Testing Linearity against Non-Signaling Strategies Alessandro Chiesa - - PowerPoint PPT Presentation

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Testing Linearity against Non-Signaling Strategies Alessandro Chiesa - - PowerPoint PPT Presentation

Feb 09, 2023 672 likes •2.31k views

Testing Linearity against Non-Signaling Strategies Alessandro Chiesa Peter Manohar Igor Shinkar UC Berkeley What is linearity testing? 1 Linearity Testing 2 Linearity Testing Given oracle access to f:{0,1} n {0,1} decide if: (1) f is

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SLIDE 1

Testing Linearity against Non-Signaling Strategies

Alessandro Chiesa Peter Manohar Igor Shinkar

UC Berkeley

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SLIDE 2

What is linearity testing?

1

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SLIDE 3

Linearity Testing

2

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SLIDE 4

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions.

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SLIDE 5

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

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SLIDE 6

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

f:{0,1}n→{0,1} Verifier

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SLIDE 7

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

f:{0,1}n→{0,1}

x,y ← {0,1}n

Verifier

slide-8
SLIDE 8

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

f:{0,1}n→{0,1}

x y x+y

x,y ← {0,1}n

Verifier

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SLIDE 9

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

f:{0,1}n→{0,1}

x y x+y

x,y ← {0,1}n

f(x) f(y) f(x+y)

Verifier

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SLIDE 10

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

f(x) + f(y) ?= f(x+y)

f:{0,1}n→{0,1}

x y x+y

x,y ← {0,1}n

f(x) f(y) f(x+y)

Verifier

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SLIDE 11

Linearity Testing

2

Given oracle access to f:{0,1}n → {0,1} decide if: (1) f is linear, or (2) f is far from all linear functions. A simple and natural test:

f(x) + f(y) ?= f(x+y)

f:{0,1}n→{0,1}

x y x+y

x,y ← {0,1}n

f(x) f(y) f(x+y)

The test works: Prx,y[f passes] ≥ 1 - 𝜁 → Δ(f, LIN) ≤ 𝜁

[BLR93, BCHKS96]

Verifier

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SLIDE 12

What are non-signaling strategies?

3

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SLIDE 13

Non-Signaling Strategies

4

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SLIDE 14

Non-Signaling Strategies

4

Multiplayer games:

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SLIDE 15

Non-Signaling Strategies

4

Multiplayer games:

V P1 P2 P3

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SLIDE 16

Non-Signaling Strategies

x y z

4

Multiplayer games:

V P1 P2 P3

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SLIDE 17

Non-Signaling Strategies

x y z b c a

4

Multiplayer games:

V P1 P2 P3

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SLIDE 18

Non-Signaling Strategies

x y z b c a

4

Multiplayer games:

V P1 P2 P3

accept/reject

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SLIDE 19

Non-Signaling Strategies

x y z b c a

4

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 20

Non-Signaling Strategies

Classical

x y z b c a

4

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 21

Non-Signaling Strategies

Classical

x y z b c a

4

non-communicating

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 22

Non-Signaling Strategies

Classical Communicating

x y z b c a

4

non-communicating

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 23

Non-Signaling Strategies

Classical Communicating

x y z b c a

4

any joint strategy non-communicating

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 24

Non-Signaling Strategies

Quantum Classical Communicating

x y z b c a

4

any joint strategy non-communicating shared quantum state 𝜔

Multiplayer games:

V P1 P2 P3

𝜔

Classes of players:

accept/reject

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SLIDE 25

Non-Signaling Strategies

Non-Signaling Quantum Classical Communicating

x y z b c a

4

any joint strategy non-communicating shared quantum state 𝜔

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 26

Non-Signaling Strategies

Non-Signaling Quantum Classical Communicating

x y z b c a

4

any joint strategy any joint strategy, but not allowed to signal non-communicating shared quantum state 𝜔

Multiplayer games:

V P1 P2 P3

Classes of players:

accept/reject

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SLIDE 27

Why non-signaling?

5

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SLIDE 28

Why non-signaling?

