1. Randomness and Computation
Oded Goldreich
Dept. of Computer Science and Applied Math.
Weizmann Institute
Slides for MINERVA presentation
MINERVA MINI-SYMPOSIA (Natural Sciences) – 7/10/13, Session IB

2. The Interplay of Randomness and Computation
Odd: Seems that computation (as a deterministic process) stands in contrast to the notion of randomness. But if you consider the output of a deterministic process on a randomly distributed input, then… Or an algorithm that tosses coins:
Probabilistic Proof Systems: E.g., IP, ZK, and PCP.
Pseudorandomness and De-randomization.
Property Testing and Sub-linear Time Algorithms
PPS and PRG are related to the Foundations of Crypto.
Comment: ZK and PRG are closely related to Cryptography.

3. Probabilistic Proof Systems
Odd: Don’t we want certainty in proofs? Well, we’ll get certainty up to an explicitly bounded error probability, and this will allow us advantages over the traditional notion (of a deterministically verifiable) proof.
IP (interactive proof): convincing via interaction.
ZK (zero-knowledge): w.o. revealing anything (beyond).
PCP (Prob. Checkable Pf.): while reading three bits.
PCP: Trade-off between the number of bits read and the confidence parameter!

4. Pseudorandomness (and PRGs)
What does this mean (in CS)? A deterministic process that stretches short random seeds into (much) longer sequences that “look random”. What does “look random” mean? See 1st parameter (below).
PRG is a “family name”; members differ by parameters:
Computational indistinguishability w.r.t a specific class of observers (defined by their resources).
Comput. complexity of generation (i.e., of the PRG).
The amount of stretch.
Re (1): E.g., all efficient observers, all small-space observers, all linear tests, all tests that look at five bits.

5. Property Testing (sublinear-time approx. decision)
What is an approximate decision?
Distinguishing objects that have a (predetermined) property from objects that are “far” from having this property.
Advantage: the algorithm may inspect a small portion of the tested object! That’s typically the goal in PT.
An example you all know: Estimating the average value of a function defined over a huge domain.
This simple example refers to “unstructured” objects and properties. Our focus is on “structured” ones.
E.g., testing that a graph is bipartite or cycle-free. One-sided error vs two-sided error. Adaptive vs non-adaptive.

6. One concrete research project (i.e., recent, ongoing)
Context: De-randomization. Known: Given a ``very simple algorithm’’ (i.e., constant-depth circuit) that evaluates to the value 1 on at least half of its inputs, we can determinstically find an input that evaluates to 1 “almost efficiently” (i.e., quasi-poly-time). N.B.: It is easy to find such an input probabilistically.
New: If we are guaranteed that the number of bad inputs is sub-exponential (in the input length), then we can find a good input efficiently. If the class of circuits is mildly extended, then this relaxation is not helpful.
Constant-depth circuits = the most simple model of algorithms.

7. End
The slides of this talk are available at