Non-commutative computation
with division
Avi Wigderson IAS, Princeton
Pavel Hrubes U. Washington
Arithmetic complexity – why?
-
Can’t deal with Boolean complexity
What can be computed with + − × ÷ ?
Linear algebra, polynomials, codes, FFT,…
Helps Boolean complexity
(arithmetization)
- ………
Arithmetic complexity – basics
+
S(f) – circuit size
“P”: S is poly(n)
L(f) – formula size
“NC”: L is poly(n) X
×
−
+
X = (X)ij an n×n matrix.
- Detn (X) = Σσ sgn(σ) Πi Xiσ(i)
- Pern (X) = Σσ
n variables,
f degree <n
÷
Xj
i
f
Πi Xiσ(i)
- (X)-1 : n2 rational functions
×
Xi
F field
c
 “P”
 “NP”
 “P”
Commutative computation
X1, X2,… commuting variables: XiXj = XjXi
F[X1, X2,…] polynomial ring: p, q.
F(X1, X2,… ) field of rational functions: pq-1
[Strassen’73] Division can be efficiently eliminated
when computing polynomials
(eg from Gauss elimination for computing Det).
Since then, arithmetic complexity focused on , , 
We’ll restore division to its former (3rd grade) glory!
State-of-the-art
F[X1,X2,…]
comm, no ÷
Circuit lb
Formula lb
S> nlog n [BS]
L> n2
[K]
NC-hard
NP-hard
Det [V]
Per [V]
NC = P?
P = NP?
P=NC [VSBR]
Pern ≤ Detp(n)
PIT (Word BPP [SZ,DL]
Problem)
FX1, X2,…
F(X1, X2,…)
non-comm, no ÷ non-comm
Non-commutative computation
(groups, matrices, quantum, language theory,…)
X1, X2,… non-commuting vars: XiXj  XjXi
FX1, X2,… non-commut. polynomial ring: p, q.
- Order of variables in monomials matter! E.g.
Detn (X) = Σσ sgn(σ) X1σ(1) X2σ(2)    Xnσ(n)
is just one option (Cayley determinant)
- Weaker model. E.g. X2-Y2 costs 2 multiplications,
but just 1 in the commut. case: X2-Y2 = (X-Y)(X+Y)
State-of-the-art
F[X1,X2,…]
comm, no ÷
Circuit lb
Formula lb
S> nlog n [BS]
L> n2
[K]
NC-hard
NP-hard
Det [V]
Per [V]
NC = P?
P = NP?
P=NC [VSBR]
Pern ≤ Detp(n)?
PIT (Word BPP [SZ,DL]
Problem)
F<X1,X2,…>
non-comm, no ÷
L(Detn)>2n[N]
Per [HWY]
Det [AS]
F{X1,X2,…}
non-comm
L(X-1 )>2n[HW]
X-1
[HW]
P  NC [N]
P  NC [HW]
BPP [AL,BW]
BPP?
The wonderful wierd world of
non-commutative rational functions
x−1 + y−1 , yx−1y
have no expression fg−1 for polys f,g
(x + xy−1x)−1
= x−1 - (x + y)−1
Hua’s identity
Can one decide equivalence of 2 expressions?
(x + zy−1w)−1
can’t eliminate this nested inversion!
Reutenauer Thm: Inverting an nxn generic
matrix requires n nested inversions.
Key to the formula lower bound on X-1
The free skew field (I) [Amitsur]
A “circuit complexity” definition!
Field of fractions F(X1, X2,…) of FX1, X2,…
Take all formulae r(X1, X2,…) with , , , ÷
r~s if for all matrices M1, M2,…of all sizes
r(M1, M2,…) = s(M1, M2,…)
whenever they make sense (no zero division)
Amitsur Thm: F(X1, X2,…) is a skew field –
every nonzero element is invertible!
Word problem (RIT): Is r = 0?
The free skew field (II) [Cohn]
Matrix inverse definition
R an nxn matrix with entries in FX1, X2,…
R is full if R ≠ AB with A nr, B rn, r<n.
Ex:
0 X Y
Singular if vars commute
-X 0 Z
Invertible if vars non-commut.
-Y –Z 0
Cohn’s Thm: F(X1, X2,…) is the field of entries of
inverses of all full matrices over FX1, X2,…
Key to formula completeness of X-1
Word problem: Is R invertible (full)?
Cohn’s Thm: Decidable (via Grobner basis alg).
Minimal dimension problem
Ex:
0 X Y
-X 0 Z
-Y –Z 0
Singular under M1(F)-substitutions
Invertible with M2(F) substitutions
Conjecture: Every full nxn R with entries in {Xi}, F,
is invertible under Md(F) substitutions, d=poly(n).
- Conjecture true for polynomials [Amitsur-Levizky]
- Conjecture implies:
1) RIT  BPP
2) Efficient elimination of division gates from noncommutative formulas computing polynomials
3) Degree bounds in Invariant Theory (& GCT )
÷