(n/log n))-search problem.

4. On Quantified Derandomization Problems
(C,B)-search problem = given an n-bit input circuit C from C such that |x:C(x)=0}| < B(n), find an input x such that C(x)=1.
THM1: The (AC0,exp(n0.999))-search problem is solvable in (deter.) poly-time.
THM2: Standard derandomization of AC0 is reducible to the (AC0,exp(n/log n))-search problem.
AC0[2] = constant-depth poly-size circuits with (unbounded fan-in) parity gates (in addition to the standard unbounded AND and OR gates)
THM3: Standard derandomization of AC0[2] is reducible to the (AC0[2],exp(n0.001))-search problem.

5. On the proof of THM 1
(C,B)-search problem = given an n-bit input circuit C from C such that |x:C(x)=0}| < B(n), find an input x such that C(x)=1.
THM1: The (AC0,exp(n0.999))-search problem is solvable in (deter.) poly-time.
Idea: Hit the circuit with a pseudorandom restriction (generated based on a seed of log length) such that (i) at least 2n0.999 variables survive, and (ii) the circuit simplifies to a constant.
The PR-restriction may not preserve the acceptance probability of a generic AC0 circuit (not even approximately), but this suffices for us since the number of surviving variables exceeds the error bound B(n). When designing the PR-restriction we focus on the simplification, which is obtained by repeated applications of a PR switching lemma.

6. On the proof of THM 3
(C,B)-search problem = given an n-bit input circuit C from C such that |x:C(x)=0}| < B(n), find an input x such that C(x)=1.
THM3: Standard derandomization of AC0[2] is reducible to the (AC0[2],exp(n0.001))-search problem.
Idea: Given a circuit for the standard problem, obtain a circuit for the quantified problem via “radical” error reduction (using a randomness extractor).
Extractors for min-entropy n0.001 that use a seed of log length and extract n bits can be computed in AC0[2], whereas approximate majority can be computed in AC0.
C’(x) = invoke C(E(x,s)) on all poly-many seeds, and take approx.-majority. The “bad” inputs yields a flat source that violates extraction.

7. On the proof of THM 2
(C,B)-search problem = given an n-bit input circuit C from C such that |x:C(x)=0}| < B(n), find an input x such that C(x)=1.
THM2: Standard derandomization of AC0 is reducible to the (AC0,exp(n/log n))-search problem.
We cannot perform error-reduction in AC0 since we do not know of an adequate extractor computable in AC0.
But it suffices to perform error-reduction for AC0! (i) Reduce randomness to polylog (via PRG), (ii) perform radical error reduction (via an extractor), (iii) straightforward error+randomness amplification.
We need to extract nOmega(1) bits from a source of (sub)constant entropy rate. It suffices that the extractor’s output fools AC0.
“in a class” = via a randomness extractor, whereas “for a class” allows using a “pseudo-extractor” that only fools the class.

8. Summary: Quantified Derandomization
(C,B)-search problem = given an n-bit input circuit C from C such that |x:C(x)=0}| < B(n), find an input x such that C(x)=1.
Q: Is the (P/poly,nlog n)-search problem solvable in (deter.) poly-time?
Results regarding quantified derand’ may improve over the known for standard derand’ (see Thm1), but in some cases quantified derand’ implies standard derand’ (see Thm2 and Thm3). Hope: A smooth transition. E.g., for AC0, from B(n)=exp(n0.999) as in Thm1 to B(n)=exp(n/log n) as in Thm2.
Mention results regarding MA0 and AM0=AM (which preserves error)?

9. END
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