2 (In-)Computability in Linear Algebra and Geometry

1 Recap on Recursive Analysis

1.1 Notions of computability for real numbers

Definition 1.1. a) Call x∈Rbinarily computableiff there exists a computable sequence bn∈ {0,1}, n≥ −N, (i.e. a function b :{−N,−N+1, . . . ,0,1,2, . . .} → {0,1}) such that∑^∞n=−Nb_n2⁻ⁿ. b) Call x∈Rcomputableiff there exists a computable integer sequence(cn)_n

such that|x−c_n/2ⁿ⁺¹| ≤2⁻ⁿ.

c) LetD_n:={c/2ⁿ|c∈Z}andD:=^S_nD_ndenote the set ofdyadic rationals.

d) Call x∈RCauchy-computableiff there exist computable sequences q_n,εn∈Q such that|x−qn| ≤εn→0 as n→∞.

e) Call x∈Rnaively computableiff there exists a computable sequence q_n∈Q such that qn→x as n→∞.

f) A sequence s_n∈ {1,0,¯1}, n≥ −N, is called asigned digit expansionof∑^∞_n=−Ns_n2⁻ⁿ. Encoded over{0,1}^ω, bin(N),(sn)_n

is aρsd–nameof x.

Call a sequence(bn)_n as in a) (encoded over{0,1}^ω) aρb–nameof x;

and(cn)_n as in b) aρ–nameof x.

A pair(qn)_n and(εn)_n of sequences as in d) is aρC–nameof x.

A sequence(qn)_n as in e) is aρn–nameof x.

Lemma 1.2. a) Every binarily computable real has a computable signed digit expansion.

b) Every real with a computable signed digit expansion is computable.

c) Every computable real is Cauchy-computable d) and vice versa.

e) (Cauchy-)computability implies naive computability,

Example 1.3 a) Every rational number x∈Qis binarily computable.

b) √

2 andπare (Cauchy-)computable real numbers.

c) For H⊆Nthe Halting problem,∑n∈H2⁻ⁿis not binarily computable d) but naively computable.

1.2 Computing functions and relations on a continuous universe

Definition 1.4. a) Amultivalued(possibly partial) function f :⊆X⇉Y (akarelationormul- tifunction) is a subset of X×Y .

We write dom(f):={x∈X | ∃y∈Y :(x,y)∈ f}and f(x) ={y∈Y |(x,y)∈ f}.

b) AType-2 Machinehas an infinite read-only input tape, an infinite one-way output tape, and an unbounded work tape.

It computes a (possibly partial) function F :⊆ {0,1}^ω→ {0,1}^ω.

c) Arepresentationof a set X is a partial surjective mappingα:⊆ {0,1}^ω→X .

d) Fix representations α of X and β of Y and a (possibly partial and multivalued) function f :⊆X⇉Y .

A(α→β)–realizerof f is a (partial but single-valued) function F :⊆ {0,1}^ω→ {0,1}^ωwith f α(σ)¯

∋β F(σ)¯

for every ¯σ∈dom(F):={σ¯ |α(σ)¯ ∈dom(f)}.

e) A function as in d) is(α→β)–computableif it has a computable realizer in the sense of b).

(We simply saycomputableifα,βare clear from context.) It is(α→β)–continuousif it has a continuous realizer.

f) Letαibe representations for X_i, i∈I ⊆N, andh· | ·i:N×N→Na computable surjective pairing function. Then(σm)_mis a ∏i∈Iαi

–name of(xi)_i∈∏iX_i iff (σ_hi,ni)_n is anαi–name of x_i∈X_ifor every i∈I.