One can study nsMIPs (MIPs sound against ns strategies)

5

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SLIDE 29

Why non-signaling?

(1) applications to cryptography One can study nsMIPs (MIPs sound against ns strategies)

5

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SLIDE 30

Why non-signaling?

(1) applications to cryptography nsMIP crypto

+

One can study nsMIPs (MIPs sound against ns strategies)

5

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SLIDE 31

Why non-signaling?

(1) applications to cryptography nsMIP crypto

+

1 round delegation

  • f computation

from LWE

One can study nsMIPs (MIPs sound against ns strategies)

[KRR13]

5

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SLIDE 32

Why non-signaling?

(1) applications to cryptography nsMIP crypto

+

1 round delegation

  • f computation

from LWE

(2) applications to complexity One can study nsMIPs (MIPs sound against ns strategies)

[KRR13]

5

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SLIDE 33

Why non-signaling?

(1) applications to cryptography nsMIP crypto

+

nsMIP

1 round delegation

  • f computation

from LWE

(2) applications to complexity One can study nsMIPs (MIPs sound against ns strategies)

[KRR13]

5

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SLIDE 34

Why non-signaling?

(1) applications to cryptography nsMIP crypto

+

nsMIP

1 round delegation

  • f computation

from LWE

hardness of approximation for LPs in bounded space

[KRR14]

(2) applications to complexity One can study nsMIPs (MIPs sound against ns strategies)

[KRR13]

5

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SLIDE 35

Why non-signaling?

(1) applications to cryptography nsMIP crypto

+

nsMIP

1 round delegation

  • f computation

from LWE

hardness of approximation for LPs in bounded space

[KRR14]

(2) applications to complexity BUT: current nsMIP constructions appear sub-optimal One can study nsMIPs (MIPs sound against ns strategies)

[KRR13]

5

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SLIDE 36

State of Affairs

6

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SLIDE 37

State of Affairs

PCP Theorem: [AS98] [ALMSS98]

6

Standard MIPs

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SLIDE 38

State of Affairs

[KRR13]: PCP Theorem: [AS98] [ALMSS98]

6

Standard MIPs Non-signaling MIPs

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SLIDE 39

State of Affairs

[KRR13]: PCP Theorem: [AS98] [ALMSS98] [Ito10]:

6

Standard MIPs Non-signaling MIPs

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SLIDE 40

State of Affairs

[KRR13]: PCP Theorem: [AS98] [ALMSS98] [Ito10]:

6

Standard MIPs Non-signaling MIPs

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SLIDE 41

State of Affairs

[KRR13]: PCP Theorem: [AS98] [ALMSS98] Fundamental question: [Ito10]:

6

Standard MIPs Non-signaling MIPs Is there a nsPCP Theorem?

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SLIDE 42

State of Affairs

[KRR13]: PCP Theorem: [AS98] [ALMSS98] Fundamental question: Namely, does ? [Ito10]:

6

Standard MIPs Non-signaling MIPs Is there a nsPCP Theorem?

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SLIDE 43

8

Why non-signaling linearity testing?

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SLIDE 44

8

Why non-signaling linearity testing?

In classical setting, Property Testing Modularity/ Abstraction

+

More efficient PCPs

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SLIDE 45

8

Why non-signaling linearity testing?

In classical setting, Property Testing Modularity/ Abstraction

+

More efficient PCPs And also in the quantum setting! [IV12] [Vid13] [NV18]

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SLIDE 46

8

Why non-signaling linearity testing?

In classical setting, Property Testing Modularity/ Abstraction

+

More efficient PCPs In non-signaling setting, No property testing results are known! And also in the quantum setting! [IV12] [Vid13] [NV18]

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SLIDE 47

8

Why non-signaling linearity testing?

In classical setting, Property Testing Modularity/ Abstraction

+

More efficient PCPs We start with linearity testing In non-signaling setting, No property testing results are known! And also in the quantum setting! [IV12] [Vid13] [NV18]

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SLIDE 48

8

Why non-signaling linearity testing?

