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Deterministic identity testing for sum of read-once oblivious arithmetic branching programs

Rohit Gurjar, Arpita Korwar, Nitin Saxena,

IIT Kanpur

Thomas Thierauf

Aalen University

June 18, 2015

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 1 / 22

Introduction

Polynomial Identity Testing

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

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 2 / 22

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, Thierauf PIT for sum of ROABPs June 18, 2015 2 / 22

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, Thierauf PIT for sum of ROABPs June 18, 2015 3 / 22

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, Thierauf PIT for sum of ROABPs June 18, 2015 3 / 22

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.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 3 / 22

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.

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

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 3 / 22

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.

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

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 3 / 22

Preliminaries

Arithmetic Branching Program

−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, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22

Preliminaries

Arithmetic Branching Program

−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, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22

Preliminaries

Arithmetic Branching Program

−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, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22

Preliminaries

Arithmetic Branching Program

−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, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22

Preliminaries

Read-once Oblivious ABP

Any variable occurs in at most one layer.

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

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

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 5 / 22

Preliminaries

Previous Work

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

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22

Preliminaries

Previous Work

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22

Preliminaries

Previous Work

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders)

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22

Preliminaries

Previous Work

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders)

s2 t2 s1 t1 B(x) A(x) · · · · · · · · · · · ·

Figure: Sum of two ROABPs

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22

Preliminaries

Previous Work

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders)

· · · · · · · · · · · · t s A(x) + B(x)

Figure: Sum of two ROABPs

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22

Preliminaries

Previous Work

[Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders)

· · · · · · · · · · · · t s A(x) + B(x)

Figure: Sum of two ROABPs

Sum of two ROABPs not captured by ROABP [Nair and Saha, 2014].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22

Characterizing ROABPs

Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991]

F-linear dependence of polynomials: P1 = 1 + x1 P2 = x1x2 P3 = 1 + x1 + 2x1x2 P1 + 2P2 − P3 = 0

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22

Characterizing ROABPs

Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991]

Any multilinear polynomial A(x) can be written as: A = A0 + x1A1, where A0, A1 ∈ F[x2, x3, . . . , xn].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22

Characterizing ROABPs

Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991]

Any multilinear polynomial A(x) can be written as: A = A0 + x1A1, where A0, A1 ∈ F[x2, x3, . . . , xn]. Similarly, A = A00 + x1A10 + x2A01 + x1x2A11, where A00, A10, A01, A11 ∈ F[x3, . . . , xn].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22

Characterizing ROABPs

Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991]

Any multilinear polynomial A(x) can be written as: A = A0 + x1A1, where A0, A1 ∈ F[x2, x3, . . . , xn]. Similarly, A = A00 + x1A10 + x2A01 + x1x2A11, where A00, A10, A01, A11 ∈ F[x3, . . . , xn]. A =

• e∈{0,1}i

xe1

1 xe2 2 · · · xei i Ae, where Ae ∈ F[xi+1, . . . , xn]

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22

Characterizing ROABPs

Coefficient Dimension = Minimum width [Nisan, 1991]

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) if and only if dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 8 / 22

Characterizing ROABPs

Coefficient Dimension ≤ Width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 9 / 22

Characterizing ROABPs

Coefficient Dimension ≤ Width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s · · · · · · · · · · · · xi xi+1 x1 xn . . . v1 v2 vw

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 9 / 22

Characterizing ROABPs

Coefficient Dimension ≤ Width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s · · · · · · xi . . . Q1 Q2 Qw v1 v2 vw

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 9 / 22

Characterizing ROABPs

Coefficient Dimension ≤ Width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s xi . . . Q1 Q2 Qw xi+1 xn x1 · · · · · · v1 v2 P2 P1 Pw vw

A = w

j=1 PjQj, where Pj ∈ F[x1, . . . , xi] and Qj ∈ F[xi+1, . . . , xn] for

each j.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 9 / 22

Characterizing ROABPs

Coefficient Dimension ≤ width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s . . . Q1 Q2 Qw v1 v2 x1x2 vw 2x1 + 3x2 1 + x1 + 5x1x2 A

