[PPT] - On algebraic branching programs of small width Karl Bringmann PowerPoint Presentation

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On algebraic branching programs

f small width

Karl Bringmann

MPII Saarbr¨ ucken

Christian Ikenmeyer

MPII Saarbr¨ ucken

Jeroen Zuiddam

CWI Amsterdam

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Small width algebraic branching programs: surprisingly powerful

1. Width-2 algebraic branching programs with

approximation are as powerful as formulas

2. Width-1 algebraic branching programs with

nondeterminism are as powerful as circuits

2

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1. Definitions
Algebraic branching programs
Formulas
Complexity classes VPk and VP

e

Approximation classes VPk and VP

e

2. Historical context
3. Statement of main result
4. Proof sketch
5. Statement of nondeterminism result

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Algebraic branching program (ABP) definition

width

         s t · · · edge labels are affine linear forms: α0 + α1x1 + · · · + αnxn (αi ∈ C)

length

f(x1, . . . , xn) =

s-t paths

in graph

product of edge labels on path

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Algebraic branching program (ABP) definition

width

         s t · · · edge labels are affine linear forms: α0 + α1x1 + · · · + αnxn (αi ∈ C)

length

f(x1, . . . , xn) =

s-t paths

in graph

product of edge labels on path Example x2 + y2 + z2 =

s-t path

products

s t x y z x y z Complexity Lk(f) = minimum length of any width-k ABP computing f

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Formula definition

x1 2 x1 x2 3                     

depth

× + × + leaves variables xi constants αi ∈ C nodes +, × fan-in 2 fan-out 1

size = number of nodes

f(x1, . . . , xn) = evaluation of tree Complexity Le(f) = minimum size of any formula computing f

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Classes VPk and VP

e definition

Recall: • Lk = width-k ABP length Recall: • Le = formula size family: sequence (fn)n∈N of polynomials fn(x1, . . . , xpoly(n)) VPk :=

families (fn)n∈N with Lk(fn) = poly(n)
k ∈ N

VP

e :=

families (fn)n∈N with Le(fn) = poly(n)
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Classes VPk and VP

e definition

Recall: • Lk = width-k ABP length Recall: • Le = formula size family: sequence (fn)n∈N of polynomials fn(x1, . . . , xpoly(n)) VPk :=

families (fn)n∈N with Lk(fn) = poly(n)
k ∈ N

VP

e :=

families (fn)n∈N with Le(fn) = poly(n)
Ben-Or and Cleve (1988) inspired by Barrington’s theorem (1986)

VP3 = VP4 = · · · = VP

e

In particular: width-3 ABPs can compute any polynomial Allender and Wang (2011) Strict inclusion: VP2 VP3 No width-2 ABP computes x1x2 + · · · + x15x16

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2 = 2 − x2 + ε2

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Approximation

s-t path

products

s t ε−1x ε2 −ε−1x = 1 + 1 + ε−1x − ε−1x + εx − εx − x2 + ε2 = 2 − x2 + ε2

2 − x2 + ε2

ε → 0

− → 2 − x2

L2(2 − x2 + ε2) ≤ 4

(ε > 0) We say “ L2(2 − x2) ≤ 4 ”

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Approximation

2 − x2 + ε2

ε → 0

− → 2 − x2

L2(2 − x2 + ε2) ≤ 4

(ε > 0) “ L2(2 − x2) ≤ 4 ”

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Approximation

2 − x2 + ε2

ε → 0

− → 2 − x2

L2(2 − x2 + ε2) ≤ 4

(ε > 0) “ L2(2 − x2) ≤ 4 ” Border complexity

cp. border rank (Bini et al., Strassen)

V = C[x1, . . . , xn]≤d degree ≤ d polyn. endowed with Euclidean norm L(f) := smallest r for which there exist (gε)ε∈R>0 ⊆ V and

lim

ε→0 gε = f

L(gε) ≤ r

for all ε > 0

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Approximation

2 − x2 + ε2

ε → 0

− → 2 − x2

L2(2 − x2 + ε2) ≤ 4

(ε > 0) “ L2(2 − x2) ≤ 4 ” Border complexity

cp. border rank (Bini et al., Strassen)

V = C[x1, . . . , xn]≤d degree ≤ d polyn. endowed with Euclidean norm L(f) := smallest r for which there exist (gε)ε∈R>0 ⊆ V and

lim

ε→0 gε = f

L(gε) ≤ r

for all ε > 0 VPk =

families (fn)n∈N with Lk(fn) = poly(n)
k ∈ N

VP

e =

families (fn)n∈N with Le(fn) = poly(n)
Clearly L(f) ≤ L(f). Therefore VPk ⊆ VPk,

VP

e ⊆ VP e,

etc

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More historical context

Valiant (1979) VP

e ⊆ VP s ⊆ VP

⊆ VNP Valiant’s conjectures VP

e, VP s, VP ?

