Universal Communication
Brendan Juba (MIT)
With: Madhu Sudan (MIT)
Setting
WHAT IS BOB
GAINING FROM
THIS INTERACTION??
TO SEE IF THEY
ARE INTELLIGENT?
TO OBTAIN
WISDOM?
WHY WOULD
YOU TALK
TO AN ALIEN?
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? STRONG
ASSUMPTIONS!
• Terms?
• Strings with
interpretations
X
Observation
Some Alices are unhelpful.
I COULD HELP,
IF I WANTED.
Solution
Require Alice be helpful in some language.
xS?
xS
Observation
WHAT’S THE
PASSWORD?
Some Alices are still unhelpful.
HELLO??
@&^#*&^%$; x?
xS
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?