Script Skeleton: Algebraic Complexity Theory

Script Skeleton: Algebraic Complexity Theory

Optimal Algorithms in Computer Algebra

Martin Ziegler

ziegler@mathematik.tu-darmstadt.de

1 Motivating Examples for Algebraic Models of Computation . . . 1

2 Examples of (Almost) Tight Complexity Bounds . . . 3

2.1 Nonscalar Cost of Polynomial Multiplication: Interpolation and Dimension Bound . . . 3

2.2 Discrete Fourier Transform: Cooley–Tukey FFT and Morgenstern’s Volume Bound . . . 3

2.3 Nonuniform Polynomial Evaluation: Transcendence Degree . . . 4

3 Efficient Algorithms for Polynomials . . . 5

3.1 Multivariate Derivatives . . . 5

3.2 Univariate Arithmetic . . . 5

4 Complexity of Matrix Multiplication . . . 7

4.1 Strassen’s Algorithm . . . 7

4.2 Complexity and Tensor Rank of Bilinear Maps . . . 7

4.3 Properties of the Tensor Rank . . . 7

4.4 Exponent of Matrix Multiplication, LUP-Decomposition, and Inversion . . . 7

4.5 Multipoint Evaluation of Bivariate Polynomials . . . 7

5 Branching Complexity . . . 7

5.1 Randomized Polynomial Identity Testing . . . 7

5.2 Recap on Semi-Algebraic Geometry . . . 8

5.3 Recap on Projective Geometry . . . 8

5.4 Ben-Or’s Lower Bound and Applications . . . 9

5.5 Range Spaces and their Vapnik-Chervonenkis Dimension . . . 9

5.6 Fast Point Location in Arrangements of Hyperplanes . . . 9

5.7 Polynomial-depth Algorithms forNP–complete Problems . . . 9

6 NP–Completeness over the Reals . . . 9

6.1 Equations over the Cross Product . . . 10

6.2 Satisfiability in Quantum Logic . . . 11

6.3 Realizability of Oriented Matroids . . . 13

6.4 Stretchability of Pseudolines . . . 13

⋆Synopsis to a lecture held from mid of April to mid of July 2014 at the TU Darmstadt in reverence to PETERB ¨URGISSER

1 Motivating Examples for Algebraic Models of Computation

Question 1.1 What is the least numberℓ(n)of multiplications to calculate Xⁿfrom given X ? Let lb(n):=⌈log₂(n+1)⌉denote the length of n’s binary expansion and #1bin(n)the number of 1s in it.

• upper boundℓ(n)≤lb(n)−2+#1bin(n)≤2 log₂(n): by induction

• lower boundℓ(n)≥ ⌈log₂n⌉=lb(n−1)since deg≤2^ℓ

• upper bound with division:ℓ^′(n)≤lb(n+1)−1+#1bin(n)/2≤ ³2·log₂n

• improved upper boundℓ(n)≤log₂(n) +O ^{log n}

loglog n

: see Exercises

• improved lower boundℓ(n)≥log₂n+0.3·log₂ #₁bin(n) : Lemma 1.2. Let F0:=0, F1:=1, F_n+2:=F_n+1+Fn, γ:= (1+√

5)/2≈1.62.

a) F_n= γⁿ−(−γ)⁻ⁿ

/√

5, F_n+3≤2·γⁿ.

b) Consider an optimal sequence of multiplications Tk :=Tk₁·Tk₂, 1≤k ≤K :=ℓ(n), where T₀:=X and 0≤k₁,k₂<k. W.l.o.g. suppose deg T_k<deg T_k+1 and write G :={k : deg T_k = 2 deg T_k−1}for the giant steps, B :={k : deg T_k<2 deg T_k−1}for the baby steps.

Then #1bin(n)≤2^#Band n=deg(TK)≤2^#G·γ^#B: induction and example|g|b|b|

c) ℓ(n) =K=#G+#B≥(log₂n−#B·log₂γ) +#B, where 1−log₂γ≥0.3 See [1, EXERCISE 1.6].

Question 1.3 Fix a polynomial f ∈C[X]. What is the least number ℓ(f) of arithmetic opera- tions (additions/subtractions, multiplications) that compute f(x)from given x and some complex constants?

• upper boundℓ(f)≤2 deg(f)−1: Horner

• lower boundℓ(f)≥ ⌈log₂(deg f)⌉

• improved upper boundℓ(f)≤deg(f) +⌊deg(f)/2⌋+2 (Knuth 1962):

LetFdenote a field and f =∑^d_j=0αjX^j∈F[X]a polynomial of degree d. Suppose that h(Y):=

∑2 j+1≤dα2 j+1Y^j is either constant or a product of linear factors in F[Y]. Then there exists a straight-line program computing f inF[X]from X and X²and some elements from Fusing at most⌊d/2⌋+1 multiplications and d additions/subtractions:

Write h(Y) = (Y−ξ)·h₁(Y)and g(Y) = (Y−ξ)·g₁(Y) +ηwhere g(Y):=∑2 j≤dα2 jY^j. Then f(X) = g(X²) +X·h(X²) = (X²−ξ)· g₁(X²) +X·h₁(X²)

+ η can be calculated from X,X²,ξ,η,g₁(X²) +X·h₁(X²)using 1 multiplication and 2 additions/subtractions.

