1. Imperfectly Shared Randomness
in Communication
Madhu Sudan
Harvard
Joint work with Clément Canonne (Columbia),
Venkatesan Guruswami (CMU) and Raghu Meka (UCLA).
11/16/2016
UofT: ISR in Communication

2. Classical theory of communication
Shannon (1948)
Clean architecture for reliable communication.
Remarkable mathematical discoveries: Prob. Method, Entropy, (Mutual) Information, Compression schemes, Error-correcting codes.
But does Bob really need all of 𝑥? 𝑥
Alice
Bob
Encoder
Decoder
11/16/2016
UofT: ISR in Communication

3. [Yao’80] Communication Complexity
The model
(with shared randomness) 𝑥 𝑦
𝑓: 𝑥,𝑦 ↦Σ
𝑅=$$$
Alice
Bob
𝐶𝐶 𝑓 =# bits exchanged
by best protocol
𝑓(𝑥,𝑦)
w.p. 2/3
11/16/2016
UofT: ISR in Communication

4. Communication Complexity - Motivation
Standard: Lower bounds
Circuit complexity, Streaming, Data Structures, extended formulations …
This talk:
Communication complexity as model for communication
Food for thought: Shannon vs. Yao? Which is the right model.
Typical (human-human/computer-computer) communication involves large context and brief communication.
Contexts imperfectly shared.
11/16/2016
UofT: ISR in Communication

5. UofT: ISR in Communication
This talk
Example problems with low communication complexity.
New form of uncertainty and overcoming it.
11/16/2016
UofT: ISR in Communication

6. Aside: Easy CC Problems
Equality testing:
𝐸𝑄 𝑥,𝑦 =1⇔𝑥=𝑦; 𝐶𝐶 𝐸𝑄 =𝑂 1
Hamming distance:
𝐻 𝑘 𝑥,𝑦 =1⇔Δ 𝑥,𝑦 ≤𝑘; 𝐶𝐶 𝐻 𝑘 =𝑂(𝑘 log 𝑘 ) [Huang etal.]
Small set intersection:
∩ 𝑘 𝑥,𝑦 =1⇔ wt 𝑥 , wt 𝑦 ≤𝑘 & ∃𝑖 𝑠.𝑡. 𝑥 𝑖 = 𝑦 𝑖 =1.
𝐶𝐶 ∩ 𝑘 =𝑂(𝑘) [Håstad Wigderson]
Gap (Real) Inner Product:
𝑥,𝑦∈ ℝ 𝑛 ; 𝑥 2 , 𝑦 2 =1;
𝐺𝐼 𝑃 𝑐,𝑠 𝑥,𝑦 =1 if 𝑥,𝑦 ≥𝜖; =0 if 𝑥,𝑦 ≤0;
𝐶𝐶 𝐺𝐼 𝑃 𝑐,𝑠 =𝑂 1 𝜖 2 ; [Alon, Matias, Szegedy]
Protocol:
Fix ECC 𝐸: 0,1 𝑛 → 0,1 𝑁 ;
Shared randomness: 𝑖← 𝑁 ;
Exchange 𝐸 𝑥 𝑖 , 𝐸 𝑦 𝑖 ;
Accept iff 𝐸 𝑥 𝑖 =𝐸 𝑦 𝑖 .
𝑝𝑜𝑙𝑦 𝑘 Protocol:
Use common randomness
to hash 𝑛 →[ 𝑘 2 ]
𝑥=( 𝑥 1 , …, 𝑥 𝑛 )
𝑦= 𝑦 1 , …, 𝑦 𝑛
〈𝑥,𝑦〉≜ 𝑖 𝑥 𝑖 𝑦 𝑖
Main Insight:
If 𝐺←𝑁 0,1 𝑛 , then
𝔼 𝐺,𝑥 ⋅ 𝐺,𝑦 =〈𝑥,𝑦〉
Summary from
[Ghazi, Kamath, S.’16]
11/16/2016
UofT: ISR in Communication

7. Uncertainty in Communication
Overarching question: Are there communication mechanisms that can overcome uncertainty?
What is uncertainty?
This talk: Alice, Bob don’t share randomness perfectly; only approximately. A B f R
11/16/2016
UofT: ISR in Communication

8. UofT: ISR in Communication
Rest of this talk
Model: Imperfectly Shared Randomness
Positive results: Coping with imperfectly shared randomness.
Negative results: Analyzing weakness of imperfectly shared randomness.
11/16/2016
UofT: ISR in Communication

