• Keine Ergebnisse gefunden

Computability on metric spaces

N/A
N/A
Protected

Academic year: 2022

Aktie "Computability on metric spaces"

Copied!
28
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

deposit_hagen

Publikationsserver der Universitätsbibliothek

Mathematik und

Informatik

Informatik-Berichte 21 – 09/1981

Computability on metric spaces

(2)
(3)

It is shown that certain effective metric spaces can be reasonably embedded into (weakly) effective algebraic cpo's. The construction is similar to that one given by Lacombe, i t is, however, more

effective. Computability of points and of functions for the metric spaces can be naturally derived from computability for effective cpo's, for which a nice theory already exists.

A generalized version of a theorem of Kreisel/ Lacombe / Shoenf ield can be proved in this framework.

(4)

0. Introduction

Computability on metric spaces has been studied already

by several authors (Lacombe [1], Moschovakis [2], Bishop [3], Martin-Löf [4]). The important basic problems are how to reasonably define the computable elements and the computable functions. In this approach we solve the pro- blem by embedding the metric space under consideration into an effective cpo (Weihrauch [5]). Computability of elements and functions then automatically follows from computability theory on effective cpo's. The approach is based on Lacombe's [1] idea of the space of approxima- tions. The main difference is that Lacombe uses actual sets ordered under inclusion while here "formal" balls are ordered under "formal" inclusion. This formal inclu- sion relation is recursively enumerable even in cases where ball inclusion is not. Therefore, this approach is more general. In Section_ 1, a certain rather wide

class of metric space is considered. A contruction is given which, for any space

-

M of this class, yields an effective cpo D with weight such that the metric space

~ M can be embedded topologically into D . This is an effective version of ·a theorem proved by Schreiber and Weihrauch [6]. In Section 2, it is discussed, how com- putability on D, a cononical theory [5,7], implies

computability on the metric space: the computable elements of the metric space are the computable elements with

weight O of the cpo, and a computable function between metric spaces is the "trace" of a computable cpo-function.

The theorems of Rice, Rice/Shapiro, and Myhill/Shepherdson for effective cpo's (Weihrauch/Deil [7]) have interesting consequences. The domains of the computable functions on metric spaces are characterized. In Section 3, a theorem proved by Ceitin [8] and Kreisel/Lacombe/Shoenfield [9]

for special cases, is generalized. Our approach seems to be a very natural background also for this theorern (a

(5)

similar version of which has also been proved by Moscho- vakis [2]). In the conclusion, Section 4, it is argued that this approach to computability via cpo 1s is quite reasonable.

We shall use the following notations:

Nm(X) = set of all (total) numberings of the set X ,

~+ = set of positive rational numbers

< >

p

P ( n ) R ( n)

= standard pairing function,

= some standard numbering of ~+ ,

= set of n-ary partial recursive functions,

= set of n-ary total recursive functions,

= standard numbering of p(l) .

A partial numbering of a set X is a,possibly partial, surjective funct..ion v: lN ---? X . Let v.: 1 lN --->

x. ,

1

be partial numberings (i = 1,2). Then a partial function f:

x

1 --->

x

2 is (v 1,v

2)-computable, iff there is some gEP(l) such that (ViEdom(v

1)) fv

1(i)=v2g(i).

(6)

1. Definition of the cpo and its relation to the metric space

We first specify the metric spaces for which we shall develop our theory.

The following assumptions will be made henceforward.

F = (F,d) is a complete metric space, CSF is dense in ~ F,

y E Nm(C) is a numbering of C satisfying (M 1) (Ml) {(i,j,k)I d(y(i),y(j)) < p(k)} is r.e.

The set of open bal 1 s B(c ;E:)

=

F , where c E C , and

E: E (!)+ i s a base of the topol ogy of F . Let M be a G

0-subset of F , computable w.r.t y:

M== n U.

i E lN 1

with

Li;:= _u B(yh1(i,j);Ph2(i,j))

JEIN

where h

1 and h2 are two tot~l recursive functions.

A theory of computability will now be developed for the metric space

M= (M,d) •

(More precisely,

-

M = (M,d 1 ) where d1 =.d IM x M) •

Note that M is determined by three recursive functions;

the enumeration in (M 1), h1 , and h2 .

The class of metric spaces satisfying these conditions in large. Especially the Euclidean space, Baire1s space of all the functions JN-IN and many Polish spaces are examples.

The first step, now,is to define an appropriate effective partial order of 11finite approximate values11

Let

a : IN - A s

Ilf

u { 1. A}

be a numbering with

(7)

a(O) = 1.A

( i , j , k ) , i f 11 ( vm :s; k ) ( 3 n )

a ( 1 + < i , j , k , t>) : = d(yh 1 (m,n) ,y(i)) < gh 2 (m,n) - p(j)"

can be proved within

J. A , and

A : = range a

.

