Transcript pptx

Universal Semantic
Communication
Brendan Juba (Harvard and MIT)
with Madhu Sudan (MSR and MIT)
& Oded Goldreich (Weizmann)
HOW DO WE
DEFINE THE
“MEANING OF THE
COMMUNICTATION?
??” 11010
11010
0
0
TO BE
MAN, WHAT
THE EFF??
A FAILURE TO
COMMUNICA
TE!
Outline
I. Model of communication
II.Theory of finite
communication
III.Example: computation
IV.Model for infinite
communication
“Meaning” = Usage
=
ENVIRONME
NT
Printing, formally
GOAL OF COMMUNICATION
ENVIRONME
INTERFACE
FIXED IN
NT
ADVANCE!
Printer driver
Printer
Printer firmware
Abstract goals of communication
“G
=
(ENV,R)”
g {0,1}
environment
R: internal state
ENVIRONMEN
T
dist.
U: Ωu × {0,1}* g over Ωu × {0,1}*
σu12
“USE
dist.
S: Ωs × {0,1}* g over Ωs × {0,1}*
σs12
FINITE GOAL OF
COMMUNICATION: “USER
ACHIEVES GOAL” IF
USER “HALTS” WHEN R = 1
“SERVE
Goal of computation (function f)
x
R = “user message = f(x)?”
ENVIRONMEN
T
f(x)
Key Concepts
1.Goal of Communication
1.Universal user
2.Sensing function
3.Helpful server
Bob’s problem
I DON’T
KNOW
WHICH ONE!
P
?
?
BOB WANTS TO PRINT
SUCCESSFULLY,
REGARDLESS OF WHICH
Universal
P-Universal
useruser
for printing
ENVIRONMEN
T
P
NOTE: WE SHOULD SUCCEED FROM
ANY STATE
FROM ANY
STATE??
I SURE
BLEW
THAT…
ENVIRONME
NT
ENVIRONME
NT
11
01
11
01
THAT’S ALL
I NEEDED
TO HEAR!
I’M
THROUGH
WITH YOU
Summary: universal user
Definition. A universal user for a goal G = (ENV,R)
and a class of servers S is a user strategy s.t.
for every server S in S and every initial state of
S and ENV, the user achieves G.(w.h.p.)
WE WILL SAY THAT THE UNIVERSAL
That
is, halts
when REACH
=1
USER IS “EFFICIENT”
IF,
WITH
SERVER S IN S,
THE USER RUNS IN SOME
POLYNOMIAL TIME
Outline
I. Model of communication
II.Theory of finite
communication
III.Example: computation
IV.Model for infinite
communication
IT’S ALL ABOUT
THE
FEEDBACK!!
Key Concepts
1.Goal of Communication
1.Universal user
2.Sensing function
3.Helpful server
Sensing functions: “safety”
I CAN
STOP!
SENSING
FUNCTION:
V : user’s view g {0,1}
“V IS SAFE”:
V = 1 e R = 1 (w.h.p.)
ENVIRONMEN
T
RECALL, REFEREE:
R : environment’s view g {0,1}
Sensing functions: “viability”
ENVIRONMEN
T
I CAN
STOP!
“V IS VIABLE” IF THERE EXISTS
SOME USER STRATEGY THAT
RELIABLY OBTAINS V = 1
Achieving Universal Communication
Theorem 1. If there is an efficiently
computable S-safe and S-viable
s e n s i n g f u n c t i o n fo r a g o a l ,
t h e n t h e r e i s a n e fEach
f i algorithm
c i e nof t
length l gets
≈ 1/l 2 S-Universal user for that
goal.
share of the total
2 l
running time
ENUMERATE ALL USER
ALGORITHMS, RUN EACH WITH
CONSTANT FACTOR OVERHEAD:
SAFE & VIABLE SENSING
FUNCTION INDICATES WHEN TO
Theorem 2. There isis aa natural
natural class
of 2l servers S s.t. a S-Universal user
for any goal that requires the server
to act experiences an overhead of
Ω(2l)
rounds.
Might still consider
restricted classes
where we can be
efficient…
IT TAKES ≈2l ROUNDS TO
SEND
ALL 2l PASSWORDS
NOTE: OF
LENGTHQUALITATIVELY
l!
OPTIMAL IN TERMS
So what is Theorem 1 good for??
CHARACTERIZATION IN
TERMS OF SENSING
FUNCTIONS CAN BE
USEFUL
KEY DEF.
#4…
Helpful servers
ENVIRONMEN
T
“S IS HELPFUL” IF THERE EXISTS
SOME USER STRATEGY THAT
RELIABLY SUCCEEDS AT G
SG
SG-Universal user for G
NO COMMON
KNOWLEDGE NECESSARY!
ENVIRONMEN
T
SG
Theorem 3. If there is an efficient
S - U n i ve rs a l u s e r fo r a go a l ,
then there is an efficiently
computable S-safe and S-viable
sensing function for that goal.
