Identity Testing for constant-width, and commutative, ROABPscomputationalcomplexity.org/Archive/2016/slides/... · [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There

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


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.


Introduction

Randomized Test




Two Paradigms:





Introduction

Randomized Test




Two Paradigms:





Introduction

Randomized Test




Two Paradigms:





Introduction

Randomized Test




Two Paradigms:





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.


Preliminaries


−1

x2

s t

x1 + x2 5

x2

x1 + 2x4 x1




C (x) =∑

p∈paths(s,t)


W (e).

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



Preliminaries


−1

x2

s t

x1 + x2 5

x2

x1 + 2x4 x1




C (x) =∑

p∈paths(s,t)


W (e).

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



Preliminaries


−1

x2

s t

x1 + x2 5

x2

x1 + 2x4 x1




C (x) =∑

p∈paths(s,t)


W (e).

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



Preliminaries


−1

x2

s t

x1 + x2 5

x2

x1 + 2x4 x1


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.


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.


Preliminaries

Power of ABPs





Preliminaries

Power of ABPs





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)



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.



PIT for ROABPs






PIT for ROABPs






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.



Our Results









Our Results









Our Results









Our Results








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



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].




































[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 .



[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 .


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)




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).




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.






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.






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.





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).



n-variate ROABP

f =[

x1

] x2

x3

· · · xn−1

xn



f =[

x1

] x2

f =

w∑r=1

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




n-variate ROABP

f =[

x1

] x2

x3

· · · xn−1

xn



f =[

x1

] x2

f =

w∑r=1

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




n-variate ROABP

f =[

x1

] x2

x3

· · · xn−1

xn



f =[

x1

] x2

f =

w∑r=1

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




n-variate ROABP

f =[

x1

] x2

x3

· · · xn−1

xn



f =[

x1

] x2

f =

w∑r=1

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




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.



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.








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.








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








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








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








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








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








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








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








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].







y0 · · · y j · · · yd

x0

...x i

...xd

|

− coeff (x iy j)









y0 · · · y j · · · yd

x0

...x i

...xd

|

− coeff (x iy j)









y0 · · · y j · · · yd

x0

...x i

...xd

|

− coeff (x iy j)






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 .






Let gr = a0x0 + a1x

1 + · · ·+ adxd and hr = b0y

0 + b1y1 + · · ·+ bdy

d .

y0 y1 · · · yd

x0

x1

...xd


...... · · ·

...adb0 adb1 adbd

=⇒ rank(f ) = rank(∑w

r=1 fr ) ≤ w .






Let gr = a0x0 + a1x

1 + · · ·+ adxd and hr = b0y

0 + b1y1 + · · ·+ bdy

d .

y0 y1 · · · yd

x0

x1

...xd


...... · · ·

...adb0 adb1 adbd

=⇒ rank(f ) = rank(

∑wr=1 fr ) ≤ w .




(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 = `




(x , y) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).

x iy j 7→ t(i+j)w (1 + t−1)j .



i + j = ` − 1

y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `




(x , y) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).

x iy j 7→ t(i+j)w (1 + t−1)j .



i + j = ` − 1

y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `




(x , y) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).

x iy j 7→ t(i+j)w (1 + t−1)j .



i + j = ` − 1

y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `




(x , y) 7→ (tw , tw + tw−1) = (tw , tw (1 + t−1)).

x iy j 7→ t(i+j)w (1 + t−1)j .



i + j = ` − 1

y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `




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 }.




y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `

0

0

0

0 0

∗

∗

...

∗


Leading term: tw`.







y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `

0

0

0

0 0

∗

∗

...

∗


Leading term: tw`.







y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `

0

0

0

0 0

∗

∗

...

∗

i + j = ` − 1


Leading term: tw`.







y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `

0

0

0

0 0

∗

∗

...

∗


Leading term: tw`.


Focus on terms {tw`, tw`−1, · · · , tw(`−1)+1}.

They come only from an `-th diagonal monomial.





y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `

0

0

0

0 0

∗

∗

...

∗


Leading term: tw`.







y0 y1

x0

x1

xd

· · ·

.

.

.

yd

i + j = `

0

0

0

0 0

∗

∗

...

∗


Leading term: tw`.







(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 + · · ·




(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 + · · ·




(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 + · · ·




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.




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 + · · ·

...


0

)t`w +

(jw1

)t`w−1 + · · ·+

( jww−1

)t(`−1)w+1 + · · ·

0 ∗ · · · 0





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 + · · ·

...


0

)t`w +

(jw1

)t`w−1 + · · ·+

( jww−1

)t(`−1)w+1 + · · ·

0 ∗ · · · 0





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 + · · ·

...


0

)t`w +

(jw1

)t`w−1 + · · ·+

( jww−1

)t(`−1)w+1 + · · ·

0 ∗ · · · 0





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 + · · ·

...


0

)t`w +

(jw1

)t`w−1 + · · ·+

( jww−1

)t(`−1)w+1 + · · ·

0 ∗ · · · 0



Conclusion

Discussion

Possible improvements:

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

Connections between arithmetic and boolean pseudorandomness?


Conclusion

Discussion

Possible improvements:

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

Connections between arithmetic and boolean pseudorandomness?


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, NewYork, NY, USA. ACM.


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.J. ACM, 27(4):701–717.

Valiant, L. G. (1979).

Completeness classes in algebra.In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC ’79, pages 249–261, New York,NY, USA. ACM.


Bibliography

Zippel, R. (1979).

Probabilistic algorithms for sparse polynomials.In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM ’79, pages216–226, London, UK, UK. Springer-Verlag.


Documents

Identity Testing for constant-width, and commutative, ROABPscomputationalcomplexity.org/Archive/2016/slides/... · [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There