In classical setting, Property Testing Modularity/ Abstraction

+

More efficient PCPs We start with linearity testing In non-signaling setting, No property testing results are known! Best-understood case in classical setting And also in the quantum setting! [IV12] [Vid13] [NV18]

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SLIDE 49

Non-Signaling Functions

9

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SLIDE 50

Non-Signaling Functions

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

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SLIDE 51

Non-Signaling Functions

F: {0,1}n → {0,1} k-non-signaling S ⊆ {0, 1}n, |S| ≤ k

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

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SLIDE 52

Non-Signaling Functions

F: {0,1}n → {0,1} k-non-signaling S ⊆ {0, 1}n, |S| ≤ k

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

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SLIDE 53

Non-Signaling Functions

F: {0,1}n → {0,1} k-non-signaling S ⊆ {0, 1}n, |S| ≤ k

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

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SLIDE 54

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

0 1 1 1 1 0

1/3 1/2 1/6

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SLIDE 55

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

0 1 1 1 1 0

1/3 1/2 1/6

1 1

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SLIDE 56

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

0 1 1 1 1 0

1/3 1/2 1/6

1 1

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SLIDE 57

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

that satisfies the non-signaling property:

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SLIDE 58

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

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SLIDE 59

S⋂T

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

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SLIDE 60

S⋂T

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

1/3 1/2 1/6

1 1 0 1 1 1 1 0

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SLIDE 61

S⋂T

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

FS |S⋂T

1/3 1/2 1/6

1 1 1

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SLIDE 62

S⋂T

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

FS |S⋂T

1/3 1/2 1/6 1/3 1/2 1/6

1 1 1 1 0 0 1 1 1 1 1 0 0

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SLIDE 63

S⋂T

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

FS |S⋂T

1/3 1/2 1/6

FT |S⋂T

1/3 1/2 1/6

1 1 1 1 1 1

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SLIDE 64

S⋂T

≣

1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1

Non-Signaling Functions

FS S ⊆ {0, 1}n, |S| ≤ k

1 1

9

A k-non-signaling function F: {0,1}n → {0,1} is a collection of distributions {FS}S over functions f: S → {0,1}, ∀ S ⊆ {0, 1}n, |S| ≤ k

Definition:

∀ S, T ⊆ {0, 1}n, |S|, |T| ≤ k FS |S⋂T ≣ FT |S⋂T (the marginal distributions are equal)

0 1 1 1 1 0

1/3 1/2 1/6

1 1

FS

1/3 1/2 1/6

FT

1/3 1/2 1/6

that satisfies the non-signaling property:

FS |S⋂T

1/3 1/2 1/6

FT |S⋂T

1/3 1/2 1/6

1 1 1 1 1 1

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SLIDE 65

NS Linearity Testing

10

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SLIDE 66

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling.

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SLIDE 67

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Run the same test as before:

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SLIDE 68

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling

Verifier

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SLIDE 69

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling

x,y ← {0,1}n

Verifier

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SLIDE 70

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling S = {x,y,x+y}

x,y ← {0,1}n

Verifier

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SLIDE 71

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling S = {x,y,x+y} {F(x), F(y), F(x+y)}

x,y ← {0,1}n

Verifier

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SLIDE 72

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling S = {x,y,x+y} {F(x), F(y), F(x+y)}

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

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SLIDE 73

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Does the test do anything useful? Prx,y,F[F passes] = 1 → some global conclusion? Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling S = {x,y,x+y} {F(x), F(y), F(x+y)}

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

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SLIDE 74

NS Linearity Testing

10

Given oracle access to F:{0,1}n → {0,1}, k-non-signaling. Does the test do anything useful? Prx,y,F[F passes] = 1 → some global conclusion? Run the same test as before:

F: {0,1}n → {0,1} k-non-signaling S = {x,y,x+y} {F(x), F(y), F(x+y)}

How? NS functions are collections of local distributions.

