Non-commutative computations: lower
bounds and polynomial identity testing
Guillaume Malod
Joint work with G. Lagarde, S. Perifel
IMSc Workshop on Arithmetic Complexity
March 1st, 2017
Table of contents
1. Introduction
2. Nisan’s results
3. Unambiguous circuits
4. Other results
1
Introduction
Arithmetic circuits
y
x
×
+
z
+
×
×
π
+
+
+
2
Non-commutative circuits
• F commutative field.
• Non-commutative : xy ≠ yx → distinguish left and right arguments
in a computation gate.
• Various motivations
3
Some results
• / No better lower bound for NC circuits than for commutative
circuits
4
Some results
• / No better lower bound for NC circuits than for commutative
circuits
But for ABPs (Algebraic Branching Programs) :
• , (Nisan 1991) Exact characterization of complexity
• , (Nisan 1991) Exponential lower bounds for the permanent
• , (Arvind, Joglekar, Srinivasan 2009) Deterministic poly-time PIT
4
Some results
• / No better lower bound for NC circuits than for commutative
circuits
But for ABPs (Algebraic Branching Programs) :
• , (Nisan 1991) Exact characterization of complexity
• , (Nisan 1991) Exponential lower bounds for the permanent
• , (Arvind, Joglekar, Srinivasan 2009) Deterministic poly-time PIT
Also (Limaye,Malod,Srinivasan 2016) Exponentiel lower bounds for
skew circuits
4
Nisan’s results
ABP (Branching programs)
x1 + x2 + x3
x1
s
− x2
x3
x1
3x
2
2x 1
x1 +
2x
2
x1 −
x2
−x2
2x 3
t
x3
5
ABP (Branching programs)
x1 + x2 + x3
x1
s
− x2
x3
x1
3x
2
2x 1
x1 +
2x
2
x1 −
x2
−x2
2x 3
t
(x1 − x2 )(3x2 )(−x2 )
x3
• DAG : source s, sink t, edges with linear forms
• Weight of a path : product of edge weights
• Computed polynomial : sum of path weights from s to t.
6
Coefficient matrices
d
• f = ∑ αm .m, homogeneous,
monomials of degree ∣Y ∣
• Π = (Y , Z ) partition of [d]
monomials of degree ∣Z ∣
m2
m1
αm
m
degree d, n variables
• Define matrix M Π (f )
⎧
⎪
⎪ m∣Y = m1
m t.q ⎨
⎪
⎪
⎩ m∣Z = m2
7
Coefficient matrices
d
• f = ∑ αm .m, homogeneous,
monomials of degree ∣Y ∣
• Π = (Y , Z ) partition of [d]
monomials of degree ∣Z ∣
m2
m1
αm
m
degree d, n variables
• Define matrix M Π (f )
⎧
⎪
⎪ m∣Y = m1
m t.q ⎨
⎪
⎪
⎩ m∣Z = m2
• Complexity measure : rank(M Π (f )).
7
Exercise: the palindrome polynomial
w = (w1 , . . . , wd/2 ) ∈ [n]d/2 Ð→ w R = (wd/2 , . . . , w1 )
x̃w = xw1 xw2 . . . xwd/2
Pald X =
∑
x̃w ⋅ x̃w R
w ∈[n]d/2
n
Pald+1 X = ∑ xi ⋅ Pald X ⋅ xi
i=1
What is the matrix if we cut in the middle?
8
Exercise: the palindrome polynomial
mR
1
m
1
1
9
• Πi = ({1, 2, . . . , k}, {k + 1, k + 2, . . . , d})
k
d −k
Theorem (Nisan, 1991)
For any homogeneous polynomial f of degree d, the size of a smallest
homogeneous algebraic branching program for f is equal to
d
∑ rank(Mk (f ))
k=0
10
• Πi = ({1, 2, . . . , k}, {k + 1, k + 2, . . . , d})
k
d −k
Theorem (Nisan, 1991)
For any homogeneous polynomial f of degree d, the size of a smallest
homogeneous algebraic branching program for f is equal to
d
∑ rank(Mk (f ))
k=0
Corollary
Any homogeneous ABP computing the palindrome of degree d over n
variables has size ≥ nd/2
Any homogeneous ABP computing the permanent has size ≥ 2n
10
Proof (lower bound)
level k
v1
vi
s
Lk
vt
t
Rk
11
Proof (lower bound)
level k
v1
vi
s
t
vt
Lk
Rk
vi
m
coef(m) btw
s and vi
t
m′
nk
vi
coef(m′ ) btw
t
vi and t
nd−k
11
Proof (lower bound)
level k
v1
vi
s
t
vt
Lk
Rk
vi
m
coef(m) btw
m′
nk
vi
t
vi and t
s and vi
t
Mk (f ) = Lk Rk
coef(m′ ) btw
nd−k
and
rank(Mk (f )) ≤ t
11
Proof (upper bound)
level k
v1
level k + 1
w1
wj
s
vi
vt
t
wp
• suppose rank(Lk ) < t, then there is a column i which is a linear
combination of the others
• the polynomial computed between s and vi is a linear combination of
the polynomials computed by the other vertices vj′ s
• we could delete vi and update the weights from level k to level k + 1.