Reminder 1.4 (Asymptotic growth) Fix f,g :N→N.

• f ∈O(g) ⇔ lim sup_nf(n)/g(n)<∞

• f ∈o(g) ⇔ lim sup_nf(n)/g(n) =0

• f ∈Ω(g) ⇔ lim sup_nf(n)/g(n)>0

(Hardy–Littlewood semantics, not Knuth’s stronger lim inf_nf(n)/g(n)>0)

• f ∈Θ(g) ⇔ 0<lim infnf(n)/g(n)≤lim sup_n f(n)/g(n)<∞

Question 1.5 (Polynomial Multiplication) What is (the asymptotic growth of) the least number M(n)of arithmetic operations to produce (the coefficient list of) p·q from given (coefficient lists of any) polynomials p,q of deg(p),deg(q)≤n ?

• upper bound(n+1)·(n+2)−1: high-school method

• lower bound 2n+1

• upper boundO(n^log23)⊆O(n^1.585): Karatsuba

(a+b·x^m)·(c+d·x^m) =u+v·x^m+w·x^2m, where u :=a·c, w :=b·d,v := (a+b)·(c+d)−u−w henceM(2n)≤3·M(n) +4 andM(2^k)≤3^k·T(1) +4·³3^k−⁻1¹.

• upper boundO(n^1+ε)for any fixedε>0: Exercises

• upper boundO(n·log n)overCusing FFT

Question 1.6 (Matrix Multiplication) What is (the asymptotic growth of) the least number of arithmetic operations to produce A·B from given n×n–matrices?

• upper bound 2n³

• lower bound n²

• upper boundO(n^log27)⊆O(n^2.81):

For A= (Ai j),B= (Bi j)∈R^2×2it holds A·B=C where

C₁₁=M₁+M₄−M₅+M₇, C₁₂=M₃+M₅, C₂₁ =M₂+M₄, C₂₂=M₁−M₂+M₃+M₆ M₁:= (A12+A₂₂)·(B11+B₂₂), M₂:= (A21+A₂₂)·B₁₁,

M3:=A11·(B12−B21), M4:=A22·(B21−B11), M₅:= (A11+A12)·B22, M₆:= (A21−A₁₁)·(B11+B₁₂), M₇:= (A12−A₂₂)·(B21+B₂₂)

• upper boundO(n^2.373): world record,de GallarXiv:1401.7714 Definition 1.7 (Straight-Line Program).

a) LetS= S,(ci),(f_j)

denote a structure with constants c_i∈S and (possibly partial) functions f_j:⊆S^aj→S of arities a_j∈N. A Straight-Line Program P (over the signature of this structure and in variables X1, . . . ,Xn) is a finite sequence of assignments Z_k:=ciand Z_k:=X_ℓ(1≤ℓ≤ n) and Z_k:= f_j(Zk₁, . . . ,Zk_{a j}), 1≤k₁, . . . ,k_aj <k.

b) When assigned values x₁, . . . ,x_n∈S to X₁, . . .,Xn, the programcomputes(the set of results consisting of (x1, . . .,x_n) =:~x and of) Z₁, . . . ,Z_K; the final result is Z_K =: P(~x). However if any intermediate operation fj(Zk₁, . . . ,Zk_{a j})happens to be undefined, then so is P(~x):=⊥. c) Acost functionC assigns to each f_jsome cost C(f_j)≥0. The cost of a straight-line program

P is the sum of the costs of the f_j occurring. The length|P|of P means its cost with respect to constant cost function f_j7→1.

d) The (straight-line)complexityC_C(F)of a familyFof functions f :⊆S^af →S with respect to a cost function C is the least cost of a straight-line program P overS. computingF.

2 Examples of (Almost) Tight Complexity Bounds

2.1 Nonscalar Cost of Polynomial Multiplication: Interpolation and Dimension Bound In Karatsuba’s Algorithm and its generalizations, the total asymptotic cost is governed by the number of multiplications of the smaller polynomials; see Exercise 1. So we now investigate the complexity of polynomial multiplication when charging only multiplications among the coeffi- cient algebra while additions and scaling by constants are considered free.

Theorem 2.1. Fix someF–algebraAwith binary addition+:A×A→Aand unary scalings

×^c:A∋a7→c·a∈Aby constants c from the infinite fieldF.

a) There is a straight-line program overS:= A,(),(+,×^c: c∈F)

which, for arbitrary but fixed distinct x₁, . . . ,x_n∈Fand on input of y₁, . . .,yn∈A, calculates (the unique) a₀, . . . ,a_n−1∈ Awith∑ⁿ_k=0⁻¹a_k·x^k_ℓ=y_ℓforℓ=1, . . .,n.

b) Consider the algebraA:=F[A0, . . . ,A_n,B₀, . . . ,B_m]in n+m+2 variables A₀, . . .A_n,B₀, . . .B_m. The set

∑i+j=ℓA_i·B_j: 0≤ℓ≤n+m can be calculated from A₀, . . . ,B_m−1by a straight-line program overSusing n+m+1 operations “×” (and arbitrary many “+” and “×^c”).

c) For x₁, . . .,x_N,y₁, . . . ,y_M∈Aconsider theF–vector spaces X :={λ1x₁+···+λNx_N:λi∈F} and Y :={µ₁y₁+···+µ_My_M : µ_j ∈F}. Then any straight-line program over S computing {y₁, . . .,y_M}from(x1, . . . ,x_N)contains at least dim_F(X+Y+F)−dim_F(X+F)algebra mul- tiplications “×”.

d) The straight-line program from Item b) is optimal!

