1. of 28 Probabilistically Checkable Proofs Madhu Sudan Microsoft Research June 11, 2015TIFR: Probabilistically Checkable Proofs1

2. of 28 Can Proofs be Checked Efficiently? June 11, 2015TIFR: Probabilistically Checkable Proofs2 The Riemann Hypothesis is true (12 th Revision) By Ayror Sappen # Pages to follow: 15783

3. of 28 Proofs and Theorems Conventional belief: Must read whole proof to verify it. Modern Constraint: Don’t have time to (do anything, leave alone to) read proofs. This talk: – New format for writing proofs. – Extremely efficiently verifiable probabilistically, with small error probability. – Not much longer than conventional proofs. June 11, 2015TIFR: Probabilistically Checkable Proofs3

4. of 28 Outline of talk Quick primer on the Computational perspective on theorems and proofs (proofs can look very different than you’d think). Definition of Probabilistically Checkable Proofs (PCPs). Why (computer scientists) study proofs/PCPs. (Time permitting) Some overview of “ancient” (~25 year old) and “modern” (~10 year old) PCPs. June 11, 2015TIFR: Probabilistically Checkable Proofs4

5. of 28 Part I: Primer June 11, 2015TIFR: Probabilistically Checkable Proofs5

6. of 28 What is a proof? June 11, 2015TIFR: Probabilistically Checkable Proofs6

7. of 28 Philosophy & Computing - 101 Theorems vs. Proofs? – Theorem: “True Statement” – Proof: “Convinces you of truth of Theorem” – Why is Proof more “convincing” than Theorem? Easier to verify? – Computationally simple (mechanical, “no creativity needed”, deterministic?) – Computational complexity provides formalism! – Advantage of formalism: Can study alternate formats for writing proofs that satisfy basic expectations, but provide other features. June 11, 2015TIFR: Probabilistically Checkable Proofs7

8. of 28 The Formalism June 11, 2015TIFR: Probabilistically Checkable Proofs8

9. of 28 Theorems: Deep and Shallow June 11, 2015TIFR: Probabilistically Checkable Proofs9 talk

10. of 28 June 11, 2015TIFR: Probabilistically Checkable Proofs10

11. of 28 Aside: P & NP P = Easy Computational Problems – Solvable in polynomial time – (E.g., Verifying correctness of proofs) NP = Problems whose solution is easy to verify – (E.g., Finding proofs of mathematical theorems) NP-Complete = Hardest problems in NP Is P = NP? – Is finding a solution as easy as specifying its properties? – Can we replace every mathematician by a computer? – Wishing = Working! June 11, 2015TIFR: Probabilistically Checkable Proofs11

12. of 28 New Formats for Proofs? June 11, 2015TIFR: Probabilistically Checkable Proofs12

13. of 28 Part II: Prob. Checkable Proof June 11, 2015TIFR: Probabilistically Checkable Proofs13

14. of 28 0 11 June 11, 2015TIFR: Probabilistically Checkable Proofs14 0101001010111010101110 1.Reads Theorem 2.Tosses random coins 3.Determines proof query locations 4.Reads locations. Accepts/Rejects V HTHHTH Does such a PCP Verifier, making few queries, exist?

15. of 28 Features of interest #queries: Small! Constant? 3 bits? Length (compared to old proof): – Linear? Quadratic? Exponential? Transformer: Old proofs => New Proofs? – (Not essential, but desirable) [Arora,Lund,Motwani,S.,Szegedy’92]: PCPs with constant queries exist. [Dinur’06]: New construction [Large body of work]: Many improvements (to queries, length) June 11, 2015TIFR: Probabilistically Checkable Proofs15

16. of 28 Part III: Why Proofs/PCPs? June 11, 2015TIFR: Probabilistically Checkable Proofs16

17. of 28 Complexity of Optimization June 11, 2015TIFR: Probabilistically Checkable Proofs17

18. of 28 Approximation Algorithms June 11, 2015TIFR: Probabilistically Checkable Proofs18

19. of 28 Theory of Approximability June 11, 2015TIFR: Probabilistically Checkable Proofs19 Proof does not fit this margin

20. of 28 Part IV: PCP Construction Ideas June 11, 2015TIFR: Probabilistically Checkable Proofs20

21. of 28 Aside: Randomness in Proofs June 11, 2015TIFR: Probabilistically Checkable Proofs21

22. of 28 Essential Ingredient of PCPs Locality of error – Verifier should be able to point to error (if theorem is incorrect) after looking at few bits of proof. Abundance of error – Errors should be found with high probability. How do get these two properties? June 11, 2015TIFR: Probabilistically Checkable Proofs22

23. of 28 3Coloring is NP-complete: June 11, 2015TIFR: Probabilistically Checkable Proofs23 T P a d c e f g b a b g c d e f Error is 2-local Abundance?

24. of 28 Abundance I: via Algebra June 11, 2015TIFR: Probabilistically Checkable Proofs24

25. of 28 Abundance II: via Graph Theory [Dinur’06] Amplification: Constant Factor more edges Double fraction of violated edges (in any coloring) Repeat many times to get fraction upto constant. June 11, 2015TIFR: Probabilistically Checkable Proofs25 a d c e f g b a d c e f g b a d c e f g b a d c e f g b

26. of 28 Wrapping up PCPs – Highly optimistic/wishful definition – Still achievable! – Very useful Understanding approximations (Hugely transformative) Checking outsourced computations Unexpected consequences: Theory of locality in error- correction June 11, 2015TIFR: Probabilistically Checkable Proofs26

27. of 28 Back to Proofs: Philosophy 201 So will math proofs be in PCP format? NO! – Proofs *never* self-contained. Assume common language. – Proofs also rely on common context Repeating things we all know is too tedious. – Proofs rarely intend to convey truth. More vehicles of understanding/knowledge. Still PCP theory might be useful in some contexts: – Verification of computer assisted proofs? June 11, 2015TIFR: Probabilistically Checkable Proofs27

28. of 28 Thank You! June 11, 2015TIFR: Probabilistically Checkable Proofs28