Example 1.5 a) Letα,β,γdenote representations of X,Y,Z, respectively. If f :⊆X⇉Y is(α→ β)–computable and g :⊆Y ⇉Z is(β→γ)–computable, then so is their composition

g◦ f :=

(x,z)x∈X,z∈Z,f(x)⊆dom(g),∃y∈Y :(x,y)∈ f∧(y,z)∈g} . (1) b) A single-valued total real function f :[0,1]→Ris(ρ→ρ)–computable if some Type-2 ma-

chine can map everyρ–name(cn)_n of some x∈[0,1]to aρ–name(c^′_m)_m of f(y).

c) Addition and multiplication are(ρ×ρ→ρ)–computable;

inversionR\ {0} ∋x7→1/x is(ρ→ρ)–computable.

d) Every polynomial with computable coefficients is computable; and vice versa.

e) Let(an)_n denote a computable sequence, R :=1/lim sup_npⁿ

|a_n|and 0<r<R. Then[−r,r]∋ x7→∑

n

anxⁿ is computable. In particular exp,sin,cos,ln(1+x)are computable.

f) Fixε>0. The multifunctionsgnf_ε :R⇉{−1,+1} with ε>x7→ −1 and −ε<x7→+1 is computable.

g) Any x∈Ris binarily computable iff it is computable.

Theorem 1.6. a) Every (oracle-)computable F :⊆ {0,1}^ω→ {0,1}^ωis continuous.

b) To every continuous F :⊆ {0,1}^ω→ {0,1}^ω, there exists an oracle relative to which F be- comes computable.

c) Every oracle-computable f :[0,1]→Ris continuous!

d) There exists a computable sequence of (degrees and coefficient lists of) univariate dyadic polynomials P_n∈D[X]withP_n(x)− |x|≤2⁻ⁿon[−1,+1].

e) Fix an oracleO. Continuous (total) f :[0,1]→Ris computable relative toO iff there exists a sequence Pn∈D_n+1[X]computable relative toOsuch thatkf−Pnk∞≤2⁻ⁿ.

f) To every continuous f :R→Rthere is an oracle relative to which f becomes computable.

1.3 Encoding functions and closed subsets

Definition 1.7. a) A[ρ^d→ρ]–name of f ∈C(R^d)is a double sequence P_n,m∈D[X1, . . . ,X_d]with|f(x)−P_n,m(x)| ≤2⁻ⁿfor allkxk ≤m.

b) A closed set A⊆R^d is computable if the function dist_A:R^d ∋ x 7→ min

kx−ak: a∈A ∈ R∪ {∞} =: R (2) is computable. Aψ^d–name of A∈A^(d) is a[ρ^d→ρ]–name of distA,

whereA^(d)denotes the space of closed subsets ofR^d. c) Aψ^d<–name of A is a ∏m∈Nρ^d

–name of some sequence x_m∈A dense in A.

d) Aψ^d>–name of A are two sequences qn∈Q^d andεn∈Qsuch that R^d\A = ^[

nB(qn,εn) where B(x,r):={y :kx−yk<r} . (3) e) Aρ^<–name of x∈Ris a sequence(qn)⊆Qwith x=sup_nq_n;

aρ>–name of x∈Ris a sequence(qn)⊆Qwith x=inf_nq_n.

f) For representationsα,βof X letα⊓β:= (α×β)^∆X, where∆X :={(x,x)|x∈X}. g) Writeαβif id : X →X is(α→β)–computable.

h) We say that U⊆X is α–r.e. if there exists a Turing machine which terminates precisely on input of allα–names of x∈U and diverges on allα–names of x∈X\U .

Theorem 1.8. a) It holdsρρ<⊓ρ>ρ.

b) Every(ρ→ρ<)–computable f :[0,1]→Ris lower semi-continuous.

c) A set A∈A^(d)isψ>^d–computable iff R^d\A isρ^d–r.e.

d) Letk · kin Equation 3 denote any fixed computable norm. Let k · k^′denote some other norm andψ>^′dthe induced representation. Thenψ^d>ψ>^′d.

e) It holdsψ^d ψ<^d⊓ψ^d>ψ^d.

Moreover A isψ<^d–computable iff dist_A is(ρ^d→ρ^>)–computable;

and A isψ^d>–computable iff dist_Ais(ρ^d→ρ^<)–computable.

