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Universal Semantic Communication

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???”. 110100. 110100. TO BE CONTINUED…. A FAILURE TO COMMUNICATE!. MAN, WHAT THE EFF??. Outline.

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Universal Semantic Communication

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  1. Universal Semantic Communication Brendan Juba (Harvard and MIT) withMadhu Sudan (MSR and MIT) & OdedGoldreich (Weizmann)

  2. HOW DO WE DEFINE THE “MEANING OF THE COMMUNICTATION???” 110100 110100 TO BE CONTINUED…

  3. A FAILURE TO COMMUNICATE! MAN, WHAT THE EFF??

  4. Outline • Model of communication • Theory of finite communication • Example: computation • Model for infinite communication

  5. “Meaning” = Usage = ENVIRONMENT

  6. Printing, formally GOAL OF COMMUNICATION ENVIRONMENT Printer INTERFACE FIXED IN ADVANCE! Printer driver Printer firmware

  7. Abstract goals of communication “G = (ENV,R)” ENVIRONMENT BEHAVIOR DEFINED WITH GOAL environment internal state R: g{0,1} dist.over dist.over U: Ωu× {0,1}* gΩu× {0,1}* S: Ωs× {0,1}* gΩs× {0,1}* σu1 σs1 σs2 σu2 FINITE GOAL OF COMMUNICATION: “USER ACHIEVES GOAL” IF USER “HALTS” WHEN R = 1 “USER” “SERVER”

  8. Goal of computation (function f) x ENVIRONMENT R = “user message = f(x)?” f(x)

  9. Key Concepts • Goal of Communication • Universal user • Sensing function • Helpful server

  10. Bob’s problem I DON’T KNOW WHICH ONE! P ? ? BOB WANTS TO PRINT SUCCESSFULLY, REGARDLESS OF WHICH PRINTER HE IS USING

  11. Universal user P-Universal user for printing ENVIRONMENT P NOTE: WE SHOULD SUCCEED FROM ANY STATE

  12. FROM ANY STATE?? I SURE BLEW THAT… ENVIRONMENT ENVIRONMENT 1101 1101 I’M THROUGH WITH YOU THAT’S ALL I NEEDED TO HEAR!

  13. 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 USER IS “EFFICIENT” IF, WITH EACH SERVER SIN S,THE USER RUNS IN SOME POLYNOMIAL TIME DEPENDING ON S, WITH THE GOAL-SPECIFIC SIZE PARAMETER DEPENDING ON ENV. That is, halts when R = 1

  14. Outline • Model of communication • Theory of finite communication • Example: computation • Model for infinite communication

  15. IT’S ALL ABOUT THE FEEDBACK!!

  16. Key Concepts • Goal of Communication • Universal user • Sensing function • Helpful server

  17. Sensing functions: “safety” ENVIRONMENT I CAN STOP! RECALL, REFEREE:R : environment’s view g{0,1} SENSING FUNCTION:V : user’s view g{0,1} “VIS SAFE”:V = 1eR = 1 (w.h.p.)

  18. Sensing functions: “viability” ENVIRONMENT I CAN STOP! M “VIS VIABLE” IF THERE EXISTS SOME USER STRATEGY THAT RELIABLY OBTAINS V = 1

  19. Achieving Universal Communication Theorem 1. If there is an efficiently computable S-safe and S-viable sensing function for a goal, then there is an efficient S-Universal user for that goal. Each algorithm of length l gets ≈ 1/l22l-share of the total running time ENUMERATE ALL USER ALGORITHMS, RUN EACH WITH CONSTANT FACTOR OVERHEAD: SAFE & VIABLE SENSING FUNCTION INDICATES WHEN TO HALT

  20. Theorem 2. There is a 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. Theorem 2. There is a 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 ALL2l PASSWORDS OF LENGTH l! NOTE: QUALITATIVELY OPTIMAL IN TERMS OF PROGRAM LENGTHS!