See [1, THEOREM 2.2].

2.2 Discrete Fourier Transform: Cooley–Tukey FFT and Morgenstern’s Volume Bound Consider the N-dimensional discrete Fourier-transform

F_N:C^N ∋ (x0, . . .,xN−1) 7→

∑

k=0,...,N−1 ∈ C^N . Theorem 2.2. Fix C≥1 and consider the structureS_C:= C,C,(+,×λ:|λ| ≤C)

where×^c: C∋z7→c·z∈Cdenotes unary complex multiplication by constants c of modulus at most C.

a) For N=2ⁿ,F_N can be computed by a straight-line program overS₁of lengthO(N·log N).

b) Consider a straight-line program P overS_Cin N variables.

Each ‘line’ℓof P computes an affine linear functionϕℓ:C^N→C;

and P computes an affine linear mapΦP:C^N ∋~x7→AP·~x+~b∈C^N+|P|, where|P|denotes the length of P and the first N components are the identity.

c) For~a1, . . . ,~am∈Cⁿwith m≥n write

∆(~a₁, . . . ,~am) := max

|det(~aj₁, . . . ,~aj_n)|: 1≤ j₁, . . .,jn≤m . Then, for 1≤k, ℓ≤m andλ∈Cwith|λ| ≥1, it holds

∆(~a₁, . . . ,~a_m,λ·~a_k) ≤ |λ| ·∆(~a₁, . . . ,~a_m) and ∆(~a₁, . . . ,~a_m,~a_k+~a_ℓ)≤2∆(~a₁, . . .,~a_m) .

d) The homogeneous linear map AP:Cⁿ→C^N+|P|from b) satisfies∆(AP)≤(2C)^|P|. e) Subject to scaling by 1/√N, the matrix exp(2πi·k·ℓ/N)

0≤k,ℓ<N is unitary and therefore has determinant of absolute value N^N/2.

See [6,§8] and [1, p.10].

The straight-line program from Item a) is thus optimal up to a constant factor!

2.3 Nonuniform Polynomial Evaluation: Transcendence Degree

Consider fieldsF⊆Eand recall that e₁, . . . ,e_n∈Eare called algebraically dependent (over F) iff there exists a non-zero polynomial p∈F[X1, . . .,Xn] with p(e1, . . . ,e_n) =0. (For example, {√2π+1,π}is algebraically dependent overQ.) A set E⊆Eis algebraically independent (over F) iff no finite subset of it is algebraically dependent. By definition, trdeg_F(E) is the largest cardinality of any subset ofEalgebraically independent (overF).

Fact 2.3 a) Any two maximal algebraically independent subsets E,E^′ofE(overF) have the same cardinality: exchange lemma + Zorn’s Lemma.

b) Eis algebraic overF iff trdeg_F(E) =0.

c) πand e are transcendental. In particular trdeg_Q{√2π+1,π}=1.

It is unknown whether{π,e}is algebraically independent overQ.

d) If x₁, . . .,x_d∈Aare linearly independent overQ,

then exp(x1), . . . ,exp(x_d)∈Care algebraically independent overQ: Lindemann–Weierstraß e) It holds trdeg_F F(X1, . . . ,Xn)

=n,

whereF(X1, . . . ,X_n)denotes the field of rational functions in n variables overF.

f) ForF⊆E⊆Dfields, it holds trdeg_F(D) =trdeg_F(E) +trdeg_E(D).

In particular trdeg_F E(x)

≤trdeg_F(E) +1 for x∈D.

g) There exist uncountable subsets ofRalgebraically independent overQ.

Theorem 2.4 (Motzkin’55+Belaga’61). Let F⊆E denote fields of characteristic 0 and F ⊆ E(~X) a finite set of rational functions in indeterminates (X1, . . . ,X_n) =~X . For p_j,q_j ∈ E[~X]

coprime overFand q_j monic (meaning at least one monomial has coefficient 1), define

CoeffF(p₁/q1, . . .,p_m/qm)⊆Eas the field overFgenerated by the coefficients from p₁, . . . ,q_m. a) Coeff_F(F)is well-defined and coincides with the field extensionF

f(~x):~x∈Fⁿ,f ∈F . b) For a_j,b_j,c_j,w_j∈E[~X]with b_j6=0, Coeff_F(wj+c_j·a_j/bj: j)⊆Coeff_F(wj,c_j,a_j,b_j: j).

c) Consider the structureS^′= E,F,(E,+,×,÷)

. Any straight-line program computingFover S^′contains at least trdeg_F Coeff_F(F)

constants fromE.

d) Consider a straight-line program P overS:= E,E,(+,−,×,÷)

computing (intermediate) results f1, . . .,fN∈E(X1, . . . ,Xn).

i) There exist 06=b_j,a_j∈E[~X], cj∈E(j=1,. . . ,N) such that f_j=c_j·a_j/bj and trdeg_F Coeff_F(a1, ..aN,b₁, ..bN)

is at most the number of additions/subtractions in P.

ii) There exist 06=v_j,u_j∈E[~X], wj∈E(j=1,. . . ,N) such that f_j=w_j+u_j/vjand trdeg_F Coeff_F(u1, . . .,vN)

is at most twice P’s number of multiplications/divisions.

e) Any straight-line program computingF over S contains at least trdeg_F Coeff_F(F)

− |F| additions/subtractions and trdeg_F Coeff_F(F)

− |F|

/2 multiplications/divisions.