⊇ VNP

9

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More historical context

Valiant (1979) VP

e ⊆ VP s ⊆ VP

⊆ VNP Valiant’s conjectures VP

e, VP s, VP ?

⊇ VNP Strassen, Mulmuley-Sohoni (GCT), B¨ urgisser Extended conjectures VP

s, VP ?

⊇ VNP

9

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More historical context

Valiant (1979) VP

e ⊆ VP s ⊆ VP

⊆ VNP Valiant’s conjectures VP

e, VP s, VP ?

⊇ VNP Strassen, Mulmuley-Sohoni (GCT), B¨ urgisser Extended conjectures VP

s, VP ?

⊇ VNP Proving e.g. VP

e ⊇ VNP using any geometric technique

(e.g. shifted partial derivatives or geometric complexity theory) automatically implies VP

e ⊇ VNP.

We study VP

e

Recent work on closures of classes: Forbes (2016), Grochow-Mulmuley-Qiao (2016)

9

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Statement of main result

Main theorem: VP2 = VP

e

VP2 VP3 VP

e

VP2 VP3 VP

e

= =

⊆

⊆

=

Ben-Or–Cleve Allender–Wang Corollary: strict inclusion VP2 VP2

10

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Ben-Or and Cleve construction

To prove: VP

e ⊆ VP3

x1 2 x1 x2 3 × + × + size s formula

s

t · · ·

edge labels: affine linear forms

size poly(s) width-3 ABP Brent (1974) depth reduction: size poly(s) depth O(log s) formula

11

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To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

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To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

SLIDE 30

To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

SLIDE 31

To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

SLIDE 32

To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

SLIDE 33

To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

SLIDE 34

To prove: VP

e ⊆ VP3

goal

s t f

base

x

addition

f g ∼

addition

f +g

multiplication

f g −1 f g −1 ∼

addition

fg

permute

→

addition

fg

12

SLIDE 35

Our construction

To prove: VP

e ⊆ VP2

(then VP

e ⊆ VP2 follows)

13

SLIDE 36

Our construction

To prove: VP

e ⊆ VP2

(then VP

e ⊆ VP2 follows)

Recall: computational model

s-t path

products

s t ε−1x ε2 −ε−1x = 2 + x2 + ε

ε → 0

− → 2 + x2 We need = f + εf1 + ε2f2 + · · ·

O(ε)

ε → 0

− → f

13

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Our construction

To prove: VP

e ⊆ VP2

goal

s t f

+O(ε)

base

x

addition

f

+O(ε)

g

+O(ε)

∼

addition

f +g

+O(ε)

squaring (idea)

ε−1f

+O(ε2)

ε2 −ε−1f

+O(ε2)

∼

addition

−f2

+O(ε)

multiplication

fg = 1

2

(f + g)2 − f2 − g2

14

SLIDE 38

Our construction

To prove: VP

e ⊆ VP2

goal

s t f

+O(ε)

base

x

addition

f

+O(ε)

g

+O(ε)

∼

addition

f +g

+O(ε)

squaring (idea)

ε−1f

+O(ε2)

ε2 −ε−1f

+O(ε2)

∼

addition

−f2

+O(ε)

multiplication

fg = 1

2

(f + g)2 − f2 − g2

14

SLIDE 39

Our construction

To prove: VP

e ⊆ VP2

goal

s t f

+O(ε)

base

x

addition

f

+O(ε)

g

+O(ε)

∼

addition

f +g

+O(ε)

squaring (idea)

ε−1f

+O(ε2)

ε2 −ε−1f

+O(ε2)

∼

addition

−f2

+O(ε)

multiplication

fg = 1

2

(f + g)2 − f2 − g2

14

SLIDE 40

Our construction

To prove: VP

e ⊆ VP2

goal

s t f

+O(ε)

base

x

addition

f

+O(ε)

g

+O(ε)

