Alan M. Turing 1936

Martin Ziegler

decision decision problem problem

•Logicians Tarski, Alonzo Church (PhD advisor)

•Kurt Gödel (1931): There exist arithmetical

statements which are true but cannot be proven so.

Alan M. Turing 1936

• • first first scientific scientific calculations calculations on digital on digital computers computers

• • What What are are its its fundamental fundamental limitations limitations ? ?

• • Uncountably Uncountably many many P P ⊆ ⊆

• • but but countably countably many many ' ' algorithms algorithms ' '

• • Undecidable Undecidable Halting Problem H : : No No algorithm algorithm B B can can always always correctly correctly answer answer the the following following question question

Given

Given 〈 〈 A, A, x〉 x 〉 , , does does algorithm algorithm A A terminate terminate on on input input x x ? ? Proof (by contradiction): Consider Consider algor algor . . B B ' ' that that , , on on input input A _A , , executes executes B B on on 〈 〈 A,A A,A 〉 〉

1941

Halting

Halting Problem Problem H H

Proof

Proof ( ( by by contradiction contradiction ): ):

A A

x x B B + +

− −

A A

A A

B' B'

∞ ∞

How How does does B B ' ' behave behave on on B' B' ? ? answer

answer , , loops loops infinitely infinitely . .

and,

and, upon upon a positive a positive

simulator

simulator / / interpreter interpreter B B ? ? B' B' B' B'

B' B'

Martin Ziegler

Formalities & Tools

''Definition:Definition:' ' AlgorithmAlgorithm

A A

decidesdecides setset

L L ⊆ ⊆ { { 0 0 , , 1 1 }* }*

ifif

•• on on inputsinputs

x x ∈ ∈ L L

printsprints 11 and and terminatesterminates,,

•• on on inputsinputs

x x ∉ ∉ L L

printsprints 00 and and terminatesterminates..

A A

semisemi--decidesdecides ifif terminatesterminates on on

x x ∈ ∈ L L

,, elseelse divergediverge..

all finite all finite

binary binary sequences sequences e.ge.g. "Turing . "Turing

machine machine""

Consider

Consider algor algor . . B B ' ' that that , on , on input input A _A , , executes executes B B on on

〈 〈 A,A A,A 〉 〉 and, and, upon upon a positive a positive answer answer , , loops loops infinitely infinitely . .

countable countable!! Techniques

Techniques:: a) a) simulationsimulation c) c) dovetailingdovetailing

Theorem:

Theorem:

L L

decidabledecidable iffiff bothboth

L L , , L L

^CC semisemi-decidable-decidable Infinite

Infinite

L L ⊆ ⊆ { { 0 0 , , 1 1 }* }*

isis semisemi-decidable-decidable iffiff

L L =range( =range( f f ) )

forfor somesome computablecomputable injectiveinjective

f f : : N N → → { { 0 0 , , 1 1 }* }*

b) b) diagonalizationdiagonalization egeg. . UU={ algorithms={ algorithms } } ×× { { inputsinputs }} Universes

Universes UU otherother thanthan

{ { 0 0 , , 1 1 }* }*

((e.ge.g. . NN):): encodeencode..

Halting

Halting Problem Problem H H only only semi semi - - decidable decidable

Hilbert Hotel Hilbert Hotel

d) d) reductionreduction (in/(in/outputoutput translationtranslation))

Martin Ziegler

Some Undecidable Problems

''Definition:Definition:' ' AlgorithmAlgorithm

A A

decidesdecides setset

L L ⊆ ⊆ { { 0 0 , , 1 1 }* }*

ifif

•• on on inputsinputs

x x ∈ ∈ L L

printsprints 11 and and terminatesterminates,,

•• on on inputsinputs

x x ∉ ∉ L L

printsprints 00 and and terminatesterminates..