Let II :s; II be a relation a s; b <=>

for a l l a, b E A 1.A<(i,j,k) and

( a = b , Where

on or

at most t steps otherwise

A defi ned by : a < b)

( i , j , k) < ( i ' , j ' , k' ) : <=> ( d (y (i) , y ( i ' ) ) < p ( j ) - p ( j ' ) a nd k < k') for all (i,j,k)EA and (i',j',k')EA.

Then define

~ A: = (A,:s;,a) .

These definitions are the crucial ones. An element (i,j,k) corresponds to the ball B(y(i);p(j))~F, and (i,j,k) E A means that B(y(i) ;p(j)) is a subset of some ball used in the definition of Um for all m :s; k, espesially B(y(i) ;p(j) c n U

m:s;k m The relation (i,j,k)<(i',j',k') implies

B(y(i'),p(j')+s)s:B(y(i);p(j)) for some s>O.

Lacombe [1] and Martin-Löf [4] introduced a 11space of approximations11 An approximation is a set of balls

satisfying certain properties using actual ball inclusions.

Ball inclusion, however, may be in rr ~

2 or for M in rr 3 of Kleene's hierachy and is not r.e. in general.

For abtaining a sufficiently general effective theory, instead of balls we use "formal" balls and add using . the third component k an information "'= n U 11 , and

m:s; k m

instead of inclusion we use the stronger property 11<11

(8)

The numbering et and the order 11::;;11 an A are defined such that A is an ~ 11effective11 partial order:

Lemma 1:

(1) (A,s;) is a partial order with minimum .LA . (2) {(m,n)I a(m) < et(n)} is r.e.

Proof:

(1) (use triangle inequality for d )

(2) et is defined such that {nl et(n) = .LA} decidable.

Therefore, if et(n)

*

.LA , the three components

i,j,k with et(n) = (i,j,k) can be determined easily from n . Axiom (M 1) now immediately implies that et(m) < et(n) is r.e.

Q.E.D.

Any partial order with minimum can be completed into an algebraic cpo (see e.g. Markowski/Rosen [10] or Weihrauch/Oeil [7]). The completion of A then corres- ponds to the 11space of approximations11 considered by Lacombe [1] and Martin-Löf [4]. One advantage of the cpo- approach is that we can use the full power of the theory of effective cpo-s (Weihrauch/Deil [7], Weihrauch [5]), Here, we shall only give short definitions. and summarize properties which we need.

Let (B,s;) be a partial order, X!:; B is directed, iff X=t=0 and (Yx,yEX)(3zEX)(xs;z and ys;z). XSB is d O W n Ward S c 1 o Se d , i ff (\IX , y E ß ) ( X ::5 y E X => X E X ) . ( ß , ::5)

is complete, iff UX exists for any directed X=B A cpo is a complete partial order with minimum.

Now the completion

~ D = (D,B,s;,.L,ß)

of ~ A (see above) is defined. The construction, of course, is a general one, however we only need it for the case of Ä here.

(9)

D = { X :: A 1 X directed and downwards closed} ,

l. ( a ) = { b E A b:::;a}ED for a l l a E A ,

B = 1 ( A) ,

X:::; y : = ~> y S X

.L = {.LA} ,

s (

i ) = 1 Cl ( i )

.

~ D is algebraic and 11effective11 as defined by [5]. Note, that D is not boundedly complete ß(i):::; ß(j) is not decidable in general, i.e.

Weihrauch and that

~ D is not necessarily effective in the sense of Kanda/Park [11] or Egli/Constable [12] .

D is complete, and UX=UX for any directed subset

xs

D. B is the algebraic basic of D: {bEB I b:::;x}

is directed and x = U{b I b:::; x} for any x ED , and (b :::;UX ~ (3x E X)b:::; x) for any directed subset of X s D

~ ~

The mapping 1 embeds A into D Obviously {(i,j) 1 ß(i)=::;ß(j)} is r.e.

The author [7] has shown that a very natural and rich theory of computability can be developped for effective cpo's. This will be discussed later. Here we shall study

~ ~

the relation between the metric space M and the cpo D For formulating the main theorem, we define weight for

~ ~

A and D .

