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 Euclid: argument based on axioms A committee decides (Fourier 1825) Frege System (1880) Some paradoxes Hilberts program (1900) Godel: completeness, incompleteness (1930) Turing: machine verifiable(1936) Machine-generate (1976) Turing Machine Theoretical computer (1936) Has inputs and outputs TM Example: Traveling Salesman All cities traversable, length < X US: cities Given proof: easy to verify Finding route: virtually impossible! Applications Verify mathematics Check homework Check long Proofs (four color theorem) TM Checking Faster Cook-Levin Theorem (70s) Each checker reads 3 letters! Problem: reliability TM Fixing Reliability Problem TM Fixing Reliability Problem TM Fixing Reliability Problem TM Fixing Reliability Problem TM Fixing Reliability Problem TM PCP Theorem [1992]! Probabilistic Checking TM Choose few checkers randomly!! Probabilistic Checking TM Choose few checkers randomly!! Computation in Cloud 42 Proof Problems Constructing checkers takes time Proof is long. Many other real-life constraints. But work is still in progress! Thank You