Identity Testing for constant-width, and commutative, ROABPs Rohit - - PowerPoint PPT Presentation

Identity Testing for constant-width, and commutative, ROABPs

Rohit Gurjar∗, Arpita Korwar, Nitin Saxena†

Aalen University and IIT Kanpur

June 1, 2016

∗supported by TCS research fellowship †supported by DST-SERB Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 1 / 26

Introduction

Polynomial Identity Testing

PIT: given a polynomial P(x) ∈ F[x1, x2, . . . , xn], P(x) = 0?

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 2 / 26

Introduction

Polynomial Identity Testing

PIT: given a polynomial P(x) ∈ F[x1, x2, . . . , xn], P(x) = 0? Input Models:

Arithmetic Circuits Arithmetic Branching Programs

× × + x2 − 2xy x y −2 x2 −2xy

Figure : An Arithmetic circuit

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 2 / 26

Introduction

Randomized Test

Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 3 / 26

Introduction

Randomized Test

Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 3 / 26

Introduction

Randomized Test

Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Two Paradigms:

Whitebox: one can see the input circuit. Blackbox: circuit is hidden, only evaluations are allowed (hitting-sets).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 3 / 26

Introduction

Randomized Test

Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Two Paradigms:

Whitebox: one can see the input circuit. Blackbox: circuit is hidden, only evaluations are allowed (hitting-sets).

Derandomizing PIT has connections with circuit lower bounds [Kabanets and Impagliazzo, 2003, Agrawal, 2005].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 3 / 26

Introduction

Randomized Test

Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Two Paradigms:

Whitebox: one can see the input circuit. Blackbox: circuit is hidden, only evaluations are allowed (hitting-sets).

Derandomizing PIT has connections with circuit lower bounds [Kabanets and Impagliazzo, 2003, Agrawal, 2005]. An efficient test is known only for restricted classes of circuits, e.g., Sparse polynomials, set-multilinear circuits, ROABP.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 3 / 26

Preliminaries

Arithmetic Branching Programs

−1 x2 s t x1 + x2 5 x2 x1 + 2x4 x1

Figure : An Arithmetic branching program.

ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 4 / 26

Preliminaries

Arithmetic Branching Programs

−1 x2 s t x1 + x2 5 x2 x1 + 2x4 x1

Figure : An Arithmetic branching program.

ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. C(x) =

• p∈paths(s,t)

W (p), where W (p) =

• e∈p

W (e).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 4 / 26

Preliminaries

Arithmetic Branching Programs

−1 x2 s t x1 + x2 5 x2 x1 + 2x4 x1

Figure : An Arithmetic branching program.

ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. C(x) =

• p∈paths(s,t)

W (p), where W (p) =

• e∈p

W (e). C(x) = (x1 + 2x4)x2x1 − (x1 + 2x4)x2 + (x1 + x2)5x2

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 4 / 26

Preliminaries

Arithmetic Branching Programs

−1 x2 s t x1 + x2 5 x2 x1 + 2x4 x1

Figure : An Arithmetic branching program.

ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. C(x) =

• p∈paths(s,t)

W (p), where W (p) =

• e∈p

W (e). C(x) = (x1 + 2x4)x2x1 − (x1 + 2x4)x2 + (x1 + x2)5x2 Width: maximum number of nodes in a layer.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 4 / 26

Preliminaries

Arithmetic Branching Programs

−1 x2 s t x1 + x2 5 x2 x1 + 2x4 x1

Figure : An Arithmetic branching program.