!.LAI = oo ,

l(i,j,k)I = p(j) for (i,j,k)EA

lxl = inf{ lal laEx} for xEO Oefine:

M': = {xEDI lxl = 0}

M' i s the set of the "most precise" elements of D We also need the canonical topology of a cpo

For ~ D = (D,B,:5,.L,ß) the canonical topology i s de- fined by the topological base

where

Ob = { X E O I b:::; X}

(10)

Let ,

0 be the topology induced by ,

0 on M' • Finally, let "M be the :opology induced on M by M=(M,d). Let o

1 an o

2 betwocpo's.Then f:o 1-o

2 is called coutinuous (more precisely (

1

,5

2)-continuous) iff f is isotone and preserves limits of directed sets.

Recall that

(0

1

,5

2)-continuity coincides with (, 0 ,,

0 )-

1 2

continuity (see e.g. Weihrauch/Oeil [7]).

The next lemma will lead to our main theorem connecting M with ~ 0 •

Lemma 2:

(1) Suppose, xEM. Then

y: = {1.A}u{(i,j,k)EA I xEB(y(i),p(j))}EM' (2)Suppose yEM' .Then

n{B(y(i),p(j)) 1 (3k)(i,j,k) Ey} = {x}

for some x EM . Proof:

(1) First, we prove that y is directed. Since 1.A is the minimum for 1.A there is nothing to prove. Suppose, (i,j,k)Ey and ( i ' , j ' , k ' ) E y . Oefine k: =l+max(k,k').

Si n c e x E B ( y ( i ) , p (j ) ) n B ( y ~ i ' ) , P ( j 1 ) ) n n {_Um ~ m ~ k} , and since C is dense in F , there are i , j with

--- - - - ---

(i,j,k)EA, (i,j,k)<(i,j,k), and ( i ' , j ' , k ' ) < ( i , j , k ) . Therefore, y is directed. y is also downwards closed, since (i,j,k)<(i',j',k')Ey implies

XE B(y(i'),p(j')) S B(y(i),p(i)) =;> (i,j,k) Ey.

(2) Since y is directed and IYI = 0 ,

(v(i',j',k') Ey)(3(i,j,k) EY)(i',j',k')< (i,j,k),

-

-

and therefore B(y(i),p(j))::B(y(i'),p(j')). (B denotes the closed ball.)

(11)

Then:

ri {B( ... )

-

(:lk)(i,j,k) Ey}

s:ri {B( ... ) (:lk)(:li 1 , j ' ,k' )(i 1 ,j 1 ,k1 ) < (i,k,j) Ey}

s=ri {B( ... ) (:lk)(i,j,k) Ey}

::ri {B( ... )

-

(:lk)(i,j,k) Ey}

Therefore, we can consider the intersection of the corres- ponding closed balls. Since y is directed, the set of the closed balls is a directed set of closed subsets of F . Since ~ F is complete, and there are balls with arbitrarily small radius, the intersection contains exactly one point XE F •

Since (V(i 1 ,j 1 ,k1 ) Ey)(:l(i,j,k) Ey)(i 1 ,j 1 ,k 1)< ( i , j , k ) , xEB(y(i),p(j)):: ri Uk holds for arbitrarily large k ,

m:s;k therefore x EM . Q. E. D.

Define K: M-D by

K(X): = {.1A}U{(i,j,k)EA I xEB(y(i);P(j))}

and

>.. : M 1 - M by

{>..(y)}: =ri {B(y(i); p(j)) 1 (:lk)(i ,j,k) Ey}

Theorem 3:

>.. is bijective and A -1 = K

( 1 )

( 2 ) K is (TM,T

0)-continuous.

Proof:

1 MI •

(1) We prove (vxEM)>..K(x)=x and (vyEM1 )K>..y=y.

Suppose, x EM . Then

(12)

{;I.K(x)} = n{B(y (i); p(j)) (3k) ( i ,j ,k) E K(X)}

=n{B(y(i);p(j)) (3k)[(i,j,k)EA and

X E ß ( y ( i) ; P (j) ) ]}

Obviously, xEn{ ... } . By Lemma 2, the intersection consitsts of exactly one point, therefore 11.K(X) = x Now, suppose y EM' . Then

K11.(Y)

= {.LA}U {(i,j,k) EA I 11.(y) Eß(y(i);P(j))}

= {.LA} u {(i,j,k) E A 1 (3(i1,jl'kl) Ey)(i,j,k) < (il'jl'kl)}

(Use the definition of "<", use that B(y(i);p(j)) is open, that yEM', and that C is dense in F .) If (i,j,k)Ey then there is (i

1,j1,k1)EA with

(i,j,k) < (il'jl'kl) since y is directed and IYI = 0 , therefore (i ,j ,k) E K\(y) . If (i ,j ,k) E K11.(y) , then

(i,j,k) < (i

1,jl'kl) for some (i 1,j

1,k1J Ey. Since y is downwards closed, we have (i,j,k)Ey. Therefore K\(y) =y.