See [1, THEOREMS 5.1+5.9].

Knuth’s answer to Question 1.3 is thus optimal up to an additive constant!

3 Efficient Algorithms for Polynomials

Recall the total degree, deg(X³·Y²) =5. LetF[X]_<ddenote the vector space of polynomials over Fof total degree less than d; andF[X]=dthose homogeneous of degree d. Moreover writeF[[X]]

for the algebra of formal power series overF.

3.1 Multivariate Derivatives

Theorem 3.1 (Baur–Strassen). Fix a fieldFof characteristic 0, 0,1∈C⊆F, and let P denote a straight-line program in n variables overS= F,C,(+,−,×,÷)

computing f ∈F(X1, . . . ,X_n).

Then there exists a straight-line program P^′in n variables overSof length|P^′| ≤5· |P|simulta- neously computing all f,∂1f, . . . ,∂nf .

See [1,§7.2].

Lemma 3.2 (Taylor and Leibniz). For f ∈F(X1, . . . ,X_N)define

f⁽⁰⁾:= f(~0)∈F, f^(d) :=

∑

∂n1···∂n_df

(~0)·X_n1···X_nd/d! ∈ F[X1, . . .,X_N]=d

a) For f ∈F[X1, . . . ,X_N]<Dit holds f =∑^D_d=0⁻¹f^(d).

b) (f·g)⁽⁰⁾= (f·g)(~0) = f⁽⁰⁾·g⁽⁰⁾∈F, (f·g)⁽¹⁾= f⁽¹⁾·g⁽⁰⁾+ f⁽⁰⁾·g⁽¹⁾,

(f·g)⁽²⁾= f⁽²⁾·g⁽⁰⁾+f⁽¹⁾·g⁽¹⁾+f⁽⁰⁾·g⁽²⁾, and(f·g)^(D)=∑^D_d=0f^(d)·g^(D−d). c) In case g(~0)6=0, u := f/g has u⁽⁰⁾= f⁽⁰⁾/g⁽⁰⁾, u⁽¹⁾= f⁽¹⁾−u⁽⁰⁾·g⁽¹⁾

/g⁽⁰⁾, u⁽²⁾= f⁽²⁾−u⁽¹⁾·g⁽¹⁾−u⁽⁰⁾·g⁽²⁾

/g⁽⁰⁾, and u^(D)= f^(D)−∑^D_d=0⁻¹u^(d)·g^(D−d) /g⁽⁰⁾. Theorem 3.3 (Strassen’73). LetAdenote anF–algebra. SupposeF⊆F[X1, . . . ,X_N]<Dcan be computed (on a Zariski–dense subset ofA^N) by a straight-line program P over (A,C,+,×,÷) can also be computed by a straight-line program Q over(A,C,+,×)of length|Q| ≤O(D²)· |P|. See [1,§7.1].

3.2 Univariate Polynomial Arithmetic AbbreviateS^′:= C,C,+,×,÷

.

Theorem 3.4 (Polynomial Multiplication).

a) The product of two polynomials ¯p,q¯∈C[X], given by their lists of coefficients (dense repre- sentation), can be computed by a straight-line program overS^′of lengthO(N·log N), where N :=deg(p) +¯ deg(q).¯

b) The (coefficients of the) product of k given polynomials ¯p1, . . . ,p¯_k∈C[X]_<d,

can be computed by a straight-line program overS^′of lengthO(N·log²N), where N :=d·k.

See [1,§2.3].

Lemma 3.5. a) ¯p=∑n≥0p_nXⁿ∈F[[X]]has a multiplicative inverse 1/p¯∈F[[X]]iff p₀6=0;

in which case ¯q=∑n≥0q_nXⁿ:=1/p is given by q¯ ₀=1/p0and inductively q_n=−∑ⁿ_m=1p_m· qn−m/p0.

b) Suppose ˜q∈F[[X]]satisfies ¯p·q˜≡1 (mod Xⁿ). Then ˜˜q :=q˜·(2−p¯·q)˜ has ¯p·˜˜q≡1 (mod X²ⁿ).

c) Fix polynomials ¯a=∑ⁿ_i=0a_iXⁱ, ¯b=∑^m_j=0b_jX^j, ¯q=∑^m_k=0⁻ⁿq_kX^k, and ¯r=∑^m_ℓ=0⁻¹r_ℓX^ℓ with ¯a=¯b·q¯+¯r, where n :=deg(a)¯ ≥deg(¯b) =: m>deg(¯r). Then

∑

/

∑

≡

∑

d) For x₁, . . .,x_N ∈Fand ¯a∈F[X], ¯r :=a rem¯ (X−x₁)···(X−x_N)satisfies ¯a(xn) =¯r(xn).

e) It holds ¯a rem ¯p= (a rem ¯¯ p·q)¯ rem ¯p.