• so rank(Lk ) = rank(Rk ) = t = rank(Mk (f )).
12
Unambiguous circuits
Parse trees
a
z
w
×
c
b
×
×
+
+
×
×
+
13
Parse trees
a
z
w
×
c
b
×
×
+
+
×
×
+
14
Parse trees
a
z
w
×
c
b
×
×
+
+
×
×
+
15
Parse trees
a
z
w
×
c
b
×
×
+
+
×
×
+
16
Parse trees
a
z
w
×
c
b
×
×
+
+
×
×
+
a
b
×
+
z
×
+
Figure 1: val(T ) = zab
• Each parse tree computes a monomial.
Lemma
f = ∑ val(T )
T
17
Parse trees of ABPs
ABP
Right-skew Circuits
18
Parse trees of ABPs
ABP
Right-skew Circuits
Définition
A circuit is unambiguous if all its parse trees are isomorphic.
18
Unambiguous circuits
a
c
b
×
w
×
2
1
2
1
−3
z
+
+
×
×
1
×
×
+
×
−1
+
+
19
Canonical circuits
a
c
b
1
+
+
×
×
1
1
+
b
1
1
1
+
a
1
+
c
b
1
1
+
+
+
×
1
×
1
+
+
20
Canonical circuits
a
c
b
1
+
+
×
×
1
1
+
b
1
1
1
+
a
1
+
c
b
1
1
+
+
+
×
1
×
1
+
+
Any unambiguous circuit can be rendered canonical at a polynomial cost.
20
Type of a gate
p
i
d −p−i
deg. i
f
β1
+
deg. p
α
+
+
×
+
βk
+
×
+
21
Results
Theorem
Let P be a homogeneous polynomial of degree d and T a shape with d
leaves. Then the minimal number of addition gates needed to compute P
by a canonical unambiguous circuit with shape T is exactly equal to
(i,p)
(P)) ,
∑ rank (M
(i,p)∈S
where S is the set of all existing types of +-gates in the shape T .
Corollary
Any UC computing the permanent has size 2Ω(n)
22
Proof (lower bound)
Parse tree shape
p
i
d −p−i
Π(p,i) =
• rg (M Π(p,i) (f )) ≤ number of gates of type (p, i)
23
Proof (upper bound)
24
Proof (upper bound)
Clearly it works
24
Other results
PIT
Hadamard product (Arvind, Joglekar, Srinivasan)
Given two polynomials P = ∑x⃗ ax⃗ x⃗ and Q = ∑x⃗ bx⃗ x⃗, the Hadamard
product of P and Q, written P ⊙ Q, equals ∑x⃗ ax⃗ bx⃗ x⃗.
Hadamard product of two unambiguous circuits
Let C and D be two unambiguous circuits in canonical form, of the same
shape, and of size s and s ′ , that compute two polynomials P and Q. Then
P ⊙ Q is computed by an unambiguous circuit of size at most ss ′ ;
moreover, this circuit can be constructed in polynomial time.
Theorem
There is a deterministic polynomial-time algorithm for PIT for polynomials
computed by non-commutative unambiguous circuits over R (or C).
25
Relationship to other classes
ABP ⊊ UC
There are polynomials computed by polynomial-size UC that need
exponential-size ABPs.
UC and skew are incomparable
There are polynomials computed by polynomial-size UC that need
exponential-size skew circuits.
There are polynomials computed by polynomial-size skew circuits that
need exponential-size UC.
26
The future
• Lower bounds for circuits with “similar” shapes.
• Lower bounds for circuits with not too many shapes.
• Poly-time PIT for a sum of UC circuits.
27
The future (much later?)
Lower bounds for general non-commutative circuits?
28
The future (much later?)
Lower bounds for general non-commutative circuits?
Theorem (Limaye, Malod, Srinivasan 2016)
There exists a polynomial computed by a small non-commutative circuit
which is full rank for any partition.
28
Thank you!
29