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of 1401/03/2015 ISCA-2015: Reliable Meaningful Communication 1
Reliable Meaningful Communication
Madhu SudanMicrosoft, Cambridge,
USA
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Reliable Communication?
Problem from the 1940s: Advent of digital age.
Communication media are always noisy But digital information less tolerant to noise!
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Alice Bob
We are not
ready
We are now
ready
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Coding by Repetition
Can repeat (every letter of) message to improve reliability:
WWW EEE AAA RRR EEE NNN OOO WWW …
WXW EEA ARA SSR EEE NMN OOP WWW … Calculations:
repetitions Prob. Single symbol corrupted To transmit symbols, choose Rate of transmission as Belief (pre-1940s): Rate of any scheme as
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Shannon’s Theory [1948]
Sender “Encodes” before transmitting Receiver “Decodes” after receiving
Encoder/Decoder arbitrary functions.
Rate = Requirement: w. high prob.
What are the best (with highest Rate)?
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Alice BobEncoder Decoder
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Shannon’s Theorem
If every bit is flipped with probability Rate can be achieved.
This is best possible. Examples:
Monotone decreasing for Positive rate for ; even if
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Challenges post-Shannon
Encoding/Decoding functions not “constructive”. Shannon picked at random, brute force. Consequence:
takes time to compute (on a computer). takes time to find!
Algorithmic challenge: Find more explicitly. Both should take time to compute
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Explicit Codes: Reed-Solomon Code
Messages = Coefficients of Polynomials. Example:
Message = Think of polynomial Encoding: First four values suffice, rest is redundancy!
(Easy) Facts: Any values suffice where = length of message. Can handle erasures or errors.
Explicit encoding Efficient decoding? [Peterson 1960]
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More Errors? List Decoding
Why was the limit for #errors? is clearly a limit – right?
First half = evaluations of Second half = evaluations of What is the right message: or ?
(even ) is the limit for “unique” answer. List-decoding: Generalized notion of decoding.
Report (small) list of possible messages. Decoding “successful” if list contains the
message polynomial.
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Reed-Solomon List-Decoding Problem
Given: Parameters: Points: in the plane
(finite field actually) Find:
All degree poly’s that pass thru of points i.e., all s.t.
: Answer unique; [Peterson 60] finds it. [S. 96, Guruswami+S. ‘98]: ; small list
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Decoding by example + picture [S’96]
Algorithm idea:
Find algebraic explanation of all points.
Stare at the solution (factor the polynomial)
Ssss
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𝑛=14 ;𝑘=1;𝑡=5
(𝑥+𝑦 )(𝑥− 𝑦 )(𝑥2+𝑦2−1)
𝑥4−𝑦 4−𝑥2+ 𝑦2=0
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Decoding by example + picture [S’96]
Algorithm idea:
Find algebraic explanation of all points.
Stare at the solution (factor the polynomial)
Ssss
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𝑛=14 ;𝑘=1;𝑡=5
(𝑥+𝑦 ) (𝑥−𝑦 )(𝑥2+𝑦2−1)
𝑥4−𝑦 4−𝑥2+ 𝑦2=0
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Decoding Algorithm
Fact: There is always a degree polynomial thru points Can be found in polynomial time (solving linear
system).
[80s]: Polynomials can be factored in polynomial time [Grigoriev, Kaltofen, Lenstra]
Leads to (simple, efficient) list-decoding correcting fraction errors for
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Summary and conclusions
(Many) errors can be dealt with: Pre-Shannon: vanishing fraction of errors Pre-list-decoding: small constant fraction Post-list-decoding: overwhelming fraction
Future challenges? Communication can overcome errors introduced by
channels. Can communication overcome errors in
misunderstanding between sender and receiver? [Goldreich,Juba,S. ‘2011]; [Juba,Kalai,Khanna,S.’2011]
….
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Thank You!
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