Towards a theory of representations

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Towards a theory of representations

Abstract

f/lffN

TOWARDS A THEORY OF REPRESENTATIONS ..bM-

,/\\r"I?~

f_.-?,,ioc,i/.J.) \

Christoph Krei .... ,:7and;;. _~ tt, aus Weihrauch

~1:).

a theory of construci:.i•.rity

:.n

rnathernatics. Computabili ty and continuity w.r.t. gJ.ven representations are-defined and S°½ud- died in connection wit~ reducibility. We investigate topo-

logical properties of ~2presentations and introduce (continuous--' i

ly-); admissible representatibns of seperable T

0 -spaces.

It is shown that the continuity theory induced by (continuously-) admissible representations corresponds to the topological con- tinuity theory. Hence these representations are very appreciate to study construtivity on all kinds • of seperable T -spaces.

0 .

..

Keywords: representations, cornputability and continuity theory

AMS Classification: 03F 65, 03D 45

UB Hagen

1,,a;.

1111111111

831679901

Introduction

At the beginning of this century the displeasure with the traditional "idealistic" mathematics which studies objects

(sets, functions) that are not constructive in general and which uses nonconstructive proofs led to a radical criticism of the foundations of mathematics. L.E.J. Brouwer was one

of the most resolute representatives of this rnovement. While the criticism was widely accepted by the rnathmaticians, the attempts for solving the problerns usually rnet with only few approval.

Brouwer himself developed foundations for an "intuitunistic"

rnathematics [4] based on some very unclear definitions ("Wahlfolgen", "Species", ... ) .

Nevertheless he obtained some interesting results on constructivity in analysis. His works also gave rise to the development of

intuitionistic logic in which e.g. the law of "excluded middle"

does not hold. More convincing than Brouwer's papers are books of P. Lorenzen [1~ and of E. Biskop (2] which show that many parts of classical analysis are accessible by constructive methods. These approaches have in common the use of only con- structive proofs.

The development of recursion theory led to (at least two) different kinds of recursive analysis. In recursive analysis computable objects, sets, and functions are defined, and i t is studied without restriction of proof rnethods which objects

(sets, .•. ) are computable.

The "Russian school" (G.S. Ceitin [5], B.A. Kushner [ 13), O. Aberth (1], and many others) starts.with an "effective"

partial nu.mbering of the computable real numbers IR

C by

which computability on JN is transferred to computability

on IR . The "Polish school" (A. Grzegorczyk ['9], D. Klaua [10],

C

and many others) starts (essentially) with an effective repre- sentation of all real nu.mbers by ^{JF (=} JNJI:-.J) by which computa- bility on ^JF is transferred to computability on JR. Remarkable is a book by P. Martin-Löf [16] where concepts of Brouwer's are presented in.a precise recursion theoretical context.

The main goal of a "constructive" mathematics should be to study which functions from traditional mathematics can be

computed by a rea~ world computer. A cornputer cannot understand abstract mathematical objects. The input/output set of a com- puter is w.l.g. the set JN of natural numbers or the set ^JF of sequences of natural nu.mbers. The elements of the set ^JN or ^JF then may serve as names for abstra~t objects. This means the computer can only handle narnes of mathematical objects.

The elements of a denumerable set M can be narned by numbers, where the naming is fixed by a numbering \) : JN - - M (onto) Computability on M then is determined by the nu.mbering \) of M under consideration. Numberings are widely used, and

a nice theory of nu.mberings is available (Mal'cev [15], Ershov [8]) The elements of a set with cardinality not greater than that

of the continuum e.g. IRn, effective cpo's (Weihrauch und

Schäfer [22]), LP-spaces (Pour El and Richards (17]), or general seperable T

0-spaces, can be represented by sequences of numbers. The foundation of a theory of representations is atheory of compu-

tability on JF = JN ^JN •

The representation then determines computability on M. This is the basic idea of the "Polish" approach to recursive analy- sis. Going into details of this theory one rather quickly ob- serves that many fundamental questions do not concern compu- tability but mainly continuity. Functions which are not com- putable generally are not even continuous. Therefore continuity and as specialization computablity w.r.t. representations

should be studied.

In this paper mainly topological questions concerning repre- sentations are investigated. After summarizing some definitions and facts of ordinary recursion theory (chapter1) and the baic concepts for numberings (chapter 2) we shortly introduce a theory of continuity and computability on JF (chapter 3).

