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Probabilistically Checkable Proofs What Theoretical Computer Science Discovered About Proofs

Probabilistically Checkable Proofs What Theoretical Computer Science Discovered About Proofs. Dana Moshkovitz The Institute For Advanced Study. My Reflections About Theoretical Computer Science and Mathematics. Algebra. Mathematics. Analysis. Probability. Combinatorics. Logic.

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Probabilistically Checkable Proofs What Theoretical Computer Science Discovered About Proofs

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  1. Probabilistically Checkable ProofsWhat Theoretical Computer Science Discovered About Proofs Dana Moshkovitz The Institute For Advanced Study

  2. My Reflections About Theoretical Computer Science and Mathematics Algebra Mathematics Analysis Probability Combinatorics Logic

  3. Mathematical Proofs Checkability! • P0 • P0 → (P1 → P2) • P1 → P2 • …

  4. Mathematical Proofs Checkability! Checking Algorithm Y/N

  5. The Probabilistically Checkable Proofs Theorem [BFLS,AS,ALMSS, 1992] The PCP Theorem:Every proof can be efficientlyconverted to a proof that can be checked probabilistically by querying only two symbols in the proof.

  6. Probabilistic Checking of Proofs Checking algorithm V Checking algorithm V’ V’ makes two probabilistic queries to its proof! • Completeness: A proof that satisfies V can be efficiently converted to a proof ‘ that V’ accepts with probability 1. • Soundness: If V’ accepts a proof ‘ with probability >, then there exists a proof that satisfies V. Remark:‘ over alphabet where||1/.

  7. Should We Referee This Way? PCP Theorem !? Almost-linear conversion! [GS02,BSVW03,BGHSV04, BS05,D06,MR07,MR08] Completely formal proof Locally testable proof

  8. Theoretical Computer Science Angle: Hardness of Approximation Big Open Problem in Theoretical Computer Science until 1991: Show that some approximation problem is NP-hard. 1991-2: The PCP Theorem resolves this! The approximation problem: Approximate how many of the checker’s local tests can be satisfied simultaneously.

  9. What Gets Inside? • Low degree testing Low degree approximations and restrictions to lines/planes in Fqn[RS90,…,AS97,RS97,MR06] • Combinatorial PCP Random walks on expanders [D06] • Parallel repetition Information theory [R94,H07] • Parallel repetition tightness Foam Tiling of Rn by Zn [R08,FKO07,KORW08] • Long-Code testing Isoperimetric inequalities in Gaussian space [KKMO04,MOO05] • UGC-based reductions Counterexamples for metric embedding [KV05,…]

  10. Research on PCP Today • Realization: The type of check matters! • Projection games • Unique games • Biggest open problems: • “The Sliding-Scale Conjecture” smallest possible error (n)=1/n[BGLR93,AS97,RS97,DFKRS99,MR07] • for projection games [R94,MR08] • “The Unique Games Conjecture” arbitrarily small constant error for unique games [K02] • More open problems: minimize size, alphabet, conversion time, checking time, more hardness of approximation results, more connections…

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