Theorem 3.6 (Polynomial Division and Multipoint Evaluation).

a) There exists a straight-line program over S^′ of length O(N·log N) computing, given (the coefficients of) p∈C[X]<N with p(0)6=0, (the coefficients of) 1/p mod X^N.

b) Given (the coefficients of) a,b∈C[X]of N :=deg(a)≥deg(b) =: M≥1, (the coefficients of) a div b and a rem b can be computed by a straight-line program overS^′of lengthO(N·log N).

c) A straight-line program overS^′ of lengthO(N·log²N)can compute, given (the coefficients of) p∈C[X]<N and x₁, . . .,x_N∈C, the values p(x1), . . .,p(xN).

d) A straight-line program overS^′ of length O Nd·log²(Nd)

can compute, given (the coef- ficients of) p₁, . . . ,p_N,q₁, . . .,q_N ∈C[X]_<d and z₁, . . . ,z_Nd ∈ C with q_j(zi)6=0, the values

∑^N_j=1^p_q^j_j^(z_(ziⁱ⁾₎, 1≤i≤Nd.

e) A straight-line program overS^′of lengthO(N·log N+log M)can compute, given p∈C[X]<N, p^M mod X^N.

See [6,§9+§10.10.1], [1,§2.4], and [10, THEOREM 2].

4 Complexity of Matrix Multiplication

4.1 Strassen’s Algorithm

4.2 Complexity and Tensor Rank of Bilinear Maps 4.3 Properties of the Tensor Rank

4.4 Exponent of Matrix Multiplication, LUP-Decomposition, and Inversion 4.5 Multipoint Evaluation of Bivariate Polynomials

5 Branching Complexity

Question 5.1 (Sorting) Given x₁, . . . ,x_nin a fixed linearly ordered set, how many comparisons are asymptotically sufficient and necessary to produce a permutationπ:[n]→[n] with x_π(1) ≤ x_π(2)≤ ··· ≤x_π(n) ?

• upper bound n·(n+1)/2: Bubble Sort

• upper boundO(n·log n): Merge Sort

• lower boundΩ(log₂n!) =O(n·log n):

Definition 5.2 (Decision Tree). Let S= S,R

denote a structure withRa family of relations R :⊆S^ak of arities a_R ∈N and Σsome arbitrary set. A Decision Tree T (over S andΣ and in variables X₁, . . . ,X_n) is an ordered full binary tree with each internal node u labelled by one of the above relations R_uand by an a_u:=a_Ru–tuple(Xu1, . . . ,X_uau)of the variables; while leaves v are labelled with elementsσv∈Σ.

When assigned values x1, . . . ,xn∈S to X1, . . . ,Xn, T starts at its root and for each internal node u iteratively proceeds to its left or right child depending on R_u(xu1, . . .,x_uau). Upon ending up in a leaf v it outputs T(x1, . . . ,x_n):=σv.

5.1 Randomized Polynomial Identity Testing

Definition 5.3. Polynomial Identity Testingis the following decision problem:

Given an expression p composed from variables X1, . . . ,Xnand integer constants using addition +and multiplication×; does this p represent the zero function onQ//R/C?

Any such expression p represents a multivariate integer polynomial; but expanding it into mono- mials can blow up its size:

For instance the determinant of a given n×n–matrix A= (ai j) is an n²–variate polynomial of total degree n in A’s entries. Expanded into monomials it consists of n! terms (Leibniz Formula) yet can be evaluated (on a Zariski–dense subset ofF^n×n) inO(n³) steps by means of Gaussian Elimination.

Lemma 5.4 (Schwartz,Zippel). LetFdenote a field, S⊆Ffinite, and p∈F[X1, . . . ,X_n]a non- zero polynomial of total degree d ∈N. Then, for r₁, . . .,r_n∈S chosen independently and uni- formly from S at random,

P[p(r1, . . . ,r_n) =0] ≤ d/|S| .

5.2 Recap on Semi-Algebraic Geometry Definition 5.5. Fix a ringF⊆Rand d∈N.

a) A set A of real solutions to a system of polynomial equalities (overF) is algebraic (overF):

~x∈R^d : p₁(~x) =. . .= p_k(~x) =0}, p₁, . . .,p_k∈F[X1, . . .,X_d] b) A constructible set is a finite Boolean combination of algebraic sets.

c) A set of solutions to a finite system of polynomial in-/equalities

~x∈R^d : p₁(~x) =. . .=p_k(~x) =0 ∧ q₁(~x)>0∧. . .∧q_ℓ(~x)>0 with p₁, . . . ,p_k,q₁, . . .,q_ℓ∈F[X1, . . . ,X_d]is called basic semi-algebraic (overF).

d) A subset ofR^dsemi-algebraic is a finite union of basic semi-algebraic ones.

e) It is countably semi-algebraic overFif the union involves countably many members, all being basic semi-algebraic overF.

Example 5.6 a) A circle is algebraic overZ. A disc is basic semi-algebraic overZ.

Every integer polytope is basic semi-algebraic overZ.

b) Every constructible subset ofRis finite or co-finite;

every semi-algebraic subset ofRis a finite union of intervals.

c) Every semi-algebraic set is the projection of a constructible set.

Fact 5.7 (Tarski–Seidenberg) The projection of a semi-algebraic set is again semi-algebraic!