Details can be found in a report by Dettki and Schuster [71.

In chapter 4 continuity and computability w.r.t. given represen- tations is defined and studied in connection with reducibility.

Two representations define the same (continuity-) computability- theory if and only if they are (continuously-) equivalent.

The main part is chapter 5, where topological properties of representations are investigated. Final topologies play a fundamental role. The essential part is the definition and study of a standard representation for a given seperable T₀ -

space which has very important topological properties.

A standard representation of a seperable T -space

M

is con- o

tinuous and open and every continuous representation of M can continuously be reduced to it. Furthermore a function between seperable T -spaces is (topological) continuous if

0

and only if i t is continuous w.r.t. standard representations.

This result holds also holds if the representation of M is (continuously-) admissible, i.e. (continuously-) equivalent<

to some standard representation. Hence (continuously-) admis- sible representations are very appreciate for studying con- structivity.

In the last chapter we finally give a short discussion con- cerning effectivity of representations.

The authors believe that the approach presented in this paper may serve as intermediator between idealistic and intuitionistic mathematics and will help formulating and answering the

questions of what is constructive in mathematics and what is not.

1. Recursion Theory on the Natural Numbers

Fora reader, who is not sufficiently familiar with basic recursion theory, in this chapter very shortly some defi- nitions and facts are summarized.

The basic consept of recursion theory is that of a computable function. A (possibly partial) function p : JN k --- JN is called computable (or partial recursive), iff there is a Turing-program which with input (n

1, ••• ,nk) halts with out- put p(n

1, ••• ,nk) E JN, if (n

1, ••• ,nk) E dom(p) and never halts if (n

1, ••• ,nk) (j: dom(p).

By Church's thesis a function p : JN ^k -- - JN is partial recur- sive iff i t is intuitively computable.

Writing Turing-programs is rather laborious. However, every PASCAir(ALGOL-, FORTRAN-, ... ) program which computes a function p : JN k - - JN can be translated into a Turing-program and vice- versa. For showing computability of a function p i t is suf-

ficient to prove that there exists a PASCAL-(ALGOL-, ... ) pro- gram which computes p . This way many basic_properties of the computable functions ("The computable functions are closed un- der substitution^{11 ,} "the inverse of an injective computable function is computable", .•. ) can be proved very clearly.

By P(k) we denote the set of all computable functions p : :N k - - - JN , by the subset of all computable total k-ary functions p:JN ^k - J N (total recursive functions).

A subset Ac JNk is decidab le or recursive iff A "'.' p - 1 {O}

for some total recursi ve p : JNk - JN (note that p must be total). The function p (or a program for

x E A or whether x

$

A by the value p (x)

p) decides whether f or any x E JN k • is called provable or recursively enumerable (shortly r. e.), iff A = dom( p) for some p E P (k) . The func- tion p (or better a p.rogram for p) can be considered as a

f t f A F XEJNk ' f

proo sys em or • or any , i x E A the hal ting

computation of p (x) is a proof for x E A , if x ~ A no such a pröof exists. The term recursively enumerable derives from the following property:

AC JN is provable <=> ( 3p ^{E R (} 1 )) A

It is easy to show that Ac JN'k JN'k\ A are provable.

={il Oj)p(j)=i+1}=:11.

p is decidable iff A and

The function ^{'ff :} JN2

- JN , ^'ff (m, n) : =n + z: { i \ i :5 n + m}

is called Cantors bijection. The standard tupeling functions

(k)

'ff (k = 1,2, .•• ) are defined inductively by

(1) _ (k+1) __ (k)

'ff ( n ) - n ' ^{'ff (} n 1 '. . . • , nk + 1 ) . -'ff ( 'ff ( n 1 , • • • , nk ) , nk + 1 ) • The functions rT(k) are computable and bijective. For con- venience we write <n

1, ••• ,nk>: =rT(k) (n

1, ••• ,nk) . The inversC;:s ^{'ff. (k)}

]. <n₁, .•• ,nk> i - - - ni are also computable.

