Representations of the Real Numbers and of the Open Subsets of the Set of Real Numbers

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Mathematik und

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Informatik-Berichte 43 – 02/1984

Representations of the Real Numbers

and of the Open Subsets of the Set of

Real Numbers

by

Klaus Weihrauch and Christoph Kreitz Fernuniversität

5800 Hagen W-Germany

theory of computability and continuity and a theory of representations. In this paper the concepts developed so far are used for the foundation of a new kind of construc- tive analysis. Different standard representations of the real numbers are compared. It turns out that the crucial differences are of topological nature and that most of the representations (e.g. the decimal representation) are not reasonable for topological reasons. In the second part some effective representations of the open subsets of the real numbers are introduced and compared.

1. Introduction

In the past repeatedly different representations of the set JR of real numbers have been investigated w.r.t. this utili- ty for recursive analysis. We only mention a few authors here: Turing [13], Specker L12], Grzegorczy [3J, Hauck [4], KO [6]; a very good survey is given by Deil [2].

As typical examples we shall consider the (unrestricted) Cauchy Representations, the normed Cauchy representations, the decimal representations, the representations by enume- ration of cuts and the representations by characteristic functions of cuts. The previous authors have compared these representations (mainly) w.r.t. computability aspects. We shall show here that the essential differences between the representations are already topological ones and do not depend on Church's thesis or some model of computation.

In the third chapter we shall investigate some natural re- presentations of the open subsets of the real numbers. This paper is based on two previous papers (Weihrauch [14],

Kreitz and Weihrauch [8]).

2. Representations of the Real Numbers

By JR we denote the set of real numbers. The standard topology 'R ^{on JR} can be defined by the basis

{ (x;y) ~ JR

I

x,y E JR ,x < y} of open intervals on JR. The space ( ^{JR , T} R) is a seperable T ^O -space. From Krei tz and Weihrauch [8] we know that there is an admissible represen- tation o of JR • Especially, , R is the final topology t 0 of o • For obtaining a reasonable theory of computabi- lity and computational complexity, we shall define an admis- sible standard representation p of the real numbers JR •

2.1 Definition

For nE ::N let ^<D :.={m • 2-n ¹

n mE~ ^} and let _<.12D: = i'i<Dn be the set of dyadic numbers. Let _vD be a bijective num- bering of _<.12D

,

which is equivalent to \)

'

^with

v'<i,j,k>

=

( i - j) ^{• 2} -k Define a numbering I of the open intervals with dyadic end points by

-k -k

I < i , k> :

=

I < i , k> :

= (

^{v D ( i ) - 2 ; v D ( i ) + 2 ) .}

Obviously

m

₀ is dense in JR and I is a numbering of a base of the open subsets of JR (w.r.t. ,R). The numberings

and I are "c-effective" insofar as all the basic function are computable and all the basic predicates are decidable

w.r.t. v

0 and I • In a previous paper [8] we already have shown how an admissible representation of a complete metric space can be defined using Cauchy sequences of normed conver- gence. Because of its importance we repeat the construction for our special case here.

2 . 2 De·f ini tion

11!1

Let a representation p : IF -- - JR be def ined as follows.

dom(p) := {pEIF ¹ (Vk)(v

0p(k) EWkAJ v

0p(k)-v

0p(k+1) j s;2-(k+' p (p) : = 1~m v

0p (n) for all p E dom(p) .

The representations öI of JR is defined as follows (c.f.

Kreitz and Weihrauch [8] , Def. 19):

dom (öI)

: =

{p ¹ ( 3x E JR) JM ={jJxEI.}} I

p J

c5I(p)

: =

the single ^X in {I,ljEJMp} for x E domcSI J

We shall call 6 the standard representation of JR •

The elements of dom(p) can be considered as names for normed Cauchy sequences. In the following diagram every descending path corresponds to a p E dom ( ~) and vice versa.

-1 0 1

We can see immediately that for any x E JR the paths conver- ging to x form a fini tely branched graph (tree) • This pro- perty is expressed in (4) of the following theorem which summarizes some important properties of ^{p •}

2.3 Theorem

(1) p=coI'

(2) p is admissible, (3) dom(p) is closed ( 4) -1

P K is compact if

Proof:

K C: JR is compact .

