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Explore non-commutative computation with division, focusing on arithmetic complexity, linear algebra, polynomials, and more. Discover the impact of division elimination in polynomial computations and delve into the basics of arithmetic complexity. Uncover the significance of non-commutative computation in various fields such as groups, matrices, quantum mechanics, and language theory.
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Non-commutative computationwith division Avi Wigderson IAS, Princeton PavelHrubesU. Washington
Can’t deal with Boolean complexity • What can be computed with + − × ÷ ? • Linear algebra, polynomials, codes, FFT,… • Helps Boolean complexity (arithmetization) • ……… Arithmetic complexity – why?
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”
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!
F[X1,X2,…] FX1, 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]
Non-commutative computation(groups, matrices, quantum, language theory,…) • X1, X2,… non-commuting vars: XiXjXjXi • FX1, 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)
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?
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
Field of fractions F(X1, X2,…) of FX1, 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!
R an nxn matrix with entries in FX1, X2,… R is fullif R ≠ AB with Anr, Brn, 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 FX1, 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
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