9. Model: Imperfectly Shared Randomness
Alice ←𝑟 ; and Bob ←𝑠 where
𝑟,𝑠 = i.i.d. sequence of correlated pairs 𝑟 𝑖 , 𝑠 𝑖 𝑖 ;
𝑟 𝑖 , 𝑠 𝑖 ∈{−1,+1};𝔼 𝑟 𝑖 =𝔼 𝑠 𝑖 =0;𝔼 𝑟 𝑖 𝑠 𝑖 =𝜌≥0 .
Notation:
𝑖𝑠 𝑟 𝜌 (𝑓) = cc of 𝑓 with 𝜌-correlated bits.
𝑐𝑐(𝑓): Perfectly Shared Randomness cc.
𝑝𝑟𝑖𝑣 𝑓 : cc with PRIVate randomness
Starting point: for Boolean functions 𝑓
𝑐𝑐 𝑓 ≤𝑖𝑠 𝑟 𝜌 𝑓 ≤𝑝𝑟𝑖𝑣 𝑓 ≤𝑐𝑐 𝑓 + log 𝑛
What if 𝑐𝑐 𝑓 ≪log 𝑛? E.g. 𝑐𝑐 𝑓 =𝑂(1)
=𝑖𝑠 𝑟 1 𝑓
=𝑖𝑠 𝑟 0 𝑓
𝜌≤𝜏⇒
𝑖𝑠 𝑟 𝜌 𝑓 ≥𝑖𝑠 𝑟 𝜏 𝑓
11/16/2016
UofT: ISR in Communication

10. Distill Perfect Randomness from ISR?
Agreement Distillation:
Alice ←𝑟; Bob ←𝑠; 𝑟,𝑠 𝜌-corr. unbiased bits
Outputs: Alice →𝑢; Bob →𝑣; 𝐻 ∞ 𝑢 , 𝐻 ∞ 𝑣 ≥𝑘
Communication = 𝑐 bits;
What is max. prob. 𝜏 of agreement (𝑢=𝑣)?
Well-studied in the literature!
[Ahlswede-Csiszar ‘70s]:
𝜏→1⇒𝑐=Ω 𝑘 (one-way)
[Bogdanov-Mossel ‘2000s]: 𝑐=0⇒𝜏≤exp(−𝑘)
[Our work]: 𝜏= exp(−𝑘+𝑂 𝑐 ) (two-way)
Summary: Can’t distill randomness!
11/16/2016
UofT: ISR in Communication

11. UofT: ISR in Communication
Results
Model first studied by [Bavarian,Gavinsky,Ito’14] (“Independently and earlier”).
Their focus: Simultaneous Communication; general models of correlation.
They show 𝑖𝑠𝑟 Equality =𝑂 1 (among other things)
Our Results:
Generally: 𝑐𝑐 𝑓 ≤𝑘⇒𝑖𝑠𝑟 𝑓 ≤ 2 𝑘
Converse: ∃𝑓 with 𝑐𝑐 𝑓 ≤𝑘 & 𝑖𝑠𝑟 𝑓 ≥ 2 𝑘
11/16/2016
UofT: ISR in Communication

12. Equality Testing (our proof)
Key idea: Think inner products.
Encode 𝑥↦𝑋=𝐸(𝑥);𝑦↦𝑌=𝐸 𝑦 ;𝑋,𝑌∈ −1,+1 𝑁
𝑥=𝑦⇒ 〈𝑋,𝑌〉=𝑁
𝑥≠𝑦⇒ 〈𝑋,𝑌〉≤𝑁/2
Estimating inner products:
Building on sketching protocols …
Alice: Picks Gaussians 𝐺 1 ,… 𝐺 𝑡 ∈ ℝ 𝑁 ,
Sends 𝑖∈ 𝑡 maximizing 𝐺 𝑖 ,𝑋 to Bob.
Bob: Accepts iff 𝐺′ 𝑖 ,𝑌 ≥0
Analysis: 𝑂 𝜌 (1) bits suffice if 𝐺 ≈ 𝜌 𝐺′
Gaussian
Protocol
11/16/2016
UofT: ISR in Communication

13. General One-Way Communication
Idea: All communication ≤ Inner Products
(For now: Assume one-way-cc 𝑓 ≤ 𝑘)
For each random string 𝑅
Alice’s message = 𝑖 𝑅 ∈ 2 𝑘
Bob’s output = 𝑓 𝑅 ( 𝑖 𝑅 ) where 𝑓 𝑅 : 2 𝑘 → 0,1
W.p. ≥ 2 3 over 𝑅, 𝑓 𝑅 𝑖 𝑅 is the right answer.
11/16/2016
UofT: ISR in Communication