By [8] Definition 21, (2) follows from (1) . We have to prove p :::;;c cSI and öI :::;;c p . There is some computable r : JF -JF such that for all p E dom ( p) :

JMr(p)={j

= { j

-i - i c

( 3 i) [ v oP ( i) - 2 ; v oP ( i) + 2 ] _ I j } p(p) EI.} ,

J

hence p(p) = oir(p). Conversely i t is easy to define a compu- table b. : JF -- - JF (via an oracle Turing machine) such that for any p E dom ( ö I) the following holds: b. (p) E dom ( p) and for all n there is some j E JMP with

c -n -n

Ij _ [v

0^{1.:l (p) (n) -} 2 ; v

0^{1.:l (p) (n)} + 2 ] , hence pt:.(p)=ör(p).

Therefore we have proved p = o I . By Definition 2. 2 ,

C .

,1:

I -

^(k+1

JF\dom(p)=U{[a

0 • • • an]I (3k<n) (v

0(¾)Falkvl(v

0(ak)-v

0 (ak+

1, >2 Therefore dom ( p) is closed. Let A ~ dom ( p)

<:.

JF • Notice that A is compact in dom (p) , iff A is compact in JF . A subset A ~ JF is compact, iff i t is closed and growth-bounded (i .e.

(Vp EA) (Vn)p(n):::;; q(n) for some bound q E JF)

.

^{Let K ~ JR}

be compact. Then ^K is closed, therefore p -1K is closed dom(p) i.e. -1

for some B closed in in

'

^{p K} = dom ( p ) n B

JF • By _{( 3) ' p} -1 K is closed. I t remains to show that p -1K is growth-bounded. Since K is compact, K is bounded. Let n E JN wi th K <; [ -n + 1 ; n - 1 ] <; JR •

Define q E IF by q (k) : =max { i Then (Vp E p-¹K) (Vk)p(k):;:; p(k) Q.E.D.

v D ( i ) E [ -n ; n ] n Olk } •

Therefore p-^1K is compact.

The domains of p and 6I are also very simple w.r.t. com- putabili ty : the sets { i ^{1 [} vN ( i) ] n dom ( p) } :j= 0 and

are decidable. Since p is admissible also and ^{p (CO)} are adrnissible.

Rice's theorern holds for

2.4 Corollary (Rice)

p n

Let X

S

JRn be not trivial.

since

'fuen p - 1x is not clopen or decidable.

p n is adrnissible.

In this case, Rice's theorern is the well-known fact that JRn has no nontrivial open and closed subsets. Roughly speaking, no property on JRn which is not trivial is decidable. For exarnple there is no cornputable (even no continuous) r : IF -JN such that for all p,qEdorn(p) r<p,q>= (0 i f p(p) =p(q) , 1 otherwise); sirnilar for p (p) < p (q) etc. But notice that the set {(x,y) EJR² x<y} is provable, while {(x,y) ¹ x:;:;y}

is not provable. The representation p(or öI) will be used almest exclusively in constructive and recursive analysis.

Two other representations derived from the enurnerations of left (right) Dedekind cuts are of interest.

2.5 Definition

Define representations and of JR as follows.

dom(p<) : ={p E JF v

0Mp has an upper bound} , dom(p>) : ={p E JF v

0JMP has a lower bound} , p < (p) : =sup vDMP (p ^E dom p <)

p>(p) : =inf vDJMP ^{(p E} dom p>)

.

p< is the left side representation, p> the right side re- presentation of JR •

We characterize the final topologies of show that they are admissible.

and and

2.6 Theorem

Let

'<

^{and '>} be the final topologies of

respectively.

(1) T<

= {

(x;eo) ¹ XE ]RU {-eo}} 1

T

> = { ( -

^{00 ; X ) 1 X} E ]R U { eo } } •

and

(2) Let Bi: =(v

0 (i) ;eo) , Ci: =(-⁰⁰;v

0⁽ⁱ⁾⁾ be numberings of a base of

'<

^{and T>} respectively. Then

P< =c oB and p> =c oc • Especially p< and p> are admissible w.r.t. their final topologies.

Proof:

We only consider P < • The proof for P,> is similar. From the definition of oB we conclude

dom ( o B) = { p E JF ^{J (} 3x E JR ) JM p = { j

I

v D ( j) < X}} ,

hence dom(öB) ~dom(p<) and öB(p) =p<(p) for pEdom(öB) . This proves oB ~c p < • Conversely there is a computable func- tion

r :

^{JF ... JF} such that

]M

r (

^p)

= {

j ^{1 (} 3 i ) \} D (j )

< \} nP (

i ) } = { j

I

^p

< (

^{p) E B} j }

for every p E dom(p<) .· Therefore p<(p) = öBr (p) , i.e.

