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

Motivation I: Binary Search SS 2 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 5 ? Yes. a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 7 ? No. a6a6 a7a7 Is x>a 6 ? …

Motivation I: Binary Search  Search question: which half of surviving list might x be in?  f(M)= d lg M e rounds to search length M list 3 a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 5 ? Yes. a6a6 a7a7 a8a8 a9a9 a 10 Is x>a 7 ? No. a6a6 a7a7 Is x>a 6 ? …

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

Motivation I: Binary Search with Errors  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. 5

Motivation II: Random Walk on Z>=0  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) 6

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

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 8

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, deep-space communication 9

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, deep-space communication 10

 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 ) Received: Decoded: blockwise majority vote Richard Hamming 11

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

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 A packing of 2 radius-1 Hamming balls in the 3-cube 13

A (5,1) -Packing Code as a 2-Player Game  (5,1)-code: 11111, 10100, 01010, What 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 >1 # errors

Covering Codes  Covering is the companion problem to packing  Packing: (n,e) -code  Covering: (n,R) -code length packing radius covering radius (3,1) -packing code and (3,1) -covering code “perfect code” (5,1)-packing code(5,1)-covering code 15

Optimal Length 5 Packing & Covering Codes (5,1) -packing code (5,1) -covering code 16 Sphere bound:

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.=

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 18

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 Message Original encoding Adapted encoding A B C D **** 11*** 10*** 1000* 101**100** 1000* Y $ 1, N $ 0 19

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 >1 # bad predictions (# lies) Bet 2 Bet 1 20

Optimal (5,1)-Codes 21 Code typeOptimal size (5,1)-code4 (5,1)-adaptive code4 Sphere bound5 1/3 (= 2 5 /(5+1) ) (5,1)-adaptive covering code6 (5,1)-covering code7

Adaptive Codes: Results and Questions 22 Sizes of optimal adaptive packing codes Binary, fixed e ≥ sphere bound - c e (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 + C e Binary, 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)

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 23

Linear Machine on Z

Linear Machine on Z 5.5

Linear Machine on Z Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips

Liar Machine on Z 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 Approximates linear machine Preserves indivisibility of chips

Liar Machine on Z 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

Liar Machine on Z 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

Liar Machine on Z 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

Liar Machine on Z 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

Liar Machine on Z 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

Liar Machine on Z 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

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

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.

Proof of Liar Machine Pointwise Discrepancy

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

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 + C e (Delsarte,Piret `86) For e/n 2 (0,1/2), M * (n,e) · sphere bound ¢ n ln 2. (C,E `09+) For e/n 2 (0,1/2), using the liar machine, M * (n,e) = sphere bound ¢.

Liar Machine vs. (6,1)-Pathological Liar Game chips t=0 disqualified

t=1 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

t=2 disqualified Liar Machine vs. (6,1)-Pathological Liar Game

Liar Machine vs. (6,1)-Pathological Liar Game t=3 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=4 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=5 disqualified

Liar Machine vs. (6,1)-Pathological Liar Game t=6 disqualified No chips survive: Paul loses

Comparison of Processes 46 ProcessOptimal #chips Linear machine9 1/7 (6,1)-Pathological liar game10 (6,1)-Liar machine (6,1)-Liar machine started with 12 chips after 6 rounds disqualified

Loss from Liar Machine Reduction t=3 disqualified disqualified Paul’s optimal 2-coloring:

Reduction to Liar Machine

Saving One Chip in the Liar Machine 49

Summary: Pathological Liar Game Theorem

Further Exploration  Tighten the discrepancy analysis for the special case of initial chip configuration f 0 =M  0.  Generalize from binary questions to q-ary questions, q ¸ 2.  Improve analysis of the original liar game from Spencer and Winkler `92; solve the optimal rate of q-ary adaptive block codes for all fractional error rates.  Prove general pointwise and interval discrepancy theorems for various discretizations of random walks. 51

Reading List  This paper: Linearly bounded liars, adaptive covering codes, and deterministic random walks, preprint (see homepage).homepage  The liar machine –Joel Spencer and Peter Winkler. Three thresholds for a liar. Combin. Probab. Comput.1(1):81-93,  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): ,  Discrepancy of deterministic random walks –Joshua Cooper and Joel Spencer, Simulating a Random Walk with Constant Error, Combin. Probab. Comput. 15 (2006), no. 06, –Joshua Cooper, Benjamin Doerr, Joel Spencer, and Gabor Tardos. Deterministic random walks on the integers. European J. Combin., 28(8): ,