Probabilistically Checkable Proofs What Theoretical Computer Science Discovered

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 Logic Probability Combinatorics

Mathematical Proofs Checkability! 1. P 0 2. P 0 → (P 1 → P 2) 3. P 1 → P 2 4. …

Mathematical Proofs Checkability! Checking Algorithm Y/N

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

Probabilistic Checking of Proofs 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/.

Should We Referee This Way? PCP Theorem !? Almost-linear conversion! [GS 02, BSVW 03, BGHSV 04, BS 05, D 06, MR 07, MR 08] Completely formal proof Locally testable proof

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.

What Gets Inside? • Low degree testing Low degree approximations and restrictions to lines/planes in Fqn [RS 90, …, AS 97, RS 97, MR 06] • Combinatorial PCP Random walks on expanders [D 06] • Parallel repetition Information theory [R 94, H 07] • Parallel repetition tightness Foam Tiling of Rn by Zn [R 08, FKO 07, KORW 08] • Long-Code testing Isoperimetric inequalities in Gaussian space [KKMO 04, MOO 05] • UGC-based reductions Counterexamples for metric embedding [KV 05, …]

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 [BGLR 93, AS 97, RS 97, DFKRS 99, MR 07] • for projection games [R 94, MR 08] – “The Unique Games Conjecture” arbitrarily small constant error for unique games [K 02] • More open problems: minimize size, alphabet, conversion time, checking time, more hardness of approximation results, more connections…