Since

oB

is admissible i t has the final topology T< . Since

=

_C ⁰ _B ^{, also} is admissible and has the final topolo- gy T< •

Q.E.D.

The following theorem describes the reducibiltity relation between the three representations p,p<' and p< •

2.7 Theorem

( 1 ) p E Inf c ( p

<,

p >) ,

(2) p< and p> are incomparable w.r.t. t-reducibility,

(3) p<$tp , P>$tP •

Proof:

Let the representation o : JF - - - JR be defined by dom ( i) = { <p, q> p E dorn ( p <) ^A q E dorn ( p >) " p < ( p) = p > ( q) } i<p,q>: =p<(p) if <p,q> E dom(i) .

(see [8] Theorem 8). One can easily define cornputable operators r and r with

p ( p) = .§_

r (

p) i f p E dom ( p ) and

i (

p) = p

r (

p) i f p E dorn (

o ) •

This implies p

=ci

and pEinfc(p<,p>) . Properties (2) and

( 3) immediately follow from the general f act o ~ ö'

=>

T

O, - T

0

and the characterizations of the final topologies of p>.

Q.E.D.

and

Fora given representation ^cS of a set M we can ask which informations about x = ö (p) EM are finitely (or continuously) accessible ( f. a.) from the name p E dom ( o) of x .

In the case of JM : JE' - P every true information n E JM (p)

w

is f.a. but no true information n ~ JM (p) is f .a. frorn a name p . The con.trary holds for the representation JMc wi th JMc (p): =N\JM . Frorn Kreitz and Weihrauch [8] we know that

p

o f E Inf ( JM , JMc ) . This means ( and i t can be clearly seen)

C C

that every true inf.ormation n E o cf (p) and every-:true information m (\: ocf(p) is f.a. frorn p . The.intuitive concept of finite accessibility can·8asLly be forrnalized.

The set M+: =. t-,i,A) E N x P w

!

^{i E A} _ is [id_::!N ,JM} ]-provable (especially open~but not [ idN , JM ],-clopen, and

is.- [id ,o f]-decidabk (given sorne

JN C

ene can decide whether i E o cf ( p) If U is the nurnbering of a basis

p E JE' and or

B of a T -space M and

0

o0 is the admissible standard representation derived frorn U

(c.f. (8] Definition 19) then for any mapping o: JE' ---M , o :s;t o

0 <=> o is continuous. It is easy to show that o ::;;t o

0 <=> { (b ,x) E B x M

I

x E b} is [ U, o] -open. This property corresponds to the theorem frorn recursion theory that a f unc.tion :N - - - N is cornputable iff its graph is r.e. For the representations p ,_P<' and p> we obtai..n the following fundamental characterizations of P,P<' and p> by "finite accessibility" properties.

2.8 Lemma

(1) P is the greatest (up to t - (c-) equivalence) represen-

<

tation O of ^JR such that M< : ={ (d,x) E

m

₀ ^{x JR}

I

d < x} is [v0,ö]-t-(c-) open.

(2) • (correspondingly for p>)

(3) o is the greatest (up to t-(c-)equivalence) represen- tation o of JR such that M< and ~ are [vD,iS}-t-{c)- open.

The proof follows from'{8.-L Corrolary Z-2.

I t follows frorn-Theorern 2. 7 ( 3) that M> is not

Now we discuss representations which are definitely inappropriate for analysis (w.r.t. (JR ,,R)) by purely

topological reasons. The first of these is the (unrestricted) Cauchy representation.

2.9 Theorem (Cauchy representation) Def ine o : IF -- - JR as follows:

C

dorn(oc) = {p E IF 1 (vDp(i)) i E N is a Cauchy sequence}

äc (p) : = lirnvDp(i) for p E dorn(oc).

(1) The final topology

'c

^of is trivial, i.e.

'c={i,JR}.

(2) oc$tP<, oc$tP>, oc$tP.

Proof:

( 1 ) Suppose, X S JR , X = ~ , in o C -open. Then

0~1x= u{[w] 1 wEA} r1dorn(oc) for sorne AS.W(E}, A=I=~ . Let v E A be arbi trary, let y E JR be arbi trary.

Then there is sorne ·p E [w]

n

dorn(oc) with oc(p) = y , hence y E X . Since y was arbi trary chosen we obtain x

=

^{JR ,} theref ore , C = { ~, JR } •

(2) This follows from

T< i

'c etc.

