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 X = (X) ij an n×n matrix. -Det n (X) = Σ σ sgn (σ) Π i X i σ (i)  “P” -Per n (X) = Σ σ Π i X i σ (i)  “NP” -(X) -1 : n 2 rational functions  “P” F field ÷ ÷ × × + + − − × × XiXi XjXj XiXi c + + S(f) – circuit size “P”: S is poly(n) L(f) – formula size “NC”: L is poly(n) n variables, f degree <n f

X 1, X 2,… commuting variables: X i X j = X j X i F[X 1, X 2,…] polynomial ring: p, q. F(X 1, X 2,… ) 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 (3 rd grade) glory! Commutative computation

State-of-the-art F[X 1,X 2,…] F  X 1, X 2,…  F(  X 1, X 2,…  ) comm, no ÷ non-comm, no ÷ non-comm Circuit lb Formula lb NC-hard NP-hard NC = P? P = NP? PIT (Word Problem) S> nlog n [BS] L> n 2 [K] Det [V] Per [V] P=NC [VSBR] Per n ≤ Det p(n) BPP [SZ,DL]

X 1, X 2,… non-commuting vars: X i X j  X j X i F  X 1, X 2,…  non-commut. polynomial ring: p, q. -Order of variables in monomials matter! E.g. Det n (X) = Σ σ sgn (σ) X 1 σ (1) X 2 σ (2)    X n σ (n) is just one option (Cayley determinant) -Weaker model. E.g. X 2 -Y 2 costs 2 multiplications, but just 1 in the commut. case: X 2 -Y 2 = (X-Y)(X+Y) Non-commutative computation (groups, matrices, quantum, language theory,…)

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

The wonderful wierd world of non-commutative rational functions x −1 + y −1, yx −1 y have no expression fg −1 for polys f,g (x + xy −1 x) −1 = x −1 - (x + y) −1 Hua’s identity Can one decide equivalence of 2 expressions? (x + zy −1 w) −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(  X 1, X 2,…  ) of F  X 1, X 2,…  Take all formulae r(X 1, X 2,…) with , , , ÷ r~s if for all matrices M 1, M 2,…of all sizes r(M 1, M 2,…) = s(M 1, M 2,…) whenever they make sense (no zero division) Amitsur Thm: F(  X 1, X 2,…  ) 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  X 1, X 2,…  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(  X 1, X 2,…  ) is the field of entries of inverses of all full matrices over F  X 1, X 2,…  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 Singular under M 1 (F)-substitutions -X 0 Z Invertible with M 2 (F) substitutions -Y –Z 0 Conjecture: Every full nxn R with entries in {X i }, F, is invertible under M d (F) substitutions, d=poly(n). - Conjecture true for polynomials [Amitsur-Levizky] - Conjecture implies: 1)RIT  BPP 2)Efficient elimination of division gates from non- commutative formulas computing polynomials 3)Degree bounds in Invariant Theory (& GCT )

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