THE FUNCTION THAT TELLS A
UNIVERSAL USER WHEN TO
HALT IS A SAFE & VIABLE
SENSING FUNCTION
Main Theorem. There is an efficient
S - U n i v e r s a l u s e r fo r a g o a l
if and only if there is an efficiently
computable S-safe and S-viable
sensing function for the goal.
MORAL: SAFE & VIABLE SENSING
FUNCTIONS ARE PRECISELY THE
FUNCTIONS THAT TELL UNIVERSAL
USERS WHEN TO HALT!
Theorem 4. If a sensing function
is SG-safe for a goal G, then it is
safe for G with all servers, even
malicious and unhelpful ones.
CAN CONSTRUCT A HELPFUL
SERVER
THAT BREAKS SAFETY
WHENEVER SOME
ADVERSARY CAN
Proof sketch: Theorem 4
I CAN
STOP!
NOT
SG-SAFE
FOR G
ENVIRONMEN
T
SG
RECAP: 1. Sensing is necessary
and sufficient
2. Sensing with helpful
servers must also be
safe with all servers
We’ll see a more concrete
interpretation of these theorems
next…
Outline
I. Model of communication
II.Theory of finite
communication
III.Example: computation
IV.Model for infinite
communication
Goal of computation (function f)
x
R = “user message = f(x)?”
ENVIRONMEN
T
f(x)
For which problems can
solutions be
communicated without
common knowledge?
Competitive Proof Systems
(Bellare-Goldwasser ‘94)
S
“x  S”
PROVE
YOU AREN’T
WELL, I’M
FOOLING
IT!
CONVINCED!
ANYONE!
EFFICIENT,
GIVEN
ORACLE FOR
S
COMPLETENESS
SOUNDNE
(“COMPETITIVE
SS
Theorem 5. Let G be the goal of
deciding membership in a set S.
Then there is a SG-universal user for
G iff there are competitive proof
systems for both S and Sc.
Corollary. If there is a SG-universal
user for G then S is in PSPACE.
Theorem 5: obtaining a competitive
proof system from a universal user
x
“x  S”
TIME’S UP…
ENVIRONMEN
T
S(x)
NOT
FOOLED:
SG
S
Theorem 5: obtaining a universal user
from a competitive proof system
x
HELPFUL
SERVER
“x  S”
I WON’T BE
FOOLED!
S
Computational problems with
universal users
•
•
•
•
Any PSPACE-complete problem [Shamir’92]
Any #P-complete problem [LFKN’92]
Graph Isomorphism [GMW’91]
Total functions in NP (solvable by
Levin’s universal search algorithm [Levin’73])
– Integer Factoring
– Discrete Logarithm
– many more…
Outline
I. Model of communication
II.Theory of finite
communication
III.Example: computation
IV.Model for infinite
communication
Multi-session goals
REPEATING FINITE COMMUNICATION
STRATEGY:
INFINITE SESSION
STRATEGY:
PROBABILITY
p OF FAILURE
EACH ZERO
ERRORS AFTER FINITE NUMBER OF
SESSION…
ROUNDS
EN
V
SESSION 1
SESSION 2
SESSION 3
…
Sensing for infinite goals
SAFETY: ERRORS
DETECTED WITHIN
I’D BETTER TRY
FINITE # OF ROUNDS
VIABILITY: FAILURESSOMETHING
CEASE WITHIN FINITE # ELSE!!
OF ROUNDS FOR AN
EN
APPROPRIATE
V
COMMUNICATION
STRATEGY
SESSION 1
SESSION 2
SESSION 3
…
This weaker version of sensing
suffices to construct universal
users for infinite goals.
But is it necessary??
An impossible finite goal
I WONDER IF
IT PRINTED…
ENVIRONMEN
T
11110
0110
110
RECALL: WE SHOULD STOP IN
FINITE TIME
A possible infinite goal
ENVIRONMEN
T
11110
0110
110
PASSWORD FOUND IN FINITE # OF
ROUNDS
MORAL: FEEDBACK IS
We saw a model for capturing
problems of misunderstanding in
communications systems.
We also saw some limits of “strong”
solutions to this problem.
Key Concepts
1.Goal of Communication
G = (ENV,R:
environment
internal state
g {0,1})
V : user’s view g {0,1}
1.Helpful server SAFETY: ERRORS
SAFETY:
V =RELIABLY
1eR=
DETECTED
WITHIN
THERE EXISTS SOME USER STRATEGY
THAT
FINITE
SUCCEEDS
AT G # OF1ROUNDS
VIABILITY: FAILURES
VIABILITY:
CEASE
WITHINTHERE
FINITE #
EXISTS
SOME
FORUSER
AN
FOR EVERY SERVER S INOF
S ROUNDS
AND EVERY
STRATEGY THAT
APPROPRIATE
INITIAL STATE OF S AND ENV, THE USER
RELIABLY
OBTAINS V = 1
COMMUNICATION
ACHIEVES G
STRATEGY
2.Universal user
3.Sensing function