Q.E.D.

The representation cc has the property that no information is f.a. from a name p of ce(p). Any initial segment of a eauchy sequence gives no information about its limit.

Because of Theorem 23 in [8] we understand continuity w.r.t.

p,p< and p> very well. Since oe is not admissible the Theorem cannot be applied to oe. It is not difficult to show that any continuous. furrction (w.r.t.p} · is (cSe,cSc)- continuous. We do not·know whether the converse is true.

The most commonly used representations are the r-adic r ~ 2 representations (e.g. r

=

2,8, or 10). The finite r-adic fractions are dense in JR , therefore they are used as

approximate values for real numbers (example:

m

₀ for r

=

2).

the finite r-adic fractions, however, are not appropriate for representing the real numbers. As an example we shall

study the case r= 10, the general case is treated similarly.

2 . 1 O' Theo:uem

Define cDEZ : JF-. •- - JR as follows:

dom cSDEZ = {PE JF 1 ('vn

>

O)p(n)

<

10}

cSDEZ (p)

=

{7rlp(o) - 1r

2p(o)) E{P(i) • 10-i

I

i

>

o}

Then the following properties hold.

(1) The final topology of oDEZ is 'R •

(2) The function xi-3x is not (oDEZ'oDEZ)-continuous.

(3) oDEZ- is not admissible (w.r.t. its final topology) ( 4 ) ⁰

=l\

^P DEZ ' ⁰ DEZ :::;; c P •

Proof:

It is an easy programming exercise to show oDEZ :::;;c P •

This implies T R

f

TDEZ . Suppose, X

s;

JR is oDEZ-open.

Let x EX • If x is not a finite decimal fraction then

there is a single p with oDEZ (p) = x such that p

f

(woo ••. ) and p =I= (w99 ••• ) for any w E W(N) • Since 0DEZ(x) ^-1 is open,

o [p[n)]~X

DEZ for some n . 0DEZ p ^{[ [ n) ]} is a TR-neighbour- hood of x • If x is a finite decimal fraction then there are two functions p,q (one with period

o,

the other with

-1 . 9) with oDEZ (p) = tSDEZ (q)

=

^{x .} Since oDEZ

[ [nJ

1

^{c [ [n]]c}

oDEZ P - X and oDEZ q _ X for scme

(X) .. is n E N . period

open,

The set o DE

z

^{[ [n)] p U 8 DE}

z

^{[ [n] q ) i s a} TR-neighbourhood

of x • Therefore X is TR-open, and we have shown TR=TDEZ • Suppose f : x ^1-o,, 3x is ( tSDEZ, oDEZ )-continuous. Then there

is some continuous r lF -- .. JF with tSDEZr (p) = 3 • tSDEZ (p) for all p E dom ( tSDEZ) Consider the converging sequences

(q ) in JF defined by n

P = (0033 ••• 300 ••• ) , n

n-times

q = ( 00 3 3 ••• 3 40 ••• ) • n

n-times

Then r(p )

=

(0099 ••. 900 .•• ) n ---.,.-,

n-times

( 0 1 00 • . . 0 2 0 • • • )

___,,_,

n-times

for all n . We obtain lim p = limq = (0033 ••• )

n n but

limr (p ) ^=f= liIJtf (q ) • Therefore r is not continuous.

n n

This proves (2). Suppose oDEZ is admissible. By (1)

this implies oDEZ =t p • Since f : x - 3x is ( p, p) -continuous, f is also (oDEZ'oDEZ)-continuous, a contradiction to (2) . We know already oDEZ ~c P • If P ~t oDEZ , then oDEZ is admissible, a contradiction to (3)

Q.E.D.

Let o r (r ^2:: 2) be the r-adic representation. It is known (see e.g. Deil [2]) that o ~ o iff r divides some

r C S

power of s • Again the negative result has a purely topo- logical reason. Although 'R is the final topology of oDEZ , many functions important in analysis are not even continuous w.r.t. oDEZ . Therefore, oDEZ is not appropriate for

analysis. The ~arue statement hol<ls fol'7:a11y r-adic repre- sentation.

Another representation of :IR is that by characteristic functions of dedekind cuts.

