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An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th , 2011 ISMAA 2011, North Central College

An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th , 2011 ISMAA 2011, North Central College Joint work with Ruoran Wang. 2. Motivation I: Binary Search. S. Is x > a 5 ? Yes. Is x > a 7 ? No. Is x > a 6 ? … . Motivation I: Binary Search. 3.

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An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th , 2011 ISMAA 2011, North Central College

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  1. An identity for dual versions of a chip-moving game Robert B. Ellis April 8th, 2011 ISMAA 2011, North Central College Joint work with Ruoran Wang

  2. 2 Motivation I: Binary Search • S Is x>a5? Yes. Is x>a7? No. Is x>a6? …

  3. Motivation I: Binary Search 3 Is x>a5? Yes. Is x>a7? No. Is x>a6? … Search question: which half of surviving list might x be in? f(M)=d lg Me rounds to search length M list

  4. Motivation I: Binary Search on Z>=0 4 Redisplay binary search as on Z with e=0. Go a couple of rounds Straight reformulation, no difference

  5. Motivation I: Binary Search with Errors 5 Let e>=0 and assume up to e responses are erroneous We can’t be sure to have found x unless other candidates have e+1 “no” votes.

  6. Motivation II: Random Walk on Z>=0 6 M chips at origin. Each round, at each position, half of the chips stay in place and half move to the right. A (good) search algorithm is a discretization of this random walk. Our search algorithm from now on: number chips left-to-right 1,…,M; split chips into odds and evens Define P*(n,e), K*(n,e)

  7. Game tree and tabular data 7 A (5,1) game tree, M=4 chips for P* tree, 3 chips for K* tree. Plus implication for P* and K*. Maybe tables?

  8. Outline of Talk 8 • 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

  9. Coding Theory Overview 9 Codewords:fixed-length strings from a finite alphabet Primary uses: Error-correction for transmission in the presence of noiseCompression of data with or without loss Viewpoints:Packings and coverings of Hamming balls in the hypercube2-player perfect information games Applications:Cell phones, compact disks, deep-space communication

  10. 10 Coding Theory Overview • Codewords:fixed-length strings from a finite alphabet • Primary uses: Error-correction for transmission in the presence of noiseCompression of data with or without loss • Viewpoints:Packings and coverings of Hamming balls in the hypercube2-player perfect information games • Applications:Cell phones, compact disks, deep-space communication

  11. x1…xn  (x1+1)…(xn+ n) Received: 110 010 000 101 blockwise majority vote Decoded: 111 000 000 111 11 Coding Theory: (n,e)-Codes • 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 Richard Hamming

  12. 1 1 1 0 0 0 0 1 1 0 0 0 1 0 3 errors: incorrect decoding 1 error: correct decoding 12 Coding Theory – A Hamming (7,1)-Code Length n=7, corrects e=1 error received 1 1 0 1 0 0 1 decoded

  13. 111 110 101 011 100 010 001 000 13 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 toa packing of Hamming ballsof radius e in the n-cube 111 110 101 011 100 010 001 000 A packing of 2 radius-1 Hamming balls in the 3-cube

  14. 01111 00100 00010 00011 14 A (5,1)-Packing Code as a 2-Player Game • (5,1)-code: 11111, 10100, 01010, 00001 Paul Carole 11111 What is the 1st bit? 0 10100 01010 What is the 2nd bit? 0 00001 What is the 3rd bit? 0 0 1 >1 What is the 4th bit? 1 What is the 5th bit? 0 # errors 11111 11111 10100 01010 00001 01010 00001 00001 00001 10100 01010 00001 00100 00010 00010 00010

  15. 111 110 101 011 111 100 010 001 110 101 011 000 100 010 001 000 15 Covering Codes • Covering is the companion problem to packing • Packing: (n,e)-code • Covering: (n,R)-code packing radius length covering radius (3,1)-packing code and(3,1)-covering code“perfect code” 11111 11111 00100 10100 01111 00010 01010 10111 00001 00001 11000 (5,1)-packing code (5,1)-covering code

  16. 16 Optimal Length 5 Packing & Covering Codes (5,1)-packing code 11111 11110 11101 11011 10111 01111 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 10000 01000 00100 00010 00001 (5,1)-covering code 11111 00000 11110 11101 11011 10111 01111 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111 Sphere bound: 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 10000 01000 00100 00010 00001 00000

  17. 17 A (5,1)-Covering Code as a Football Pool Round 1 Round 2 Round 3 Round 4 Round 5 Bet 1 W W W W W 11111 Bet 2 L W W W W 01111 Bet 3 W L W W W 10111 Bet 4 W W L L L 11000 Bet 5 L L W L L 00100 Bet 6 L L L W L 00010 Bet 7 L L L L W 00001 Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=7

  18. 18 Codes with Feedback (Adaptive Codes) • FeedbackNoiseless, delay-less report of actual received bits • Improves the number of decodable messagesE.g., from 20 to 28 messages for an (8,1)-code 1, 1, 1, 1, 0 1, 0, 1, 1, 0 receiver sender Noise 1, 1, 1, 1, 0 Elwyn Berlekamp Noiseless Feedback