(2) It is sufficiant to prove, that K -1 (Oda)) is o p e n f o r a ny a E A . ~ f a = .LA , t h e n K-1 0 =K-1D=M

l ( a)

which is open. Furthermore, for any (i,j,k)EA:

xEK -1 a ( . . k)<-'=>K(x)Ea ( . . k) l 1,J, t 1,J,

<=;> (i ,j ,k) E K(X)

<.=;>XEB(y(i);p(j)),

therefore K-lai(i,j,k) = B(y(i) ;P(j)) , which is open.

(3) Suppose U E 'M . Then U = u* n M for some u* which is open w. r.t.

F.

We show that K(U) E 'M' .

For this, we prove

K(U)=M'n Ü{a ( " . k) 1 (i,j,k)EA and B(y(i);P{j))sU*}

t l 'J ' Suppose x EU .

By Lemma 2, there is (i,j,k) E K(x) with

(13)

ß(y(i);P(j))::U*. But (i,j,k)EK(X)=;>

1.(i,j,k)::s;K(x) ;>K(x)EO ( ' . k). Therefore we have

t l 'J '

proved 11'.::11

Suppose, yEM'nU{ .... } . It suffices to show that

;i._(y)EU. There is (i,j,k)EA with yEO ( ' . k),

. . . . * 1 l,J,

i.e. (1,J,k)Ey, and B(y(1};P(J))SU .

8 U t ( i , j , k) e Y =;> A ( Y) E 8 ( y ( i ) ; P ( j ) ) by t h e definit i On of ;i._ , therefore ;i._(y) Eu* and ;i._(y) Eu* n M = U

Q. E. D.

Theorem 3 guarantees that the cpo ~ D is suitable for studying continuity on the metric space ~ M

It says~that TM , the topology induced by space M , and T

0

1

, the topology induced the subset M1 s D , are homeomorphic.

~

the metric by

To

on For i=l,2

and let

o.

l

let M; be. metric spaces defined as above be the corresponding cpo's with weight-0- subsets M . .'

l

For f:

o

1 -

o

2 define

Def (f): = M 1

1 n f-1M21

and Tr(f): Def (f) -M2

1

, the trace of f , by

(vx E.Def (f)) Tr(f) (x) = f(x) .

If f is continuous then Tr(f) obviously is continuous.

Weihrauch/Schreiber [6] have proved that for any continuous g: X-M

2

1

, where X:: M21 , there is a continuous f:

o

1

-o

2

such that Tr(f) extends g , under the assumption that

~ D2 is boundedly complete. ~ We have already defined weight for D •

It is easy to define some kind of distances on D such that ;i._ becomes an isometric mapping.

Define two distances d> and d< on A as follows:

d>(.LA,a): =eo, d<(.LA,a): = 0

d> ( ( i , j , k) , ( i 1 , j 1 , k 1 ) ) : = d ( y ( i ) , Y ( i 1 ) ) + P ( j ) + P ( j 1 )

(14)

d<((i,j,k),(i 1 ,j 1 ,k 1) ) =d (y(i),y(i 1 ))-P(j) -p(jJ Then d>(a.b) = d>(b,a) and d<(a,b) = d<(b,a) .

Suppose a::; a 1 Then

and

d<(a,b)::; d<(a 1 ,b) ,

i.e. d> is an antitone and d< is an isotone function into (lR ,::;)

d> and d< can be 11extended11 to D as follows:

d>(x,y) = inf{d>(a,b) aEx,bEy}

d<(x,y) = sup{d<(a,b) aEx,bEy}

The following lemma connects d>' d< and d Lemma 4:

(vx,y EM') d> (x,y) = d<(x,y) = d(;\. (X) ,A (y)) Proof: (easy)

This means, d> and d< coincide on M' ' and ;\. i s an isometric mapping w.r.t. d> (or d< ) and d

.

We may, therefore, forget the metric space ~ M and study only the 11metric c po 11 (D, 11 ,d>,d<) ~

(15)

2. Computability on the cpo and on the metric space As we have proved, D = (D,B,:5,.L,ß) ~ is an effective algebraic cpo. For such cpo 1s there is a standard theory of computability. We refer to the author's papers [5] or [7] for this theory.

An element x ED is computable, iff

{i I ß(i) :5 x} is r.e . . The set of computable elements De of D can be numbered admissibly.

A numbering 11 of De is admissible, iff (A 1) and (A2) hold.

(Al) {(i,j) ß(i)::;11(j)} is r.e.

(A 2) There is some ,5 E R(l) with 11-6(i) =UßW; , whenever ßW.

l is directed.

