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Universal Communication

Universal Communication. Brendan Juba (MIT) With: Madhu Sudan (MIT). Setting. 101011110001010. 001100101011101. . . . 1111010110001. WHAT IS BOB GAINING FROM THIS INTERACTION??. TO SEE IF THEY ARE INTELLIGENT ?. TO OBTAIN WISDOM?. WHY WOULD YOU TALK TO AN ALIEN ?.

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Universal Communication

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  1. Universal Communication Brendan Juba (MIT) With: Madhu Sudan (MIT)

  2. Setting 101011110001010 001100101011101 . . . 1111010110001 WHAT IS BOB GAINING FROM THIS INTERACTION??

  3. TO SEE IF THEYARE INTELLIGENT? TO OBTAINWISDOM? WHY WOULD YOU TALKTO AN ALIEN? TO ASK THEM TO STOP BOMBARDINGUS WITH DANGEROUS RADIATION??

  4. Motivation WHAT CAN BOB LEARN FROM ALICE?

  5. Setting • Fix a setS and a stringx • Bob wishes to learn “xS?” • WANT: protocol that terminates with a verdict that is CORRECT (whp) • Also: efficient in length of x

  6. Outline • Definition: Universal protocol • Analysis of communicating wisdom • Generalizing goals

  7. We want a theorem of the form ??? “Here is a Bob s.t. for every alien languageand every instance x,Bob efficiently learns if xS”

  8. Language??? X • Grammar? • Terms? • Strings with interpretations STRONG ASSUMPTIONS!

  9. I COULD HELP,IF I WANTED. Observation Some Alices are unhelpful. x∈S? y∈S? z∈S?

  10. Solution Require Alice be helpful in some language. xS? xS

  11. WHAT’S THE PASSWORD? Observation Some Alices are stillunhelpful. HELLO?? @&^#*&^%$; x? xS I’M NOT TALKINGTO YOU ANYMORE.

  12. Revision • Require that some B’ can efficientlydecide “xS?” with Alice’s assistance,independent of prior message history • Henceforth, such Alices will be called S-helpful

  13. Definition: S-Universal Bob is S-Universal if S-helpfulA polynomialp x (of lengthn)whp Bobdecides “xS?” when conversing withA, within p(n)steps in expectation

  14. Outline • Definition: Universal protocol • Analysis of communicating wisdom • Generalizing goals

  15. MAIN IDEA #1 • We can efficientlyenumerate and runallefficient protocols • If A is S-Helpful, she helps an efficient protocolB’ that appears in the enumeration

  16. MAIN IDEA #2 • If we can get a proof of either xS or xS, we can guarantee correctness • If SIP, such proofsexist • If S is PSPACE-complete, we can reduce proving (non)membershipto other instances of S

  17. Theorem For any PSPACE-completeS,there is a S-Universalprotocol

  18. For how large a class of setscan we exhibit a universal protocol?

  19. Limitation 1: main observation • Suppose that for somex,some malicious alien Alicecan misleadBob (whp) • We can convert Alice into a “helpful” A’who stillmisleadsBob: pad the useful queries • Recall: a S-UniversalBobshould not be misled by a S-HelpfulAlice!

  20. Limitation 1: finishing up • Thus: a S-UniversalBob satisfies a strong soundness condition • In PSPACE we can find the messages that maximize the probability that Bob halts quickly • Since Bob is sound, his verdict on these messages decideS

  21. First limitation If an S-Universalprotocol exists,SPSPACE

  22. Second limitation (Assuming BPP ≠ PSPACE)For any PSPACE-completeS,if Alice helps a protocol of lengthlthe running time of a S-UniversalBobmust include a constant factorthat isexponential in l

  23. Outline • Definition: Universal protocol • Analysis of communicating wisdom • Generalizing goals

  24. What about efficiency? • Our construction obtained wisdom from an Alice who could decide PSPACE • We obtain analogous results with efficientAlices: limit resources used by our interpreter • Depending on resources used to verify, may only be meaningful in an online sense: “Bob converges to a non-trivial interpreter”

  25. General setting • SOME interactions are successful, others are NOT. • We seek a protocol that tells us how to engage in successfulinteractions (whp)

  26. Define: “goal” • Efficiently verifiablesufficientconditions on Bob’s view of interaction • E.g., effective, efficient protocols! • Easy generalization of our definitions and universal protocol for the computational goal to any such goal

  27. (technical) CONCLUSION UNIVERSAL COMMUNICATIONis (only) possible for VERIFIABLE GOALS.

  28. Practical motivation • Designing protocols for individual devices. (cf. sets, pairs, etc.) • Simpler, more robust networks

  29. Practical technical challenges • Design suitable “goals” (think: “program checking”) • Find a restricted class of protocols that permits “length-efficient” setup

  30. Thank you! Questions?

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