Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity ofXOR functions

Arkadev Chattopadhyay Nikhil Mande

TIFR, Mumbai

NMI Workshop on Complexity Theory, IIT Gandhinagar

November 04, 2016

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Outline

1 Model of Computation

2 Main Theorem and Proof Outline

3 An Upper Bound

4 Conclusions

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity [PS’86]

2 parties, Alice and Bob.

Receive inputs X ∈ X , Y ∈ Y respectively.

Wish to evaluate a function f : X × Y → 0, 1 on the pair(X,Y ) by using a communication protocol Π.

Access to unlimited amount of private randomness. RequirePr[Π(X,Y ) = f(X,Y )

]> 1/2 ∀(X,Y ) ∈ X × Y.

cost(Π) = maxr,X∈X ,Y ∈Y

# bits transmitted by Π on r,X, Y .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity [PS’86]

2 parties, Alice and Bob.

Receive inputs X ∈ X , Y ∈ Y respectively.

Wish to evaluate a function f : X × Y → 0, 1 on the pair(X,Y ) by using a communication protocol Π.

Access to unlimited amount of private randomness. RequirePr[Π(X,Y ) = f(X,Y )

]> 1/2 ∀(X,Y ) ∈ X × Y.

cost(Π) = maxr,X∈X ,Y ∈Y

# bits transmitted by Π on r,X, Y .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity [PS’86]

2 parties, Alice and Bob.

Receive inputs X ∈ X , Y ∈ Y respectively.

Wish to evaluate a function f : X × Y → 0, 1 on the pair(X,Y ) by using a communication protocol Π.

Access to unlimited amount of private randomness. RequirePr[Π(X,Y ) = f(X,Y )

]> 1/2 ∀(X,Y ) ∈ X × Y.

cost(Π) = maxr,X∈X ,Y ∈Y

# bits transmitted by Π on r,X, Y .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity [PS’86]

2 parties, Alice and Bob.

Receive inputs X ∈ X , Y ∈ Y respectively.

Wish to evaluate a function f : X × Y → 0, 1 on the pair(X,Y ) by using a communication protocol Π.

Access to unlimited amount of private randomness. RequirePr[Π(X,Y ) = f(X,Y )

]> 1/2 ∀(X,Y ) ∈ X × Y.

cost(Π) = maxr,X∈X ,Y ∈Y

# bits transmitted by Π on r,X, Y .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity [PS’86]

2 parties, Alice and Bob.

Receive inputs X ∈ X , Y ∈ Y respectively.

Wish to evaluate a function f : X × Y → 0, 1 on the pair(X,Y ) by using a communication protocol Π.

Access to unlimited amount of private randomness. RequirePr[Π(X,Y ) = f(X,Y )

]> 1/2 ∀(X,Y ) ∈ X × Y.

cost(Π) = maxr,X∈X ,Y ∈Y

# bits transmitted by Π on r,X, Y .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Unbounded Error Communication Complexity [PS’86]

2 parties, Alice and Bob.

Receive inputs X ∈ X , Y ∈ Y respectively.

Wish to evaluate a function f : X × Y → 0, 1 on the pair(X,Y ) by using a communication protocol Π.

Access to unlimited amount of private randomness. RequirePr[Π(X,Y ) = f(X,Y )

]> 1/2 ∀(X,Y ) ∈ X × Y.

cost(Π) = maxr,X∈X ,Y ∈Y

# bits transmitted by Π on r,X, Y .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

XOR functions

Definition (XOR functions)

A function F : 0, 1n × 0, 1n → −1, 1 is said to be an XORfunction if there exists a function f : 0, 1n → 0, 1 such thatfor all x1, . . . , xn, y1, . . . yn ∈ 0, 1, we haveF (x1, . . . , xn, y1, . . . , yn) = f(x1 ⊕ y1, . . . , xn ⊕ yn). We use thenotation F = f XOR.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

XOR functions

Definition (XOR functions)

A function F : 0, 1n × 0, 1n → −1, 1 is said to be an XORfunction if there exists a function f : 0, 1n → 0, 1 such thatfor all x1, . . . , xn, y1, . . . yn ∈ 0, 1, we haveF (x1, . . . , xn, y1, . . . , yn) = f(x1 ⊕ y1, . . . , xn ⊕ yn). We use thenotation F = f XOR.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