14. General One-Way Communication
For each random string 𝑅
Alice’s message = 𝑖 𝑅 ∈ 2 𝑘
Bob’s output = 𝑓 𝑅 ( 𝑖 𝑅 ) where 𝑓 𝑅 : 2 𝑘 → 0,1
W.p. ≥ 2 3 , 𝑓 𝑅 𝑖 𝑅 is the right answer.
Vector representation:
𝑖 𝑅 ↦ 𝑥 𝑅 ∈ 0,1 2 𝑘 (unit coordinate vector)
𝑓 𝑅 ↦ 𝑦 𝑅 ∈ 0,1 2 𝑘 (truth table of 𝑓 𝑅 ).
𝑓 𝑅 𝑖 𝑅 =〈 𝑥 𝑅 , 𝑦 𝑅 〉; Acc. Prob. ∝ 𝑋,𝑌 ;𝑋= 𝑥 𝑅 𝑅 ;𝑌= 𝑦 𝑅 𝑅
Gaussian protocol estimates inner products of unit vectors to within ±𝜖 with 𝑂 𝜌 1 𝜖 2 communication.
11/16/2016
UofT: ISR in Communication

15. Two-way communication
Still decided by inner products.
Simple lemma:
∃ 𝐾 𝐴 𝑘 , 𝐾 𝐵 𝑘 ⊆ ℝ 2 𝑘 convex, that describe private coin k-bit comm. strategies for Alice, Bob s.t. accept prob. of 𝜋 𝐴 ∈ 𝐾 𝐴 𝑘 , 𝜋 𝐵 ∈ 𝐾 𝐵 𝑘 equals 〈 𝜋 𝐴 , 𝜋 𝐵 〉
Putting things together:
Theorem: 𝑐𝑐 𝑓 ≤𝑘⇒𝑖𝑠𝑟 𝑓 ≤ 𝑂 𝜌 ( 2 𝑘 )
11/16/2016
UofT: ISR in Communication

16. Main Technical Result: Matching lower bound
Theorem: There exists a (promise) problem 𝑓 s.t.
𝑐𝑐 𝑓 ≤𝑘
𝑖𝑠 𝑟 𝜌 𝑓 ≥exp⁡(𝑘)
The Problem:
Gap Sparse Inner Product (G-Sparse-IP).
Alice gets sparse 𝑥∈ 0,1 𝑛 ; wt 𝑥 ≈ 2 −𝑘 ⋅𝑛
Bob gets 𝑦∈ 0,1 𝑛
Promise: 〈𝑥,𝑦〉 ≥ −𝑘 ⋅𝑛 or 𝑥,𝑦 ≤ −𝑘 ⋅𝑛.
Decide which.
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

17. 𝒑𝒔𝒓 Protocol for G-Sparse-IP
Note: Gaussian protocol takes 𝑂 2 𝑘 bits.
Need to get exponentially better.
Idea: 𝑥 𝑖 ≠0 ⇒ 𝑦 𝑖 correlated with answer.
Use (perfectly) shared randomness to find random index 𝑖 s.t. 𝑥 𝑖 ≠0 .
Shared randomness: 𝑖 1 , 𝑖 2 , 𝑖 3 , … uniform over [𝑛]
Alice → Bob: smallest index 𝑗 s.t. 𝑥 𝑖 𝑗 ≠0.
Bob: Accept if 𝑦 𝑖 𝑗 =1
Expect 𝑗≈ 2 𝑘 ; 𝑐𝑐≤𝑘.
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

18. UofT: ISR in Communication
Lower Bound Idea
Catch: ∀ Dist., ∃ Det. Protocol w. comm. ≤𝑘.
Need to fix strategy first, construct dist. later.
Main Idea:
Protocol can look like G-Sparse-Inner-Product
But implies players can agree on common index to focus on …
Agreement is hard
Protocol can ignore sparsity
Requires 2 𝑘 bits
Protocol can look like anything!
Invariance Principle [MOO, Mossel … ] 𝑥
11/16/2016
UofT: ISR in Communication

