1. Adaptive Coding from a Diffusion Process on the Integer Line Robert Ellis October 26, 2009 Joint work with Joshua Cooper, University of South Carolina

2. Outline of Talk  Coding theory overview –Packing (error-correcting) & covering codes –Coding as a 2-player game –Liar game and pathological liar game  Diffusion processes on Z –Simple random walk (linear machine) –Liar machine –Pathological liar game, alternating question strategy  Improved pathological liar game bound –Reduction to liar machine –Discrepancy analysis of liar machine versus linear machine  Concluding remarks 2

3. Coding Theory Overview  Codewords: fixed-length strings from a finite alphabet  Primary uses: Error-correction for transmission in the presence of noise Compression of data with or without loss  Viewpoints: Packings and coverings of Hamming balls in the hypercube 2-player perfect information games  Applications: Cell phones, compact disks, Mars Reconnaissance Orbiter 3

4.  Transmit blocks of length n  Noise changes ≤ e bits per block ( ||  || 1 ≤ e )  Repetition code 111, 000 – length: n = 3 – e = 1 –information rate: 1/3 Coding Theory: (n,e) -Codes  x1…xnx1…xn (x 1 +  1 )…(x n +  n ) 110010000 101 000111 Received: Decoded: blockwise majority vote Richard Hamming 4

5. 0010011 3 errors: incorrect decoding Coding Theory – A Hamming (7,1)-Code 10001110110110 01000110101101 00101010011011 00011101110001 00000001101010 11001001011100 10100100111000 10010011111111 Length n=7, corrects e=1 error 1001011 received decoded 1001001 1 error: correct decoding 5

6. A Repetition Code as a Packing  (3,1)-code: 111, 000  Pairwise distance = 3  1 error can be corrected  The M codewords of an (n,e) -code correspond to a packing of Hamming balls of radius e in the n -cube 110011101 111 000 010001100 000 010001100 110011101 111 A packing of 2 radius-1 Hamming balls in the 3-cube 6

7. A (5,1) -Packing Code as a 2-Player Game  (5,1)-code: 11111, 10100, 01010, 00001 0What is the 5 th bit? 1What is the 4 th bit? 0What is the 3 rd bit? 0What is the 2 nd bit? 0What is the 1 st bit? CarolePaul 11111 00001 10100 01010 0 1 >1 # errors 11111000011010001010 01111001000001000011 00100 01010 00010 00001 11111 101000101000001 7

8. Covering Codes  Covering is the companion problem to packing  Packing: (n,e) -code  Covering: (n,R) -code length packing radius covering radius 110011101 111 000 010001100 000 010001100 110011101 111 (3,1) -packing code and (3,1) -covering code “perfect code” 11111 00001 10100 01010 11111 11000 01111 1011100001 00100 00010 (5,1)-packing code(5,1)-covering code 8

9. Optimal Length 5 Packing & Covering Codes 01001 01100 01110 01101 00100 11100 01000 111101110101111 00000 01010 11000 10100 00110 00101 10110 10011 10001 10010 11011 00011 10111 000010001010000 11111 10101 00111 01011 11001 11010 0111001101 0100101100 00100 11100 01000 111101110101111 00000 0101011000101000011000101 1011010011 1000110010 11011 00011 10111 000010001010000 11111 1010100111010111100111010 (5,1) -packing code (5,1) -covering code 9 Sphere bound:

10. A (5,1) -Covering Code as a Football Pool WLLLLBet 7 LWLLLBet 6 LLWLLBet 5 LLLWWBet 4 WWWLWBet 3 WWWWLBet 2 WWWWWBet 1 Round 5Round 4Round 3Round 2Round 1 Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=7 00100 01111 11000 10111 00001 00010 11111 10

11. Codes with Feedback (Adaptive Codes)  Feedback Noiseless, delay-less report of actual received bits  Improves the number of decodable messages E.g., from 20 to 28 messages for an (8,1) -code sender receiver Noise Noiseless Feedback Elwyn Berlekamp 1, 0, 1, 1, 0 1, 1, 1, 1, 0 11

