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

Setting WHAT IS BOB GAINING FROM THIS INTERACTION??

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??

Motivation WHAT CAN BOB LEARN FROM ALICE?

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

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

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” ???

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

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

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

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

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

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

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

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

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

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

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

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!

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

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

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

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

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”

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)

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

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

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

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

Thank you! Questions?