ProbabilisticallyCheckableProofs

Guy Kindler

The Hebrew University

Proofs – why?Proofs problems all around…• Math : Homework• Identity: passwords suck• Credit cards: chip (zero knowledge)• Smartness!!• Cloud computation• Sensitive computation (fission decay)• Etc…

What is a proof• Euclid: argument based on axioms• A committee decides (Fourier 1825)• Frege System (1880)• Some paradoxes• Hilbert’s program (1900)• Godel: completeness, incompleteness (1930)• Turing: machine verifiable(1936)• Machine-generate (1976)

Turing Machine• Theoretical computer (1936)• Has inputs and outputs

TMInput Output

Example: Traveling Salesman

• All cities traversable, length < X • US: 13509 cities

• Given proof: easy to verify• Finding route: virtually impossible!

Applications• Verify mathematics• Check homework• Check long Proofs (four color theorem)

Statement Accept/RejectTM

Proof

Checking Faster• Cook-Levin Theorem (70’s)• Each checker reads 3 letters!

• Problem: reliability

Proof

Statement, k TMCheck 1

Check 2

Check 3

Fixing Reliability ProblemProof

Statement, k TMCheck 1Check 2

Check 3Check 4Check 5Check 6Check 6Check 6Check 6

Fixing Reliability Problem

Statement, k TM

Correct Proof

Check 1Check 2

Check 3Check 4Check 5Check 6Check 7Check 8Check 9

Fixing Reliability Problem

Statement, k TM

Correct Proof

Check 1Check 2

Check 3Check 4Check 5Check 6Check 7Check 8Check 9

Fixing Reliability Problem

Wrong Statement TM

Proof

Check 1Check 2

Check 3Check 4Check 5Check 6Check 7Check 8Check 9

Fixing Reliability Problem

Wrong Statement TM

Proof

Check 1Check 2

Check 3Check 4Check 5Check 6Check 7Check 8Check 9

PCP Theorem [1992]!

Probabilistic Checking

Wrong Statement TM

Proof

Check 1Check 2

Check 3Check 4Check 5Check 6Check 7Check 8Check 9

Choose few checkers randomly!!

Probabilistic Checking

Wrong Statement TM

Proof

Check 2

Check 5

Check 8

Choose few checkers randomly!!

Computation in CloudLife the universe and everything?

42Prove!

Check 1,3,8,9 Proof

Problems

• Constructing checkers takes time

• Proof is long.

• Many other real-life constraints.

• But work is still in progress!

Thank You