12. A (5,1) -Adaptive Packing Code as a 2-Player Liar Game A D B C 0 1 >1 # lies YIs the message C? NIs the message D? NIs the message B? NIs the message A or C? YIs the message C or D? CarolePaul 00101 Message Original encoding Adapted encoding A B C D 0111001010 11000 10011 1**** 11*** 10*** 1000* 101**100** 1000* 1000010001 Y $ 1, N $ 0 12

13. A (5,1)-Adaptive Covering Code as a Football Pool LWLLW Carole L Bet 6 L Bet 5 L Bet 4 W Bet 3 W L L WW Bet 2 L W W W W W L L WW Bet 1 Round 5Round 4Round 3Round 2Round 1 Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6 Bet 3 Bet 6 Bet 4 Bet 5 0 1 >1 # bad predictions (# lies) Bet 2 Bet 1 13

14. Optimal (5,1)-Codes 14 Code typeOptimal size (5,1)-code4 (5,1)-adaptive code4 Sphere bound5 1/3 (5,1)-adaptive covering code6 (5,1)-covering code7

15. 9-9-8-7-6-5-4-3-2012345678 15 11 Linear Machine on Z

16. 9-9-8-7-6-5-4-3-2012345678 Linear Machine on Z 16 5.5

17. 9-9-8-7-6-5-4-3-2012345678 Linear Machine on Z 17 2.755.52.75 Time-evolution is proportional to rows of Pascal’s triangle

18. Liar Machine on Z 18 9-9-8-7-6-5-4-3-2012345678 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 11 chips t=0

19. Liar Machine on Z 19 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=1

20. Liar Machine on Z 20 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=2

21. Liar Machine on Z 21 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=3

22. Liar Machine on Z 22 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=4

23. Liar Machine on Z 23 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=5

24. Liar Machine on Z 24 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 t=6

25. Liar Machine on Z 25 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) 9-9-8-7-6-5-4-3-2012345678 Height of linear machine at t=7 t=7

26. (6,1)-Liar Game 26 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 t=0 disqualified Paul bipartitions Carole moves purple 9 chips

27. (6,1)-Liar Game 27 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 t=1 disqualified Paul bipartitions Carole moves green

28. (6,1)-Liar Game 28 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 t=2 disqualified Paul bipartitions Carole moves green

29. (6,1)-Liar Game 29 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 t=3 disqualified Paul bipartitions Carole moves purple

30. t=4 (6,1)-Liar Game 30 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 disqualified Paul bipartitionsCarole moves purple

31. t=5 (6,1)-Liar Game 31 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 disqualified Paul bipartitionsCarole moves green

32. t=6 (6,1)-Liar Game 32 Liar game time step Paul bipartitions chips: green, purple Carole moves one color to right Paul’s goal: disqualify all but ≤1 chip after t=6 time steps 012 disqualified Two chips survive: Paul loses

33. A Liar Game Strategy for Carole  Weight function for n rounds left; x i = #chips with i lies:  Lemma (Berlekamp)  Refined sphere bound Liar game. Carole keeps half of weight every step. Initial weight > 2 n ) Final weight >1 ) Carole wins. Pathological variant. Carole reduces half of weight every step. Initial weight < 2 n ) Final weight <1 ) Carole wins. 33

34. (6,1)-Pathological Liar Game 34 Paul’s goal: preserve ¸ 1 chip after t=6 time steps 012 t=0 disqualified Paul bipartitions Carole moves green 9 chips wt 6-t (x)=wt 6 (x)=2 6 -1

35. Carole moves green (6,1)-Pathological Liar Game 35 012 t=1 disqualified wt 5 (x)=2 5 -3 Paul bipartitions Paul’s goal: preserve ¸ 1 chip after t=6 time steps

36. Paul bipartitions (6,1)-Pathological Liar Game 36 012 t=2 disqualified Carole moves green wt 4 (x)=2 4 -2 Paul’s goal: preserve ¸ 1 chip after t=6 time steps

37. Paul bipartitions Carole moves purple (6,1)-Pathological Liar Game 37 012 t=3 disqualified wt 3 (x)=2 3 -1 Paul’s goal: preserve ¸ 1 chip after t=6 time steps