2.11 Theorem

Let S ~:IR be a dense set (w.r.t. 'R) • let v be a total bijective numbering of S • The left-cut representation

(w. r. t. v) of :IR is def ined as follows:

dom = {pEJF !(3xER)(Vi)(p(i)=O<=>v(i)<x)}, oLv (p) = sup vp -1 {0} for all p E dom oLv . Then

B: ={ (x;yJ · 1 x,y ES , x < y} is a basis of the final topo- logy 'Lv of oLv , and oLv is admissible.

This rneans that TL\l is strictly finer than TR ⁱ the sets

{ Z

I

X< Z ~ y} with x,y ES

'

^x<y

'

are not in TR

.

The topology obviously depends on the set s

.

^Therefore _{8 1 = CDD}

and s₂

=

CD yield incornparable left-cut representations.

A representation which is so sensitive against unirnportant changes cannot be natural. Notice that sorne ö t-equivalent to p is obtained if in Definition ² is substituted by sorne arbitrary nurnbering v of a-dense (w.r.t. TR) subset of JR •

Proof:

Let Since

(if

T be the topology on JR defined by the basis B.

ö

~~

( v ( i) ; v ( j ) ]

= {

^{p E dorno Lv}

I

p ( i)

=

^{O, p ( j ) =l= O}}

v ( i) < v ( j) ) is open in dorn ( o L) , o Lv -1 ^X is open in

dorn ( öLv) for any XE T • oLv satisfies the following recur- sion equations:

0 Lv [ E:]

= (

^{-00; 00) ,}

o [ wk]

=

^v

{

oL [w] n (vlg(w) ;00 ) if k =0 L v o L v [ w ] n ( -00 , v 1 g ( w) ] i f k

'f

⁰

for w E W ( N ) and k E E • Hence, oLv (A) ET for any open subset Therefore T is the final topology of oLv" It is easy to show that oLv is adrnissible.

Q.E.D.

The representation oLv is adrnissible (whatever v is chosen). Right-cut representations oRv can be defined

correspondingly. In a recent paper K.Skandalis [11] defines a bijective representation which is c-equivalent to sorne right -cut representation (w.r.t. sorne effective nurnbering

v of a subset <O'

'=

^(J} which is dense in JR ) . Since i ts final topology depends on <O' i t is not very natural.

Let öCF be the continued fractions representation of JR (Deil [2]). Then öCFEinfc(öLv'öRv) where v is some standard numbering of the rational numbers <O.

Therfore, öCF is admissible and its final topology has the basis { [a;b] ¹ a,b E <O,a :s; b} •

We conclude this chapter with some remarks about computable real numbers. If 6 : JE' -- - M is a representations, then v": = ö~ is the numbering of the ö-computable elements of

0

M induced by ö . It can be shown that the representations p,öDEZ'öLC and öRC (w.r.t. an effective numering of <O)

and of öCF define the same kind of computable numbers, while the corresponding nurnberings are not equivalent (see e.g.

Deil [2]) • For JR the representations p,p<' and the corresponding numberings are of interest.

2.12 Definition

with

i;: =p~; ,;<: =p<~;~>. - p>~. JRc: =range(,;) (JR<c: =range(,;<) , JR>,: =range(,;>)) is called the set of computable (left- computable, right-computable) real numbers.

The numberings ,;,,;<' and ,;> are related as follows.

2.13 Lemma

( 1 ) JR <c :j; R>c ' R>c $ R<c ' JR c = JR <c n JR >c · (2) ,;Einf(,;<,,;>),

Proof:

Let K~JN be an r.e. set which is not recursive. Then -i .

xK: =E{2 1. EK} EJR<c \R>c and -xK EJR>c \R<c • This proves (1). Property (2) follows iromediately from

p E inf c ( ^p

<, P)

(Theorem 7) . Q.E.D.

Since is precomplete, no non-trivial property on is ,-decidable.

JR C

3 Representations of the Open Subsets of JR

The set of all subsets of JR has greater cardinality than

JF , hence there is no representation of 2JR . Consider the

theorem: "Every bounded subset A <;; JR has a least upper bound".

Since also the set of bounded subsets of JR is too large the theorem cannot be formulated constructively in our the- ory of representations. But different constructive versions

can be proved. In this chapter we only define and compare some standard representations of classes of subsets of JR which are used in analysis. First open sets are considered.

The (topological) standard representation of the set of open subsets of JR is defined as follows_ (c.f. Def. 2.1).