Equivalent representation:

• x1 + 2x4

x1 + x2 x2 −1 5 x1 x2

• C(x) = (x1 + 2x4)x2x1 − (x1 + 2x4)x2 + (x1 + x2)5x2

Width: maximum dimension of the matrices.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 5 / 26

Preliminaries

Power of ABPs

Almost as powerful as arithmetic circuits [Valiant, 1979, Berkowitz, 1984].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 6 / 26

Preliminaries

Power of ABPs

Almost as powerful as arithmetic circuits [Valiant, 1979, Berkowitz, 1984]. Width-3 ABPs have the same expressive power as arithmetic formulas [Ben-Or and Cleve, 1992].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 6 / 26

Preliminaries

Power of ABPs

Almost as powerful as arithmetic circuits [Valiant, 1979, Berkowitz, 1984]. Width-3 ABPs have the same expressive power as arithmetic formulas [Ben-Or and Cleve, 1992]. Deterministic PIT: only for special ABPs, e.g. read-once oblivious ABP.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 6 / 26

Read-once Oblivious ABP

Read-once Oblivious ABP

Any variable occurs in at most one layer.

4x3 − 3 x3 x2 x3 4 − 1 2x4 + 1 3x1 x2 1 + 1 2 1 − x3 3x4 3 x3 + 1 x2 3 + 5x3 1 − x2 x2 + 3x2 2 x4 2x3 1 + 3

Figure : A Read-once oblivious ABP with variable order (x1, x3, x2, x4)

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 7 / 26

Read-once Oblivious ABP

PIT for ROABPs

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 8 / 26

Read-once Oblivious ABP

PIT for ROABPs

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Blackbox test: nO(log n) time [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2015].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 8 / 26

Read-once Oblivious ABP

PIT for ROABPs

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Blackbox test: nO(log n) time [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2015]. Nothing better known even for constant width.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 8 / 26

Read-once Oblivious ABP

Our Results

1 Polynomial time blackbox test for constant width ROABPs*. Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 9 / 26

Read-once Oblivious ABP

Our Results

1 Polynomial time blackbox test for constant width ROABPs*.

* known variable order. * zero characteristic field (or large enough).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 9 / 26

Read-once Oblivious ABP

Our Results

1 Polynomial time blackbox test for constant width ROABPs*.

* known variable order. * zero characteristic field (or large enough).

2 Commutative ROABP: where matrices commute (no variable order). Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 9 / 26

Read-once Oblivious ABP

Our Results

1 Polynomial time blackbox test for constant width ROABPs*.

* known variable order. * zero characteristic field (or large enough).

2 Commutative ROABP: where matrices commute (no variable order).

dO(log w)(nw)O(log log w)-time blackbox test [Forbes et al., 2014]

– for n variables, width w and individual degree d.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 9 / 26

Read-once Oblivious ABP

Our Results

1 Polynomial time blackbox test for constant width ROABPs*.

* known variable order. * zero characteristic field (or large enough).

2 Commutative ROABP: where matrices commute (no variable order).

dO(log w)(nw)O(log log w)-time blackbox test [Forbes et al., 2014]

– for n variables, width w and individual degree d.

We improve it to (dnw)O(log log w)-time.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 9 / 26

Read-once Ordered Branching Programs

Read-once Ordered Branching Programs

x1 = 0 x1 = 1 x2 = 0 x2 = 1 x2 = 1 x3 = 0 x3 = 1 x3 = 1 x4 = 0 x4 = 1 x2 = 0 x4 = 0 x3 = 1 x3 = 0

Figure : An ROBP

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 10 / 26

Read-once Ordered Branching Programs

Pseudorandomness for ROBP

Comes from the RL versus L question.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 11 / 26

Read-once Ordered Branching Programs

Pseudorandomness for ROBP

Comes from the RL versus L question. A distribution is pseudorandom if any ROBP cannot distinguish it from the uniform random distribution.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 11 / 26

Read-once Ordered Branching Programs

Pseudorandomness for ROBP

Comes from the RL versus L question. A distribution is pseudorandom if any ROBP cannot distinguish it from the uniform random distribution. Goal: construct a PRG with O(log n) seed length (polynomial size sample space).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 11 / 26

Read-once Ordered Branching Programs

Pseudorandomness for ROBP

Comes from the RL versus L question. A distribution is pseudorandom if any ROBP cannot distinguish it from the uniform random distribution. Goal: construct a PRG with O(log n) seed length (polynomial size sample space). Best known result: O(log2 n) seed length [Nisan, 1990, Impagliazzo et al., 1994, Raz and Reingold, 1999].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 11 / 26