Using the bijective transformations ^{A -} rT (k) ^A recursion theoretic properties of JNk and P (k_) formulated within JN or P(1

) respectively.

and (k) -1

p -p(rT ) can be

A canonical numbering of all the Turing-programs for functions from P ( 1) yields a standard numbering _cp : JN - P ( 1) , where cp,: = cp(i) is the function computed by the i-th Turing-pro-

1

gram. The numbering cp is (up to equivalence, see chapter 2)

uniquely determined by the following axioms:

( 1 )

( 2)

(3uEP(1_{))('v'i, jEJN)} ( Vp E P ( 1 ) ( 3 s E R ( 1 ) ) ( Vi lation lemma") .

u<i,j> = <p. (j) ("utm-theorem"),

].

, j E JN) p<i,j> = cp

5 (i') (j) ("trans-

For any (program-)number i let 4i. : JN ---JN

]. be the step-

counting function for i : 4ii(n) is the number of steps the i-th Turing-program needs for computing ^<p. (n} . The mapping

l.

i 1,

1 1 II;. _,,

qi : JN - P ( 1 ) 1 , qi. :

=

^{qi (} i) , has the following properties:

].

_(1) (ViEJN) (2) the set

dorn ( ^{((l . )}

=

dorn qi . ,

]. ].

{<i,n,t>

!

^{1l. (n) =t}}

]. is decidable .

A consequence is the projection theorern. For AC JN define

pr (A) : ={ i ¹ (3 j) <i, j> E A} • Then : A provable => pr (A) provable and A provable => ( 3B ':= JN , recursi ve) A = pr (B) •

Using PASCAL-(ALGOL-, ... ) prograrns instead of Turing-prograrns would yield a nurnbering equivalent to ((l •

By W . : =· w ( i ) : = dom ( <P • )

]. ]. a standard nurnbering of the provable subsets of JN is defined. A typical set which is provable but not decidable is

K : = { i \ i E W . } = { i \ i E dorn ( ((l • ) } •

]. ].

Then JN\ K is not decidable and not provable. Many other im- portant sets are even rnore complicated than not decidable and not provable, e.g.

problem),

{ i 1 ((l. = p} where p E P ( 1 )

]. (correctness

(equivalence problem), or

For rnore detailed information the reader should consult a book like Rogers [19], Mal'cev [15], Brainerd and Landweber [3] , or Cutland [ 6] .

2. Numberings

In this chapter some basic concepts for numberings are intro- duced. The material is presented slightly more detailed in order to demonstrate that the definitions in Chapter 4 for representations have their roots in ordinary recursion theory.

i' i;

Computablity on a (denumerable) set S different from 1'J

can be reduced to computability on 1'J by using a numbering of S . A numbering of S is a surjective (not necessarily total) function v : JN' - - S . For any element s of the "ab- stract" set S there is at least on number (or name) i E 1'J

wi th s ^{= v (} i) ^{=: v. •} For rnany sets standard numberings are

].

used which are "ef fecti ve ^{11 (} "cornputable ^II is not def ined, if s

f

^JN) in a certain sense. This rneans that ^II important ^II proper- ties of e---1ements can be computed.frorn their numbers.

Usually these properties determine the numbering ^v uniquely up to equivalence (see below). We shall not discuss this im- portant question in rnore detail.

2.1 Some Standard Numberings

( 1 ) i dJN : JN - - JN numEers.

is the standard numbering of the natural

(2) Let E(JN) be the set of all finite subsets of JN • For ME E (JN) define 11(M): =I{2i

I

iEM} . Then 11 is bijective and D : =-D(n) : =11 ^-1 (n)

n defines the standard numbering

D of E (JN) • We have i E D , iff the binary expansion n

of n has a 1 at position i . ( 3) v (k) : JN - JN k wi th

C

v ^(k)

=

(TI ^{(k)) -1}

C is the Cantor num-

bering of ^{JNk •} It is bijecti ve. ^{(k 2!:} 1) .

(4) Let W(JN) be the set of all finite words over the alpha- bet ^{JN •} A standard numbering v:lil: JN -

w

(1'J) is defined by ^V _JN (0) :

=

^E (the empty word), and

vN (<k,<n

0 , ••• ,nk>> + 1) : =n

0n

1 .•. nk. ^vN is bijective.

(5) Let ^{(0D :} ={XE ^{ID ) (} 3 z E Zl) ( ^{- .} 3n E ^{JN )} x = z • 2 ^-n } be the set of all dyadic rationals. By v

0 <i, j ,n> : = ( i - j) • 2 -n a standard numbering of

m

₀ is defined. The numbering v0 is total but not bijective. However equivalence is decidable: {<i,j> ¹ v

0 (i) =v

0 (j)} is recursive.