3.1 Definition

Let w : JF - t R be defined by wp =w(p) : =U{Ik

i

k E JMP}

By w a topology, the final topology , of w , on the

w

set , R of open subsets of JR is defined. We show that w is admissible (w.r.t., ) •

3.2 Theorem Let U.

J (where defined hold:

w

={XE,R

i

^Kj

<::x}

^{where K.=U{I}

J uj = -rR if K. = 9)) let ^'t be

J

by the. basis ^{U.

i

J jEJN}. Then

( 1) (tR' ,) is a seperable T

0 -space.

n

i

n ED.}

J

the topology on 'R ( 1 ) ' ^{( 2)} and ^{( 3)}

(2) w =c cSU, where cSU is the standard representation of 'R w.r.t.U.

(3) w is admissible.

Proof:

The set { U j

I

j E JN } is closed w. r. t. finite in tersection.

Hence i t is a basis of a topology on ' R .

Let A,B open subsets of JR ,A=!=B . There is (w.l.g.) some x E JR with x E A \ B • Then x EI S I = K. SA for some j, n E JN , and AE U.

a seperable T -space.

0

J but

The definition of cSU implies dom(cSU) = {p·-1 {:lX E ,:R}':!Mp·= {j and for p E dom ( cSU)

n n J B

lf

U . • Hence

J

XEU.}}

J

j E JM <=>K.

s

cu{p)

p J (for all j

There is some computable r : JF - JF with

is

JM r ( p) = { n E JN

i (

³ j E JMP ) n E D j } • S ince A =

U {

IK

i I

^K

S

A}

for any A E 'R , wr (p) = cS

0 (p) for any p E dom(cS 0) , i.e. cSU=s;cw.

Conversely def ine ti. : JF - JF by

f

^j+1 i f K.SU{I nEJM . }

t.(p)<i,j>: = J n p,1.

lo

^otherwise

where JM . = {n 1 (:im=:;; i) p (m) = n + 1} • Then Li is compu- p, 1.

table, and since K.

J is compact for any j ,

Kj S w(p) = u{In in E JMP} <=> (3i)Kj ~ U{In n E JMp,i} <=> j E JM t.(p) holds f or any p E JF and j E N •

Hence, w(p) =cS

0t.(p) , i.e. w=s;ccSU. This proves (2) and also ( 3} •

Q.E.D.

The characterization of , is a special case of a result

w

obtained by Hay and Miller [5]. The characteristic property of w (and of

ou)

is that any true information ^-I._-w(p) ^C

J

if f.a. from p . The representation w corresponds to the definition of r.e. sets by ranges of total recursive functions. The r.e. sets can also be characterized as the domains of the partial recursive functions. A corresponding representation of is the representation .w

p

(c.f. Definition 4(5) in [8])

p E dom ( w ) <=> ( Vq, q' E dom ( p) ) [ p ( q) = p ( q' ) => X ( q) = X ( q' ) ] ,

p p p

w ( p) : = p ( dom ( x ) ) f o r p E dom ( w ) •

p p p

3.3 Theorem ( 1 ) w

=

_C ^w _p ^,

(2) w is admissible.

p

Proof:

Consider the proof of Lemma 24 in [8] . It can casily be shown that r and r are computable in our case.

Q.E.D.

Let A be a closed set, let MA : ={k Ik n A ::j: {21} • Since JR is complete, A= n U {I<. >l<j,n>EMA} ,

n J J,n

which means that A is uniquely determined by MA.

For the open subsets of JR we obtain the following new representation.

3.4 Definition

Define w c IF -- - ^T R by

dom(wc) : ⁼ {p E IF

I

JMP ⁼ {k

i

Ik \ B

f

0} for some BETR}.

w {p)

=

JR\

n u.

{I<. > \ <j

,n>

E JM }

c n J J,n p

= the set BETR with JMP = {k

I

Ik \ B =j= 0} (pEdom(wc)) The following theorem surnmarizes the main properties of

3.5 Theorem:

w C

Let V j = { X E T R 1 ( Vk E D j ) Ik \ X

f

0 } for j :2: 1 and V : =T

o R

let T be the topology on TR defined by the basis {Vj j E JN} • Then (1), (2) and (3) hold

Proöf:

is a seperable T -space,

0

is admissible)

The set {V. 1 j E JN } is closed w. r. t. finite intersection, J

hence i t is a basis of a topology T on TR. Let A and B open subsets of JR , A ^=!= B . Then there is (w .1. g.) some XE JR

Then Hence wc{p) öv(q) There

with x E A \ B and some n wi th x E I

s;;

A • n

for j with D. = {k} we obtain A

$

^V. and

J J

T is a seperable T

0 -space. The definitions

=X<=> JM = {k

I

Ik <;EX}

p

=X<=> JM _q = { j 1 ( Vk E D j ) Ik

i

^{X }}

are computable functions ^~

, r

: IF -IF with

BE V.