Techniques

Techniques:: a) a) simulationsimulation c) c) dovetailingdovetailing

b) b) diagonalizationdiagonalization

For For

L,L' L,L' ⊆ ⊆ { { 0 0 , , 1 1 }* }*

writewrite

L L ≼ ≼ L' L'

ifif therethere isis a computablea computable

f f : : { { 0 0 , , 1 1 }* }* → → { { 0 0 , , 1 1 }* }*

such such thatthat

∀ ∀ x x : : x x ∈ ∈ L L ⇔ ⇔ f f ( ( x x ) ) ∈ ∈ L L '. '.

a) a)

L' L'

decidabledecidable ⇒⇒ so so

L L

. . b) b)

L L ≼ ≼ L' L' ≼ ≼ L'' L''

⇒⇒

L L ≼ ≼ L'' L''

Universes

Universes UU otherother thanthan

{ { 0 0 , , 1 1 }* }*

((e.ge.g. . NN):): encodeencode.. d) d) reductionreduction (in/(in/outputoutput translationtranslation)) Halting

Halting problemproblem: :

H H = { = { 〈 〈 A A , , x x 〉 〉 : : A A

terminatesterminates onon

x x } }

Hilbert's

Hilbert's 10th:10th: TheThe followingfollowing setset isis undecidableundecidable::

{ { 〈 〈 p p 〉 〉 | | p p ∈ ∈

^NN

[ [ X X

₁₁

, , … … X X

_nn

], ], n n ∈ ∈

^NN

, , ∃ ∃ x x

₁₁

… … x x

_nn

∈ ∈ N N p p ( ( x x

₁₁

, , … … x x

_nn

)=0 } )=0 }

Word Problem

Word Problem forfor finitelyfinitely presentedpresented groupsgroups Mortality

Mortality Problem Problem forfor twotwo 2121××21 21 matricesmatrices Homeomorphy

Homeomorphy of 2 finite of 2 finite simplicialsimplicial complexescomplexes

Martin Ziegler

integer integer

Exercise Questions

Which

Which of of thethe followingfollowing areare unun--/semi/semi--//decidabledecidable?? a) a) GivenGiven an integer, an integer, isis itit a prime a prime numbernumber??

b) b) GivenGiven a finite a finite stringstring overover ++,,××,(,(,,)),,00,1,1,,

X X

₁₁

, , … … X X

_{n n}

isis itit syntacticallysyntactically correctcorrect??

c) c) GivenGiven a a BooleanBoolean formulaformula

ϕ ϕ ( ( X X

₁₁

, , … … X X

_nn

) )

,, doesdoes itit havehave a a satisfyingsatisfying assignmentassignment?? d) d) GivenGiven

M M ∈ ∈

^{nn××nn and and}

b b ∈ ∈

^{nn, ,}

doesdoes therethere existsexists a a vectorvector

x x

s.t. s.t.

M M · · x x ≤ ≤ b b

?? e) e) GivenGiven an an algorithmalgorithm AA, , inputinput xx, and integer , and integer NN,,

doesdoes AA terminateterminate on on inputinput xx withinwithin NN stepssteps ??

f) Doesf) Does a givena given algorithmalgorithm terminateterminate on all on all inputsinputs?? g) g) DoesDoes givengiven algorithmalgorithm terminateterminate on on somesome inputinput??

realreal

Martin Ziegler

Computable Real Numbers

Theorem:

Theorem: For For r r ∈ ∈ , ,

t t he he following following are are equivalent equivalent : :

a) a) r r has a has a computable computable binary binary expansion expansion

b) b) There There is is an an algorithm algorithm printing printing , on , on input input n n ∈ ∈ , , some some a a ∈ ∈ with with | | r r - - a a /2 /2

ⁿ⁺¹ⁿ⁺¹

| | ≤ ≤ 2 2

^--nn

. .

c) c) There There is is an an algorithm algorithm printing printing two two sequences

sequences ( ( q q

_nn

) ) ⊆ ⊆ and ( and ( ε ε

_nn

) ) with with | | r r - - q q

_nn

| | ≤ ≤ ε ε

_nn

→ → 0 0

H := { 〈 B,x 〉 : algorithm B terminates on input x } ⊆

There is an algorithm which, given n∈, prints b_n∈{0,1} where r=∑_n b_n 2^-n

b) ⇔ c) holds uniformly,

⇔ a) does not [Turing'37]

numerators+

denominators

Ernst

Ernst Specker Specker (1949): (c) (1949): (c) ⇔ ⇔ Halting Halting problem problem plus (d) plus (d) d) d) There There is is an an algorithm algorithm printing printing ( ( q q

_nn

) ) ⊆ ⊆ with with q q

_nn

→ → r r . .

interval arithmetic

⇔ r∈ [q_n±ε_n]

