Identity Testing for constant-width, and commutative,...
Transcript of Identity Testing for constant-width, and commutative,...
Identity Testing for constant-width, andcommutative, ROABPs
Rohit Gurjar∗, Arpita Korwar, Nitin Saxena†
Aalen University and IIT Kanpur
June 1, 2016
∗supported by TCS research fellowship†supported by DST-SERBGurjar, 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?
Input Models:
Arithmetic CircuitsArithmetic Branching Programs
Figure : An Arithmetic circuit
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 CircuitsArithmetic 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 itcomputes 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
Introduction
Randomized Test
Rephrasing the question: Given an arithmetic circuit decide if itcomputes 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
Introduction
Randomized Test
Rephrasing the question: Given an arithmetic circuit decide if itcomputes 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
Introduction
Randomized Test
Rephrasing the question: Given an arithmetic circuit decide if itcomputes 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
Introduction
Randomized Test
Rephrasing the question: Given an arithmetic circuit decide if itcomputes 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].
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.
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.
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.
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 −10 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].
Width-3 ABPs have the same expressive power as arithmeticformulas [Ben-Or and Cleve, 1992].
Deterministic PIT: only for special ABPs, e.g. read-once obliviousABP.
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 arithmeticformulas [Ben-Or and Cleve, 1992].
Deterministic PIT: only for special ABPs, e.g. read-once obliviousABP.
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 arithmeticformulas [Ben-Or and Cleve, 1992].
Deterministic PIT: only for special ABPs, e.g. read-once obliviousABP.
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
x2x34 − 1
2x4 + 1
3x1
x21 + 1
2
1 − x3
3x43
x3 + 1
x23 + 5x3
1 − x2
x2 + 3x22
x4
2x31 + 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 forROABP.
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
PIT for ROABPs
[Raz and Shpilka, 2005] gave a polynomial time whitebox test forROABP.
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
PIT for ROABPs
[Raz and Shpilka, 2005] gave a polynomial time whitebox test forROABP.
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*.
* 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 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 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 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 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.
A distribution is pseudorandom if any ROBP cannot distinguish itfrom the uniform random distribution.
Goal: construct a PRG with O(log n) seed length (polynomial sizesample 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
Pseudorandomness for ROBP
Comes from the RL versus L question.
A distribution is pseudorandom if any ROBP cannot distinguish itfrom the uniform random distribution.
Goal: construct a PRG with O(log n) seed length (polynomial sizesample 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
Pseudorandomness for ROBP
Comes from the RL versus L question.
A distribution is pseudorandom if any ROBP cannot distinguish itfrom the uniform random distribution.
Goal: construct a PRG with O(log n) seed length (polynomial sizesample 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
Pseudorandomness for ROBP
Comes from the RL versus L question.
A distribution is pseudorandom if any ROBP cannot distinguish itfrom the uniform random distribution.
Goal: construct a PRG with O(log n) seed length (polynomial sizesample 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
Pseudorandomness for ROBP
Comes from the RL versus L question.
A distribution is pseudorandom if any ROBP cannot distinguish itfrom the uniform random distribution.
Goal: construct a PRG with O(log n) seed length (polynomial sizesample 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
Sample space size: poly(w)× 2r instead of trivial 2r × 2r .
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
f (x1, x2) =∑w
r=1 gr (x1) hr (x2)
Claim: f (tw , tw + tw−1) 6= 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
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) 6= 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
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) 6= 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
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) 6= 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
Claim: f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
Proof: treat x3, x4, . . . , xn as constants.
f =[
x1
] x2
f =
w∑r=1
gr (x1) hr (x2, x3, . . . , xn)
f (tw1 , tw1 + tw−11 ) 6= 0 (bivariate ROABP).
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 (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
Proof: treat x3, x4, . . . , xn as constants.
f =[
x1
] x2
f =
w∑r=1
gr (x1) hr (x2, x3, . . . , xn)
f (tw1 , tw1 + tw−11 ) 6= 0 (bivariate ROABP).
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 (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
Proof: treat x3, x4, . . . , xn as constants.
f =[
x1
] x2
f =
w∑r=1
gr (x1) hr (x2, x3, . . . , xn)
f (tw1 , tw1 + tw−11 ) 6= 0 (bivariate ROABP).
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 (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
Proof: treat x3, x4, . . . , xn as constants.
f =[
x1
] x2
f =
w∑r=1
gr (x1) hr (x2, x3, . . . , xn)
f (tw1 , tw1 + tw−11 ) 6= 0 (bivariate ROABP).
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 (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
Proof: treat x3, x4, . . . , xn as constants.
f =[
x1
] x2
f =
w∑r=1
gr (x1) hr (x2, x3, . . . , xn)
f (tw1 , tw1 + tw−11 ) 6= 0 (bivariate ROABP).
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 15 / 26
Hitting-set for ROABP
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 0.
