1. P robabilistically C heckable P roofs Guy Kindler The Hebrew University

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

3. 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)

4. Turing Machine Theoretical computer (1936) Has inputs and outputs TM

5. Example: Traveling Salesman All cities traversable, length < X US: 13509 cities Given proof: easy to verify Finding route: virtually impossible!

6. Applications Verify mathematics Check homework Check long Proofs (four color theorem) TM

7. Checking Faster Cook-Levin Theorem (70’s) Each checker reads 3 letters! Problem: reliability TM

8. Fixing Reliability Problem TM

9. Fixing Reliability Problem TM

10. Fixing Reliability Problem TM

11. Fixing Reliability Problem TM

12. Fixing Reliability Problem TM PCP Theorem [1992]!

13. Probabilistic Checking TM Choose few checkers randomly!!

14. Probabilistic Checking TM Choose few checkers randomly!!

15. Computation in Cloud 42 Proof

16. Problems Constructing checkers takes time Proof is long. Many other real-life constraints. But work is still in progress!

17. Thank You