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Understanding Probabilistically Checkable Proofs: Concepts and Challenges

This overview delves into the fascinating world of Probabilistically Checkable Proofs (PCP), exploring their significance in computational theory and mathematical verification. We discuss foundational concepts from Euclidean proofs to Gödel's completeness and incompleteness theorems, while highlighting the challenges of verifying long proofs in practical scenarios. The lecture addresses the evolution of theoretical computing, including Turing machines and the Cook-Levin theorem, and emphasizes the application of PCP in fields such as cloud computation and sensitive computation. Whether for homework checks or complex theorem validations, PCPs represent a pivotal step in the future of computational verification.

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Understanding Probabilistically Checkable Proofs: Concepts and Challenges

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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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