(6) The rnapping vQ: JN-W defined by vQ<i,j,k> =(i -j) / (k+ 1)

is a nurnbering of the rational nurnbers.

(7) The rnapping <P : JN - P ( 1) is a nurnbering of ^{P ( 1) •} Although <P has several effectiveness properties (see Chapter 1) the equivalence problem · {<i,j>

provable.

<P. = ^{(l). }} is not

1. 1.

(8) W with W. : = dom ^(1).

]. ]. is the standard nurnbering of the provable subsets of JN •

There· are sets with different "standard" nurnberings which are not equivalent (for equivalence see below). There are sets

for which no total nurnbering exists which satisfies the desired effectivity properties. There are operations on nurnberings, which produce "effective" nurnberings from effective ones.

We introduce one of it, tupling.

2. 2 Definition

Let ^{JN - - -} s. ( i = 1, ••• ,n) be numberings. Then

1.

[ v 1 , ..• , v ] : JN - - - S. x ••• x S is defined by

n 1. n

dom[v

1, ... ,vn) : ={<i

1, •.. ,in> ¹ ik E dom (vk) for k = 1, ... ,n}

[v1, .•. ,v )<i

1, •.• ,i > :=(v 1(i

1), •.. ,v ( i ) )

· n n n n

if <i

1, .•. ,in>Edom[v

1, ••. ,vn).

Computability, decidability and provability for sets w.r.t.

given numberings are defined as follows.

2.3 Definition: (computability, decidability, provability) Let v . : JN - - - S . ( j

=

1 , 2)

J J be numberings. Let F

be a function and A ~ S

1 be a subset of S 1 •

(1) F isweakly (v 1,v

2)-computable, iff there is some p E P ( 1 ) wi th

Fv1(i) =v

2p(i) for all iEdom (Fv 1) (2) F is (v

1^,v2)-computable, iff there is some pEP( 1 ) wi th Fv

1 ( i) = v

2 P ( i) i $ dom (p) for all

for all i E dom ( Fv 1 ) , i E dom ( v

1 ) ~ dom ( Fv 1 ) ( 3) A is v

1-provable, iff there is some function F : s

1 - - - JN which is ( o

1 , idJN) -cornputable such that A = dom (F) • (4) A is v

1-decidable, iff there is some total function F : s

1- JN which is (v

1, idN) -computable such that A=F -1 {O} •

The following diagram for (a) shows that the computable function transforms names.

p

JN - - - JN

1 1

1 1

1 1

\)1 1 1 \)2

1 1

s _____ ^{r _____ t}

1 2

The equation Fv

1 (i)

=

v

2p(i) expresses that p transforms a number of an element x: =v

1 (i) E dom (F) into a number w.r.t.

v2 of F(x) . The additional condition in (b) can be formulated as follows : dom(p) n dom(v

1) = dom(Fv 1)

Thus p forces F to have a "natural" domain. This natural domain is used in (c). The following characterizations follow irnmediately: A is v

1-provable, iff -1

v₁ (A)

=

^B n dom(v

1) for some provable set B ~ JN ; · A is v

1-decidable, iff there is some p E p ( 1 ) w i th dom ( v

1 ) ~ dom ( p ) and v ^{-1 -1}

1 (A)

=

p { o} n dom ( v 1 ) •

By b1e following lernma the concept of v-computability is closed under composition.

2.4 Lemma

Let v i : JN - - - Si ( i = 1, 2, 3) numberings, let F : M

1 - - - M 2 be (weakly) ( v

1 , v

2) -computable, let G : M

2 - - - M

3 be (weakly) ( v 2, v

3) -computable. Then GF : M

1 - - - M

3 is (weakly) ( v 1, v

3) - computable.

The proof is easy and formally equivalent to the corresponding lernma for representations (see Chapter 4).

A numbering may be changed in a certain way without changing the kind of computability induced by i t .

2.5 Definition (reducibility, equivalence)

Let vi : JN' - - - Si be numberings, We define

( n V

1

and

is reducible to is

\)

"

) 2

(v1,v

2)-computable)

( "equi valence" )

Therefore, v

1 ^{~ v}2 , iff (Vi E dom(v

1)) v 1 ⁼ v

2p(i) for some p E P (1) ;p "translates" v

1 into v

2 • By the following lernma equivalent numberings define the same kind of computability.