Read-once Ordered Branching Programs

Pseudorandomness for ROBP

Comes from the RL versus L question. A distribution is pseudorandom if any ROBP cannot distinguish it from the uniform random distribution. Goal: construct a PRG with O(log n) seed length (polynomial size sample space). Best known result: O(log2 n) seed length [Nisan, 1990, Impagliazzo et al., 1994, Raz and Reingold, 1999]. Nothing better known even for constant width.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 11 / 26

Read-once Ordered Branching Programs

[Impagliazzo et al., 1994]

PRG r bits r bits r + O(log w) bits

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 12 / 26

Read-once Ordered Branching Programs

[Impagliazzo et al., 1994]

PRG r bits r bits r + O(log w) bits

Sample space size: poly(w) × 2r instead of trivial 2r × 2r.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 12 / 26

Hitting-set for ROABP

Hitting-set for Bivariate ROABP

g1(x1) g2(x1) h2(x2) hw (x2) gw (x1) h1(x2) . . .

f (x1, x2) =

w

• r=1

gr(x1) hr(x2)

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 13 / 26

Hitting-set for ROABP

Hitting-set for Bivariate ROABP

HSG g1(x1) g2(x1) h2(x2) hw (x2) gw (x1) h1(x2) . . .

tw tw + tw−1 t Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 14 / 26

Hitting-set for ROABP

Hitting-set for Bivariate ROABP

HSG g1(x1) g2(x1) h2(x2) hw (x2) gw (x1) h1(x2) . . .

tw tw + tw−1 t

f (x1, x2) = w

r=1 gr(x1) hr(x2)

Claim: f (tw, tw + tw−1) = 0.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 14 / 26

Hitting-set for ROABP

Hitting-set for Bivariate ROABP

HSG g1(x1) g2(x1) h2(x2) hw (x2) gw (x1) h1(x2) . . .

tw tw + tw−1 t

f (x1, x2) = w

r=1 gr(x1) hr(x2)

Claim: f (tw, tw + tw−1) = 0. Degree= 2wd, where deg(gr), deg(hr) = d.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 14 / 26

Hitting-set for ROABP

Hitting-set for Bivariate ROABP

HSG g1(x1) g2(x1) h2(x2) hw (x2) gw (x1) h1(x2) . . .

tw tw + tw−1 t

f (x1, x2) = w

r=1 gr(x1) hr(x2)

Claim: f (tw, tw + tw−1) = 0. Degree= 2wd, where deg(gr), deg(hr) = d. Hitting-set size: 2wd + 1, instead of trivial (d + 1) × (d + 1).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 14 / 26

Hitting-set for ROABP

n-variate ROABP

f =

• x1
• 

 x2     x3   · · ·   xn−1    xn  

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 15 / 26

Hitting-set for ROABP

n-variate ROABP

f =

• x1
• 

 x2     x3   · · ·   xn−1    xn   Claim: f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 15 / 26

Hitting-set for ROABP

n-variate ROABP

f =

• x1
• 

 x2     x3   · · ·   xn−1    xn   Claim: f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. Proof: treat x3, x4, . . . , xn as constants.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 15 / 26

Hitting-set for ROABP

n-variate ROABP

f =

• x1
• 

 x2     x3   · · ·   xn−1    xn   Claim: f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. Proof: treat x3, x4, . . . , xn as constants. f =

• x1
• 

x2   f =

w

• r=1

gr(x1) hr(x2, x3, . . . , xn)

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 15 / 26

Hitting-set for ROABP

n-variate ROABP

f =

• x1
• 

 x2     x3   · · ·   xn−1    xn   Claim: f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. Proof: treat x3, x4, . . . , xn as constants. f =

• x1
• 

x2   f =

w

• r=1

gr(x1) hr(x2, x3, . . . , xn) f (tw

1 , tw 1 + tw−1 1

) = 0 (bivariate ROABP).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 15 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t1     t2   · · ·   tn/2    tn/2  

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t2   · · ·  tn/2  

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t2   · · ·  tn/2  

• no. of variables = n → n/2, individual degree = d → 2wd.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t2   · · ·  tn/2  

• no. of variables = n → n/2, individual degree = d → 2wd.