  19. 10*** 1**** 10*** 1000* 11*** 1**** 101** 100** 10001 10000 19 A (5,1)-Adaptive Packing Code as a 2-Player Liar Game Paul Carole A Is the message C or D? Y B Is the message A or C? N C Is the message B? N D Is the message D? N 0 1 >1 Is the message C? Y # lies A B D Message C Originalencoding 00101 01110 01010 11000 10011 Adaptedencoding 1000* Y $ 1, N $ 0

  20. W W L W W L L L L W W W W 20 A (5,1)-Adaptive Covering Code as a Football Pool Round 1 Round 2 Round 3 Round 4 Round 5 Bet 1 W Bet 1 Bet 2 Bet 2 W Bet 3 Bet 3 W Bet 4 Bet 5 Bet 4 L Bet 6 Bet 5 L 0 1 >1 # badpredictions(# lies) Bet 6 L Carole W L L W L Payoff: a bet with ≤ 1 bad prediction Question. Min # bets to guarantee a payoff? Ans.=6

  21. 21 Optimal (5,1)-Codes

  22. Adaptive Codes: Results and Questions 22 • Sizes of optimal adaptive packing codes • Binary, fixed e≥ sphere bound - ce (Spencer `92) • Binary, e=1,2,3 =sphere bound - O(1), exact solutions (Pelc; Guzicki; Deppe) • Q-ary, e=1 =sphere bound - c(q,e), exact solution (Aigner `96) • Q-ary, e linear unknown if rate meets Hamming bound for all e. (Ahlswede, C. Deppe, and V. Lebedev) • Sizes of optimal adaptive covering codes • Binary, fixed e·sphere bound + CeBinary, e=1,2 =sphere bound + O(1), exact solution (Ellis, Ponomarenko, Yan `05) • Near-perfect adaptive codes • Q-ary, symmetric or “balanced”, e=1 exact solution (Ellis `04+) • General channel, fixed e asymptotic first term (Ellis, Nyman `09)

  23. Outline of Talk 23 • 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

  24. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 24 Linear Machine on Z 11

  25. Linear Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 5.5 5.5

  26. Linear Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 2.75 5.5 2.75 Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips

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

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

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

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

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

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

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

  34. Liar Machine on Z -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 Liar machine time-step Number chips left-to-right 1,2,3,… Move odd chips right, even chips left (Reassign numbers every time-step) t=7 Height of linear machine at t=7 l1-distance: 5.80 l∞-distance: 0.98

  35. Discrepancy for Two Discretizations Liar machine: round-offs spatially balanced Rotor-router model/Propp machine: round-offs temporally balanced The liar machine has poorer discrepancy… but provides bounds to the pathological liar game.

  36. Proof of Liar Machine Pointwise Discrepancy

  37. The Liar Game as a Diffusion Process A priori: M=#chips, n=#rounds, e=max #lies Initial configuration: f0 = M¢0 Each round, obtain ft+1 from ft by: (1) Paul 2-colors the chips (2) Carole moves one color class left, the other right Final configuration: fn Winning conditions Original variant (Berlekamp, Rényi, Ulam) Pathological variant (Ellis, Yan)

  38. Pathological Liar Game Bounds Fix n, e. Define M*(n,e) = minimum M such that Paul can win the pathological liar game with parameters M,n,e. Sphere Bound (E,P,Y `05) For fixed e,M*(n,e) · sphere bound + Ce (Delsarte,Piret `86) For e/n2 (0,1/2), M*(n,e) · sphere bound ¢n ln 2 . (C,E `09+) For e/n2 (0,1/2), using the liar machine, M*(n,e) = sphere bound ¢ .

  39. Liar Machine vs. (6,1)-Pathological Liar Game 39 9 chips -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=0 9 chips disqualified

  40. Liar Machine vs. (6,1)-Pathological Liar Game 40 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=1 disqualified

  41. Liar Machine vs. (6,1)-Pathological Liar Game 41 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=2 disqualified

  42. Liar Machine vs. (6,1)-Pathological Liar Game 42 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=3 disqualified

  43. Liar Machine vs. (6,1)-Pathological Liar Game 43 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=4 disqualified

  44. Liar Machine vs. (6,1)-Pathological Liar Game 44 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=5 disqualified

  45. Liar Machine vs. (6,1)-Pathological Liar Game 45 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 t=6 No chips survive: Paul loses disqualified

  46. Comparison of Processes 46 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 disqualified (6,1)-Liar machine started with 12 chips after 6 rounds

  47. Loss from Liar Machine Reduction 47 t=3 -9 -9 -9 -8 -8 -8 -7 -7 -7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 Paul’s optimal 2-coloring: disqualified disqualified

  48. Reduction to Liar Machine

  49. Saving One Chip in the Liar Machine 49

  50. Summary: Pathological Liar Game Theorem

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