There is an admissible numbering 11 and it is deter- mined uniquely up to isomorphism. An admissible num-

bering is complete in the sense defined by Ershov [13].

Some consequences are the following ones:

The recursion theorem holds for 11 , i.e. there is some frER(l) with (vi)(<P;ER(1)=>11fr(i)=11<P;fr(i))

Rice's theorem holds for 11, i.e. 11-lx is not recursive whenever 0cxcDc.

The theorem of Rice/Shapiro holds for 11 . Especially, 11 -1 (Dc nY) is not r.e. and not co-r.e., whenever

0 ~ Y '.: M 1

The elements of M' represent the elements of M by Theorem 3. The elements of M' n De represent the computable elements of M . The definition generalizes the explicit definition of computable real numbers (Aberth [14]).

Let X:= 11 -1 (M'). Then 11

0 : X-M defined by 11 (i): = 11.11 (i) is a parial numbering cf the com-

o

putable elements of M . 11 generalizes the commonly

0

used partial numberings of the computable real numbers (Aberth [14]).

(16)

The advantage of the cpo-approach is that a very elegant theory is obtained by adding an appropriate set of approximate values. These values allow to extend the partial numbering into a total numbering which is complete. In the special case of functions on lN , by adding all the partial recursive functions, the partial canonical numbering of all the total

recursive functions can be extended into a numbering with very pleasant properties, namely ~ .

Hauck [15] studied representations of the set

m.

The canonical (most useful) representation of real numbers by total functions on lN also used by Grzegorczyk for defining computabil ity on

m

is essentially a restriction of an admissible represen- tation o defined and studied in_ full ge~erality by Weihrauch/Schäf~r [16].

There is a canonical definition of computable

functions between effective cpo's (Weihrauch/Deil [7]).

- -

Let o1 and D2 be two effective algebraic cpo's.

f:

o

1 -

o

2 is (D1

,o

2 )-computable, iff f is continuous and

{(i,~) 1 s 2(j):s;fs 1(i)} is r.e ..

Roughly speaking, the graph of the continuous function f must be r.e.

The generalized Myhill/Sheperdson theorem (Weirauch/

Deil [7]) states that f c: = ( f restricted to o1c) is (n

1,n

2)-computable. Clearly, also

f0 = ( f restricted to o1cnTr(f)) is (n 1,n 2 )-

computable.

The Myhill/Sheperdson theorem also states that when- ever some g: o1c-Dzc is (nl'n 2 )-computable then g = fc for some

(0

1

,5

2 )-computable f . Especially, g is continuous on Tr(f) n D1c.

(17)

We define computability of functions between metric spaces now.

Definition 5:

X M i h xcM, .

g: - 2 , w ere -

1 , 1s compu- A functions

table, iff function f

g = Tr(f) for some (51

2 )-computable This definition implies that computable functions on metric spaces have natural domains.

The class of these natural domains is characterized in Theorem 6.

It is not known, whether a more general definition

is useful. One could, e.g., require 11Tr(f) extends 911 But this generalization is not essential. There might, however, be intuitively computable functions which cannot be derived from a (total) computable cpo- function.

Weihrauch/Schreiber [6] have proved that a subset X'= M1 is a G

0-subset iff X= Def(f) for some continuous (strong) f . In our special case, every function is strong. We shall prove here the recursive version of this theorem.

A subset Ts D is an effective G ~

0-subset (w.r.t. D iff

T= 01--! Osg(i,j)

1 J

for some total recursive function g (remember:

Oa={xEDia:S;x}).

s

s M1 is an effective G

0-subset iff S = T n M' for some effective G

0-subset T of D .

(18)

Theorem 6:

( 1) Let f :

o

1 -

o

2 be (01

,o

2)-computable. Then 0ef(f) = 0 or 0ef(f) is an effective G

0-subset of M

1' •

( 2 ) L et S '= 0 1 , b e a n e ff e c t i v e ~ G O -~s u b s e t o f M 1

1

Then S = 0ef(f) for some (01' o

2) - computabl e function f , provided

M/

n o2c :t= 0 .

