A Brief Introduction To The Theory of Computer Science

and The PCP Theorem

By Dana Moshkovitz

Faculty of Mathematics and Computer Science

The Weizmann Institute of Science

Theory of CS

Applied CS

Computer Science (CS)

Where Are We?

NetworksBio-

informatics

Software Engineering

Algorithms

Complexity

Learning

Computational

Geometry

Theory of CS

Goal: Understand the mathematics of Computing.

Computing = Solving Problems

•Algorithms: Solve problems efficiently • [small #operations, memory-usage, randomness, #processors,…]

•Complexity: Understand the limits of algo. • [how much #operations, memory-usage, randomness, #processors do we really need?]

Example of Problem: Sudoku

• Input: An n£n Sudoku table.

• Task: Solve!

Theorem (Cook, Levin, Karp, 1972):

Solving Sudoku efficiently is equivalent to solving efficiently ~100,000 other problems:– related to mapping the human genome,

routing, designing railroads, etc.

Theorem (Cook, Levin, Karp, 1972):

Solving Sudoku efficiently is equivalent to solving efficiently ~100,000 other problems:– related to mapping the human genome,

routing, designing railroads, etc.

What’s So Special In Sudoku??

• if I know a solution for Sudoku, I can prove this to you!

• Efficient Checking: Given my solution, you can check its validity efficiently.

• The class of all problems having this property is called NP (= non-deterministic

polynomial time).

Solving vs. Checking

So, what’s easier, solving or checking?

CheckingSolving

Solving vs. Checking

• Solving ) Checking: If you can solve Sudoku yourself, you can compare to my solution, and thus check.

• Checking Solving?? Seems not!• We let P (= polynomial time) be the class of all

problems that can be solved efficiently.

The biggest question in the Theory of CS:

Is Checking equivalent to Solving? Is P=NP?

How The Area Evolves From Here?

P=NP?If not, then you

also cannot do… (hardness results)

If not, then you can use this to…

(cryptography)

concentrate on what you can solve (algorithms,

approximations)

relax the question (lower

bounds)

is it the right question?

(average-case complexity, quantum

computing,…)

equivalent to other questions? (pseudo-

randomness)

understand NP better (PCP)

My Research: Understanding NP Better

PCP (=probabilistically checkable proofs): Surprising characterization of NP.

Even more important than first seems…

• Cannot do: gives hardness of approximation.

• Can do: gives new constructions.

Sudoku Standard Verification

Do I have a solution?

Probabilistically Checkable Proof

Yes! With confidence

99.9% I think it’s a solution.

PCP Theorem

Theorem ([AS,ALMSS], 1992):

Given any Sudoku riddle, given any solution, one can write the solution in a way that allows extremely fast verification:

• verification by picking at random 2 places and checking only them!

Theorem ([AS,ALMSS], 1992):

Given any Sudoku riddle, given any solution, one can write the solution in a way that allows extremely fast verification:

• verification by picking at random 2 places and checking only them!