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 10 / 22

Characterizing ROABPs

Coefficient Dimension ≤ width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s . . . Q1 Q2 Qw v1 v2 1x1x2 vw 1 + x1 + 5x1x2 A 2x1 + 3x2

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 10 / 22

Characterizing ROABPs

Coefficient Dimension ≤ width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s . . . Q1 Q2 Qw v1 v2 1 vw 5 A11

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 10 / 22

Characterizing ROABPs

Coefficient Dimension ≤ width

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) = ⇒ dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

t s . . . Q1 Q2 Qw v1 v2 vw α2 αw Ae α1

Ae =

w

• j=1

αjQj, where αj ∈ F and Qj ∈ F[xi+1, . . . , xn] for each j.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 10 / 22

Characterizing ROABPs

Coefficient Dimension = Minimum width [Nisan, 1991]

A multilinear polynomial A is computed by an ROABP of width-w and variable order (x1, x2, . . . , xn) ⇐ = dim{Ae | e ∈ {0, 1}i} ≤ w, ∀i ∈ [n].

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 11 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s 1 x1 A1 A0

A = A0 + x1A1

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s 1 x1 A00 1 1 x2 x2 A10 A01 A11

A = A00 + x1A10 + x2A01 + x1x2A11

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . Ae1 Ae2 Aew

A = P1Ae1 + P2Ae2 + · · · + PwAew

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . 1 1 1 (Ae1 )0 (Ae1 )1 (Aew )0 (Aew )1 xi+1 xi+1 xi+1 (Ae2 )0 (Ae2 )1

Aej = (Aej)0 + xi+1(Aej)1

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . 1 1 1 xi+1 xi+1 xi+1 Ab1 Ab2 Ab3 Ab2w

Suppose Ab3 = 3Ab1 + 2Ab2

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . 1 1 xi+1 xi+1 xi+1 Ab1 Ab2 Ab3 Ab2w 2 3

Suppose Ab3 = 3Ab1 + 2Ab2

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . 1 1 xi+1 xi+1 xi+1 Ab1 Ab2 Ab2w 3 2

Suppose Ab3 = 3Ab1 + 2Ab2

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . Ab1 Ab2 xi+1

Note that these dependencies essentially define the ROABP.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Characterizing ROABPs

Width ≤ Coefficient Dimension

For a polynomial A, small Coefficient dimension = ⇒ small width ROABP.

t s · · · · · · xi . . . Ab1 Ab2 xi+1

Dependency support at most w + 1.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 12 / 22

Sum of two ROABPs

Equivalence of two ROABPs

A = B?, where A has an ROABP in variable order (x1, x2, . . . , xn).

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 13 / 22

Sum of two ROABPs

Equivalence of two ROABPs

A = B?, where A has an ROABP in variable order (x1, x2, . . . , xn). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 13 / 22

Sum of two ROABPs

Equivalence of two ROABPs

A = B?, where A has an ROABP in variable order (x1, x2, . . . , xn). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if Ab3 = 3Ab1 + 2Ab2, then verify whether Bb3 = 3Bb1 + 2Bb2.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 13 / 22

Sum of two ROABPs

Equivalence of two ROABPs

A = B?, where A has an ROABP in variable order (x1, x2, . . . , xn). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if Ab3 = 3Ab1 + 2Ab2, then verify whether Bb3 = 3Bb1 + 2Bb2. If any dependency fails, then A = B.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 13 / 22

Sum of two ROABPs

Equivalence of two ROABPs

A = B?, where A has an ROABP in variable order (x1, x2, . . . , xn). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if Ab3 = 3Ab1 + 2Ab2, then verify whether Bb3 = 3Bb1 + 2Bb2. If any dependency fails, then A = B. If all the dependencies hold then A = cB for some constant c.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 13 / 22

Sum of two ROABPs

Equivalence of two ROABPs

A = B?, where A has an ROABP in variable order (x1, x2, . . . , xn). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if Ab3 = 3Ab1 + 2Ab2, then verify whether Bb3 = 3Bb1 + 2Bb2. If any dependency fails, then A = B. If all the dependencies hold then A = cB for some constant c. Question: how to verify dependencies for B?