Repeat log n times. 1 variable, indvidual degree = (2w)log nd.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t2   · · ·  tn/2  

• no. of variables = n → n/2, individual degree = d → 2wd.

Repeat log n times. 1 variable, indvidual degree = (2w)log nd. Hitting-set size: O(ndwlog n).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t2   · · ·  tn/2  

• no. of variables = n → n/2, individual degree = d → 2wd.

Repeat log n times. 1 variable, indvidual degree = (2w)log nd. Hitting-set size: O(ndwlog n). Hitting-set size: poly(n, d), if w is constant.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

n-variate ROABP

f (tw

1 , tw 1 + tw−1 1

, x3, x4, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , xn) = 0. f (tw

1 , tw 1 + tw−1 1

, tw

2 , tw 2 + tw−1 2

, . . . , tw

n/2, tw n/2 + tw−1 n/2 ) = 0.

f ′ =

• t1
• 

 t2   · · ·  tn/2  

• no. of variables = n → n/2, individual degree = d → 2wd.

Repeat log n times. 1 variable, indvidual degree = (2w)log nd. Hitting-set size: O(ndwlog n). Hitting-set size: poly(n, d), if w is constant. Known variable order.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 16 / 26

Hitting-set for ROABP

Proof of the bivariate case

Claim: If f (x, y) = w

r=1 gr(x)hr(y), then f (tw, tw + tw−1) = 0.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 17 / 26

Hitting-set for ROABP

Proof of the bivariate case

Claim: If f (x, y) = w

r=1 gr(x)hr(y), then f (tw, tw + tw−1) = 0.

Coefficient Matrix for f (x, y) [Nisan, 1991] y0 · · · yj · · · yd x0 . . . xi . . . xd       | − coeff (xiyj)      

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 17 / 26

Hitting-set for ROABP

Proof of the bivariate case

Claim: If f (x, y) = w

r=1 gr(x)hr(y), then f (tw, tw + tw−1) = 0.

Coefficient Matrix for f (x, y) [Nisan, 1991] y0 · · · yj · · · yd x0 . . . xi . . . xd       | − coeff (xiyj)       Define rank(f ) as the rank of this matrix.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 17 / 26

Hitting-set for ROABP

Proof of the bivariate case

Claim: If f (x, y) = w

r=1 gr(x)hr(y), then f (tw, tw + tw−1) = 0.

Coefficient Matrix for f (x, y) [Nisan, 1991] y0 · · · yj · · · yd x0 . . . xi . . . xd       | − coeff (xiyj)       Define rank(f ) as the rank of this matrix. Claim: rank(f ) ≤ w [Nisan, 1991].

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 17 / 26

Hitting-set for ROABP

Proof of the bivariate case

Define fr = gr(x)hr(y). Claim: rank(fr) ≤ 1.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 18 / 26

Hitting-set for ROABP

Proof of the bivariate case

Define fr = gr(x)hr(y). Claim: rank(fr) ≤ 1. Let gr = a0x0 + a1x1 + · · · + adxd and hr = b0y0 + b1y1 + · · · + bdyd. y0 y1 · · · yd x0 x1 . . . xd      a0b0 a0b1 a0bd a1b0 a1b1 a1bd . . . . . . · · · . . . adb0 adb1 adbd     

