Probabilistically Checkable Proofs

What Theoretical Computer Science Discovered About Proofs

Dana MoshkovitzThe Institute For Advanced Study

My Reflections About Theoretical Computer Science and Mathematics

MathematicsMathematicsAnalysisAnalysis

AlgebraAlgebra

LogicLogic

ProbabilityProbability

CombinatoricsCombinatorics

Mathematical Proofs

1. P0

2. P0 → (P1 → P2)

3. P1 → P2

4. …

Checkability!

Mathematical ProofsCheckability!

Checking Algorithm

Y/N

The Probabilistically Checkable Proofs Theorem [BFLS,AS,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 ProofsChecking 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/.

Should We Referee This Way?

Completely formal proof

PCP Theorem

Locally testable proof

Almost-linear conversion!

[GS02,BSVW03,BGHSV04,BS05,D06,MR07,MR08]

!?

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 [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,…]

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…