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

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SLIDE 75

Let’s first understand non-signaling functions

11

slide-76
SLIDE 76

Examples of NS Functions

12

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SLIDE 77

Examples of NS Functions

12

1 1 1 1 1 1 1 1 1

  • A function:
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SLIDE 78

Examples of NS Functions

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

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SLIDE 79

Examples of NS Functions

F: {1,2,3} → {0,1}, k = 2 FS defined as:

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:
  • A more interesting example:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

slide-80
SLIDE 80

Examples of NS Functions

F: {1,2,3} → {0,1}, k = 2 FS defined as:

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:
  • A more interesting example:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

F{1,2}

1 1

1/2 1/2

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SLIDE 81

Examples of NS Functions

F: {1,2,3} → {0,1}, k = 2 FS defined as:

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:
  • A more interesting example:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

slide-82
SLIDE 82

Examples of NS Functions

F: {1,2,3} → {0,1}, k = 2 FS defined as:

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:
  • A more interesting example:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

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SLIDE 83

Examples of NS Functions

Cannot be explained by a distribution… F: {1,2,3} → {0,1}, k = 2 FS defined as:

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:
  • A more interesting example:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

slide-84
SLIDE 84

Examples of NS Functions

Cannot be explained by a distribution… F: {1,2,3} → {0,1}, k = 2 FS defined as:

12

1 1 1 1 1 1 1 1 1

  • A function:
  • A distribution over functions:
  • A more interesting example:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2 1/6 1/3

F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

But can try anyways!

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SLIDE 85

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

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SLIDE 86

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways.

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SLIDE 87

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. System of linear eqs:

slide-88
SLIDE 88

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} System of linear eqs:

slide-89
SLIDE 89

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: System of linear eqs:

slide-90
SLIDE 90

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: 1/2 = Pr[F{1,2} = (0,0)] = q(0,0,0) + q(0,0,1) System of linear eqs:

slide-91
SLIDE 91

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: 1/2 = Pr[F{1,2} = (0,0)] = q(0,0,0) + q(0,0,1) System of linear eqs: 1/2 = Pr[F{1,2} = (1,1)] = q(1,1,0) + q(1,1,1)

slide-92
SLIDE 92

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: 1/2 = Pr[F{1,2} = (0,0)] = q(0,0,0) + q(0,0,1) System of linear eqs: 1/2 = Pr[F{1,2} = (1,1)] = q(1,1,0) + q(1,1,1) 0 = Pr[F{1,2} = (0,1)] = q(0,1,0) + q(1,0,1)

slide-93
SLIDE 93

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: 1/2 = Pr[F{1,2} = (0,0)] = q(0,0,0) + q(0,0,1) System of linear eqs: 1/2 = Pr[F{1,2} = (1,1)] = q(1,1,0) + q(1,1,1) 0 = Pr[F{1,2} = (0,1)] = q(0,1,0) + q(1,0,1) 0 = Pr[F{1,2} = (1,0)] = q(1,0,0) + q(1,0,1) …

slide-94
SLIDE 94

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: 1/2 = Pr[F{1,2} = (0,0)] = q(0,0,0) + q(0,0,1) System of linear eqs: 1/2 = Pr[F{1,2} = (1,1)] = q(1,1,0) + q(1,1,1) 0 = Pr[F{1,2} = (0,1)] = q(0,1,0) + q(1,0,1) 0 = Pr[F{1,2} = (1,0)] = q(1,0,0) + q(1,0,1) … Fact: system of linear equations has a solution.

slide-95
SLIDE 95

Example cont.

