1. Efficient Semantic Communication via Compatible Beliefs Brendan Juba (MIT CSAIL & Harvard) with Madhu Sudan (MSR & MIT)

2. 1.Motivation 2.Beliefs model 3.Sketch of result 2

3. Miscommunication happens… Got that? Q: CAN COMPUTERS COPE WITH MISCOMMUNICATION AUTOMATICALLY?? 3

4. Printer Defining (mis)communication Printer driver Printer firmware ENVIRONME NT INTERFACE FIXED IN ADVANCE 4

5. ENVIRONMEN T Goal of computation (function f) x x f(x) “user returns f(x)?” “user” “server” This talk: f will be PSPACE- complete 5

6. ENVIRONMEN T S -universal user (for computing f) x x f(x) “user returns f(x)?” S 6

7. Formally, U is a S -universal user for computing f if  servers S  S  poly. running time for U  inputs x, initial states of U & S Pr [ ] ≥ ⅔ S -universal user (for computing f) On input x with S, U runs poly(|x|) steps and U’s final output is f(x) [JS’08]: can construct universal users for computing PSPACE-complete f for maximally large class S 7

8. 110 0110 11110 Problem: Password-protected servers ENVIRONMEN T 11110 x x f(x) 8

9. Theorem [JS08,GJS09]. S -Universal users for classes S containing password-protected servers and goals that require the server to act must run for Ω(2 l ) rounds with servers with passwords of length l. PROMISES MORE THAN WE WANTED! CAN WE REFINE AWAY PASSWORDS?? 9

10. 1.Motivation 2.Beliefs model 3.Sketch of result 10

11. Server’s Beliefs Q x x f(x) t U,S (|x|) Def’n: Q -Benchmark running time T Q,S (n) = E U  Q [t U,S (n)] I’VE CHOSEN AN S WITH POLYNOMIAL Q - BENCHMARK RUNNING TIME SO, NO PASSWORD!! 11

12. Q Q WHY DOESN’T IT WORK?? PREFIXING A MESSAGE WITH %!PS-Adobe IS THE MOST NATURAL THING IN THE WORLD. x x MORAL: NEED “SIMILAR” BELIEFS… THAT ISN’T A “PASSWORD.” 12

13. Compatibility P Q  ( P, Q ) =   min {P(  ),Q(  )} = 1-|P- Q| TV 13

14. Compatibility controls overhead of universal communication Theorem. Let P be a sampleable distribution, suppose every server S  S has a belief distribution Q S. For PSPACE-complete , there exist polys r & w such that if strategies from Q S decide  with S, there is a user strategy U P that computes  with any S  S on x of length n in time w( 1 /  ( P, Q ),n) × (T Q,S o r)(n) RECALL: “benchmark time” T Q,S (n) = E U  Q [t U,S (n)], “compatibility”  ( P, Q ) =   min {P(  ),Q(  )} Can recover [JS’08] by taking P = length-weighted uniform Q S = δ U(S) (for U(S) helped by S) DEPENDENCE ON SERVER? DEPENDENCE ON BENCHMARK TIME W.R.T SERVER BELIEFS DEPENDENCE ON COMPATIBILE BELIEFS 14

15. Key points 1.Server designers can evaluate benchmark time w.r.t. their beliefs 2.Compatible beliefs lead to low overhead (beyond benchmark time) 3.Beliefs capture natural approaches 15

16. 1.Motivation 2.Beliefs model 3.Sketch of result 16

17. Starting point: [JS’08] 1.Enumerate algorithms U’: give each constant share of running time, repeatedly double running time (cf. Levin’73) 2.Use U’ with S to simulate interactive proof system for , return answer if successful (exploits efficient prover strategy using  ) SAMPLE FROM P REPEATEDLY DOUBLE # OF SAMPLES PER TIME BOUND, INTRODUCE DOUBLED MAXIMUM TIME BOUND High weight in P corresponds to short programs in enumeration 17

18. Analysis in three easy steps 1.Markov’s inequality: For poly. r from proof system, U’  Q S, see success w.p. 1-γ if we use U’ run for ( 1 / γ )(T Q,S o r)(n) steps 2.See success w.p. 1-γ- |P-Q| TV =  ( P, Q S )-γ for U’  P instead 3.So, if we run 2/  ( P, Q S ) samples from P for ( 2 /  ( P, Q ) )(T Q,S o r)(n) steps each, see success with constant probability So, try using γ =  ( P, Q S )/2 w also contains: overhead from simulating proof system, logarithmic overhead in 1/  ( P, Q S ) (i distinct bounds in phase i) 18

19. Compatibility controls overhead of universal communication Theorem. Let P be a sampleable distribution, suppose every server S  S has a belief distribution Q S. For PSPACE-complete , there exist polys r & w such that if strategies from Q S decide  with S, there is a user strategy U P that computes  with any S  S on x of length n in time w( 1 /  ( P, Q ),n) × (T Q,S o r)(n) RECALL: “benchmark time” T Q,S (n) = E U  Q [t U,S (n)], “compatibility”  ( P, Q ) =   min {P(  ),Q(  )} Actual dependence: Õ( 1 /  ( P, Q ) 2 ) 19

20. RECAP: We refined the semantic communication model to capture natural settings in which flexible communication is possible with low overhead. 20

21. Open problem Construct a server with low benchmark running time for a natural goal and belief distribution! 21

22. Key points 1.Server designers can evaluate benchmark time w.r.t. their beliefs 2.Compatible beliefs lead to low overhead (beyond benchmark time) 3.Beliefs capture natural approaches RECALL: “benchmark time” T Q,S (n) = E U  Q [t U,S (n)] RECALL: “compatibility”  ( P, Q ) =   min{P(  ),Q(  )}, overhead is Õ( 1 /  ( P, Q ) 2 ) FIN. 22