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 THEYARE INTELLIGENT?

TO OBTAINWISDOM?

TO ASK THEM TO STOP BOMBARDINGUS WITH DANGEROUS RADIATION??

Motivation

WHAT CAN BOB LEARN FROM ALICE?

Setting

• Fix a set S and a string x• Bob wishes to learn “xS?”

• 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 xS”

???

Language???

• Grammar?

• Terms?

• Strings with interpretations

X STRONG ASSUMPTIONS!

ObservationSome Alices are unhelpful.

I COULD HELP,IF I WANTED.

SolutionRequire Alice be helpful in some language.

xS?xS

ObservationSome Alices are still unhelpful.

WHAT’S THE PASSWORD?

@&^#*&^%$; x?

xS

HELLO??

I’M NOT TALKINGTO YOU ANYMORE.

Revision

• Require that some B’ can efficiently decide “xS?” 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 “xS?” 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 xS or xS, we can guarantee correctness

• If SIP, 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,SPSPACE

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 Bobmust 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?