5.3 Recap on Projective Geometry Definition 5.8. Fix a fieldF⊇Qand d∈N.

a) Projective spaceP^d(F)is the set{[~v]:~06=~v∈F^d+1}of lines through the origin, where[~v]:={λ~v :λ∈F}denotes a projective point.

b) The Grassmannian Gr_k(F^d)is the set of k–dimensional linear subspaces ofF^d; Gr(F^d):=^S_kGr_k(F^d). (So Gr₁(F^d+1) =P^d(F). . . )

c) For(~a1, . . .,~a_d)^†=B= (~b1, . . .,~b_k)∈F^d×k a matrix of full rank, the family of its maximal minors

Det span(B)

:= det(~a_i1, . . . ,~a_ik)

1≤i₁<i₂<...<i_k≤d

is called the Pl¨ucker Coordinates of span(B)∈Gr_k(F^d).

Lemma 5.9. Det : Gr_k(F^d)→P(^dk)−1

(F)is well-defined and injective (but not surjective).

See [17, PROPOSITION 14.2].

5.4 Ben-Or’s Lower Bound and Applications

5.5 Range Spaces and their Vapnik-Chervonenkis Dimension 5.6 Fast Point Location in Arrangements of Hyperplanes 5.7 Polynomial-depth Algorithms forNP–complete Problems

6 NP –Completeness over the Reals

A BCSS machine M (over R) can in each step add, subtract, multiply, divide, and branch on the result of comparing two reals. Its memory consists of an infinite sequence of cells, each capable of holding a real number and accessed through an index register (similar to a one-head Turing machine). A program for M may store a finite number of real constants. The notions of decidability and semi-decidability translate straightforwardly from discrete L⊆ {0,1}^∗ and L⊆N^∗to real languagesL⊆R^∗. Computing a function f :⊆R^∗→R^∗means that the machine, given~x ∈dom(f), outputs f(~x) within finitely many steps and terminates while diverging on inputs~x6∈dom(f).

Example 6.1 a) rank :R^n×m→Nis uniformly BCSS–computable in timeO(n³+m³) b) The graph of the square root function is BCSS–decidable.

c) Qis BCSS semi-decidable; and so is the setAof algebraic reals.

d) The algebraic degree function deg :A→Nis BCSS–computable.

e) A languageL⊆R^∗is BCSS semi-decidable iff L=range(f)for some total computable f :R^∗→R^∗. f) The real Halting problemHis not BCSS–decidable, where

H :=

hM,~xi: BCSS machineMterminates on input~x g) Every discrete language L⊆ {0,1}^∗is BCSS–decidable!

h) The following discrete problems (i) FEAS⁰_Rand (ii) QUART⁰_R can be verified in polynomial time by a BCSS machine without constants:

i) Given (the degrees and coefficients in binary of) a system of multivariate

polynomial in-/equalities with integer coefficients, does it admit a real solution?

ii) Given a multivariate polynomial of total degree at most 4, does it admit a real root?

Definition 6.2. LetNP⁰

Rdenote the family of discrete decision problems of the form ~x∈ {0,1}ⁿ: n∈N,∃~y∈R^p(n):h~x,~yi ∈V

where p∈N[N] and V⊆R^∗ can be decided in polynomial time by a BCSS machine without constants.

Theorem 6.3. FEAS⁰_Rand QUART⁰_Rare complete forNP⁰

R

(with respect to many-one reduction by a polynomial-time Turing machine).

Fact 6.4 (Grigoriev’88,Canny’88,Heintz&Roy&Solern´o’90,Renegar’92) NP⊆NP⁰

R⊆PSPACE.

6.1 Equations over the Cross Product

The cross product inR³is well-known due to its many applications in physics such as torque or electromagnetism. Mathematically it constitutes the mapping

×:R³×R³∋ (v0,v₁,v₂),(w0,w₁,w₂)

7→ (v1w₂−v₂w₁,v₂w₀−v₀w₂,v₀w₁−v₁w₀) ∈R³ (1) It is bilinear (thus justifying the name “product”) but anti-commutative~v×~w=−~w×~v and non-associative and fails the cancellation law:

~v×w=~u×w 6⇒ ~v=~u 6⇐ ~w×~v=~w×~u .

Fact 6.5 a) For linearly independent~v,~w, their cross product~v×~w=:~u is uniquely determined by the following:~u⊥~v, ~u⊥~w (where “⊥” denotes orthogonality), the triplet~v,~w,~u is right- handed, and lengths satisfyk~uk=k~vk · k~wk ·cos∠(~v,~w).

In particular, anti-/parallel~v,~w are mapped to~0.

b) Cross products commute with simultaneous orientation preserving orthogonal transforma- tions: For O∈R^3×3 with O·O^†=id and det(O) =1 it holds(O·~v)×(O·~w) =O·(~v×~w), where O^†denotes the transposed matrix.

Definition 6.6. a) Atermt(V1, . . . ,Vn)(over “×”, in variables V₁, . . . ,Vn) is either one of the variables, or(s×t)for terms s,t (in variables V₁, . . . ,Vn).

b) For~v1, . . .,~vn∈R³thevaluet(~v1, . . . ,~vn)is defined inductively via Equation (1).

c) Fix a fieldF⊆Qand recall from Definition 5.8 thatP²(F) ={[~v]:~06=~v∈F³} denotes the projective plane (overF), where[~v]:={λ~v :λ∈F}.

