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Speaking of Proofs: Context and Complexity in Communication

Explore the role of context in efficient communication and its impact on proof complexity. Discover the challenges and models in uncertain communication contexts. Learn about compression, uncertain functionality, and interactive proofs. Unravel the truth behind mathematical proofs and the necessity of context.

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Speaking of Proofs: Context and Complexity in Communication

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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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