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 18 / 26

Hitting-set for ROABP

Proof of the bivariate case

Define fr = gr(x)hr(y). Claim: rank(fr) ≤ 1. Let gr = a0x0 + a1x1 + · · · + adxd and hr = b0y0 + b1y1 + · · · + bdyd. y0 y1 · · · yd x0 x1 . . . xd      a0b0 a0b1 a0bd a1b0 a1b1 a1bd . . . . . . · · · . . . adb0 adb1 adbd      = ⇒ rank(f ) = rank(w

r=1 fr) ≤ w.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 18 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw + tw−1) = (tw, tw(1 + t−1)).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 19 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw + tw−1) = (tw, tw(1 + t−1)). xiyj → t(i+j)w(1 + t−1)j.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 19 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw + tw−1) = (tw, tw(1 + t−1)). xiyj → t(i+j)w(1 + t−1)j. leading-term(xiyj) = t(i+j)w.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 19 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw + tw−1) = (tw, tw(1 + t−1)). xiyj → t(i+j)w(1 + t−1)j. leading-term(xiyj) = t(i+j)w. Same for all xiyj with i + j = ℓ.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 19 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw + tw−1) = (tw, tw(1 + t−1)). xiyj → t(i+j)w(1 + t−1)j. leading-term(xiyj) = t(i+j)w. Same for all xiyj with i + j = ℓ.

i + j = ℓ − 1 y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 19 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗

Leading nonzero Diagonal: at most w nonzero entries.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗

Leading nonzero Diagonal: at most w nonzero entries. Leading term: twℓ.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗ i + j = ℓ − 1

Leading nonzero Diagonal: at most w nonzero entries. Leading term: twℓ. Leading term from the next diagonal: tw(ℓ−1).

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗

Leading nonzero Diagonal: at most w nonzero entries. Leading term: twℓ. Leading term from the next diagonal: tw(ℓ−1). Focus on terms {twℓ, twℓ−1, · · · , tw(ℓ−1)+1}.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗

Leading nonzero Diagonal: at most w nonzero entries. Leading term: twℓ. Leading term from the next diagonal: tw(ℓ−1). Focus on terms {twℓ, twℓ−1, · · · , tw(ℓ−1)+1}. They come only from an ℓ-th diagonal monomial.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

y0 y1 x0 x1 xd · · · . . . yd i + j = ℓ ∗ ∗ . . . ∗

Leading nonzero Diagonal: at most w nonzero entries. Leading term: twℓ. Leading term from the next diagonal: tw(ℓ−1). Focus on terms {twℓ, twℓ−1, · · · , tw(ℓ−1)+1}. They come only from an ℓ-th diagonal monomial. ℓ-th diagonal nonzero monomials: {xℓ−j1yj1, xℓ−j2yj2, · · · , xℓ−jw yjw }.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw(1 + t−1)). xℓ−j1yj1 → tℓw(1 + t−1)j1.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 21 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw(1 + t−1)). xℓ−j1yj1 → tℓw(1 + t−1)j1. xℓ−j1yj1 → tℓw j1

• +

j1 1

• t−1 + · · · +

j1 j1

• t−j1
• .

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 21 / 26

Hitting-set for ROABP

Proof of the bivariate case

(x, y) → (tw, tw(1 + t−1)). xℓ−j1yj1 → tℓw(1 + t−1)j1. xℓ−j1yj1 → tℓw j1

• +

j1 1

• t−1 + · · · +

j1 j1

• t−j1
• .