Proof:

(1) Suppose, 0ef(f) :t= 0 . For any x E o

1 we have:

f X E M 2

1

<=>

1 fx 1 = 0

<=>

(vn)(:im)( 1s -n

2(m)1<2 and s2(m)sfx)

<=>

( v n ) ( :l m ) ( :l k ) ( 1 ß 2 ( m ) 1 < 2 -n a n d ß

2 ( m ) s f ß

1 ( k ) a n d ß 1 ( k ) _s x ) . There is some h E R(l) such that

range and

(J)h(n)={kl(:Jm)( 1s2(m)1<2-n and

R(l) ,·f { } ~ w,·th

(J)h(n)E .... :t=p.

we obtain

<=>

Since Def(f) :t= 0 , there is some with Therefore, (vn)(j)h(n) E R(l) and

f X E M 2

1

<=>

( vn) x E u { 0 1 i E IN}

ßl(j)h(n)

<=>

s2 (m) s fs

1 (k)) b s X<=> XE C\

(19)

0efine g E R( 2 ) by Therefore f-lM '

2

g(n,i): = (j)h(n)(i) . is a G0-subset of o1

0ef(f) = M 1

1 n f-1M

2

1 is a G

0-subset of M 1 (2) Suppose, gER( 2 ) and

and

S = M1 ' n n u O (. . ) • Choose some fi xed i j ßlg l ,J

yEM2'no2c .Thereissome pER(l) with

s

2 p(i)::;s 2 p(i+l) and Y=[-:1s2p(i) 1i~eihrauch/0eil [ ]).

l

0efine f:

o

1 ....

o

2 as follows:

f(b). - U{ß2P(i) 1 (vk<i)(:3j)ß1g(k,j)::;b}

for bEB

1 , and f(x): = U{fb I b::s;x}

for x E o1 .

f is well defined on B

1 since any subset of a chain is directed. f is isotone on B

1 , therefore f is continuous.

f is computable, since s2(m)::;fs

1(n) holds iff·

(3i)(vk<i)(:3j)(s

2(m)::;s

2p(i) and s

1g(k,j)::;s

1(n)), and this is a r.e. relation.

Now, suppose, x E M

1' . Then

XE S

<=>

<=>

(vi)(vk<i)(:3j) xEOß1g(k,j) (vi)(vk<i)(:3j) s1g(k,j)::;x

<=>

<=>

<=>

y=U{fbib::;x}

(20)

<=>

y = f(x)

<=>

XE Oef(f)

Q. E. D.

Under certain circumstances it can be proved that if a function h: X-M2

1 with X s M1 ' is (nl'n 2)-compu- table, then h is continuous on X . For the real

numbers essentially this has been proved by Ceitin [8], and for functionals over ~ this is the theorem of Kreisel/Lacombe/Shoenfield [9]. A more general version is proved in an ad hoc approach by Moschovakis [2].

We shall prove this theorem in our framework under

assumptions similar to those of Moschovakis in Section 3.

(21)

3. The CKLS - Theorem

We shall prove now a theorem whieh generalizes two theorems proved by Ceitin [8] and by Kreisel/Laeombe/

Shoenfield [9]. It is very similar to a theorem proved by Mosehovakis [2].

Suppose M. and ~ D. are defined as in Seetion 1

1 1

for i = 1,2 . Suppose, the numbering also satisfies Property (M2):

(M2) {(i,j,k) 1 d(y

2(i),y

2(j))>p(k)} is r.e.

Axiom (M 1) impl ies that {(i,j,k) 1 d>(n

2(i),n

2(j))<p(k)} is r.e., and Axiom (M 2) impl ies that

· {(i,j,k) 1 d<(n

2(i),n

2(j))>p(k)} is r.e., . where ni is an admissible numbering of Die , the eomputable elements of Di (i = 1,2) .

Theorem 7:

Suppose X

s

M

1

1 n Die , suppose n2(E) is dense in X for some r.e. set E ~ lN. Suppose F: X-M2

1 n D2e is (n1,n

2)-eomputable. Then F is 11eomputably-eontinuous 11 on X, i.e. there is some rEP(2

) with ( vm) ( v y E n

1 -1

(X) ) [ F ( 0 ß

1 r ( Y , m) ) s B ( F n 1 ( y) ; P ( m) ) and

n1(Y) E Oß1r(y,m)]

(Balls are defined using the metrie d< on M21 . ) The proof generalizes the proof of the Kreisel/Laeombe/

Shoenfield- theorem given by Rogers ([17], P.362)

(22)

Proof:

Since

5

1 is an effective cpo, there is some q E R( 2 ) with

(vx,y)s1q(y,x) ~ s

1q(y,x + 1) and (vy)n

1(y) =Us

1q(y,x)

X

If Z

*

0 is an r.e. set then Z = range(f) for some fixed total computable function f . We shall say

11zEZ can be proved within i steps11 iff i=µi'[f(i')=z]

Since F is (n 1 n

2)-computable, there is some 't'EP(l) with

(vy E n 1 . X)Fn 1 (y) = n2'±'(Y) -1 Define a function h JN 4

---->lN as follows:

q(y,x) if 11d>(n

24'(z) ,n24'(y)) <jp(m) 11

cannot be proved within less then x steps q(k,x) if 11d>(n 24'(Z),n 24'(Y))<{p(m) 11

can be proved within w steps where- .,,., < x , h(x,y,z,m) = and k , the first number for which

11 kEE and

s1q(y,w) ~ n

1( k) and

d < ( n 2 4' (y) , n 2 4' ( k) ) >

j

p ( m) 11

can be proved, exists.

div otherwise.