In particularψ^d–computability is invariant under a change of computable norms.

f) UnionA^(d)×A^(d) ∋(A,B)7→A∪B∈A^(d) is(ψ^d×ψ^d→ψ^d)–computable;

but intersection is not.

g) Closed image C(R^d→R^k)×A^(d)∋(f,A)7→f[A]∈A^(k) is([ρ^d→ρ^k]×ψ^d<,ψ^k<)–computable.

h) Preimage C(R^d→R^k)×A^(k)∋(f,B)7→ f⁻¹[B]∈A^(d) is([ρ^d→ρ^k]×ψ>^k,ψ^d>)–computable.

j) {/0}isψ^d>

^[0,1]d–r.e.

2 (In-)Computability in Linear Algebra and Geometry

Common algorithms (e.g. Gaussian Elimination) generally pertain to the Blum-Shub-Smale model (equivalently: real-RAM) of real computation — and lead to difficulties when imple- mented.

Definition 2.1. a) For a set S⊆R^d, its convex hull is the least convex set containing S:

chull(S) := ^\

C : S⊆C⊆R^d,C convex .

A polytope is the convex hull of finitely many points, chull({p1, . . . ,pN}). For a convex set C, point p∈C is called extreme (written “p∈ext(C)”) if it does not lie on the interior of any line segment contained in C:

p=λ·x+ (1−λ)·y ∧ x,y∈C ∧ 0<λ<1 ⇒ x=y .

b) For a set X , let ^X_k :=

{x1, . . . ,x_k}: xi∈X pairwise distinct .Convex Hull, as understood in computational geometry, is the problem

extchull_N : R^d

N

∋ {x₁, . . .,xN} 7→

y extreme point of chull(x1, . . . ,x_N) (4) of identifying the extreme points of the polytope C spanned by given pairwise distinct x₁, . . . ,x_N. c) For 1≤ j ≤n let g_j :R^d ∋x 7→a_j0+∑ix_i·a_ji ∈R ((aj1, . . .,ajd)6=0) denote an affine

linear function and Hj=g⁻_j¹(0)its induced oriented hyperplane, H⁺_j =g⁻_j¹(0,∞)the positive halfspace and H⁻_j =g⁻_j¹(−∞,0)the negative one. For x∈R^d,

π(x) = sgn g_j(x)

j=1,...,n ∈ {−1,0,+1}ⁿ

is called itsposition vector. AcellofH={H1, . . . ,Hn}is a subset π⁻¹(σ) ofR^d for some σ∈ {−1,+1}ⁿ. Point Location is the problem of identifying, to fixedH and upon input of some point x, the cell that x lies in, i.e., of computing the function

Pointloc_H:R^d\^[H→ {−1,+1}ⁿ, x7→π(x) . (5) d) Let det_d:R^d×d→Rdenote the determinant mapping and rank_d,k:R^d×k→Nthe rank.

Moreover abbreviate Gr_k(V):=

U⊆V lin.subspace, dim(U) =k and Gr(V) =^S_kGr_k(V).

Finally,GramSchmidt_d,k : rank⁻_d,k¹[k]→R^d×k is the mapping induced by the Gram–Schmidt orthonormalization process.

e) Consider the multifunctions

LSolve_d,k: rank⁻_d,k¹[{0,1, . . .,k−1}]∋AZ⇒ker(A)\ {0} andLSolve^′

d,k:⊆R^d×(k+1)∋(A,b)Z⇒

x|A·x=b .

f) Consider the multifunctions SomeEVec_d :R^d×(d+1)/2 ∋AZ⇒

x6=0| ∃λ: A·x=λx and EVecBase_d:R^d×(d+1)/2∋AZ⇒

O∈O(R^d)|O^†·A·O=diag(···) .

p

Z

x Z

1

2

4 3

5

6

Z

Z₁₉Z₁₅

2

Z₅ Z₇

Z

Z6

Z

Z

Z Z

3 9

Z

Z

Z

Z

Z₁₂ Z₈

Proposition 2.2. a) extchull_N is discontinuous, hence incomputable;

b) For anyH6= /0,Pointloc_H is discontinuous, hence incomputable;

c) det_d andGramSchmidt_d,k are computable;

d) rank_d,k is discontinuous (hence incomputable) but(ρ^d×k,ρ^<)–computable.

e) Linear independence

(x1, . . . ,x_k)∈R^d×klinearly independent isρ^d×k–r.e.

f) LSolve_d,kandSomeEVec_dare uncomputable.