F: {1,2,3} → {0,1}, k = 2 FS defined as: F{1,3}

1 1

1/2 1/2

F{2,3}

1 1

1/2 1/2

F{1,2}

1 1

1/2 1/2

13

Let’s try to write it as a distribution anyways. Variables: qf for every f: {1,2,3} → {0,1} Constraints: 1/2 = Pr[F{1,2} = (0,0)] = q(0,0,0) + q(0,0,1) System of linear eqs: 1/2 = Pr[F{1,2} = (1,1)] = q(1,1,0) + q(1,1,1) 0 = Pr[F{1,2} = (0,1)] = q(0,1,0) + q(1,0,1) 0 = Pr[F{1,2} = (1,0)] = q(1,0,0) + q(1,0,1) … Fact: system of linear equations has a solution. Solution has negative entries, but marginals on “queryable sets” are non-negative.

slide-96
SLIDE 96

Quasi-Distributions

14

slide-97
SLIDE 97

Quasi-Distributions

14

A quasi-distribution is a distribution, only “probabilities” are allowed to be negative.

slide-98
SLIDE 98

Quasi-Distributions

14

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  • 1/3

2/3 2/3

A quasi-distribution is a distribution, only “probabilities” are allowed to be negative.

slide-99
SLIDE 99

Quasi-Distributions

14

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  • 1/3

2/3 2/3

A quasi-distribution is a distribution, only “probabilities” are allowed to be negative. A quasi-distribution is k-local if every marginal on k points is a (standard) distribution.

slide-100
SLIDE 100

Quasi-Distributions

14

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  • 1/3

2/3 2/3

A quasi-distribution is a distribution, only “probabilities” are allowed to be negative. A quasi-distribution is k-local if every marginal on k points is a (standard) distribution. Observation: k-non-signaling functions k-local quasi-distributions

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SLIDE 101

Quasi-Distributions

Does the reverse direction hold?

14

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  • 1/3

2/3 2/3

A quasi-distribution is a distribution, only “probabilities” are allowed to be negative. A quasi-distribution is k-local if every marginal on k points is a (standard) distribution. Observation: k-non-signaling functions k-local quasi-distributions

slide-102
SLIDE 102

Theorem 1

15

k-non-signaling functions k-local quasi-distributions

slide-103
SLIDE 103

Theorem 1

15

k-non-signaling functions k-local quasi-distributions

Proof sketch: direction: easy

slide-104
SLIDE 104

Theorem 1

15

k-non-signaling functions k-local quasi-distributions

Proof sketch: direction: easy direction:

slide-105
SLIDE 105

Theorem 1

15

k-non-signaling functions k-local quasi-distributions

Proof sketch: direction: easy direction:

slide-106
SLIDE 106

Theorem 1

15

Fourier analysis: Quasi-dist is a function Q: funcs → ℝ.

k-non-signaling functions k-local quasi-distributions

Proof sketch: direction: easy direction:

slide-107
SLIDE 107

Theorem 1

15

Fourier analysis: Quasi-dist is a function Q: funcs → ℝ. A basis element 𝛙S for each S ⊆ {0, 1}n,

k-non-signaling functions k-local quasi-distributions

Proof sketch: direction: easy direction:

slide-108
SLIDE 108

Theorem 1

15

Fourier analysis: Quasi-dist is a function Q: funcs → ℝ. A basis element 𝛙S for each S ⊆ {0, 1}n,

k-non-signaling functions k-local quasi-distributions

Proof sketch:

which determines

direction: easy direction:

slide-109
SLIDE 109

k size

  • f S

Fourier spectrum of Q

Theorem 1

15

Fourier analysis: Quasi-dist is a function Q: funcs → ℝ. A basis element 𝛙S for each S ⊆ {0, 1}n,

k-non-signaling functions k-local quasi-distributions

Proof sketch:

which determines

direction: easy direction:

slide-110
SLIDE 110

k size

  • f S

Fourier spectrum of Q

Theorem 1

15

fixed by F

Fourier analysis: Quasi-dist is a function Q: funcs → ℝ. A basis element 𝛙S for each S ⊆ {0, 1}n,

k-non-signaling functions k-local quasi-distributions

Proof sketch:

which determines

direction: easy direction:

slide-111
SLIDE 111

k size

  • f S

Fourier spectrum of Q

Theorem 1

15

fixed by F free

Fourier analysis: Quasi-dist is a function Q: funcs → ℝ. A basis element 𝛙S for each S ⊆ {0, 1}n,

k-non-signaling functions k-local quasi-distributions

Proof sketch:

which determines

direction: easy direction:

