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Randomness and Computation

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

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Randomness and Computation

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  1. Randomness and Computation Oded Goldreich Dept. of Computer Science and Applied Math. Weizmann Institute 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 deterministicprocess 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 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.

  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.

  5. Property Testing (sublinear-time approx. decision) What is an approximate decision? Distinguishing objects that have a (predetermined) propertyfrom 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.

  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.

  7. End The slides of this talk are available at http://www.wisdom.weizmann.ac.il/~oded/T/minerva.ppt

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