MOD functions

Definition (MOD functions)

A function f : 0, 1n → −1, 1 is called a MOD function if thereexists a positive integer m < n and an ‘accepting’ set A ⊆ [m]such that

f(x) =

−1n∑i=1

xi ≡ k mod m for some k ∈ A

1 otherwise

We write f = MODAm.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

MOD functions

Definition (MOD functions)

A function f : 0, 1n → −1, 1 is called a MOD function if thereexists a positive integer m < n and an ‘accepting’ set A ⊆ [m]such that

f(x) =

−1n∑i=1

xi ≡ k mod m for some k ∈ A

1 otherwise

We write f = MODAm.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

MOD functions

Definition (MOD functions)

A function f : 0, 1n → −1, 1 is called a MOD function if thereexists a positive integer m < n and an ‘accepting’ set A ⊆ [m]such that

f(x) =

−1n∑i=1

xi ≡ k mod m for some k ∈ A

1 otherwise

We write f = MODAm.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Boolean Fourier Analysis

Consider the vector space of functions from 0, 1n to R, equippedwith the following inner product.

< f, g >= Ex∈0,1nf(x)g(x) =1

2n

∑x∈0,1n

f(x)g(x)

Define χS(x) = (−1)∑

i∈S xi for all S ⊆ [n].

The set χS : S ⊆ [n] forms an orthonormal basis for thisvector space.

Thus, every f : 0, 1n → R can be uniquely written asf =

∑S⊆[n]

f(S)χS where

f(S) =< f, χS >= Ex∈0,1nf(x)χS(x)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Boolean Fourier Analysis

Consider the vector space of functions from 0, 1n to R, equippedwith the following inner product.

< f, g >= Ex∈0,1nf(x)g(x) =1

2n

∑x∈0,1n

f(x)g(x)

Define χS(x) = (−1)∑

i∈S xi for all S ⊆ [n].

The set χS : S ⊆ [n] forms an orthonormal basis for thisvector space.

Thus, every f : 0, 1n → R can be uniquely written asf =

∑S⊆[n]

f(S)χS where

f(S) =< f, χS >= Ex∈0,1nf(x)χS(x)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Boolean Fourier Analysis

Consider the vector space of functions from 0, 1n to R, equippedwith the following inner product.

< f, g >= Ex∈0,1nf(x)g(x) =1

2n

∑x∈0,1n

f(x)g(x)

Define χS(x) = (−1)∑

i∈S xi for all S ⊆ [n].

The set χS : S ⊆ [n] forms an orthonormal basis for thisvector space.

Thus, every f : 0, 1n → R can be uniquely written asf =

∑S⊆[n]

f(S)χS where

f(S) =< f, χS >= Ex∈0,1nf(x)χS(x)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Boolean Fourier Analysis

Consider the vector space of functions from 0, 1n to R, equippedwith the following inner product.

< f, g >= Ex∈0,1nf(x)g(x) =1

2n

∑x∈0,1n

f(x)g(x)

Define χS(x) = (−1)∑

i∈S xi for all S ⊆ [n].

The set χS : S ⊆ [n] forms an orthonormal basis for thisvector space.

Thus, every f : 0, 1n → R can be uniquely written asf =

∑S⊆[n]

f(S)χS where

f(S) =< f, χS >= Ex∈0,1nf(x)χS(x)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Boolean Fourier Analysis

Consider the vector space of functions from 0, 1n to R, equippedwith the following inner product.

< f, g >= Ex∈0,1nf(x)g(x) =1

2n

∑x∈0,1n

f(x)g(x)

Define χS(x) = (−1)∑

i∈S xi for all S ⊆ [n].

The set χS : S ⊆ [n] forms an orthonormal basis for thisvector space.

Thus, every f : 0, 1n → R can be uniquely written asf =

∑S⊆[n]

f(S)χS where

f(S) =< f, χS >= Ex∈0,1nf(x)χS(x)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Boolean Fourier Analysis

Consider the vector space of functions from 0, 1n to R, equippedwith the following inner product.