J imply:

JM ( ) ={j \D.~JM}

r

P J P and

for q E

öv ,

hence wc =c

ov

and ^w _C

( 3 j E JMq) k E D j} . is admissible.

C

w and wc are incomparable w.r.t. t-reduction. This proves

( 3)

Q.E.D.

From the distance functions dA : X - d (x,A) for A =!= 0 further representations of open subsets of JR can be derived. The pro- perty

A

=

B

<=> d = d

A B (where A is the ^A)

induces a bijective relation between the open subsets E E ' R , E =I= JR , and the distance functions E -dJR \E .

Any distance function is continuous, therefore (p,p)-continuous, (p,p<)-continuous and (p,p>)-continuous. The standard repre- sentations of the (p,p)-, the (p,p<), and the (p,p>)-contin- uous functions ( [8] Definition 4(4))induce the following definition.

3.6 Definition

Define representations

w< , w>

and w

1 of 'R \ {JR} as follows:

dom ( w <) : = { p E JF

w<(p) ^{: =} {xEJR Correspondingly,

(3X~JR,X=j=0)(VqEdom(p))d(P(q),X) =p<~ (q)} ' p (VqEp -1 {x})p<qip{q) >0} for pEdom(w<) .

w>

is defined with and is defined

with p instead of

We show that is essentially equivalent to w and that

Finally,

essentially is equivalent to defined above.

is the infimum of and ~w> . In Bishop¹s [1]

constructive analysis a set A for which dA "exists" is called "located" . The representation w

1 in our theory corresponds to locatedness (of complements).

3.7 Theorem

Let T : =T R \ { JR_} • Then the following holds.

(1) w< =cwlT, (2) w>=cwclT, ( 3 ) w l E inf c ( w < , w > )

Proof:

( 1) There is some total recursive function q:·lN - JN with v0 (i) =p~q(i) . It is not difficult to construct an oracle Turing machine which computes a function 6. = JF - JF such that for any p E dom(w<):

JM6.(p) = {<i,j> j 2-j<d(v 0 (i),JR \w<(p)) =p_<iµp(~q(i))}

1

- C

= {<i,j> _. I<. '>-w<(p)} • _J.,J

Therefore w6. (p) = w< (p) . on·the other hand there is a computable function 6. : JF - JF such that for all p E JF with w (p)

f

^JR:

JM6.(p,r) = {i

I

v0 (i) <d(p(r), JR \w(p)},

hence p<6.(p,r) =d(p (r) ,JR \w(p)) for all r Edom(p) • The smn-theorem for iµ yields some computable L : JF - JF

with 6. ( p , r) = iµ L ( p) ( r) • Then w ( p) = w <

-r (

p) • (2) (similar to (1))

(3) This follows from p E infc (p<,p>) by using the smn-theorem.

Q.E.D.

4 Conclusion

In Chapter 2 we have compared several representations of the set of all real numbers. The representation p is (up

to equivalence) the only one which is satisfactory for topologi- cal reasons. The representations

p<

and

p>

(by cut enu-

meration) are admissible but do not have the final topology 'R, the decimal representation (more gerierally r-adic representation) has the final topology 'R but is not

admissible with the consequence that many elementary functions become noncontinuous. The unrestricted Cauchy representation has a bad final topology, {~,JR} , and the cut representations oLv have final topologies which depend on the (arbitrary)

choice of a set s which is dense in JR • Therefore, there is sufficient evidence that ^p is (up to equivalence) the only reasonable representation of JR , and we can confirm Mostowski's [9] statement:"The other definitions represent merely a mathematical curiosity".

In Chapter 3 we have investigated some representations of the open subsets of JR , i. e. of , R , which very likely can be used in constructive analysis. Again topological considerations are the basic ones.