∼

addition

f +g

+O(ε)

squaring (idea)

ε−1f

+O(ε2)

ε2 −ε−1f

+O(ε2)

∼

addition

−f2

+O(ε)

multiplication

fg = 1

2

(f + g)2 − f2 − g2

14

SLIDE 41

Our construction

To prove: VP

e ⊆ VP2

goal

s t f

+O(ε)

base

x

addition

f

+O(ε)

g

+O(ε)

∼

addition

f +g

+O(ε)

squaring (idea)

ε−1f

+O(ε2)

ε2 −ε−1f

+O(ε2)

∼

addition

−f2

+O(ε)

multiplication

fg = 1

2

(f + g)2 − f2 − g2

14

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Statement of nondeterminism result

Recall: (gn) ∈ VP1 means gn is product of poly(n) many affine linear forms Definition: (fn) ∈ VNP1 if

∃ (gn) ∈ VP1
fn(x1, . . . , xp(n)) =
b∈{0,1}poly(n)

gn(x1, . . . , xp(n), b1, . . . , bpoly(n)) Naturally generalises to VNP

e and VNP

15

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Statement of nondeterminism result

Recall: (gn) ∈ VP1 means gn is product of poly(n) many affine linear forms Definition: (fn) ∈ VNP1 if

∃ (gn) ∈ VP1
fn(x1, . . . , xp(n)) =
b∈{0,1}poly(n)

gn(x1, . . . , xp(n), b1, . . . , bpoly(n)) Naturally generalises to VNP

e and VNP

Valiant (1980): VNP

e = VNP

Theorem: VNP1 = VNP Corollary: strict inclusions VP1 VNP1 and VP2 VNP2

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VP1 VP2 VP

e

VP VP1 VP2 VP

e

VP VNP1 VNP2 VNP

e

VNP VP1 VP2 VP

e

VP VP1 VP2 VP

e

VP VNP1 VNP2 VNP

e

VNP

=

⊆ =

⊆

⊆

⊆
⊆

⊆ = = =

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SLIDE 45

VP1 VP2 VP

e

VP VP1 VP2 VP

e

VP VNP1 VNP2 VNP

e

VNP VP1 VP2 VP

e

VP VP1 VP2 VP

e

VP VNP1 VNP2 VNP

e

VNP

=

⊆ =

⊆

⊆

⊆
⊆

⊆ = = = Thank you!

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Proof sketch VNP1 = VNP

1. We know VP

e ⊆ VP3 (Ben-Or–Cleve).

2. We prove VP3 ⊆ VNP1. Construction: let nondeterminism select

s-t paths in width-3 ABP.

3. This shows VP

e ⊆ VNP1. This implies VNP e ⊆ VNP1.

We know VNP = VNP

e (Valiant).

17

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Side result: the continuant

Definition continuant F0 = 1 F1(x1) = x1 Fn(x1, . . . , xn) = xn · Fn−1(x1, . . . , xn−1) + Fn−2(x1, . . . , xn−2) Example: Fn(1, 1, . . . , 1) = nth Fibonacci number Continuant complexity LF (f) = smallest n such that f(x1, . . . , xn) = Fn(ℓ1, . . . , ℓn) LF induces classes VPF and VPF Proposition: VPF = VP

e

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SLIDE 48

VNPwst

1

VNPw

1

VNPg

1

VNPwst

2

VNPw

2

VNPg

2

VNP

e

VNP

s

VNP VNPwst

1

VNPw

1

VNPg

1

VNPwst

2

VNPw

2

VNPg

2

VNP

e

VNP

s

VNP VPwst

1

VPw

1

VPg

1

VPwst

2

VPw

2

VPg

2

VP

e

VP

s

VP VPwst

1 poly

VPw

1 poly

VPg

1 poly

VPwst

2 poly

VPw

2 poly

VPg

2 poly

VP

e poly

VP

s poly

VPpoly VPwst

1

VPw

1

VPg

1

VPwst

2

VPw

2

VPg

2

VP

e

VP

s

VP

=

= = = = = = = ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆

=

= = = = = = =

=

= = ⊆ ⊆ = = = ⊆ ⊆ ⊆ ⊆ ⊆ ⊆

=

= = ⊆ ⊆ = = =

=

= =

⊆
⊆

⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