< f, g >= Ex∈0,1nf(x)g(x) =1

2n

∑x∈0,1n

f(x)g(x)

Define χS(x) = (−1)∑

i∈S xi for all S ⊆ [n].

The set χS : S ⊆ [n] forms an orthonormal basis for thisvector space.

Thus, every f : 0, 1n → R can be uniquely written asf =

∑S⊆[n]

f(S)χS where

f(S) =< f, χS >= Ex∈0,1nf(x)χS(x)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Sign Rank and Unbounded Error CommunicationComplexity

Definition (Sign Rank)

sr(M) = minArk(A) : sgn(Aij) = sgn(Mij)

We will overload notation and use sr(f) to denote sr(Mf ).

Theorem (PS’86)

UPP(f) = log sr(A)±O(1)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Sign Rank and Unbounded Error CommunicationComplexity

Definition (Sign Rank)

sr(M) = minArk(A) : sgn(Aij) = sgn(Mij)

We will overload notation and use sr(f) to denote sr(Mf ).

Theorem (PS’86)

UPP(f) = log sr(A)±O(1)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Sign Rank and Unbounded Error CommunicationComplexity

Definition (Sign Rank)

sr(M) = minArk(A) : sgn(Aij) = sgn(Mij)

We will overload notation and use sr(f) to denote sr(Mf ).

Theorem (PS’86)

UPP(f) = log sr(A)±O(1)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Sign Rank and Unbounded Error CommunicationComplexity

Definition (Sign Rank)

sr(M) = minArk(A) : sgn(Aij) = sgn(Mij)

We will overload notation and use sr(f) to denote sr(Mf ).

Theorem (PS’86)

UPP(f) = log sr(A)±O(1)

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Prior Work

Forster [For’01] showed that it suffices to show an upperbound on ||Mf || to give a sign rank lower bound for f .

Sherstov [She’07] characterizes unbounded error complexity off ∧ for symmetric f , in terms of the number of sign changesof the underlying predicate.

Zhang and Shi [ZS’09] characterize bounded error andquantum complexity of all symmetric XOR functions.

Zhang and Shi conjecture that the unbounded errorcomplexity of f XOR for symmetric f is essentially|i : Df (i) 6= Df (i+ 2)|.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Prior Work

Forster [For’01] showed that it suffices to show an upperbound on ||Mf || to give a sign rank lower bound for f .

Sherstov [She’07] characterizes unbounded error complexity off ∧ for symmetric f , in terms of the number of sign changesof the underlying predicate.

Zhang and Shi [ZS’09] characterize bounded error andquantum complexity of all symmetric XOR functions.

Zhang and Shi conjecture that the unbounded errorcomplexity of f XOR for symmetric f is essentially|i : Df (i) 6= Df (i+ 2)|.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Prior Work

Forster [For’01] showed that it suffices to show an upperbound on ||Mf || to give a sign rank lower bound for f .

Sherstov [She’07] characterizes unbounded error complexity off ∧ for symmetric f , in terms of the number of sign changesof the underlying predicate.

Zhang and Shi [ZS’09] characterize bounded error andquantum complexity of all symmetric XOR functions.

Zhang and Shi conjecture that the unbounded errorcomplexity of f XOR for symmetric f is essentially|i : Df (i) 6= Df (i+ 2)|.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Prior Work

Forster [For’01] showed that it suffices to show an upperbound on ||Mf || to give a sign rank lower bound for f .

Sherstov [She’07] characterizes unbounded error complexity off ∧ for symmetric f , in terms of the number of sign changesof the underlying predicate.

Zhang and Shi [ZS’09] characterize bounded error andquantum complexity of all symmetric XOR functions.

Zhang and Shi conjecture that the unbounded errorcomplexity of f XOR for symmetric f is essentially|i : Df (i) 6= Df (i+ 2)|.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Prior Work

Forster [For’01] showed that it suffices to show an upperbound on ||Mf || to give a sign rank lower bound for f .

Sherstov [She’07] characterizes unbounded error complexity off ∧ for symmetric f , in terms of the number of sign changesof the underlying predicate.

Zhang and Shi [ZS’09] characterize bounded error andquantum complexity of all symmetric XOR functions.