Note, that w is a partial recursive function in Obviously, hEP( 4 ) by (Ml) and (M2) for D2 by (A 1) for o

1 . There is some p E R( 3

) with t~ ( ) = {h(x,y,z,m) x E lN} .

p y,z,m

We consider 3 cases.

(z,y,m) and

Case 1: 11d>(n24'(Z) ,n 24'(y)) <jp(m) 11 Then

does not hold.

W ( p y,z,m )={q(y,x)lxElN}

Case 2: 11d>( ... ) and k

< ... 11 can be proved vlithin w (not depending on x ) exits.

steps

(23)

Then

Wp(y,z,m) = {q(y,x) 1 x:::; w} u {q(k,x) 1 w < x}

Case 3: 11d>( ... ) < ... 11 can be proved within w steps but k does not exist.

Then

Wp(y,z,m) = {q(y,x) 1 x:::;w}

In Case 1 and Case 3, ß1W ( p y,z,m ) is directed.

In Case 2, ß1q(y,w)5n1(k), which also implies that ß 1 p(y,z,m)

w

is directed. By Axiom (A 2) for n

1 , n 1sp(y,z,m) =Uß 1Wp(y,z,m)

The recursion theorem for n

1 yields some t E R( 2 ) with

n 1 s p ( y , t (y, m) , m) = n

1 t ( y, m) Therefore,

where w'

Now, assume

n1(Y) if 11d>(n 2ft(y,m) ,n 2f(Y)) <jp(m) 11 does not hold,

n1(k) if 11d>(n

2ft(y,m),n 2f(Y))<jp(m) 11 can be proved within w' steps and k, the first number for which

11 kEE and

ß1q(y,w'):::;n1(k) and d<(n2f(Y) ,n 2f(k)) >jp(m)11 can be proved, exists ß1(y,w') otherwise,

stands for w(t (y,m) ,y,m) yEnl -1 (X).

In Case 1, n

1(y) =n

1t(y,m), i .e. n2f(y) = n2ft(y,m), which contradicts Case 1.

In Case 2 n1t(y,m) = n 1 (k) , i .e. n 2ft(y,m) = n21f'(k) Since d>=d< on M

2

1

, there is a contradiction.

(24)

Case 2 is, therefore, not possible. This means

( v k E E ) ( ß l q ( y, w 1 ) ::; n l ( k) ;> d < ( n 2 '±' (y) , n 2 '±' ( k) ) ::; j p ( m))

Therefore, only Case 3 is possible. Especially w1 exists.

Define r(y,m) = q(y,w(t(y,m) ,y,m))

We prove that r has the desired properties.

If y E n1 -1 (X) , then r(y,m) always exits. Clearly

s

1r(y,m)::; n

1 (y) , therfore n

1 (y) E 0 . . .

SL ppose, v E n

1-1

x .

Then substituting y by v we obtain (Vk E E) (ß1q(v,~)::; n1(k) ;:, d<(n2'±'(V) ,n2'!'(k)) ::;jp(m))

for some w .

Suppose, n 1v E os

1r(y,m) . Then s1q(y,w1 ) : : ; n 1 (v) , and, since n

1(E) is dense, there is some k with ß1q(y,w1 ) : : ; n1(k) and ß1q(v,w)::; n1(k) . The triangle unequation immediately yields

d<(Fn 2

1 (y) ,Fn1 (V))= d<(n 2'±'(Y) ,n 2'!'(v)) ::;"Jp(m) < p(m) This prove the theorem.

Q . E . D .

(25)

4. Conclusion

A general definition of computability for metric spaces has been given by reduction to computability on effective cpo's. This definition is applicable to almost all 11effictive11 metric spaces considered so far and yields the standard

kind of computability. Any computable function has a computable G0-set as it natural domain. We suggest that this natural domain should be respected in the more detailed study of computability, e.g. in recursive analysis.

This definition generalizes the definition from the literature using 11computations on intervals11 Let us call it the 11 interval -definition11 Let us call the

k i n d o f c o m p u t ab i l i t y de f i n e d v i a II e ff e c t i v e 11 n um b er i n g s on the computabl e el ements the 11numbering - definition11 By the Myhill/Shepherdson theorem the interval - definition implies the numbering definition. Therefore, the num- bering - definition is at least as powerful as the interval - definition restricted to the computabl e elements.

