Non-commutative computation with division

1. Non-commutative computationwith division Avi Wigderson IAS, Princeton PavelHrubesU. Washington

2. Can’t deal with Boolean complexity • What can be computed with + − × ÷ ? • Linear algebra, polynomials, codes, FFT,… • Helps Boolean complexity (arithmetization) • ……… Arithmetic complexity – why?

3. f n variables, f degree <n + S(f) – circuit size “P”: S is poly(n) L(f) – formula size “NC”: L is poly(n) × ÷ Arithmetic complexity – basics − + × F field Xi Xj Xi c • X = (X)ij an n×n matrix. • Detn (X) = Σσsgn(σ) ΠiXiσ(i) “P” • Pern(X) = ΣσΠiXiσ(i)  “NP” • (X)-1: n2 rational functions  “P”

4. 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!

5. F[X1,X2,…] FX1, X2,… F(X1, X2,…) comm, no ÷ non-comm, no ÷ non-comm Circuit lb Formula lb NC-hard NP-hard NC = P? P = NP? PIT (Word Problem) State-of-the-art S> nlogn [BS] L> n2[K] Det[V] Per [V] P=NC [VSBR] Pern≤ Detp(n) BPP [SZ,DL]

6. 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) = Σσsgn(σ) X1σ(1) X2σ(2)Xnσ(n) • is just one option (Cayley determinant) • Weaker model. E.g. X2-Y2costs 2 multiplications, • but just 1 in the commut. case: X2-Y2 = (X-Y)(X+Y)

7. F[X1,X2,…] F<X1,X2,…> F{X1,X2,…} comm, no ÷ non-comm, no ÷ non-comm Circuit lb Formula lb NC-hard NP-hard NC = P? P = NP? PIT (Word Problem) State-of-the-art L(Detn)>2n[N] Per [HWY] Det [AS] P NC [N] BPP [AL,BW] S> nlogn [BS] L> n2[K] Det[V] Per [V] P=NC [VSBR] Pern≤ Detp(n)? BPP [SZ,DL] L(X-1)>2n[HW] X-1[HW] P NC [HW] BPP?

8. 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’teliminatethisnested inversion! ReutenauerThm: Inverting an nxngeneric matrix requiresnnested inversions. Key to the formula lower bound on X-1 The wonderful wierd world of non-commutative rational functions

9. 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,…) whenevertheymakesense (no zero division) AmitsurThm: F(X1, X2,…) is a skew field – every nonzero element is invertible! Word problem (RIT): Is r = 0? The free skew field (I) [Amitsur]A “circuit complexity” definition!

10. R an nxn matrix with entries in FX1, X2,… R is fullif R ≠ AB with Anr, Brn, r<n. Ex: 0 X Y Singular if vars commute -X 0 Z Invertible if vars non-commut. -Y –Z 0 Cohn’sThm: 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 Rinvertible (full)? Cohn’sThm: Decidable (via Grobner basis alg). The free skew field (II) [Cohn]Matrix inverse definition

11. Ex: 0 X Y Singularunder M1(F)-substitutions -X 0 Z Invertiblewith M2(F) substitutions -Y –Z 0 Conjecture: Every full nxnR with entries in {Xi}, F, is invertible under Md(F) substitutions, d=poly(n). - Conjecture true for polynomials [Amitsur-Levizky] - Conjecture implies: RIT  BPP Efficient elimination of division gates from non-commutative formulas computing polynomials Degree bounds in Invariant Theory (& GCT ) Minimal dimension problem

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