Zhang and Shi conjecture that the unbounded errorcomplexity of f XOR for symmetric f is essentially|i : Df (i) 6= Df (i+ 2)|.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Main Theorem

Theorem (Main)

For any integer m ≥ 3, express m = j2k uniquely, where j is eitherodd or 4, and k is a positive integer. Then for any non-trivial A,

UPP(MODAm XOR) ≥ Ω

(n

jm

)

Corollary

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Main Theorem

Theorem (Main)

For any integer m ≥ 3, express m = j2k uniquely, where j is eitherodd or 4, and k is a positive integer. Then for any non-trivial A,

UPP(MODAm XOR) ≥ Ω

(n

jm

)

Corollary

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Main Theorem

Theorem (Main)

For any integer m ≥ 3, express m = j2k uniquely, where j is eitherodd or 4, and k is a positive integer. Then for any non-trivial A,

UPP(MODAm XOR) ≥ Ω

(n

jm

)

Corollary

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 0

Forster’s Theorem relates the unbounded error communicationcomplexity to the spectral norm of the correspondingcommunication matrix.

Spectral norm of an XOR function’s communication matrixare just the Fourier coefficients.

Fourier analysis of MODAm functions for odd m.

Shifting and XORing to reduce the period of any periodicpredicate to 4 or an odd length period.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 0

Forster’s Theorem relates the unbounded error communicationcomplexity to the spectral norm of the correspondingcommunication matrix.

Spectral norm of an XOR function’s communication matrixare just the Fourier coefficients.

Fourier analysis of MODAm functions for odd m.

Shifting and XORing to reduce the period of any periodicpredicate to 4 or an odd length period.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 0

Forster’s Theorem relates the unbounded error communicationcomplexity to the spectral norm of the correspondingcommunication matrix.

Spectral norm of an XOR function’s communication matrixare just the Fourier coefficients.

Fourier analysis of MODAm functions for odd m.

Shifting and XORing to reduce the period of any periodicpredicate to 4 or an odd length period.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 0

Forster’s Theorem relates the unbounded error communicationcomplexity to the spectral norm of the correspondingcommunication matrix.

Spectral norm of an XOR function’s communication matrixare just the Fourier coefficients.

Fourier analysis of MODAm functions for odd m.

Shifting and XORing to reduce the period of any periodicpredicate to 4 or an odd length period.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 0

Forster’s Theorem relates the unbounded error communicationcomplexity to the spectral norm of the correspondingcommunication matrix.

Spectral norm of an XOR function’s communication matrixare just the Fourier coefficients.

Fourier analysis of MODAm functions for odd m.

Shifting and XORing to reduce the period of any periodicpredicate to 4 or an odd length period.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 1

Theorem (FKLMSS’01)

Let Mm×N be a real matrix with no 0 entries. Then,

sr(M) ≥√mN

||M ||·minx,y|M(x, y)|

Lemma

Let f : 0, 1n × 0, 1n → R be any real valued function and letM denote the communication matrix of f XOR. Then,

||M || = 2n · maxS⊆[n]

∣∣∣f(S)∣∣∣

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 1

Theorem (FKLMSS’01)

Let Mm×N be a real matrix with no 0 entries. Then,

sr(M) ≥√mN

||M ||·minx,y|M(x, y)|

Lemma

Let f : 0, 1n × 0, 1n → R be any real valued function and letM denote the communication matrix of f XOR. Then,

||M || = 2n · maxS⊆[n]

∣∣∣f(S)∣∣∣

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 1

Theorem (FKLMSS’01)

Let Mm×N be a real matrix with no 0 entries. Then,

sr(M) ≥√mN

||M ||·minx,y|M(x, y)|

Lemma

Let f : 0, 1n × 0, 1n → R be any real valued function and letM denote the communication matrix of f XOR. Then,

||M || = 2n · maxS⊆[n]

∣∣∣f(S)∣∣∣

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 2

Theorem

For odd m, and any A ⊆ 0, 1, . . . ,m− 1 which isn’t the full orempty set,∣∣∣∣MODA

m(S)

∣∣∣∣ ≤

1− 2m + 2m

(cos(π

2m

))nS = ∅

2m(cos(π

2m

))nS 6= ∅

Theorem

For any function f : 0, 1n → −1, 1, and any collection of sets

S ⊆ supp(f), if∑

S∈S

∣∣∣f(S)∣∣∣ ≤ 1− δ, and maxS/∈S

∣∣∣f(S)∣∣∣ ≤ c.