19. Invariance Principle [MOO’08]
Informally:
Analysis of prob. processes with bits often hard.
Analysis of related processes with Gaussian variables easy.
“Invariance Principle” – under sufficiently general conditions … prob. of events “invariant” when switching from bits to Gaussian
Theorem: For every convex 𝐾 1 , 𝐾 2 ⊆ −1,1 ℓ
∃ transformations 𝑇 1 , 𝑇 2 s.t.
if 𝑓: 0,1 𝑛 → 𝐾 1 and 𝑔: 0,1 𝑛 → 𝐾 2
have no common influential variable, then
𝐹= 𝑇 1 𝑓: ℝ 𝑛 → 𝐾 1 and 𝐺= 𝑇 2 𝑔: ℝ 𝑛 → 𝐾 2 satisfy
Exp 𝑥,𝑦 〈𝑓 𝑥 ,𝑔 𝑦 〉 ≈ Exp 𝑋,𝑌 〈𝐹 𝑋 ,𝐺 𝑌 〉
11/16/2016
UofT: ISR in Communication

20. UofT: ISR in Communication
Summarizing
𝑘 bits of comm. with perfect sharing
→ 2 𝑘 bits with imperfect sharing.
This is tight
Invariance principle for communication
Agreement distillation
Low-influence strategies
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

21. UofT: ISR in Communication
Conclusions
Imperfect agreement of context important.
Dealing with new layer of uncertainty.
Notion of scale (context LARGE)
Many open directions+questions:
Imperfectly shared randomness:
One-sided error?
Does interaction ever help?
How much randomness?
More general forms of correlation?
11/16/2016
UofT: ISR in Communication

22. UofT: ISR in Communication
Thank You!
11/16/2016
UofT: ISR in Communication

23. Towards a lower bound: Ruling out a natural approach
Alice and Bob use (many) correlated bits to agree perfectly on few random bits?
For G-Sparse-IP need 𝑂( 2 𝑘 log 𝑛) random bits.
Agreement Distillation Problem:
Alice & Bob exchange 𝑡 bits; generate 𝑘 random bits, with agreement probability 𝛾.
Lower bound [Bogdanov, Mossel]:
𝑡≥𝑘 −𝑂 log 1 𝛾
11/16/2016
UofT: ISR in Communication

24. UofT: ISR in Communication
Towards Lower Bound
Explaining two natural protocols:
Gaussian Inner Product Protocol:
Ignore sparsity and just estimate inner product.
Uses ~ 2 2𝑘 bits. Need to prove it can’t be improved!
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

25. Optimality of Gaussian Protocol
Problem:
𝑥,𝑦 ← 𝜇 𝑛 : 𝜇= 𝜇 𝑌𝐸𝑆 𝑜𝑟 𝜇 𝑁𝑂 supported on ℝ×ℝ
𝜇 𝑌𝐸𝑆 :𝜖-correlated Gaussians
𝜇 𝑁𝑂 : uncorrelated Gaussians
Lemma: Separating 𝜇 𝑌𝐸𝑆 𝑛 𝑣𝑠. 𝜇 𝑁𝑂 𝑛 requires Ω 𝜖 −1 bits of communication.
Proof: Reduction from Disjointness
Conclusion: Can’t ignore sparsity!
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

26. UofT: ISR in Communication
Towards Lower Bound
Explaining two natural protocols:
Gaussian Inner Product Protocol:
Ignore sparsity and just estimate inner product.
Uses ~ 2 2𝑘 bits. Need to prove it can’t be improved!
Protocol with perfectly shared randomness:
Alice & Bob agree on coordinates to focus on:
( 𝑖 1 , 𝑖 2 , …, 𝑖 2 𝑘 , …);
Either 𝑖 1 has high entropy (over choice of 𝑟,𝑠)
Violates agreement distillation bound
Or has low-entropy:
Fix distributions of 𝑥,𝑦 s.t. 𝑥 𝑖 1 ⊥ 𝑦 𝑖 1
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

27. Aside: Distributional lower bounds
Challenge:
Usual CC lower bounds are distributional.
𝑐𝑐(G-Sparse-IP) ≤𝑘, ∀ inputs.
⇒𝑐𝑐(G-Sparse-IP) ≤𝑘 ∀ distributions.
⇒ det-cc (G-Sparse-IP) ≤𝑘 ∀ distributions.
So usual approach can’t work …
Need to fix strategy first and then “identify” a hard distribution for the strategy …
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

28. UofT: ISR in Communication
Towards lower bound
Summary so far:
Symmetric strategy ⇒ 2 𝑘 bits of comm.
Strategy asymmetric; 𝑥 1 , 𝑦 1 … 𝑥 𝑘 , 𝑦 𝑘 have high influence ⇒ fix the distribution so these coordinates do not influence answer.
Strategy asymmetric; with random coordinate having high influence ⇒ violates agreement lower bound.
Are these exhaustive? How to prove this?
Invariance Principle!!
[Mossel, O’Donnell, Oleskiewisz], [Mossel] …
11/16/2016
UofT: ISR in Communication