  21. So what is Theorem 1 good for?? CHARACTERIZATION IN TERMS OF SENSING FUNCTIONS CAN BE USEFUL

  22. KEY DEF. #4… Helpful servers ENVIRONMENT “SIS HELPFUL” IF THERE EXISTS SOME USER STRATEGY THAT RELIABLY SUCCEEDS AT G

  23. SG

  24. SG-Universal user for G No Common Knowledge Necessary! ENVIRONMENT SG

  25. Theorem 3. If there is an efficient S-Universal user for a goal, 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

  26. Main Theorem. There is an efficient S-Universal user for a goal 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!

  27. 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 SERVERTHAT BREAKS SAFETY WHENEVER SOME ADVERSARY CAN

  28. Proof sketch: Theorem 4 ENVIRONMENT I CAN STOP! NOT SG-SAFE FOR G SG

  29. 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…

  30. Outline • Model of communication • Theory of finite communication • Example: computation • Model for infinite communication

  31. Goal of computation (function f) x ENVIRONMENT R = “user message = f(x)?” f(x)

  32. For which problems can solutions be communicated without common knowledge?

  33. Competitive Proof Systems (Bellare-Goldwasser ‘94) S “x S” PROVE IT! WELL, I’M CONVINCED! YOU AREN’T FOOLING ANYONE! EFFICIENT, GIVEN ORACLE FOR S COMPLETENESS(“COMPETITIVE PROVER”) SOUNDNESS(STANDARD)

  34. 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.

  35. Theorem 5: obtaining a competitive proof system from a universal user TIME’S UP… S x ENVIRONMENT “x S” S(x) SG NOT FOOLED: THEOREMS 3&4

  36. Theorem 5: obtaining a universal user from a competitive proof system S x HELPFUL SERVER “x S” I WON’T BE FOOLED!

  37. 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…

  38. Outline • Model of communication • Theory of finite communication • Example: computation • Model for infinite communication

  39. Multi-session goals REPEATING FINITE COMMUNICATION STRATEGY:PROBABILITY p OF FAILURE EACH SESSION… REPEATING FINITE COMMUNICATION STRATEGY:PROBABILITY p OF FAILURE EACH SESSION… INFINITE SESSION STRATEGY: ZERO ERRORS AFTER FINITE NUMBER OF ROUNDS ENV … SESSION 1 SESSION 2 SESSION 3

  40. Sensing for infinite goals SAFETY: ERRORS DETECTED WITHIN FINITE # OF ROUNDS I’D BETTER TRY SOMETHING ELSE!! VIABILITY: FAILURES CEASE WITHIN FINITE # OF ROUNDS FOR AN APPROPRIATE COMMUNICATION STRATEGY ENV … SESSION 1 SESSION 2 SESSION 3

  41. This weaker version of sensing suffices to construct universal users for infinite goals. But is it necessary??

  42. An impossible finite goal I WONDER IF IT PRINTED… ENVIRONMENT 11110 110 0110 RECALL: WE SHOULD STOP IN FINITE TIME

  43. A possible infinite goal ENVIRONMENT 11110 110 0110 PASSWORD FOUND IN FINITE # OF ROUNDS MORAL: FEEDBACK IS UNNECESSARY!

  44. We saw a model for capturing problems of misunderstanding in communications systems. We also saw some limits of “strong” solutions to this problem.

  45. Key Concepts • Goal of Communication • Helpful server • Universal user • Sensing function G = (ENV,R: g{0,1}) environment internal state V : user’s view g{0,1} SAFETY: ERRORS DETECTED WITHIN FINITE # OF ROUNDS SAFETY: V = 1eR = 1 THERE EXISTS SOME USER STRATEGY THAT RELIABLY SUCCEEDS AT G VIABILITY: FAILURES CEASE WITHIN FINITE # OF ROUNDS FOR AN APPROPRIATE COMMUNICATION STRATEGY VIABILITY: THERE EXISTS SOME USER STRATEGY THAT RELIABLY OBTAINS V = 1 for every server Sin S and every initial state of Sand ENV, the user achIeveSG

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