Computable Urysohn

Martin Ziegler

Computable Urysohn

Proof

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

rr_mm rr_mm++εε_mm rr_mm--εε_mm

εε_mm//22^mm

LetLet

( ( r r

) )

mm

, , ( ( ε ε

) )

mm

⊆ ⊆ Q Q

ThenThen therethere isis a a computablecomputable

f f :[0;1] :[0;1] → → [0;1] [0;1]

s.t.

s.t.

f f

[0] = [0;1] [0] = [0;1] \ \  

( ( r r

- - ε ε

^{, ,} r r

+ + ε ε

^{) )}

max(0,

max(0, ε ε

^{- - | | x x - - r r}

^{|)/ |)/ 2 2}

CC^∞∞

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

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

||tt|<1|<1

Martin Ziegler

Specker'59: Uncomputable roots/argminapproximatingapproximating aa rootroot vsvs. . approximateapproximate rootroot

Lemma:

Lemma: ThereThere areare computablecomputable sequencessequences

( ( r r

) )

mm

, , ( ( ε ε

) )

mm

⊆ ⊆ Q Q

U U := :=  

( ( r r

- - ε ε

^{, ,} r r

+ + ε ε

^{) )}

contains

contains all all computablecomputable realsreals in [0;1]in [0;1]

and has

and has measuremeasure <<½½..

Corollary

Corollary:: ThereThere isis a a computablecomputable CC^∞∞

f f :[0;1] :[0;1] → → [0;1] [0;1]

f f

[0] [0]

butbut containscontains nono computablecomputable real real numbernumber.. LetLet

( ( r r

) )

mm

, , ( ( ε ε

) )

mm

⊆ ⊆ Q Q

ThenThen therethere isis a a computablecomputable

f f :[0;1] :[0;1] → → [0;1] [0;1]

s.t.

s.t.

f f

[0] = [0;1] [0] = [0;1] \ \  

( ( r r

- - ε ε

^{, ,} r r

+ + ε ε

^{) )}

∞∞

Martin Ziegler

Singular Covering of Computable Reals

Lemma:

Lemma: ThereThere areare computablecomputable sequencessequences

( ( r r

) )

mm

, , ( ( ε ε

) )

mm

⊆ ⊆ Q Q

U U := :=  

( ( r r

- - ε ε

^{, ,} r r

+ + ε ε

^{) )}

contains

contains all all computablecomputable realsreals in [0;1]in [0;1]

and has

and has measure measure <<½½.. Proof

Proof:: DoveDove--tailingtailing w.r.tw.r.t. (. (

M M , , t t ) )

# # M M

t t

((butbut notnot

t t - - 1 1

a a

, , … … a a

s.t.

s.t.

| | a a

/2 /2

- - a a

/2 /2

| | ≤ ≤ 2 2

+2 +2

∀1 ∀ 1 ≤ ≤ k k , , ℓ ℓ ≤ ≤ M M +5 +5

thenthen letlet

r r

:= := a a

/2 /2

ε ε

:= := 2 2

r r

:= := 0 0

ε ε

:= 2 := 2

Machine

Machine computescomputes rr∈∈RR

iffiff prints seq. prints seq. aa_nn⊆⊆ withwith ||aa_nn/2/2^{n+1 n+1} -- aa_mm/2/2^{m+1 m+1} ||≤≤22^{--n n} ++22^--mm..