Call Call r r ∈ ∈ computable computable if if

Martin Ziegler

Exercises: Computable Reals

r r ∈ ∈

computable_computable iffiff an an algorithmalgorithm cancan printprint, , on on inputinput

n n ∈ ∈

, _, somesome

a a

^yetyet

∈ ∈

^{naivelynaively}with_with

| | r r

^computable

-

^computable

- a a /2 /2

ⁿ⁺¹ⁿ⁺¹

| | ≤ ≤ 2 2

^{-n-n ..}

a) a) EveryEvery rational has a rational has a computablecomputable binarybinary expansionexpansion b) Everyb) Every dyadicdyadic rational has tworational has two binarybinary expansionsexpansions c) Computablec) Computable binarybinary expansionexpansion ⇔ ⇔ computablecomputable realreal d) d) IfIf

a a , , b b

areare computablecomputable, , thenthen also also

a+b a+b , , a a · · b b , , 1/ 1/ a a ( ( a a ≠ ≠ 0) 0)

e) Fix

e) Fix

p p ∈ ∈

[ [ X X ] ]

. . ThenThen

p p

's's coefficientscoefficients areare computablecomputable

⇔ ⇔ p p ( ( x x ) )

isis computablecomputable forfor all all computablecomputable

x x

.. f) f) TheThe degreedegree of of everyevery

p p ∈ ∈

[ [ X X ] ]

isis computablecomputable..

g) g) EveryEvery algebraicalgebraic numbernumber isis computablecomputable; and so ; and so isis

π π . .

h) h) IfIf

x x

isis computablecomputable, , thenthen so so areare

exp( exp( x x ) ) , , sin( sin( x x ) ) , , log( log( x x ) )

j) For

j) For everyevery computablecomputable

x x

, ,

sign( sign( x x ) )

isis computablecomputable.. k) k) Specker'sSpecker's sequencesequence

( ( ∑ ∑

_{k>nk>n∈H∈H}

2 2

^-n-n

) )

_kk isis computablecomputable,,

itsits limitlimit isis uncomputableuncomputable

Martin Ziegler

Uniformity, Sequences and Equality Testing

In In numerics numerics , , don't don't test test for for ( ( in in - - )equality )equality ! !

Fact:

Fact: There exists a computable sequence ( There exists a computable sequence ( r r

_mm

) ) ⊆ ⊆ [0,1] [0,1]

such that {

such that { m m : : r r

_mm

≠ ≠ 0 } is the Halting problem 0 } is the Halting problem H H . .

H := { 〈 B,x 〉 : algorithm B terminates on input x } ⊆

Reminder:

Reminder: For For r r ∈ ∈ , , t t he he following following are are equivalent equivalent : : a) a) ∃ ∃ algorithm algorithm deciding deciding r r 's 's bin. bin. expansion expansion

b) b) ∃ ∃ algorithm algorithm printing printing on on input input n n some some a a ∈ ∈ with with | | r r - - a a /2 /2

ⁿ⁺¹ⁿ⁺¹

| | ≤ ≤ 2 2

^--nn

. .

c) c) ∃ ∃ algorithm algorithm printing printing ( ( q q

_nn

),( ),( ε ε

_nn

) ) ⊆ ⊆ with with | | r r - - q q

_nn

| | ≤ ≤ ε ε

_nn

→ → 0 0 Call (r

_m

) ⊆ computable iff an algorithm can print, on input 〈 n,m 〉 ∈ , some a ∈ with |r

_m

-a/2

ⁿ⁺¹

|≤2

^-n

.

a)a)⇒⇒b)b)⇔c⇔c) ) computablecomputable transformation transformation on on algorithmsalgorithms b)b)⇒⇒aa) ') 'undecidableundecidable' '

casecase splitsplit on ron r∈∈

Martin Ziegler

x x ∈ ∈ computable computable ⇔ ⇔ | | x x - - a a

_nn

/2 /2

ⁿⁿ⁺¹⁺¹

| | ≤ ≤ 2 2

^--nn

for for recursive recursive ( ( a a

_nn

) ) ⊆ ⊆

f

Uniformly Computable Real Functions

A A computable computable function

function must must be be continuous continuous

x' x

Martin Ziegler

Computable Weierstrass Theorem

Theorem:

Theorem: For For f f :[0,1] :[0,1] → → the the following following are are equivalent equivalent : : a) a) There There is is an an algorithm algorithm converting converting any any seq seq . . q q