29. ISR lower bound for GSIP.
One-way setting (for now)
Strategies: Alice 𝑓 𝑟 𝑥 ∈ 𝐾 ; Bob 𝑔 𝑠 𝑦 ∈ 0,1 𝐾 ;
Distributions:
If 𝑥 𝑖 , 𝑦 𝑖 have high influence on (𝑓 𝑟 , 𝑔 𝑠 ) w.h.p. over 𝑟,𝑠 then set 𝑥 𝑖 = 𝑦 𝑖 =0. [𝑖 is BAD]
Else 𝑦 𝑖 correlated with 𝑥 𝑖 in YES case, and independent in NO case.
Analysis:
𝑖∈𝐵𝐴𝐷 influential in both 𝑓 𝑟 , 𝑔 𝑠 ⇒ No help.
𝑖∉𝐵𝐴𝐷 influential … ⇒ violates agreement lower bound.
No common influential variable
⇒ 𝑥,𝑦 can be replaced by Gaussians
⇒ 2 𝑘 bits needed.
G-Sparse-IP:
𝒙, 𝒚 ∈ 𝟎,𝟏 𝒏 ;𝒘𝒕 𝒙 ≈ 𝟐 −𝒌 ⋅𝒏
Decide 𝒙,𝒚 ≥(.𝟗) 𝟐 −𝒌 ⋅𝒏
or 𝒙,𝒚 ≤(.𝟔) 𝟐 −𝒌 ⋅𝒏?
11/16/2016
UofT: ISR in Communication

30. Invariance Principle + Challenges
Informal Invariance Principle: 𝑓,𝑔 low-degree polynomials with no common influential variable ⇒ Exp 𝑥,𝑦 𝑓 𝑥 𝑔 𝑦 ≈ Exp 𝑋,𝑌 [𝑓(𝑋)𝑔(𝑌)] (caveat 𝑓≈𝑓;𝑔≈𝑔)
where 𝑥,𝑦 Boolean 𝑛-wise product dist.
and 𝑋,𝑌 Gaussian 𝑛-wise product dist
Challenges [+ Solutions]:
Our functions not low-degree [Smoothening]
Our functions not real-valued
𝑔: 0,1 𝑛 → 0,1 ℓ : [Truncate range to 0,1 ℓ ]
𝑓: 0,1 𝑛 → ℓ : [???, [work with Δ ℓ ]]
11/16/2016
UofT: ISR in Communication

31. Invariance Principle + Challenges
Informal Invariance Principle: 𝑓,𝑔 low-degree polynomials with no common influential variable ⇒ Exp 𝑥,𝑦 𝑓 𝑥 𝑔 𝑦 ≈ Exp 𝑋,𝑌 [𝑓(𝑋)𝑔(𝑌)] (caveat 𝑓≈𝑓;𝑔≈𝑔)
Challenges
Our functions not low-degree [Smoothening]
Our functions not real-valued [Truncate]
Quantity of interest is not 𝑓 𝑥 ⋅𝑔 𝑦 …
[Can express quantity of interest as inner product. ]
… (lots of grunge work …)
Get a relevant invariance principle (next)
11/16/2016
UofT: ISR in Communication

32. Invariance Principle for CC
Theorem: For every convex 𝐾 1 , 𝐾 2 ⊆ −1,1 ℓ
∃ transformations 𝑇 1 , 𝑇 2 s.t.
if 𝑓: 0,1 𝑛 → 𝐾 1 and 𝑔: 0,1 𝑛 → 𝐾 2
have no common influential variable, then
𝐹= 𝑇 1 𝑓: ℝ 𝑛 → 𝐾 1 and 𝐺= 𝑇 2 𝑔: ℝ 𝑛 → 𝐾 2 satisfy
Exp 𝑥,𝑦 〈𝑓 𝑥 ,𝑔 𝑦 〉 ≈ Exp 𝑋,𝑌 〈𝐹 𝑋 ,𝐺 𝑌 〉
Main differences: 𝑓,𝑔 vector-valued.
Functions are transformed: 𝑓↦𝐹;𝑔↦𝐺
Range preserved exactly ( 𝐾 1 =Δ ℓ ; 𝐾 2 = 0,1 ℓ )!
So 𝐹,𝐺 are still communication strategies!
11/16/2016
UofT: ISR in Communication