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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 0.
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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 0.
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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 0.
f ′ =[
t1
] t1
t2
· · · tn/2
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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 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(ndw log 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
n-variate ROABP
f (tw1 , tw1 + tw−11 , x3, x4, . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , xn) 6= 0.
f (tw1 , tw1 + tw−11 , tw2 , tw2 + tw−1
2 , . . . , twn/2, twn/2 + tw−1
n/2 ) 6= 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(ndw log 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) 6= 0.
Coefficient Matrix for f (x , y) [Nisan, 1991]
y0 · · · y j · · · yd
x0
...x i
...xd
|
− coeff (x iy j)
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
Claim: If f (x , y) =∑w
r=1 gr (x)hr (y), then f (tw , tw + tw−1) 6= 0.
Coefficient Matrix for f (x , y) [Nisan, 1991]
y0 · · · y j · · · yd
x0
...x i
...xd
|
− coeff (x iy j)
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
Claim: If f (x , y) =∑w
r=1 gr (x)hr (y), then f (tw , tw + tw−1) 6= 0.
Coefficient Matrix for f (x , y) [Nisan, 1991]
y0 · · · y j · · · yd
x0
...x i
...xd
|
− coeff (x iy j)
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
Claim: If f (x , y) =∑w
r=1 gr (x)hr (y), then f (tw , tw + tw−1) 6= 0.
Coefficient Matrix for f (x , y) [Nisan, 1991]
y0 · · · y j · · · yd
x0
...x i
...xd
|
− coeff (x iy j)
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.
Let gr = a0x0 + a1x
1 + · · ·+ adxd and hr = b0y
0 + b1y1 + · · ·+ bdy
d .
y0 y1 · · · yd
x0
x1
...xd
a0b0 a0b1 a0bda1b0 a1b1 a1bd
...... · · ·
...adb0 adb1 adbd
=⇒ rank(f ) = rank(
∑wr=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
Define fr = gr (x)hr (y).
Claim: rank(fr ) ≤ 1.
Let gr = a0x0 + a1x
1 + · · ·+ adxd and hr = b0y
0 + b1y1 + · · ·+ bdy
d .
y0 y1 · · · yd
x0
x1
...xd
a0b0 a0b1 a0bda1b0 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
Define fr = gr (x)hr (y).
Claim: rank(fr ) ≤ 1.
Let gr = a0x0 + a1x
1 + · · ·+ adxd and hr = b0y
0 + b1y1 + · · ·+ bdy
d .
y0 y1 · · · yd
x0
x1
...xd
a0b0 a0b1 a0bda1b0 a1b1 a1bd
...... · · ·
...adb0 adb1 adbd
=⇒ rank(f ) = rank(
∑wr=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) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).
x iy j 7→ t(i+j)w (1 + t−1)j .
leading-term(x iy j) = t(i+j)w .
Same for all x iy j with i + j = `.
i + j = ` − 1
y0 y1
x0
x1
xd
· · ·
.
.
.
yd
i + 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) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).
x iy j 7→ t(i+j)w (1 + t−1)j .
leading-term(x iy j) = t(i+j)w .
Same for all x iy j with i + j = `.
i + j = ` − 1
y0 y1
x0
x1
xd
· · ·
.
.
.
yd
i + 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) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).
x iy j 7→ t(i+j)w (1 + t−1)j .
leading-term(x iy j) = t(i+j)w .
Same for all x iy j with i + j = `.
i + j = ` − 1
y0 y1
x0
x1
xd
· · ·
.
.
.
yd
i + 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) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).
x iy j 7→ t(i+j)w (1 + t−1)j .
leading-term(x iy j) = t(i+j)w .
Same for all x iy j with i + j = `.
i + j = ` − 1
y0 y1
x0
x1
xd
· · ·
.
.
.
yd
i + 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) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).
x iy j 7→ t(i+j)w (1 + t−1)j .
leading-term(x iy j) = t(i+j)w .
Same for all x iy j with i + j = `.
i + j = ` − 1
y0 y1
x0
x1
xd
· · ·
.
.
.
yd
i + j = `
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 = `
0
0
0
0 0
∗
∗
...
∗
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
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 = `
0
0
0
0 0
∗
∗
...
∗
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
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 = `
0
0
0
0 0
∗
∗
...
∗
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
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 = `
0
0
0
0 0
∗
∗
...
∗
i + j = ` − 1
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
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 = `
0
0
0
0 0
∗
∗
...
∗
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.
They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
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 = `
0
0
0
0 0
∗
∗
...
∗
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
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 = `
0
0
0
0 0
∗
∗
...
∗
Leading nonzero Diagonal: at most w nonzero entries.
Leading term: tw`.
Leading term from the next diagonal: tw(`−1).
Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.They come only from an `-th diagonal monomial.