_nn

∈ ∈

_n+1n+1

with with | | x x - - q q

_nn

| | ≤ ≤ 2 2

^{--n n}

into into p p

_mm

∈ ∈

_m+1m+1

with with | | f f ( ( x x ) ) - - p p

_mm

| | ≤ ≤ 2 2

^-m-m

b) b) There There is is an an algorithm algorithm printing printing a a sequence sequence (of (of degrees degrees

and and coefficient coefficient lists lists of) ( of) ( P P

_nn

) ) ⊆ ⊆ [X [X ] ] with with || || f f - - P P

_nn

|| || ≤ ≤ 2 2

^-n-n

c) c) The The real real sequence sequence f f ( ( q q ), ), q q ∈ ∈ ∩ ∩ [0,1], [0,1], is is computable computable

&

& f f admits admits a a computable computable modulus modulus of uniform of uniform continuity. continuity

Call Call ( ( r r

_mm

) ) ⊆ ⊆ computable computable iff iff an an algorithm algorithm can can print print , , on on input input n,m n,m ∈ ∈ , , some some q q ∈ ∈

_n+1n+1

with with | | r r

_mm

- - q q | | ≤ ≤ 2 2

^--nn

. .

:= :=  

_nn

_nn

, ,

_nn

:= := { { a a /2 /2

ⁿⁿ

: : a a ∈ ∈ } }

| | x x - - y y | | ≤ ≤ 2 2

^--µ(µ(m)m)

⇒ ⇒ | | f f ( ( x x ) ) - - f f ( ( y y )| )| ≤ ≤ 2 2

^--mm

Proof: Proof: a) a) ⇒ ⇒ c) c) ⇒ ⇒ b) b)

Martin Ziegler

uncomputable

uncomputable in in generalgeneral Exercises: Computable Real Functions

a) a) f f computable computable

⇒⇒

same same for for any any restriction restriction

b) b) exp exp , , sin sin , , cos cos , ln(1+ , ln(1+ x x ) ) are are computable computable functions functions c) c) For For a a computable computable sequence sequence a a =( =( a a

_nn

), ),

the the power power series series x x → → ∑ ∑

_nn

a a

_nn

· · x x

ⁿⁿ

is is computable computable on on ( ( - - r,r r,r ) ) for for r r < < R R ( ( a a ) := 1/ ) := 1/ limsup limsup

_nn

| | a a

_nn

| |

^1/n1/n

d) d) Let Let f f ∈ ∈ C[0,1] C[0,1] be be computable computable . . Then Then so so are are

∫ ∫ f f : : x x → → ∫ ∫

₀₀^xx

f f ( ( t t ) ) dt dt and and max( max( f f ): ): x x → → max{ max{ f f ( ( t t ): ): t t ≤ ≤ x x }. }.

e) e) If If ( ( x,m x,m ) ) →f → f

_mm

( ( x x ) ) computable computable with with | | f f

_nn

- - f f

_mm

| |

_∞∞

≤ ≤ 2 2

^-n-n

+2 +2

^--mm

then

then

lim lim

_nn

f f

_nn is computable.is computable.

f) For

f) For computable computable a a ∈ ∈ , , f f :[0, :[0, a a ] ] → → , and , and

g To compute g To :[ :[ a a compute ,1] ,1] → → with with f f : : → → f f ( ( a a : : )= )= convert convert g g ( ( a a ) ) , , their their any any sequence sequence

join

join is is computable computable q q

_nn

∈ ∈

_{nn+1 +1}

with with | | x x - - q q

_nn

| | ≤ ≤ 2 2

^{--n n}

into into p p

_mm

∈ ∈

_m+1m+1

with with | | f f ( ( x x ) ) - - p p

_mm

| | ≤ ≤ 2 2

^--mm

Martin Ziegler

Computable Urysohn

Proof

Proof : : Let Let f f ( ( x x ):= ):= ∑ ∑

_mm

rr_mm rr_mm++εε_mm rr_mm--εε_mm

εε_mm//22^mm

Let Let ( ( r r

_mm

) )

mm

, , ( ( ε ε

_mm

) )

mm

⊆ ⊆ Q Q _{be be computable computable sequences sequences}

Then Then there there is is a a computable computable f f :[0;1] :[0;1] → → [0;1] [0;1]

s.t.