For distinct[~v],[~w]∈P²(F)(well-)define[~v]×[~w]:= [~v×~w];[~v]×[~v]is undefined.

d) For a term t(V1, . . . ,Vn)and[~v₁], . . . ,[~v_n]∈P²(F), thevalue

t([~v₁], . . .,[~v_n])is defined inductively via c), provided all sub-terms are defined.

Definition 6.7. a) XNONTRIV⁰_F3 :=

ht(V1, . . .Vn)i

n∈N,∃~v₁, . . .~v_n∈F³: t(~v₁, . . .~v_n)6=~0 . b) XNONTRIV⁰_P2(F) :=

ht(V1, . . . ,Vn)i

n∈N,∃[~v₁], . . .,[~v_n]∈P²(F): t([~v₁], . . .,[~v_n])defined . c) XUVEC⁰_F3 :=

ht(V1, . . . ,Vn)i

n∈N, ∃~v₁, . . . ,~v_n∈F³: t(~v₁, . . .,~v_n) =~e₃:= (0,0,1) . d) XNONEQUIV⁰_P2(F) :=

hs(V1, . . . ,Vn),t(V1, . . .,Vn)i

n∈N, ∃[~v₁], . . .,[~v_n]∈P²(F): s([~v₁], . . .,[~v_n])6=t([~v₁], . . .,[~v_n]), both sides defined . e) XSAT⁰_F3 :=

ht₁(V1, . . .,Vn)i

n∈N, ∃~v₁, . . . ,~v_n∈F³: t(~v₁, . . .,~v_n) =~v₁6=~0 . f) XSAT⁰_P2(F) :=

ht₁(V1, . . . ,Vn)i

n∈N, ∃[~v₁], . . .,[~v_n]∈P²(F): t([~v₁], . . .,[~v_n]) = [~v₁] . Following JOHN VON NEUMANN (who in turn credits KARL VONSTAUDT), express arithmetic overFas geometric operations onF³by identifying r∈Fwith the line _x

rx

: x∈F .

Lemma 6.8. Fix a subfieldFofR. Let~v1,~v2,~v3denote an orthogonal basis ofF³. Then Vj:=F~vj

satisfies V₁×V₂=V₃, V₂×V₃=V₁, and V₃×V₁=V₂. Moreover abbreviating V₁₂ :=F(~v₁−~v₂) and V₂₃ :=F(~v₂−~v₃)and V₁₃:=F(~v₁−~v₃), we have for r,s∈F:

a) F(~v₁−rs~v₂) = V₃×

F(~v₃−r~v₂)×F(~v₁−s~v₃)

b) F(~v1−s~v3) = V2×

V23×F(~v1−s~v2) c) F(~v₃−r~v₂) = V₁×

V₁₃×F(~v₁−r~v₂) d) F ~v1−(r−s)~v2

= V3×

[V23×F(~v1−r~v2)]×[V2×F(~v1−s~v3)]

×V3 e) V₁₃ = V₂×(V12×V₂₃).

f) For W ∈P²(F), the expression ı(W):= (W×V3)×

(W×V3)×V3

×V2

is defined pre- cisely when W =F(~v₁−r~v₂+s~v₃) for some s ∈F and a unique r∈ F; and in this case ı(W) =F(~v₁−r~v₂). Moreover, if W =F(~v₁−r~v₂)then ı(W) =W .

Theorem 6.9. a) XNONTRIV⁰_R3,XNONTRIV⁰_P2(R),XUVEC⁰_R3, andXNONEQUIV⁰_P2(R)are polytime equivalent toPolynomial Identity Testing(Definition 5.3).

b) XSAT⁰_R3 andXSAT⁰_P2(R) areNP⁰

R–complete.

c) There is a term t(V1, . . . ,Vn)s.t.~06=t(V1, . . .,Vn) =V1is satisfiable overR³but not overQ³.

6.2 Satisfiability in Quantum Logic

Definition 6.10. a) For a vector space V , the Grassmannian Gr_k(V)is the set of k–dimensional linear subspaces of V ;

Gr(V):=^S_kGr_k(V), 1 :=V is called (strong) truth, every X 6=0 :={~0}isweakly true.

b) For a finite-dimensional inner product space V , equip Gr(V)with the operations X∧Y :=X∩Y, X∨Y :=X+Y, and¬X :=X^⊥={~v∈V :∀~a∈X :~v⊥~a} . c) Alattice termis an expression over variables and∨,∧;

an (ortho)termmay in addition involve¬.

d) For a term t with variables X1, . . . ,Xn and for an assignment x1, . . . ,xn∈Gr(V), the value t_V(x1, . . . ,x_n)is defined inductively according to b).

We may omit the subscript V if it is clear from the context.

e) C(X,Y):= (X∧Y)∨(X∧ ¬Y)∨(¬X∧Y)∨(¬X∧ ¬Y)is calledcommutator(of X and Y ).

f) SAT_V :={ht(X1, . . . ,X_n)i:∃x₁, . . . ,x_n∈Gr(V): t_V(x1, . . . ,x_n) =1}, satV :={ht(X1, . . . ,Xn)i:∃x1, . . .,xn∈Gr(V): tV(x1, . . . ,xn)6=0}. g) For a term t(X1, . . .,X_n)and a fieldF⊆Clet

maxdim_F(t,d) := max

dim t_Fd(x1, . . .,x_n)

: x₁, . . .,x_n∈Gr(F^d) .

h) A d–diamondin V is a(d+1)–tuple D0,D₁, . . .,D_d∈Gr(V)such that V =D₁^k. . .^kD_d= D₀⊕D_jfor all 1≤ j≤d, where^kand⊕denote orthogonal and direct sum, respectively.