Then, sr(f XOR) ≥ δc .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 2

Theorem

For odd m, and any A ⊆ 0, 1, . . . ,m− 1 which isn’t the full orempty set,∣∣∣∣MODA

m(S)

∣∣∣∣ ≤

1− 2m + 2m

(cos(π

2m

))nS = ∅

2m(cos(π

2m

))nS 6= ∅

Theorem

For any function f : 0, 1n → −1, 1, and any collection of sets

S ⊆ supp(f), if∑

S∈S

∣∣∣f(S)∣∣∣ ≤ 1− δ, and maxS/∈S

∣∣∣f(S)∣∣∣ ≤ c.

Then, sr(f XOR) ≥ δc .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Proof Outline - 2

Theorem

For odd m, and any A ⊆ 0, 1, . . . ,m− 1 which isn’t the full orempty set,∣∣∣∣MODA

m(S)

∣∣∣∣ ≤

1− 2m + 2m

(cos(π

2m

))nS = ∅

2m(cos(π

2m

))nS 6= ∅

Theorem

For any function f : 0, 1n → −1, 1, and any collection of sets

S ⊆ supp(f), if∑

S∈S

∣∣∣f(S)∣∣∣ ≤ 1− δ, and maxS/∈S

∣∣∣f(S)∣∣∣ ≤ c.

Then, sr(f XOR) ≥ δc .

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Upper Bound

Theorem

Suppose f : −1, 1n → −1, 1 is a symmetric function definedby the predicate D : [n]→ −1, 1. Say

|i ∈ 0, 1, . . . , n− 2 : D(i) 6= D(i+ 2)| = k

Then,

UPP(f XOR) ≤ O(k log

(nk

))Proof idea:

Construct a polynomial with small number of monomials, signrepresenting f XOR.

Show that this implies an efficient unbounded error protocol.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Upper Bound

Theorem

Suppose f : −1, 1n → −1, 1 is a symmetric function definedby the predicate D : [n]→ −1, 1. Say

|i ∈ 0, 1, . . . , n− 2 : D(i) 6= D(i+ 2)| = k

Then,

UPP(f XOR) ≤ O(k log

(nk

))

Proof idea:

Construct a polynomial with small number of monomials, signrepresenting f XOR.

Show that this implies an efficient unbounded error protocol.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Upper Bound

Theorem

Suppose f : −1, 1n → −1, 1 is a symmetric function definedby the predicate D : [n]→ −1, 1. Say

|i ∈ 0, 1, . . . , n− 2 : D(i) 6= D(i+ 2)| = k

Then,

UPP(f XOR) ≤ O(k log

(nk

))Proof idea:

Construct a polynomial with small number of monomials, signrepresenting f XOR.

Show that this implies an efficient unbounded error protocol.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Upper Bound

Theorem

Suppose f : −1, 1n → −1, 1 is a symmetric function definedby the predicate D : [n]→ −1, 1. Say

|i ∈ 0, 1, . . . , n− 2 : D(i) 6= D(i+ 2)| = k

Then,

UPP(f XOR) ≤ O(k log

(nk

))Proof idea:

Construct a polynomial with small number of monomials, signrepresenting f XOR.

Show that this implies an efficient unbounded error protocol.

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Conclusions and Open Problems

Recall

Theorem

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

What if the period is more than√n?

General symmetric predicates?

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Conclusions and Open Problems

Recall

Theorem

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

What if the period is more than√n?

General symmetric predicates?

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Conclusions and Open Problems

Recall

Theorem

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

What if the period is more than√n?

General symmetric predicates?

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Conclusions and Open Problems

Recall

Theorem

If m = O(n1/2−ε), then UPP(MODAm XOR) = nΩ(1) unless f

represents a constant, or parity.

What if the period is more than√n?

General symmetric predicates?

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions

Model of ComputationMain Theorem and Proof Outline

An Upper BoundConclusions

Thank You

Arkadev Chattopadhyay Nikhil Mande Unbounded Error Communication Complexity of XOR functions