s.t. f f

^--11

[0] = [0;1] [0] = [0;1] \ \  

_mm

( ( r r

_mm

- - ε ε

_mm

^{, ,} r r

_mm

+ + ε ε

_mm

^{) ) . .} max(0,

max(0, ε ε _{m m} ^{- - | | x x - - r r}

_mm

^{|)/ |)/ 2 2}

^mm

C C

^∞∞

CC^{∞∞ ''}pulse' functionpulse' function

φφ((tt) = ) = exp(exp(--tt²²/1/1--tt²²))

||tt|<1|<1

Martin Ziegler

Specker'59: Uncomputable roots/argmin

approximating approximating a a root root vs vs . . approximate approximate root root

Lemma:

Lemma: There There are are computable computable sequences sequences

( ( r r

_mm

) )

mm

, , ( ( ε ε

_mm

) )

mm

⊆ ⊆ Q Q _{s.t. s.t.} U U := :=  

_mm

( ( r r

_mm

- - ε ε

_mm

^{, ,} r r

_mm

+ + ε ε

_mm

^{) )}

contains

contains all all computable computable reals reals in [0;1] in [0;1]

and has

and has measure measure < < ½ ½ . .

Corollary

Corollary : : There There is is a a computable computable C C

^∞∞

f f :[0;1] :[0;1] → → [0;1] [0;1] s.t. s.t. f f

^--11

[0] [0] has has measure measure > > ½ ½

but but contains contains no no computable computable real real number number . . Let Let ( ( r r

_mm

) )

mm

, , ( ( ε ε

_mm

) )

mm

⊆ ⊆ Q Q _{be be computable computable sequences sequences}

Then Then there there is is a a computable computable f f :[0;1] :[0;1] → → [0;1] [0;1]

s.t.

s.t. f f

^--11

[0] = [0;1] [0] = [0;1] \ \  

_mm

( ( r r

_mm

- - ε ε

_mm

^{, ,} r r

_m

^{C C}

_m

+ + ε ε

_mm

^{) ) . .}

∞∞

Martin Ziegler

Singular Covering of Computable Reals

Lemma:

Lemma: There There are are computable computable sequences sequences

( ( r r

_mm

) )

mm

, , ( ( ε ε

_mm

) )

mm

⊆ ⊆ Q Q _{s.t. s.t.} U U := :=  

_mm

( ( r r

_mm

- - ε ε

_mm

^{, ,} r r

_mm

+ + ε ε

_mm

^{) )}

contains

contains all all computable computable reals reals in [0;1] in [0;1]

and has

and has measure measure < < ½ ½ . . Proof

Proof : : Dove Dove - - tailing tailing w.r.t w.r.t . ( . ( M M , , t t ) ) : : If If Turing Turing machine machine # # M M within within t t

( ( but but not not t t - - 1 1 ) ) steps steps prints prints a a

₁₁

, , … … a a

_MM+5+5

s.t.

s.t. | | a a

_kk

/2 /2

^kk+1+1

- - a a

_ℓℓ

/2 /2

^ℓℓ+1+1

| | ≤ ≤ 2 2

^--kk

+2 +2

^--ℓℓ

∀1 ∀ 1 ≤ ≤ k k , , ℓ ℓ ≤ ≤ M M +5 +5

then then let let r r

_{〈〈M,t〉M,t〉}

:= := a a

_MM+5+5

/2 /2

^M+6M+6

and and ε ε

_{〈〈M,tM,t〉〉}

:= := 2 2

^-M-M--55

, , else else r r

_{〈〈M,tM,t〉〉}

:= := 0 0 and and ε ε

_{〈〈M,t〉M,t〉}

:= 2 := 2

^{--〈〈M,t〉M,t〉-3-3}

. .

Machine

Machine computescomputes

r r ∈ ∈ R R

iffiff prints seq. prints seq.

a a

_nn⊆⊆ withwith

| | a a

_nn

/2 /2

^{n+1 n+1}

- - a a

_mm

/2 /2

^{m+1 m+1}

| | ≤ ≤ 2 2

^{--n n}

+ + 2 2

^--mm

. .