SoSAT_F1 =sat_F1 coincides with the classical, Boolean satisfiability problem;

hti ∈SAT_F^d ⇔maxdim_F(t,d) =d,hti ∈sat_F^d ⇔maxdim_F(t,d)>0.

Lemma 6.11. a) Gr(V)satisfiesde Morgan’s Rules¬(X∨Y) = (¬X)∧(¬Y) and ¬(X∧ Y) = (¬X)∨(¬Y); but Gr(R²)violates the distributive law(X∨Y)∧Z= (X∧Z)∨(Y∧Z).

b) Gr(V)satisfies themodular laws

x⊆y ⇒ x∨(y∧z) =y∧(x∨z), x⊇y ⇒ x∧(y∨z) =y∨x∧z) and in particular theorthomodular laws

u⊆v ⇒ u∨(v∧ ¬u) =v, u⊇v ⇒ u∧(v∨ ¬u) =v c) For a,b∈Gr(V)it holds: C(a,b) =1 ⇔ a= (a∨b)∧(a∨ ¬b) ⇔: aC b.

In particular aC b ⇔ ¬aC b ⇔ bC a.

d) Suppose x,y,z∈Gr(V)have C(x,y) =1=C(x,z).

Then x∧(y∨z) = (x∧y)∨(x∧z)and C(x,y∨z) =1.

e) If x₁, . . .,x_n∈Gr(V)satisfy C(xi,x_j) =1 and t(x1, . . . ,x_n)6=0, then there exist y₁, . . .,y_n∈ {0,1}with t(y1, . . . ,y_n) =1.

f) For any field F⊆ C, if t(X1, . . . ,X_n) admits a weakly/strongly satisfying assignment in Gr(F²), it also admits one inMO_n:=

0,1,Q ¹₁

,Q ⁻₁¹

, . . . , ¹_n

, ⁻₁ⁿ .

Proposition 6.12. Fix a field F ⊆C and 3≤ d ∈N. Let~e₁, . . . ,~e_d ∈ F^d denote a basis and abbreviateΘ:F∋a7→F(~e₁−a~e₂)∈Gr₁(F^d)and E_j:=F~e_jand E_{i j} :=F(~e_i−~e_j).

a) F(~e₁−b~e₃)∨F(~e₃−a~e₂)

∧(E1∨E₂) = Θ(a·b);

b) F(~ei−a~e_k) = F(~ei−a~ej)∨E_jk

∧(Ei∨E_k) for pairwise distinct 1≤i,j,k≤d;

c) F(~e_j−a~e_k) = F(~e_i−a~e_j)∨E_ik

∧(Ej∨E_k) for pairwise distinct 1≤i,j,k≤d;

d)

F(~e₁−b~e₃)∨E₂

∧ Θ(a)∨E₂₃

∨E₃

∩(E1∨E₂) = Θ(a−b);

f) For pairwise distinct 1≤i,j,k≤d it holds:^Wd_i=1E_i=1, E_i∧^Wj6=iE_j=0, E_{i j}∨E_j=E_i∨E_j, E_{i j}∧E_j=0, E_ik=E_ki= (Ei∨E_k)∧(Ei j∨E_jk).

g) Conversely every choice of E_i,E_{i j}∈Gr(F^d)satisfying the conditions expressed in f) arise from a basis e_i.

Theorem 6.13. a) For any fieldF⊆C, bothSAT_F² andsat_F² areNP–complete.

b) For every d≥3,SAT_R^d andsat_R^d areNP_R–complete c) and so areSAT_Cd andsat_Cd!

d) There exists a term t that is (weakly/strongly) satisfiable over Gr(R³)but not over Gr(Q³) and a term s (weakly/strongly) satisfiable over Gr(C³)but not over Gr(R³).

Lemma 6.14. a) For terms s(X1, . . . ,Xn)and t(Y1, . . .,Ym)it holds maxdim(s∨t,d) =min{maxdim(s,d) +maxdim(t,d),d}. b) Fix V ∈Gr(W)and a term t(X1, . . . ,X_n).

For x₁, . . .,x_n∈Gr(V)it holds t_V(x1, . . .,x_n) =t_W(x1, . . . ,x_n)∩V . c) For terms s(X1, . . .,X_n) =s(X)¯ and t(Y¯)abbreviate s|^t

(X,¯ Y¯):=s X₁∧t(Y¯), . . .,X_n∧t(Y¯)

∧ t(Y¯). Then maxdim(s|^t,d) =maxdim s,maxdim(t,d)

.

d) Every d–diamond D0,D1, . . .,D_d∈Gr(V), d :=dim(V), weakly satisfies the following term g_d(Z0,Z₁, . . . ,Z_d) =g_d(Z):¯

¬Z₀∧^{^d}_j=1 Z₀∨g_d,j(Z)¯

, where g_d,j(Z)¯ :=Z_j∧^{^}_i6=j>0¬Z_i . (2)