Theorem 2.3. a) dim_d: Gr(R^d)→ {0,1, . . .,d}is(ψ^d<,ρ<)–computable b) and(ψ^d>,ρ>)–computable; that is,(ψ^d,ρ)–computable!

Lemma 2.4. a) The generalized determinant is(ρ^d×k,ρ)–computable, namely the mapping Det_d×k:R^d×k∋(a₁, . . . ,a_k)7→max

|det(aj₁, . . .,aj_d)|: 1≤ j₁≤ ··· ≤ j_d≤k . b) R^d×k∋A7→range(A)∈A^(d) is(ρ^d×k,ψ<^d)–computable.

c) R^d×k∋A7→kern(A)∈A^(k)is(ρ^d×k,ψ^k>)–computable.

d) R^d×k∩rank⁻¹[k]∋A7→range(A)∈A^(d)is(ρ^d×k,ψ>^d)–computable.

e) Orthogonal complement, i.e. the mapping Gr(R^d)∋L7→L^⊥∈Gr(R^d), is both(ψ^d<,ψ>^d)–computable and(ψ>^d,ψ^d<)–computable.

f) The multivalued mapping Basis_d,k: Gr_k(R^d)∋LZ⇒

B∈R^d×k: range(B) =L is(ψ^d<,ρ^d×k)–computable.

Theorem 2.5. For fixed integers 0≤k≤d, the following representations of Gr_k(R^d)are uniformly equivalent: A name of L∈Gr_k(R^d)is

a) aρ^d×k-name for some basis x₁, . . .,x_k∈R^d for L;

b) same for an orthonormal basis;

c) aρ^d×m-name (m∈Narbitrary) for some real d×m-matrix B with L=range(B);

d) aψ^d<–name of d_L, i.e., approximations to d_L from above e) aψ^d>–name of dL, i.e., approximations to dL from below

f) aρ^m×d-name (m∈Narbitrary) for some real m×d-matrix A with L=kern(A);

a’)–f ’) similarly, but for L^′:=L^⊥and k^′:=d−k instead of L and k.

Fact 2.6 (E. Specker 1967) Let C_d[Z]denote the vector space of monic polynomials of degree d. The mappingC^d∋(z1, . . . ,z_d)7→∏^d_j=1(Z−z_j)∈C_d[Z]has a computable multivalued inverse.

Lemma 2.7. a) For d∈N, given x,y₁, . . .,y_d∈Randν:=Card{1≤i≤d : x=y_i}, one can compute(i1, . . . ,i_ν)with 1≤i₁<···<i_ν≤d and x=y_i1 =···=y_iν. b) Given x₁, . . .,x_d∈Rand k :=Card{x₁, . . . ,x_d},

one can computeν1, . . .,νd∈Nwithνj=Card{i : 1≤i≤d,x_i=x_j}.

Theorem 2.8. a) Given aρ^d·(d−1)/2–name of a symmetric real d×d–matrix A,

a d–tuple(λ1, . . .,λd)of its eigenvalues with multiplicities is multivaluedρ^d–computable.

b) Given aρ^d·(d−1)/2–name of a symmetric real d×d–matrix A and given its number Cardσ(A) of distinct eigenvalues, one can diagonalize A in the sense of ρ^d×d–computing an orthonor- mal basis of eigenvectors.

c) Given aρ^d·(d−1)/2–name of a symmetric real d×d–matrix A and given the integer

⌊log₂m⌋, where m(A) := min

dim kern(A−λ·id):λ∈σ(A) ∈ {1, . . . ,d} denotes the multiplicity of some least-degenerate eigenvalue, one can ρ^d–compute some eigenvector of A.

Definition 2.9. For 1≤k≤d integers letClass_d,k(x1, . . . ,x_d):=

j : 1≤ j≤d,xj=x_k} and consider the multivalued mapping

SomeClass_d:R^d ∋ (x1, . . . ,x_d) Z⇒ Class_d,k(x1, . . . ,x_d): 1≤k≤d yielding, for some k, the set of all indices i with x_i=x_k.