38. Paul bipartitions t=4 (6,1)-Pathological Liar Game 38 012 disqualified wt 2 (x)=2 2 -1 Carole moves purple Paul’s goal: preserve ¸ 1 chip after t=6 time steps

39. Carole moves green Paul bipartitions t=5 (6,1)-Pathological Liar Game 39 012 disqualified wt 1 (x)=2 1 -1 Paul’s goal: preserve ¸ 1 chip after t=6 time steps

40. t=6 (6,1)-Pathological Liar Game 40 012 disqualified No chips survive: Paul loses wt 0 (x)=2 0 -1<1 Paul’s goal: preserve ¸ 1 chip after t=6 time steps

41. Optimal (6,1)-Codes 41 Code typeOptimal #chips (6,1)-code8 (6,1)-adaptive code (Liar game) 8 Sphere bound9 1/7 (6,1)-adaptive covering code (Pathological liar game) 10 (6,1)-covering code12

42. New Approach to the Pathological Liar Game Spencer and Winkler (`92) reduced the liar game to the liar machine, a discrete diffusion process on the integer line. Ellis and Yan (`04) introduced the pathological liar game. Cooper and Spencer (`06) use discrepancy analysis to compare the Propp-machine to simple random walk on Z d. Here: (1) We reduce the pathological liar game to the liar machine, (2) Use discrepancy analysis to compare the liar machine to simple random walk on Z, and thereby (3) Improve the best known pathological liar game strategy when the number of lies is a constant fraction of the number of rounds. 42

43. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 43 9-9-8-7-6-5-4-3-2012345678 9 chips t=0 disqualified

44. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 44 9-9-8-7-6-5-4-3-2012345678 t=1 disqualified

45. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 45 9-9-8-7-6-5-4-3-2012345678 t=2 disqualified

46. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 46 9-9-8-7-6-5-4-3-2012345678 t=3 disqualified

47. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 47 9-9-8-7-6-5-4-3-2012345678 t=4 disqualified

48. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 48 9-9-8-7-6-5-4-3-2012345678 t=5 disqualified

49. 9-9-8-7-6-5-4-3-2012345678 Liar Machine vs. Pathological Liar Game 49 9-9-8-7-6-5-4-3-2012345678 t=6 disqualified No chips survive: Paul loses

50. (6,1)-Pathological Liar Game, Liar Machine 50 Code typeOptimal #chips Sphere bound9 1/7 (6,1)-adaptive covering code (Pathological liar game) 10 (6,1)-liar machine12 (6,1)-liar machine optimum: Minimum number of initial chips for ¸ 1 chip to be at position · -4 when t=6 9-9-8-7-6-5-4-3-2012345678 (6,1)-Liar machine started with 12 chips after 6 rounds disqualified

51. Reduction to Liar Machine 51

52. Reduction to Liar Machine 52

53. Liar Machine Versus Linear Machine 53

54. Saving One Chip in the Liar Machine 54

55. Pathological Liar Game Theorem 55

56. Further Exploration  Tighten the discrepancy analysis for the special case of initial chip configuration f 0 (z)=M  0 (z).  Generalize from binary questions to q-ary questions, q ¸ 2.  Improve analysis of the original liar game from Spencer and Winkler `92.  Prove general pointwise and interval discrepancy theorems for various discretizations of random walks. 56

57. Reading List  This paper: Linearly bounded liars, adaptive covering codes, and deterministic random walks, preprint (see homepage).  The liar machine –Joel Spencer and Peter Winkler. Three thresholds for a liar. Combin. Probab. Comput.,1(1):81-93, 1992.  The pathological liar game –Robert Ellis, Vadim Ponomarenko, and Catherine Yan. The Renyi-Ulam pathological liar game with a fixed number of lies. J. Combin. Theory Ser. A, 112(2):328-336, 2005.  Discrepancy of deterministic random walks –Joshua Cooper and Joel Spencer, Simulating a Random Walk with Constant Error, Combinatorics, Probability, and Computing, 15 (2006), no. 06, 815-822. –Joshua Cooper, Benjamin Doerr, Joel Spencer, and Gabor Tardos. Deterministic random walks on the integers. European J. Combin., 28(8):2072-2090, 2007. 57