slide-112
SLIDE 112

16

slide-113
SLIDE 113

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling

16

Verifier

slide-114
SLIDE 114

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling

16

x,y ← {0,1}n

Verifier

slide-115
SLIDE 115

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

16

x,y ← {0,1}n

Verifier

slide-116
SLIDE 116

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)}

16

x,y ← {0,1}n

Verifier

slide-117
SLIDE 117

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)}

16

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

slide-118
SLIDE 118

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)}

16

Observation: F = quasi-dist over LIN → Pr[F passes] = 1

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

slide-119
SLIDE 119

Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)}

16

Observation: F = quasi-dist over LIN → Pr[F passes] = 1 Is the converse true?

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

slide-120
SLIDE 120

Theorem 2

Pr[F passes] = 1 ⇔ F is a quasi-dist over LIN Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)}

16

Observation: F = quasi-dist over LIN → Pr[F passes] = 1 Is the converse true?

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

slide-121
SLIDE 121

Theorem 2

Proof: characterize Fourier spectrum of quasi-dists over LIN. Pr[F passes] = 1 ⇔ F is a quasi-dist over LIN Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)}

16

Observation: F = quasi-dist over LIN → Pr[F passes] = 1 Is the converse true?

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

slide-122
SLIDE 122

Theorem 2

Proof: characterize Fourier spectrum of quasi-dists over LIN. Pr[F passes] = 1 ⇔ F is a quasi-dist over LIN Recall the linearity test:

F: {0,1}n -> {0,1} k-non-signaling S = {x,y,x+y}

{F(x), F(y), F(x+y)} This is a global conclusion!

16

Observation: F = quasi-dist over LIN → Pr[F passes] = 1 Is the converse true?

F(x) + F(y) ?= F(x+y) x,y ← {0,1}n

Verifier

slide-123
SLIDE 123

Now suppose F passes the linearity test w.p. 1 - 𝜁

17

slide-124
SLIDE 124

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s

17

slide-125
SLIDE 125

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s

17

(this is what happens in the classical case)

slide-126
SLIDE 126

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true.

17

(this is what happens in the classical case)

slide-127
SLIDE 127

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true.

17

But… (this is what happens in the classical case)

slide-128
SLIDE 128

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true. it’s true without the hypothesis!

17

But… (this is what happens in the classical case)

slide-129
SLIDE 129

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true. it’s true without the hypothesis!

17

Even for !

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

But… (this is what happens in the classical case)

slide-130
SLIDE 130

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true. Let’s try again. it’s true without the hypothesis!

17

Even for !

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

But… (this is what happens in the classical case)

slide-131
SLIDE 131

Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true. Another natural conjecture: F ≈ quasi-dist over linear f’s Let’s try again. it’s true without the hypothesis!

17

Even for !

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

But… (this is what happens in the classical case)

slide-132
SLIDE 132

This is the correct answer. Now suppose F passes the linearity test w.p. 1 - 𝜁 Natural conjecture: F = quasi-dist over almost linear f’s This is true. Another natural conjecture: F ≈ quasi-dist over linear f’s Let’s try again. it’s true without the hypothesis!

17

Even for !

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

But… (this is what happens in the classical case)

slide-133
SLIDE 133

18

slide-134
SLIDE 134

Suppose F passes the linearity test w.p. 1 - 𝜁

18

slide-135
SLIDE 135

Suppose F passes the linearity test w.p. 1 - 𝜁

18

Goal: show F ≈ quasi-dist over linear f’s

slide-136
SLIDE 136

Suppose F passes the linearity test w.p. 1 - 𝜁

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS)

slide-137
SLIDE 137

Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS)

1

slide-138
SLIDE 138

Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q!