Lemma 2.10. Let x₁, . . . ,x_d∈Rand m :=min_1≤k≤dCardClass_d,k(x)as above.

a) For each 1≤k, ℓ≤d it either holdsClass_d,ℓ(x) =Class_d,k(x)orClass_d,ℓ(x)∩Class_d,k(x) =/0.

Also,^S_kClass_d,k(x) = [d].

b) Consider I⊆[d]such that

xi6=xj for all i∈I and all j∈[d]\I . (6) Then I∩Class_d,k(x)6= /0impliesClass_d,k(x)⊆I.

Moreover 1≤Card(I)<2m implies I=Class_d,k(x)for some k.

c) Suppose k∈Nis such that k≤m<2k.

Then there existsℓsuch that I :=Class_d,ℓ(x)satisfies (6) and has k≤Card(I)<2k.

Conversely every I ⊆[d]with k≤Card(I)<2k satisfying (6) has I=Class_d,ℓ(x)for someℓ.

d) Given aρ^d–name of(x1, . . . ,x_d)and given k∈Nwith k≤m<2k, one can computably find someClass_d,ℓ(x).

3 Continuity for Multivalued Functions

Definition 3.1. Let (X,d)and (Y,e) denote metric spaces and abbreviate B(x,r):={x^′∈X : d(x,x^′)<r} ⊆X and B(x,r):={x^′∈X : d(x,x^′)≤r}; similarly for Y . Now fix some f :⊆X⇉Y and call(x,y)∈ f apoint of continuity of f if the following formula holds:

∀ε>0 ∃δ>0 ∀x^′∈B(x,δ)∩dom(f) ∃y^′∈B(y,ε)∩f(x^′) . a) Call f strongly continuousif every(x,y)∈ f is a point of continuity of f :

∀x∈dom(f) ∀y∈ f(x) ∀ε>0 ∃δ>0 ∀x^′∈B(x,δ)∩dom(f) ∃y^′∈B(y,ε)∩f(x^′).

Fig. 1. a) For a relation g (dark gray) to tighten f (light gray) means no more freedom (yet the possibility) to choose some y∈g(x) than to choose some y∈f(x)(whenever possible). b) Illustratingε–δ–continuity in(x,y)for a relation (black)

b) Call f weakly continuousif the following holds:

∀x∈dom(f) ∃y∈ f(x) ∀ε>0 ∃δ>0 ∀x^′∈B(x,δ)∩dom(f) ∃y^′∈B(y,ε)∩f(x^′).

c) Call f uniformly weakly continuousif the following holds:

∀ε>0 ∃δ>0 ∀x∈dom(f) ∃y∈ f(x) ∀x^′∈B(x,δ)∩dom(f) ∃y^′∈B(y,ε)∩f(x^′).

d) Call f nonuniformly weakly continuousif the following holds:

∀ε>0 ∀x∈dom(f) ∃δ>0 ∃y∈ f(x) ∀x^′∈B(x,δ)∩dom(f) ∃y^′∈B(y,ε)∩f(x^′).

e) Call f Henkin-continuousif the following holds:

∀ε>0 ∃δ>0

∀x∈dom(f) ∃y∈ f(x)

∀x^′∈B(x,δ)∩dom(f) ∃y^′∈B(y,ε)∩f(x^′) . (7) f) Some g :⊆X⇉Y tightens f (and f loosensg)

if both dom(f)⊆dom(g)and∀x∈dom(f): g(x)⊆ f(x)hold.

Fig. 2. a) Example of a uniformly weakly continuous but not weakly continuous relation. b) A semi-uniformly strongly continu- ous relation which is not uniformly strongly continuous. c) A compact, weakly and uniformly weakly continuous relation which is not computable relative to any oracle.

Lemma 3.2. a) Let f be uniformly weakly continuous and suppose that f is pointwise compact in the sense that f(x)⊆Y is compact for every x∈X . Then f is weakly continuous.

b) Let f be nonuniformly weakly continuous and dom(f)compact.