We have mainly studied topological properties of represen- tations. In a second step recursion and complexity theoreti- cal properties have tobe comsidered for those representations whichare topological sound. Such properties can be introduced either by explicity defining "computably effective" represen- tations (e.g. :M ,P

,w, •.• )

or by requiring effectivity

properties for certain numberings from which by standard

1

constructions representations are derived. For the numbering U of a basis of a topological space one could require that U. S U. n Uk is recursively enumerable (or recursive), for a

J_ J

numbering v of a subset p dense in a matric space (M,d) one could require that the distance is ([v,v],p~)-computable, etc. Almost all representations we have defined explicity have satisfactory computablity properties. A general concept of the definition of effective numberings and representations has been studied by Weihrauch [15]. An interesting observation can be made: whenever something is impossible i t is impossible for topological reasons (discontinuous), and if something

is continuous then i t is even easily computable. We have not yet discovered a natural correspondence (Kreitz and Weihrauch [8]) which is continuously effective but not com- putably effective w.r.t. the (computably effective) standard representations. (0f course artificial combinatorial coun- terexamples can be constructed.)

Many concepts which are introduced here for the real numbers can easily be generalized to other metric spaces (Pour-El and Richards [10]) for which our approach yields the cano- nical effectivity theory. Even constructive measure theory can be developed within this framework. For example there is an "effective" representation of the set B/N where B is the set of Borel-sets on [0;1] and N is the set

of all subsets of [0;1] with measure 0 .

Let Vk: = u{In

I

n E Dk} n [0; 1] , thus V is an effective numbering of a set T of open subsets of [0;1] .

By d(A,B) : =length(A\ BUB\ A) a metric on T is defined.

Its completion is essentially B/N. An effective repre- sentation can be defined by the standard construction with normed Cauchy sequences given in [8].

An essential feature of the constructive analysis presented here is the flexibility in the choice of the representation.

Gi ven a representation o : JF -- - M , by any p E dom ( o) the element o (p) EM is uniquely determined, i. e. p

the complete information for identifying o (p) EM •

contains

The basic idea of constructivity however, is that parts of this information should already be obtained from initial segments of p (our oc does not satisfy this requirement) and that o(p) is completely determined by the informations

given by the segments of p . This corresponds to continuity of o . Representations of a set M may differ in the amount and type of information which is finitely accessible. If o :s;t o' then the o-names contain at least as much fini tely asccessible information as the o'-names (examples:

0_LvD :s;c 0DEZ Sc o :s;c o

<

:s;c 0c ; o cf :s; :r1 )

Generally, the choice of the representation depends on the amount of finitely accessible information of the represented objects. Thus the assumpt.;i,.ons of a theorem can often be ex- pressed by an appropriate representation. Instructive examples

will be given in a further paper where constructive compact- ness on JR is studied in our framework.

References

[1] BISHOP,E., Foundations of constructive analysis, McGraw-Hill, New York 1967.

[2] DEIL, T., Darstellungen und Berechenbarkeit reeller Zahlen, Dissertation, Fernuniversität Hagen 1983.

[3] GRZEGORCZYK,A., On the definition of computable real continuous functions, Fund.Math. 44 (1957), 61-71.

[4] HAUCK,J., Berechenbare reelle Funktionen, Zeitschr.f.

math. Logik Grundl.Math. _:!.2. (1973), 121-140.

[5] HAY,L. and MILLER,D., A topological analog to the Rice- shapiro index theorem, Research report, 1982.

[6] KO,K., On the definitions of some complexity classes of real numbers, Technical report UH-CS-80-3

University of Huston, 1980.

[7] KREITZ,C. and WEIHRAUCH,K., Theory of representations, (tobe published).

[8] KREITZ,C. and WEIHRAUCH,K., Towards a theory of repre- sentations, Informatik Berichte Nr. 40, Fernuniver- sität Hagen, 1983.

[9] MOSTOWSKI,A., On computable sequences, Fund.math. 44 (1957) 37-51.

[10} POUR-EL,M.B. and RICHARDS,!., Lp-computability in recur- sive analysis, Technical report, University of Minn- sota (1983).

[11] SKANDALIS,K., Programmability in the set of real numbers and second-order recursion, Fundamenta Informaticae VI • 3- 4 ( 1 9 8 3 ) , 2 5 7- 2 7 4 •

[12] SPECKER,E., Nicht konstruktiv beweisbare Sätze der Ana- lysis. J.Symbolic Logic 14 (1949), 145-158.

[13] TURING,A.M., On computable numbers with an application to the Entscheidungsproblem. Proc.Lond.Math.Soc., ser.

2, 42 (1937), 230-265.

[14] WEIHRAUCH,K., Type 2 recursion theory,(to be published).

[15] WEIHRAUCH,K., On natural numberings and representations, Informatik Berichte 29, Fernuniversität Hagen, 1982.