1

slide-139
SLIDE 139

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

1

slide-140
SLIDE 140

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

x

1

slide-141
SLIDE 141

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

{0,1}n wx

x

1

slide-142
SLIDE 142

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

{0,1}n wx x+wx, wx

x

1

slide-143
SLIDE 143

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

{0,1}n wx x+wx, wx F(x+wx), F(wx)

x

1

slide-144
SLIDE 144

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

{0,1}n wx x+wx, wx F(x+wx), F(wx)

x

F*(x) := F(x+wx) - F(wx)

1

slide-145
SLIDE 145

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

{0,1}n wx x+wx, wx F(x+wx), F(wx)

x

F*(x) := F(x+wx) - F(wx)

F*(x)

1

slide-146
SLIDE 146

Verifier

F Fix: define F*, a “smooth” F (self-correction) Suppose F passes the linearity test w.p. 1 - 𝜁 Bad example: F =

18

Goal: show F ≈ quasi-dist over linear f’s Δ(F, Q) = maxS ΔTV(FS, QS) Then Δ(F, Q) is large for all Q! F*

{0,1}n wx x+wx, wx F(x+wx), F(wx)

x

F*(x) := F(x+wx) - F(wx)

F*(x)

If Prx,y,F[F(x) + F(y) = F(x+y)] ≥ 1 - 𝜁 then ∀x,y, PrF*[F*(x) + F*(y) = F*(x+y)] ≥ 1 - 4𝜁

Average to worst case:

1

slide-147
SLIDE 147

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN

slide-148
SLIDE 148

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN Summary:

slide-149
SLIDE 149

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN If F passes the linearity test w.h.p, then its self-correction can be well-approximated by a quasi-dist over LIN. Summary:

slide-150
SLIDE 150

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN If F passes the linearity test w.h.p, then its self-correction can be well-approximated by a quasi-dist over LIN. Property testing is possible against non-signaling strategies. Summary:

slide-151
SLIDE 151

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN If F passes the linearity test w.h.p, then its self-correction can be well-approximated by a quasi-dist over LIN. Property testing is possible against non-signaling strategies. The above is a sample of what you can prove. Summary:

slide-152
SLIDE 152

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN If F passes the linearity test w.h.p, then its self-correction can be well-approximated by a quasi-dist over LIN. Property testing is possible against non-signaling strategies. The above is a sample of what you can prove. Quasi-distributions are essential. Summary:

slide-153
SLIDE 153

Theorem 3

19

If Pr[F passes] ≥ 1 - 𝜁 then F* is Ok(𝜁)-close to a quasi-dist over LIN If F passes the linearity test w.h.p, then its self-correction can be well-approximated by a quasi-dist over LIN. Property testing is possible against non-signaling strategies. The above is a sample of what you can prove. Quasi-distributions are essential. Not just a technique! Can’t state results without them. Summary:

slide-154
SLIDE 154

Thanks!

full version available on ECCC (TR18-067)

slide-155
SLIDE 155

Non-Signaling Players

slide-156
SLIDE 156

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

slide-157
SLIDE 157

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

V P1 P2 P3

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SLIDE 158

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

x y z

V P1 P2 P3

slide-159
SLIDE 159

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

x y z

V P1 P2 P3

(a,b,c) ← F(x,y,z)

slide-160
SLIDE 160

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

x y z b c a

V P1 P2 P3

(a,b,c) ← F(x,y,z)

slide-161
SLIDE 161

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

x y z b c a

V P1 P2 P3

accept/reject

(a,b,c) ← F(x,y,z)

slide-162
SLIDE 162

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

x y z b c a

V P1 P2 P3

accept/reject

(a,b,c) ← F(x,y,z)

Analogues of all three theorems hold for ns players

slide-163
SLIDE 163

Non-Signaling Players

A k-non-signaling player P is a collection of distributions {F(x1, …, xk)}

  • ver functions f: [k] → {0,1}, ∀ tuples (x1, …, xk)

Definition:

∀ (x1, …, xk), (y1, …, yk) where S = {i : xi = yi}, then F(x1, …, xk) |S ≣ F(y1, …, yk) |S

that satisfies the non-signaling property:

x y z b c a

V P1 P2 P3

accept/reject

(a,b,c) ← F(x,y,z)

Analogues of all three theorems hold for ns players k-non-signaling players local quasi-distributions

  • ver k-tuples of f’s

Theorem 1 for players