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

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Setting WHAT IS BOB GAINING FROM THIS INTERACTION??

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WHY WOULD YOU TALK TO AN ALIEN? TO SEE IF THEY ARE INTELLIGENT? TO OBTAIN WISDOM? TO ASK THEM TO STOP BOMBARDING US WITH DANGEROUS RADIATION??

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Motivation WHAT CAN BOB LEARN FROM ALICE?

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Setting Fix a set S and a string x Bob wishes to learn “x S?” WANT: protocol that terminates with a verdict that is CORRECT (whp) Also: efficient in length of x

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Outline 1.Definition: Universal protocol 2.Analysis of communicating wisdom 3.Generalizing goals

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

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Language??? Grammar? Terms? Strings with interpretations X STRONG ASSUMPTIONS!

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Observation Some Alices are unhelpful. I COULD HELP, IF I WANTED.

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Solution Require Alice be helpful in some language. x S?xSxS

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Observation Some Alices are still unhelpful. WHAT’S THE x? xSxS HELLO?? I’M NOT TALKING TO YOU ANYMORE.

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

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Definition: S-Universal Bob is S-Universal if S-helpful A polynomial p x (of length n) whp Bob decides “x S?” when conversing with A, within p(n) steps in expectation

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Outline Definition: Universal protocol 2.Analysis of communicating wisdom 3.Generalizing goals

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MAIN IDEA #1 We can efficiently enumerate and run all efficient protocols If A is S-Helpful, she helps an efficient protocol B’ that appears in the enumeration

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MAIN IDEA #2 If we can get a proof of either x S or x S, we can guarantee correctness If S IP, such proofs exist If S is PSPACE-complete, we can reduce proving (non)membership to other instances of S

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Theorem For any PSPACE-complete S, there is a S-Universal protocol

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For how large a class of sets can we exhibit a universal protocol?

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Limitation 1: main observation Suppose that for some x, some malicious alien Alice can mislead Bob (whp) We can convert Alice into a “helpful” A’ who still misleads Bob: pad the useful queries Recall: a S-Universal Bob should not be misled by a S-Helpful Alice!

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Limitation 1: finishing up Thus: a S-Universal Bob 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 decide S

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First limitation If an S-Universal protocol exists, S PSPACE

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Second limitation (Assuming BPP ≠ PSPACE) For any PSPACE-complete S, if Alice helps a protocol of length l the running time of a S-Universal Bob must include a constant factor that is exponential in l

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Outline Definition: Universal protocol Analysis of communicating wisdom 3.Generalizing goals

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What about efficiency? Our construction obtained wisdom from an Alice who could decide PSPACE We obtain analogous results with efficient Alices: 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”

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General setting 1.SOME interactions are successful, others are NOT. 2.We seek a protocol that tells us how to engage in successful interactions (whp)

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Define: “goal” Efficiently verifiable sufficient conditions 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

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(technical) CONCLUSION UNIVERSAL COMMUNICATION is (only) possible for VERIFIABLE GOALS.

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

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Practical technical challenges 1.Design suitable “goals” (think: “program checking”) 2.Find a restricted class of protocols that permits “length-efficient” setup

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Thank you! Questions?

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