Then f is uniformly weakly continuous.

c) If f is Henkin-continuous and tightens g, then also g is Henkin-continuous.

d) If f and g :⊆Y ⇉Z are Henkin-continuous, then so is g◦ f :⊆X ⇉Z.

e) A function F :⊆ {0,1}^ω→ {0,1}^ω is an(α,β)–realizer of f iff F tightensβ⁻¹◦f◦α iff β◦F◦α⁻¹tightens f .

f) If range(f)⊆dom(g)holds and if both f and g map compact sets to compact sets, then so does g◦f .

Proposition 3.3. a) The inverseρ⁻_b¹:[0,1]⇉{0,1}^ωof the binary representation restricted to [0,1]is not weakly continuous.

b) Every x∈Rhas a signed digit expansion

x =

∑

c) For k∈N, each|x| ≤ ²3·2^−kadmits such an expansion with a_n=0 for all n≤k.

And, conversely, x=∑^∞_n=k+1a_n2⁻ⁿwith(an,a_n+1)∈ {10,¯10,01,0¯1,00}for every n requires|x| ≤ ²3·2^−k.

d) Let x=∑^∞n=−Nan2⁻ⁿbe a signed digit expansion and k∈N such that(an,a_n+1)∈ {10,¯10,01,0¯1,00}for each n>k.

Then every x^′∈[x−2^−k/3,x+2^−k/3]admits a signed digit expansion x^′=∑^∞n=−Nbn2⁻ⁿ with an=bn∀n≤k.

d) LetΣ:={0,1,¯1,.}.

The inverseρ⁻_sd¹:R⇉Σ^ω of the signed digit representation is Henkin-continuous.

Theorem 3.4. Let K⊆Rbe compact and f : K⇉Rcomputable relative to some oracle.

Then f is Henkin-continuous.

Example 3.5 A compact total Henkin–continuous but not relatively computable relation.

(Dashed lines indicate alignment and are not part of the graph)

4 Computational Complexity

Definition 4.1. Call f :[0,1]→R computable in time t(n)and space s(n) if some Turing ma- chine can, upon input of everyρsd–name of every x∈dom(f)and of n in unary, produce within these ressource bounds some c∈Zsuch that|f(x)−c/2ⁿ⁺¹| ≤2⁻ⁿ.

Lemma 4.2. If f is (even oracle-)(ρD,ρD)–computable in time t(n), then µ :N∋n7→t(n+2)∈ Nconstitutes a modulus of uniform continuity to f , i.e.,|x−x^′| ≤2^−µ(n) ⇒ |f(x)−f(x^′)| ≤2⁻ⁿ. Example 4.3 The following function is computable in exponential time, but not in polynomial time — and oracles do not help: f :(0,1] ∋ x 7→ 1/ln(e/x) ∈ (0,1], f(0) =0.

0.06 0.08 0.1 0.12 0.14 0.16 0.18

0 0.002 0.004 0.006 0.008 0.01

1/ln(e/x)

RECURSION

THEORETIC TOPOLOGICAL

NON-

COMPUTABLE f(x)≡y[H] f(x) =sgn(x)

EXPONENTIAL

COMPLEXITY f(x)≡y[E] f(x) =1/ln(e/x)

Fig. 3. a) (Part of) the graph of f(x) =1/ln(e/x)from Example 4.3 demonstrating its exponential rise from 0.

b) Lower bound techniques in real function computation; H⊆Nis the Halting problem andN⊇E∈EXP\P.

In particular functional evaluation(f,x)7→ f(x)is not computable within time bounded only in n, the output precision, even when restricting to smooth functions f :[0,1]→[0,1].

5 Recap on Blum-Shub-Smale (BSS) Machines

A BSS machine M(overR) 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 via two special index registers (similar to a two-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 5.1 a) rank :R^n×m→Nis uniformly BSS–computable (in timeO(n³+m³)) b) The multivalued mapping R^n×m ∋ A Z⇒

(b1, . . .)basis of kern(A) ∈ R^m×∗ is uniformly

BSS–computable (in timeO(n³+m³)).

c) The multivalued mapping R^n×m ∋ AZ⇒

(c1, . . .)basis of range(A) ∈ R^n×∗ is uniformly

BSS–computable (in timeO(n³+m³)).

d) The graph of the square root function is BSS–decidable.

