Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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