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Madhu Sudan Harvard University

Speaking of Proofs. Madhu Sudan Harvard University. Based on conversations with Brendan Juba, Oded Goldreich , Elad Haramaty , Badih Ghazi …. This Talk. Part I: Speaking Part II: Speaking of Proofs. Part I: Speaking. Ingredients in Human Communication.

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Madhu Sudan Harvard University

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  1. Speaking of Proofs MadhuSudan Harvard University Based on conversations with Brendan Juba, OdedGoldreich, EladHaramaty, Badih Ghazi … Shafifest: Speaking of Proofs

  2. This Talk Part I: Speaking Part II: Speaking of Proofs Shafifest: Speaking of Proofs

  3. Part I: Speaking Shafifest: Speaking of Proofs

  4. Ingredients in Human Communication • Ability to start with (nearly) zero context and “learning to communicate by communicating”. • Children learn to speak … (not by taking courses) • Focus of work with Brendan and Oded • “What is possible?” (a la computability) • Ability to leverage large uncertain context. • E.g. … This talk today … • Assumes … English, Math, TCS, Social info, Geography. • Aside … what is “self-contained”? • “How to make communication efficient (using context)?” (a la complexity) Shafifest: Speaking of Proofs

  5. Context in Communication • Empirically, Informally: • Huge piece of information (much larger than “content” of communication) • Not strictly needed for communication … • … But makes communication efficient, when shared by communicating players • … helps even if context not shared perfectly. • Challenge: Formalize? • Work so far … some (toy?) settings Shafifest: Speaking of Proofs

  6. Underlying Model: Communication Complexity (with shared randomness) Usually studied for lower bounds. This talk: CC as +ve model. Alice Bob bits exchanged by best protocol w.p. 2/3 The model Shafifest: Speaking of Proofs

  7. Communication & (Uncertain) Context Alice Bob w.p. 2/3 Simons: Mathematical Theory of Communication

  8. 1. Imperfectly Shared Randomness Alice Bob w.p. 2/3 Simons: Mathematical Theory of Communication

  9. Imperfectly Shared Randomness (ISR) • Model: (-correlated iid on each coord.) • Thm [Bavarian-Gavinsky-Ito’15]: Equality testing has -comm. comp. with ISR. • Thm [Canonne-Guruswami-Meka-S.’16]: If has CC with perfect rand., then it has ISR-CC • Thm [CGMS] This is tight (for promise problems). • Complete problem: Estimate for • -approximation needs communication • Hard problem: Sparse inner product – where is non-zero only -fraction of the times. Shafifest: Speaking of Proofs

  10. 2. Uncertain Functionality Alice Bob w.p. 2/3 w.p. 2/3 over Simons: Mathematical Theory of Communication

  11. Definitions and results • Defining problem is non-trivial: • Alice/Bob may not “know” but protocol might! • Prevent this by considering entire class of function pairs that are admissible. • Complexity = complexity of ! • Theorem[Ghazi,Komargodski,Kothari,S. 16]: • If arbitrary then there exists s.t. every has cc = 1, but uncertain-cc • If uniform and every has one-way cc , then uncertain-cc • Theorem[Ghazi-S.,18]: Above needs perfect shared randomness. Shafifest: Speaking of Proofs

  12. 3. Compression & (Uncertain) Context Alice Bob w.p. 2/3 Simons: Mathematical Theory of Communication

  13. 3. (Uncertain) Compression • Without context: CC = (where ) • With shared context, Expected-CC= • With imperfectly shared context, but with shared randomness, Expected-CC= • Where [JKKS’11] • Without shared randomness … exact status unknown! Best upper bound ([Haramaty,S’14]): • Expected-CC Shafifest: Speaking of Proofs

  14. Compression as a proxy for language • Information theoretic study of language? • Goal of language: Effective means of expressing information/action. • Implicit objective of language: Make frequent messages short. Compression! • Frequency = Known globally? Learned locally? • If latter – every one can’t possibly agree on it; • Yet need to agree on language (mostly)! • Similar to problem of Uncertain Compression. • Studied formally in [Ghazi,Haramaty,Kamath,S. ITCS 17] Shafifest: Speaking of Proofs

  15. Part II: Proofs Shafifest: Speaking of Proofs

  16. Well understood … (thanks to Shafi & co.) Interactive Proofs Zero Knowledge Proofs Multi-Prover Interactive Proofs PCPs Interactive Proofs for Muggles Pseudodeterministic Proofs … … nevertheless some challenges in understanding communication of proofs … Shafifest: Speaking of Proofs

  17. Standard Assumption A T • Small (Constant) Number of Axioms • , Peano, etc. • Medium Sized Theorem: • … • Big Proof: • Blah blahblahblahblahblahbla blah blahblahblahblahblah blahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblahblah Shafifest: Speaking of Proofs

  18. The truth • Mathematical proofs assume large context. • “By some estimates a proof that 2+2=4 in ZFC would require about 20000 steps … so we will use a huge set of axioms to shorten our proofs – namely, everything from high-school mathematics” [Lehman,Leighton,Meyer – Notes for MIT 6.042] • Context (= huge set of axioms) shortens proofs. • But context is uncertain! • What is “high school mathematics”? Shafifest: Speaking of Proofs

  19. Communicating (“speaking of”) Proofs A T • Ingredients: • Prover: • Axioms Theorem , Proof • Communicates (Claim: proves ) • Verifier: • Complete+sound: iff • Verifier efficient = Polywith oracle access to • Axioms = Context. Shafifest: Speaking of Proofs

  20. Uncertainty of Context? • Prover: works with axioms generates s.t. • Verifier: works with axioms , checks • Robust prover/verifier ? • Need measure (not symmetric). • Given prover should be able to generate such that s.t. • not much larger than • “reasonable” … • E.g. if and then tiny. • Open: Does such an efficient verifier exist? Shafifest: Speaking of Proofs

  21. Beyond oracle access: Modelling the mind • Brain (of verifier) does not just store and retrieve axioms. • Can make logical deductions too! But should do so feasibly. • Semi-formally: • Let denote the set of axioms “known” to the verifier given query-proc. time • Want , but storage space • What is a computational model of the brain that allows this? • Cell-probe – . ConsciousTM – Maybe. Etc… Shafifest: Speaking of Proofs

  22. Conclusions • Very poor understanding of “communication of proofs” as we practice it. • Have to rely on verifier’s “knowledge” • But can’t expect to know it exactly • Exposes holes in computational understanding of knowledge-processing. • Can we “verify” more given more time? • Or are we just memorizing? Shafifest: Speaking of Proofs

  23. Thank You!Happy Birthday, Shafi !! Shafifest: Speaking of Proofs

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