e) Qis BSS semi-decidable; and so is the setAof algebraic reals.

f) The algebraic degree function deg :A→Nis BSS–computable.

g) A language L⊆R^∗ is BSS semi-decidable iff L=range(f) for some total computable f :R^∗→R^∗.

h) The real Halting problemHis not BSS–decidable, where H :=

hM,xi: BSS machineMterminates on input x Definition 5.2. Fix a field F ⊆Rand d∈N. A set

B =

x∈R^d : p1(x) =. . .=p_k(x) =0 ∧ q1(x)>0∧. . .∧q_ℓ(x)>0 (9) of solutions to a finite system of polynomial (in)equalities with p₁, . . . ,p_k,q₁, . . .,q_ℓ∈F[X1, . . . ,X_d] is called basic semi-algebraic over F .

A subset ofR^d semi-algebraic over F is a finite union of ones that are basic semi-algebraic over F. It is countably semi-algebraic over F if the union involves countably many members, all being basic semi-algebraic over F.

If is known that every basic semi-algebraic set has at most finitely many connected components.

Lemma 5.3. For f :⊆R^∗→R^∗, and c₁, . . .,c_j∈R, consider the following claims:

a) f is computable by a BSS Machine with constants c₁, . . . ,c_j∈R.

b) There is an integer sequence(dn)_n such that dom(f) =^U_nB_nis the countable disjoint union of setsB_n⊆R^dn semi-algebraic over field extension F :=Q(c1, . . . ,c_j), and each restriction

f

Bn, n∈N, a quolynomial with coefficients from F.

c) There exists c_j+1∈Rsuch that f is computable by a BSS Machine with constants c₁, . . . ,c_j,c_j+1.

Then a) implies b) implies c).

Corollary 5.4. a) The square root function[0,∞)∋x7→√

x≥0 is not BSS–computable.

b) The sequenceN∋n7→e^√nis not BSS–computable.

c) QandAare not BSS–decidable

d) nor is real integer linear programming{hA,bi |A∈R^n×m,b∈Z^m,∃x∈Zⁿ: A·x=b}. Fact 5.5 (Lindemann–Weierstraß) Let a₁, . . .,anbe algebraic yet linearly independent overQ.

Then e^a1, . . . ,e^an are algebraically independent overQ.

6 Post’s Problem over the Reals

Proposition 6.1. a) Let x∈R,ε>0, N∈N. There exists a∈Aof deg(a) =N with|x−a|<ε. b) Let f : dom(f)⊆R→Rbe analytic and non-constant, T ⊆dom(f)uncountable.

Then, f maps some x∈T to a transcendental value, that is, f(x)6∈A.

c) Fix non-constant f = p/q∈R(X)with polynomials p,q of deg(p)<n, deg(p)<m.

Let a1, . . .,a_n+m∈dom(f)be distinct real algebraic numbers with f(a1), . . . ,f(an+m)∈Q.

There are co-prime polynomials ˜p,q of deg(˜ p)˜ <n, deg(q)˜ <m with coefficients in the alge- braic field extension Q(a1, . . . ,a_n+m)such that, for all x∈dom(f) ={x : q(x)6=0} ⊆R, it holds f(x) = f˜(x):=p(x)/˜ q(x).˜

d) Continuing c), let d≥max_ideg(ai). Then f(x)6∈Qfor all transcendental x∈dom(f) as well as for all x∈Aof deg(x)>D :=d^n+m·max{n−1,m−1}.

Theorem 6.2. The setQof rationals is semi-decidable and undecidable yet strictly ‘easier’

thanH: Aremains undecidable to a machine with oracle access toQ.

7 Computable Analysis vs. Algebraic Computability

Theorem 7.1. a) Let f :⊆R^k→Rbe continuous and computable by a BSS machineMwithout real constants. Then f is(ρ^k→ρ)–computable with oracle access to the Halting problem.

b) To everyℓ there exists a C^ℓtotal function f :[0,1]→Rcomputable by a constant-free BSS machine which is not(ρ→ρ)–computable.

Fig. 4. A piecewise linear and a C^kunit pulse, and a non-overlapping superposition by scaled shifts

References

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