Martin Ziegler

• • g g

_kk

( ( x x ):= ):= g g ( ( ( ( x x · · 2 2

^ψψ((kk))

- - 1) 1) · · 2 2

^kk

) ) / / 2 2

^ψψ(k(k))

h' h' := := ∑ ∑

_kk

g g

_kk

• • g g

_kk

( ( x x ):= ):= g g ( ( x x · · 2 2

^ψψ((kk))

- - 1 1 ) ) / / 2 2

^ψψ(k(k))

n n =1 =1 n n =2 =2

½½

¼¼

Fact Fact : : ∃ ∃ ^{computable computable bijection bijection} ψ ψ : : → → H H

• • g g and and ∫ ∫ g g computable computable

⅛⅛

½½

¼¼

⅛⅛ 11

00

Myhill'71:

Myhill'71: uncomputable uncomputable ∂∂∂∂ ∂∂∂∂ on on C C

¹¹

[0,1] [0,1]

yet yet h h := := ∫ ∫ h' h' ∈ ∈ C C

¹¹

[0;1] [0;1] computable computable . .

incomputable incomputable , ,

hat hat function function g g

• • g g

_nn

( ( x x ):= ):= g g ( ( x _x · · 2 2

ⁿⁿ

- - 1 1 ) ) /2 _/2

ⁿⁿ

• • ∫ ∫ g g

_kk

≤ ≤ 2 2

^--kk

continuous continuous , ,

∑ ∑ ∑

_nn∈∈HH

g g

_nn

∑

_nn

g g

_nn

e.g e.g . . H H ={2,3,5,...} ={2,3,5,...}

q.e.d

q.e.d . .

Martin Ziegler

The Case of the Wave Equation

Pour Pour - - El&Richards'81 construct a computable El&Richards'81 construct a computable ƒ∈ ƒ∈ C C

¹¹

( (

³³

) ) such that for

such that for g g :=0 the unique solution is :=0 the unique solution is in in computable at computable at t t =1 and =1 and x x =(0,0,0). =(0,0,0).

∂ ²/ ∂ t² u(x,t) = ∆ u(x,t), u(x,0)= ƒ (x), ∂ / ∂ t u(x,0)=g(x)

Church

Church - - Turing Turing Hypothesis Hypothesis ( ( Kleene Kleene ): ):

Everything

Everything that that can can be be computed computed by by a a Turing

Turing machine machine can can also also be be computed computed by by a a physical physical device device – – and and vice vice versa versa ! !

Myhill'71:

Myhill'71: computable computable h h ∈ ∈ C C ^{1 1} [0,1] [0,1]

with with uncomputable uncomputable h h '(1) '(1)

Martin Ziegler

The Case of the Wave Equation

Myhill'71:

Myhill'71: computable computable h h ∈ ∈ C C ^{1 1} [0,1] [0,1]

with with uncomputable uncomputable h h '(1) '(1)

Pour Pour - - El&Richards'81 construct a computable El&Richards'81 construct a computable ƒ∈ ƒ∈ C C

¹¹

( (

³³

) ) such that for

such that for g g :=0 the unique solution is :=0 the unique solution is in in computable. computable.

∂ ²/ ∂ t² u(x,t) = ∆ u(x,t), u(x,0)= ƒ (x), ∂ / ∂ t u(x,0)=g(x)

Kirchhoff's Kirchhoff's

formula

formula : :

Martin Ziegler

up to

up to permutationpermutation [Specker'67][Specker'67]

Example

Example fund. fund. theoremtheorem of of algebraalgebra:: Given

Given

a a

₀₀

, , … … a a

_d-d-11

∈ ∈

, _, returnreturn rootsroots

x x

₁₁

, , … … x x

_dd

∈ ∈

of _of

a a

₀₀

+a +a

₁₁

· · X+ X+ … … +a +a

_d-d-11

· · X X

^d-d-11

+X +X

^dd

∈ ∈ [ [ X X ] ]

incl. multiplicitiesincl. multiplicities

Two Effects in Real Computability a) a) Multivalued Multivalued ' ' functions functions ' '

b) b) Discrete Discrete ' ' advice advice ' '

Example

Example floorfloor functionfunction: : givengiven

x x ∈ ∈

, _, returnreturn itsits least integer

least integer upperupper boundbound Given

Given

x x

, , returnreturn somesome integer integer upper

upper bound: bound: computablecomputable!!

Example

Example matrixmatrix diagonalizationdiagonalization: : givengiven

A A ∈ ∈

^{d·d·(d(d--1)/21)/2},_, return

return a a basisbasis of of eigenvectorseigenvectors

ThmThm:: ComputableComputable knowingknowing

| | σ σ ( ( A A )| )|

..^ε·

cos(1/ε) sin(1/ε) sin(1/ε) −cos(1/ε)

―

― discontinuousdiscontinuous::

―― discontinuousdiscontinuous..