xℓ−j1yj1 → j1

• tℓw +

j1 1

• tℓw−1 + · · · +
• j1

w − 1

• t(ℓ−1)w+1 + · · ·

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 21 / 26

Hitting-set for ROABP

Proof of the bivariate case

xℓ−j1yj1 → j1

• tℓw +

j1

1

• tℓw−1 + · · · +

j1

w−1

• t(ℓ−1)w+1 + · · ·

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26

Hitting-set for ROABP

Proof of the bivariate case

xℓ−j1yj1 → j1

• tℓw +

j1

1

• tℓw−1 + · · · +

j1

w−1

• t(ℓ−1)w+1 + · · ·

xℓ−j2yj2 → j2

• tℓw +

j2

1

• tℓw−1 + · · · +

j2

w−1

• t(ℓ−1)w+1 + · · ·

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26

Hitting-set for ROABP

Proof of the bivariate case

xℓ−j1yj1 → j1

• tℓw +

j1

1

• tℓw−1 + · · · +

j1

w−1

• t(ℓ−1)w+1 + · · ·

xℓ−j2yj2 → j2

• tℓw +

j2

1

• tℓw−1 + · · · +

j2

w−1

• t(ℓ−1)w+1 + · · ·

. . . xℓ−jw yjw → jw

• tℓw +

jw

1

• tℓw−1 + · · · +

jw

w−1

• t(ℓ−1)w+1 + · · ·

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26

Hitting-set for ROABP

Proof of the bivariate case

xℓ−j1yj1 → j1

• tℓw +

j1

1

• tℓw−1 + · · · +

j1

w−1

• t(ℓ−1)w+1 + · · ·

xℓ−j2yj2 → j2

• tℓw +

j2

1

• tℓw−1 + · · · +

j2

w−1

• t(ℓ−1)w+1 + · · ·

. . . xℓ−jw yjw → jw

• tℓw +

jw

1

• tℓw−1 + · · · +

jw

w−1

• t(ℓ−1)w+1 + · · ·

∗ · · ·

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26

Hitting-set for ROABP

Proof of the bivariate case

xℓ−j1yj1 → j1

• tℓw +

j1

1

• tℓw−1 + · · · +

j1

w−1

• t(ℓ−1)w+1 + · · ·

xℓ−j2yj2 → j2

• tℓw +

j2

1

• tℓw−1 + · · · +

j2

w−1

• t(ℓ−1)w+1 + · · ·

. . . xℓ−jw yjw → jw

• tℓw +

jw

1

• tℓw−1 + · · · +

jw

w−1

• t(ℓ−1)w+1 + · · ·

∗ · · · Assuming jk = jk′ requires nonzero characteristic.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26

Conclusion

Discussion

Possible improvements:

Unknown variable order Hitting-set for all fields. Poly-time for arbitrary width.

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 23 / 26

Conclusion

Discussion

Possible improvements:

Unknown variable order Hitting-set for all fields. Poly-time for arbitrary width.

Connections between arithmetic and boolean pseudorandomness?

Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 23 / 26

Bibliography Agrawal, M. (2005). Proving lower bounds via pseudo-random generators. In FSTTCS, volume 3821 of Lecture Notes in Computer Science, pages 92–105. Agrawal, M., Gurjar, R., Korwar, A., and Saxena, N. (2015). Hitting-sets for ROABP and sum of set-multilinear circuits. SIAM J. Comput., 44(3):669–697. Ben-Or, M. and Cleve, R. (1992). Computing algebraic formulas using a constant number of registers. SIAM J. Comput., 21(1):54–58. Berkowitz, S. J. (1984). On computing the determinant in small parallel time using a small number of processors. Information Processing Letters, 18(3):147 – 150. Demillo, R. A. and Lipton, R. J. (1978). A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193 – 195. Forbes, M. A., Saptharishi, R., and Shpilka, A. (2014). Hitting sets for multilinear read-once algebraic branching programs, in any order. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 867–875. Forbes, M. A. and Shpilka, A. (2013). Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In FOCS, pages 243–252. Impagliazzo, R., Nisan, N., and Wigderson, A. (1994). Pseudorandomness for network algorithms. In Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, STOC, pages 356–364, New York, NY, USA. ACM. Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 24 / 26

Bibliography Kabanets, V. and Impagliazzo, R. (2003). Derandomizing polynomial identity tests means proving circuit lower bounds. STOC, pages 355–364. Nisan, N. (1990). Pseudorandom generators for space-bounded computations. In Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, STOC ’90, pages 204–212, New York, NY, USA. ACM. Nisan, N. (1991). Lower bounds for non-commutative computation (extended abstract). In Proceedings of the 23rd ACM Symposium on Theory of Computing, ACM Press, pages 410–418. Raz, R. and Reingold, O. (1999). On recycling the randomness of states in space bounded computation. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1-4, 1999, Atlanta, Georgia, USA, pages 159–168. Raz, R. and Shpilka, A. (2005). Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1–19. Schwartz, J. T. (1980). Fast probabilistic algorithms for verification of polynomial identities.

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