Speaking of Proofs Madhu Sudan Harvard University Based

Speaking of Proofs Madhu. Sudan Harvard University Based on conversations with Brendan Juba, Oded Goldreich, Elad Haramaty, Badih Ghazi … January 13, 2019 Shafifest: Speaking of Proofs 1 of 23

This Talk n n Part I: Speaking Part II: Speaking of Proofs January 13, 2019 Shafifest: Speaking of Proofs 2 of 23

Part I: Speaking January 13, 2019 Shafifest: Speaking of Proofs 3 of 23

Ingredients in Human Communication n n Ability to start with (nearly) zero context and “learning to communicate by communicating”. n Children learn to speak … (not by taking courses) n Focus of work with Brendan and Oded n “What is possible? ” (a la computability) Ability to leverage large uncertain context. n E. g. … This talk today … n n Assumes … English, Math, TCS, Social info, Geography. n Aside … what is “self-contained”? “How to make communication efficient (using context)? ” (a la complexity) January 13, 2019 Shafifest: Speaking of Proofs 4 of 23

Context in Communication n n Empirically, Informally: n Huge piece of information (much larger than “content” of communication) n Not strictly needed for communication … But makes communication efficient, when shared by communicating players n … helps even if context not shared perfectly. Challenge: Formalize? n Work so far … some (toy? ) settings January 13, 2019 Shafifest: Speaking of Proofs 5 of 23

Underlying Model: Communication Complexity The model (with shared randomness) n n Alice Usually studied for lower bounds. This talk: CC as +ve model. n Bob n w. p. 2/3 January 13, 2019 Shafifest: Speaking of Proofs 6 of 23

Communication & (Uncertain) Context n n n n Alice n n n Bob n w. p. 2/3 October 22, 2018 Simons: Mathematical Theory of Communication 7 of 23

1. Imperfectly Shared Randomness n n n n Alice n n n Bob n w. p. 2/3 October 22, 2018 Simons: Mathematical Theory of Communication 8 of 23

Imperfectly Shared Randomness (ISR) n January 13, 2019 Shafifest: Speaking of Proofs 9 of 23

2. Uncertain Functionality n n n n Alice n n n Bob n w. p. 2/3 n October 22, 2018 Simons: Mathematical Theory of Communication 10 of 23

Definitions and results n January 13, 2019 Shafifest: Speaking of Proofs 11 of 23

3. Compression & (Uncertain) Context n n n n Alice n n n Bob nn w. p. 2/3 October 22, 2018 Simons: Mathematical Theory of Communication 12 of 23

3. (Uncertain) Compression n January 13, 2019 Shafifest: Speaking of Proofs 13 of 23

Compression as a proxy for language n n 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? n If latter – every one can’t possibly agree on it; n Yet need to agree on language (mostly)! n Similar to problem of Uncertain Compression. n Studied formally in [Ghazi, Haramaty, Kamath, S. ITCS 17] January 13, 2019 Shafifest: Speaking of Proofs 14 of 23

Part II: Proofs January 13, 2019 Shafifest: Speaking of Proofs 15 of 23

Well understood … (thanks to Shafi & co. ) n n n n Interactive Proofs Zero Knowledge Proofs Multi-Prover Interactive Proofs PCPs Interactive Proofs for Muggles Pseudodeterministic Proofs … … nevertheless some challenges in understanding communication of proofs … January 13, 2019 Shafifest: Speaking of Proofs 16 of 23

Standard Assumption A n T January 13, 2019 n Shafifest: Speaking of Proofs 17 of 23

The truth n Mathematical proofs assume large context. n “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” n n [Lehman, Leighton, Meyer – Notes for MIT 6. 042] Context (= huge set of axioms) shortens proofs. But context is uncertain! n What is “high school mathematics”? January 13, 2019 Shafifest: Speaking of Proofs 18 of 23

Communicating (“speaking of”) Proofs n A T January 13, 2019 n Shafifest: Speaking of Proofs 19 of 23

Uncertainty of Context? n January 13, 2019 Shafifest: Speaking of Proofs 20 of 23

Beyond oracle access: Modelling the mind n January 13, 2019 Shafifest: Speaking of Proofs 21 of 23

Conclusions n n Very poor understanding of “communication of proofs” as we practice it. n Have to rely on verifier’s “knowledge” n But can’t expect to know it exactly Exposes holes in computational understanding of knowledge-processing. n Can we “verify” more given more time? n Or are we just memorizing? January 13, 2019 Shafifest: Speaking of Proofs 22 of 23

Thank You! Happy Birthday, Shafi !! January 13, 2019 Shafifest: Speaking of Proofs 23 of 23