Theorem: For For r r ∈ ∈ , ,

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 iff iff an an algorithm algorithm can can print print , ,}

on on input input n n ∈ ∈ ^{, , some some} 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∈∈