1. Dana Moshkovitz The Institute For Advanced Study
Probabilistically Checkable Proofs What Theoretical Computer Science Discovered About Proofs
Dana Moshkovitz
The Institute For Advanced Study
Goal: present PCP, give overview, “common knowledge”,
And put in context of TCS and Mathemtics

2. My Reflections About Theoretical Computer Science and Mathematics
Algebra
Mathematics
Analysis
Probability
Combinatorics
TCS – different perspective, different problems
To solve these problems, study problems for algebra, analysis, combinatorics, probability etc – whatever works!
Sometimes prove interesting theorems and solve open problems in “classical” mathematics
This talk – an example for such a case: fundamental question about math, to solve it – solve problems from all over mathematics
Logic

3. Mathematical Proofs Checkability! P0 P0 → (P1 → P2) P1 → P2 …
The logicians’ perspective on proofs: axiom systems, inference
CS perspective: a proof = can check whether statement is true

4. Mathematical Proofs Checkability! Y/N Checking Algorithm
The checking algorithm must be efficient
Checks whether the proof follows logic rules AND yields the desired conclusion
Checking Algorithm
Y/N

5. 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.
Original number of queries could have been the entire proof, which we’ll denote N.
N can be arbitrarily large, and we will care about the aymptotics
New number of queries = constant, two!
Note: the checking algorithm has to work, no matter what the proof is

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/.
Epsilon is called the “error” – if there is no proof that satisfies V, the prob V’ accepts is at most epsilon.
Alphabet must depend on epsilon! (but not on N!!)

7. Should We Referee This Way?
PCP Theorem !?
Almost-linear conversion! [GS02,BSVW03,BGHSV04,
BS05,D06,MR07,MR08]
The PCP Theorem takes a *completely formal* proof and translates it to proof one can check locally.
Doesn’t say a thing when given a “human proof”.
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 : The PCP Theorem resolves this! The approximation problem: Approximate how many of the checker’s local tests can be satisfied simultaneously.
Why TCS people were so interested in PCP? What does the PCP Thm mean in terms of TCS?
It means that approximation problems can be NP-hard!
[this was major open question before 91-2]
NP = computational problems where solution can be checked efficiently;
NP-hard = all NP problem efficiently reduce to it; if we can solve it, we can solve all NP-problems.
surprisingly reach class of problems, containing many natural, many real-life problems

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,…]
To prove the PCP theorem and to use it in TCS, we had to solve many interesting math problems.
For some, it was clear from the beginning that they boil down to interesting math problems.
For others, surprising theorems revealed their equivalency to natural math problems.

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…
Understanding that emerged in the field: it matters what is the local test the verifier makes.
Projection test = two queries; value for first query determines value for the second query.
Unique test = two queries; value for *each* query determines value for the other query.
From Projection/Unique verifiers with low error can prove hardness of approximation for other problems.
We have constructions of projection verifiers with very small error. Open problem: can achieve optimal error?
We don’t have constructions of unique verifier with low error. Open problem (“The Unique Games Conjecture”): construct such!