`-th diagonal nonzero monomials: {x`−j1y j1 , x`−j2y j2 , · · · , x`−jw y jw }.
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 20 / 26
Hitting-set for ROABP
Proof of the bivariate case
(x , y) 7→ (tw , tw (1 + t−1)).
x`−j1y j1 7→ t`w (1 + t−1)j1 .
x`−j1y j1 7→ t`w((
j10
)+
(j11
)t−1 + · · ·+
(j1j1
)t−j1
).
x`−j1y j1 7→(j10
)t`w +
(j11
)t`w−1 + · · ·+
(j1
w − 1
)t(`−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
(x , y) 7→ (tw , tw (1 + t−1)).
x`−j1y j1 7→ t`w (1 + t−1)j1 .
x`−j1y j1 7→ t`w((
j10
)+
(j11
)t−1 + · · ·+
(j1j1
)t−j1
).
x`−j1y j1 7→(j10
)t`w +
(j11
)t`w−1 + · · ·+
(j1
w − 1
)t(`−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
(x , y) 7→ (tw , tw (1 + t−1)).
x`−j1y j1 7→ t`w (1 + t−1)j1 .
x`−j1y j1 7→ t`w((
j10
)+
(j11
)t−1 + · · ·+
(j1j1
)t−j1
).
x`−j1y j1 7→(j10
)t`w +
(j11
)t`w−1 + · · ·+
(j1
w − 1
)t(`−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
x`−j1y j1 7→(j1
0
)t`w +
(j11
)t`w−1 + · · ·+
( j1w−1
)t(`−1)w+1 + · · ·
x`−j2y j2 7→(j2
0
)t`w +
(j21
)t`w−1 + · · ·+
( j2w−1
)t(`−1)w+1 + · · ·
...
x`−jw y jw 7→(jw
0
)t`w +
(jw1
)t`w−1 + · · ·+
( jww−1
)t(`−1)w+1 + · · ·
0 ∗ · · · 0
Assuming jk 6= jk ′ requires nonzero characteristic.
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26
Hitting-set for ROABP
Proof of the bivariate case
x`−j1y j1 7→(j1
0
)t`w +
(j11
)t`w−1 + · · ·+
( j1w−1
)t(`−1)w+1 + · · ·
x`−j2y j2 7→(j2
0
)t`w +
(j21
)t`w−1 + · · ·+
( j2w−1
)t(`−1)w+1 + · · ·
...
x`−jw y jw 7→(jw
0
)t`w +
(jw1
)t`w−1 + · · ·+
( jww−1
)t(`−1)w+1 + · · ·
0 ∗ · · · 0
Assuming jk 6= jk ′ requires nonzero characteristic.
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26
Hitting-set for ROABP
Proof of the bivariate case
x`−j1y j1 7→(j1
0
)t`w +
(j11
)t`w−1 + · · ·+
( j1w−1
)t(`−1)w+1 + · · ·
x`−j2y j2 7→(j2
0
)t`w +
(j21
)t`w−1 + · · ·+
( j2w−1
)t(`−1)w+1 + · · ·
...
x`−jw y jw 7→(jw
0
)t`w +
(jw1
)t`w−1 + · · ·+
( jww−1
)t(`−1)w+1 + · · ·
0 ∗ · · · 0
Assuming jk 6= jk ′ requires nonzero characteristic.
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26
Hitting-set for ROABP
Proof of the bivariate case
x`−j1y j1 7→(j1
0
)t`w +
(j11
)t`w−1 + · · ·+
( j1w−1
)t(`−1)w+1 + · · ·
x`−j2y j2 7→(j2
0
)t`w +
(j21
)t`w−1 + · · ·+
( j2w−1
)t(`−1)w+1 + · · ·
...
x`−jw y jw 7→(jw
0
)t`w +
(jw1
)t`w−1 + · · ·+
( jww−1
)t(`−1)w+1 + · · ·
0 ∗ · · · 0
Assuming jk 6= jk ′ requires nonzero characteristic.
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26
Hitting-set for ROABP
Proof of the bivariate case
x`−j1y j1 7→(j1
0
)t`w +
(j11
)t`w−1 + · · ·+
( j1w−1
)t(`−1)w+1 + · · ·
x`−j2y j2 7→(j2
0
)t`w +
(j21
)t`w−1 + · · ·+
( j2w−1
)t(`−1)w+1 + · · ·
...
x`−jw y jw 7→(jw
0
)t`w +
(jw1
)t`w−1 + · · ·+
( jww−1
)t(`−1)w+1 + · · ·
0 ∗ · · · 0
Assuming jk 6= jk ′ requires nonzero characteristic.
Gurjar, Korwar, Saxena PIT for constant-width ROABPs June 1, 2016 22 / 26
Conclusion
Discussion
Possible improvements:
Unknown variable orderHitting-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
Conclusion
Discussion
Possible improvements:
Unknown variable orderHitting-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
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