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

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

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

 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 4

errors: incorrect decoding Coding Theory – A Hamming (7,1)-Code Length n=7, corrects e=1 error received decoded error: correct decoding 5

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 6

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 8

Optimal Length 5 Packing & Covering Codes (5,1) -packing code (5,1) -covering code 9 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 11

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 12

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 13

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

Linear Machine on Z

Linear Machine on Z

Linear Machine on Z Time-evolution is proportional to rows of Pascal’s triangle

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

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) t=1

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) t=2

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) t=3

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) t=4

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) t=5

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) t=6

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) Height of linear machine at t=7 t=7

(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

(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

(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

(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

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

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

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

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

(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

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

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

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

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

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

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

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

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

Liar Machine vs. Pathological Liar Game chips t=0 disqualified

Liar Machine vs. Pathological Liar Game t=1 disqualified

Liar Machine vs. Pathological Liar Game t=2 disqualified

Liar Machine vs. Pathological Liar Game t=3 disqualified

Liar Machine vs. Pathological Liar Game t=4 disqualified

Liar Machine vs. Pathological Liar Game t=5 disqualified

Liar Machine vs. Pathological Liar Game t=6 disqualified No chips survive: Paul loses

(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= (6,1)-Liar machine started with 12 chips after 6 rounds disqualified

Reduction to Liar Machine 51

Reduction to Liar Machine 52

Liar Machine Versus Linear Machine 53

Saving One Chip in the Liar Machine 54

Pathological Liar Game Theorem 55

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

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,  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, Combinatorics, Probability, and Computing, 15 (2006), no. 06, –Joshua Cooper, Benjamin Doerr, Joel Spencer, and Gabor Tardos. Deterministic random walks on the integers. European J. Combin., 28(8): ,