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P robabilistically C heckable P roofs

P robabilistically C heckable P roofs. 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.

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P robabilistically C heckable P roofs

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  1. ProbabilisticallyCheckableProofs 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 • Acommittee 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 Input Output

  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) Proof TM Statement Accept/Reject

  7. Checking Faster • Cook-Levin Theorem (70’s) • Each checker reads 3 letters! • Problem: reliability Proof TM Check 1 Statement, k Check 2 Check 3

  8. Fixing Reliability Problem Proof TM Check 1 Statement, k Check 2 Check 3 Check 4 Check 5 Check 6 Check 6 Check 6 Check 6

  9. Fixing Reliability Problem Correct Proof TM Check 1 Statement, k Check 2 Check 3 Check 4 Check 5 Check 6 Check 7 Check 8 Check 9

  10. Fixing Reliability Problem Correct Proof TM Check 1 Statement, k Check 2 Check 3 Check 4 Check 5 Check 6 Check 7 Check 8 Check 9

  11. Fixing Reliability Problem Proof TM Check 1 Wrong Statement Check 2 Check 3 Check 4 Check 5 Check 6 Check 7 Check 8 Check 9

  12. Fixing Reliability Problem Proof TM Check 1 Wrong Statement Check 2 Check 3 Check 4 PCP Theorem [1992]! Check 5 Check 6 Check 7 Check 8 Check 9

  13. Probabilistic Checking Proof TM Check 1 Wrong Statement Check 2 Check 3 Check 4 Choose few checkers randomly!! Check 5 Check 6 Check 7 Check 8 Check 9

  14. Probabilistic Checking Proof TM Wrong Statement Check 2 Choose few checkers randomly!! Check 5 Check 8

  15. Computation in Cloud Life the universe and everything? Prove! 42 Check 1,3,8,9 Proof

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

  17. Thank You

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