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 13 / 22

Sum of two ROABPs

Verifying Dependencies for B

For example B00 + 2B10 − B11 = 0.

s t · · · B 1 − x1 2 + x1 2x2 3x2 − 1 5 + 4x2 1 + x1 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 14 / 22

Sum of two ROABPs

Verifying Dependencies for B

For example B00 + 2B10 − B11 = 0.

s t · · · B 1 − x1 2 + x1 0 + 2x2 3x2−1 5 + 4x2 1 + x1 s t · · · B00 1 1 2 −1 5 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 14 / 22

Sum of two ROABPs

Verifying Dependencies for B

For example B00 + 2B10 − B11 = 0.

s t · · · B 1−1x1 2 + 1x1 0 + 2x2 3x2−1 5 + 4x2 1 + 1x1 s t · · · B10 1 −1 1 −1 5 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 14 / 22

Sum of two ROABPs

Verifying Dependencies for B

For example B00 + 2B10 − B11 = 0.

s t · · · B 1−1x1 2 + 1x1 2x2 3x2 − 1 5 + 4x2 1 + 1x1 s t · · · B11 1 −1 1 2 3 4 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 14 / 22

Sum of two ROABPs

Verifying Dependencies

Verifying dependencies for B? For example B00 + 2B10 − B11 = 0. B00, B10 and B11 all three have an ROABP in the same variable order.

B00 t3 · · · . . . · · · t2 · · · . . . · · · t1 · · · . . . · · · B10 B11 s1 s2 s3

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 15 / 22

Sum of two ROABPs

Verifying Dependencies

Verifying dependencies for B? For example B00 + 2B10 − B11 = 0. B00, B10 and B11 all three have an ROABP in the same variable order.

· · · . . . · · · t · · · . . . · · · · · · . . . · · · s 1 2 −1

It reduces to zero testing for an ROABP of a larger width (≤ w(w + 1)).

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 15 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc).

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 16 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc). We also make this test blackbox but with quasi-polynomial cost (nw)c·2c log nw. based on the ideas of least basis isolation and low-support rank-concentration.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 16 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc). We also make this test blackbox but with quasi-polynomial cost (nw)c·2c log nw. based on the ideas of least basis isolation and low-support rank-concentration. The same techniques work in the case of higher individual degree, with essentially the same time complexity.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 16 / 22

Conclusion

Discussion

Can one bring down the time complexity from wO(2c) to wO(c)? Can we use these ideas to solve depth-3 multilinear circuits?

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 17 / 22

Conclusion

Discussion

Can one bring down the time complexity from wO(2c) to wO(c)? Can we use these ideas to solve depth-3 multilinear circuits? Our proof was inspired from a question in the boolean setting – equivalence of two ROBPs. Are there more connections?

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 17 / 22

Conclusion

Thank you

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 18 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc).

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 19 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc). A1 = A2 + A3 + · · · + Ac−1?

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 19 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc). A1 = A2 + A3 + · · · + Ac−1? Take dependencies of A1 and verify for A2 + A3 + · · · Ac−1.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 19 / 22

Conclusion

Generalizations

We generalize this test to sum of c ROABPs with time complexity poly(w2cnc). A1 = A2 + A3 + · · · + Ac−1? Take dependencies of A1 and verify for A2 + A3 + · · · Ac−1. Reduces to PIT for sum of c − 1 ROABPs. T(w, c) = nwT(c − 1, w2) + poly(n, w)

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 19 / 22

Conclusion

Dependencies for A

Say, A has variable order (x1, x2, . . . , xn).

t s . . . Q1 Q2 Qw v1 v2 vw α2 αw Ae α1

Recall that for any e ∈ {0, 1}i, Ae =

w

• j=1

αjQj, where αj ∈ F

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 20 / 22

Conclusion

Dependencies for A

Say, A has variable order (x1, x2, . . . , xn).

t s . . . Q1 Q2 Qw v1 v2 vw α2 αw Ae α1

Recall that for any e ∈ {0, 1}i, Ae =

w

• j=1

αjQj, where αj ∈ F Consider the vectors (α1, α2, . . . , αw), and find the dependencies.

Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 20 / 22

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