The CKLS - theorem can be interpreted in such a way that under certain very strong restrictions the num- bering -definition impl ies the interval - definition.

Aberth [14] in his book on recursive analysis uses the numbering - definition, but a careful study of his proofs sho\vs that he essentially uses the interval - definition, i.e. any property he proves can be proved from the

interval - definition (and then restricted to the computable elements).

We claim here, that the interval - approach is the more natural one and will result in a nicer theory. Some reasons are the following ones. The interval - approach is rather direct, while the intuitive meaning of opera- tions on numbers is not so clear, since it is hidden by a numbering which we do not understand directly. The

(26)

interval-approach defines computable functions on all the elements of the metric space and not only on the com- putable ones. The interval approach leads to theories of computational complexity of elements (Weihrauch [5]) and of functions (Ko [18]). And finally the greater generality of the numbering definition has not been used and, the author suspects, has not been fully understood until today.

(27)

References

[1] Lacombe, D., Quelques proc~des de definition en topologie recursive. Constructivity in mathematics, 129 - 158. North - Holland, Amsterdam, 1959.

[2] Moschovakis, Y.N., Recursive metric spaces, Fundamenta mathematical LV (1964) 215 - 238.

[3] Bishop, E, Foundations of constructive analyis, McGraw-Hill, New York, 1967.

[4] Martin-Löf, P., Notes on Constructive Mathematics, Almquist & Wiksel l, Stockholm, 1970.

[5] Weihrauch, K., Recursion and complexity theory on cpo1s, in: P.Deussen (ed.), Theoretical Computer Science,

5th GI - Conference 1981, pp 195 - 202.

[6] Weihrauch, K., Schreiber, U., Embedding metric spaces into cpo 1s, Jheoretical Computer Science 16 (1981) (to appear).

[7] Weihrauch, K., Deil, T., Berechenbarkeit auf cpo 1s, Informatik Bericht Nr.63 (1980) , Technische Hoch- schule Aachen.

[8] Ceitin, G.S., Algorithmic operators in constructive complete seperable metric spaces (Russian),

Doklady Akad. Nauk 128 (1959), 49 - 52.

[9] Kreisel, G., Lacombe, D., Shoenfield, J.R., Partial recursive functions and effective operations,

Constructivity in mathematics 129 - 158, North-Holland, Amsterdam, 1959.

[ 1 0 ] Mark o w s ky , G . , Rosen , B . K . , Base s f o r Ch a i n - Co m p l et e Posets, IBM Research Report RC 5363, April 1975.

[11] Kanda, A., Park, D., When are two effectively given domains identical, in: K. Weihrauch (Hrsg.),

Theoretical Computer Science 4th GI - Conference, S. 170 - 181, Springer, Berlin 1979.

(28)

[12] Egli, H., Constable, R.L., Computability Concepts for Programming, Language Semantics, Theoretical Computer Science 2, 1976, 133 - 145.

[13] Ershov, J.L., Theorie der Numerierungen I, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 19 (1973), 289-388.

[14] Aberth, 0., Computable analysis, McGrow-Hill, New York 1980.

[15] rlauck, J., Berechenbare reele Funktione, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 19 (1973), 121-140.

[16] Weihrauch, K., Schäfer, G., Admissible representations of effective cpo1s, Informatik Berichte 16 (1981)

Fernuni Hagen.

[17] Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York 1967.

[18] Ko, K., Friedman, H., Computational complexity of real functions, Technical report, 1980, Ohio State University.

Referenzen

ÄHNLICHE DOKUMENTE

The course aims to present the basic results of computability theory, including mathematical models of computability, primitive recursive and partial recursive functions,

The course aims to present the basic results of computability theory, including mathematical models of computability, primitive recursive and partial recursive functions,

Lower bounds on the distortion of embedding finite metric spaces in graphs. Metric spaces and positive

An upper bound of O(n 2/d log 3/2 n) for the distortion is obtained by first embedding the considered metric space into # n 2 (Theorem 15.8.1), and then project- ing on a

For performing k-NN queries, during peer ranking a list L q of reference object IDs i is sorted in ascending order according to d(q, c i ), i.e. The first element of L q corresponds

The course aims to present the basic results of computability theory, including mathematical models of computability, primitive recursive and partial recursive functions,

The course aims to present the basic results of computability theory, including well-known mathematical models of computability such as the Turing machine, the unlimited

The course aims to present the basic results of computability theory, including well-known mathematical models of computability such as the Turing machine, the unlimited