It follows immediately from Lemma 2.4.

2.6 Lemma:

Let v. , v ~ be numberings of S. ( i = 1 , 2) , let F :

s

1- - -

s

2 ,

i i i

be and then F

We conclude with some examples. We shall consider the standard numberings from example 2.1 and say "computable" instead of

(v1,v

2)-computable etc. For n-ary functions we use the tuple numbering from def. 2.2 .

2.7 Examples

(1) On E{JN) the operations union, intersection and

difference are computable. The functions max, min, and card: E (JN) - JN are computable. The set

{ (i,X) c JN x E(JN) 1 i EX} is decidable .

(2) On W(:N) concatenation is computable, the relation u f:: w (u is a prefix of w ) is decidable, the length lg : W(Thl) - JN is computable, etc.

(3) On

m

₀ functions such as +,-,x,: are computable, the relations <,=, and > are decidable.

The theory of (total) numberings is presented in Ershov's

is

famous paper [8]. Several interesting results al- ready can be föund in Mal'cev's book [15).

General properties of effective numberings are discussed by Reiserand Weihrauch [18] and by Weihrauch [23].

ii

3. Continuity and Computability on JF

For naming the elements of a nondenumerable set, the set JN

of natural numbers is too small. As a larger set of names the set JF:=JN ^JN of all sequences of natural numbers can be used.

In this section a unified theory of continuity and computability on JF is sketched. Most of the proofs are easy or can be found in Dettki and Schuster [ 7 ], (see also Weihrauch and Schäfer [22]).

The theory formally corresponds to ordinary recursion theory

on ^{N •} Instead of ^{Q) :} JN - p ( 1 ) a representation 1jJ : JF - [ JF - ^{JB ]} of the set of continuous functions from JF to JB (see below)

is introduced which is very "effective". The idea of represen- ting the continuous functions on M by elements of M itself is not new, Scott [20], e.g., studies the case M=P ^=2JN.

w

Let JF : = JN ^JN" and ^{JB :} = JN ^JN

u

W ( JN ) the set of all finite and infinite sequences of numbers. For v,wEW(JN'), pEJF and n E ^JN define v cw : <=> v is a prefix of w ,

vi;p: <=> (:lm) v=p(O) ... p(m-1), p[n]: = p(O) ... p(n-1) , [v]={qEJF lv1=q}. If v = a a

1 ••• a

o n-1 ^{where a.} l. ^{E JN , we}

define v[i] :

=

^{a. for} i < n , v[i]

=

div. otherwise.

l.

On JB let a topology be def ined by the basis { Ow

I

^{w E W ( JN)}}

of open sets, where O ={x E B

I

w 1= x} • (This is the topology w

corresponding to the complete partial order c on B.) On

on

]F ]B

we consider the topology induced by the above topology , i.e. {[w] 1 wEW(JN)} is a basis of the open sets for this topology. Obviously this is Baire's topology, the pro- duct of the discrete topology on JN • It should be mentioned that Baire's topology is metrizable. The function d with

d(p,p) = 0 , d(p,q)

=

2 -m where m

=

µi[p-(i)

+

^{q) i)] if p}

^t

^{q ,}

is an appropriate metric. By [ JF - JB ] we shall denote the set of all continuous total functions from JF to JB • For a subset A of a topological space we shall tacidly consider the induced

topology.

The next important step is the definition of a certain repre- sentation 1J;' of [ JF - JB] • A function ^{y :} W ( ^JN)-W ( ^JN) is mono- tone (w. r. t. S), iff u ~ v

=>

^y (u) 1= y (w) . For any monotone y

a con tin uous f unction ^{1/J (} y) : JF - B i s de f ined by

1jJ ( y) (p) :

=

sup { y (w) w

'=

p} . On the other hand, for any continuous

!:

r :

JF - JB the function

y :

W ( JN) - W ( JN) with

y (w) : = max {v

I

lg(v) ~ lg(w) Ar ( [w]) c O } is monotone and

C V

satis fies r = ijj ( y) (see Weihrauch and Schäfer [ 22]) . Therefore ^{1jJ 1 : ]F - - - [ ]F -} JB ] defined by

is a surjective function. Using the function p (or y = V p -VJN ) ^-1 ' JN

i t is easy to determine arbitrary prefixes of 1/J' (p) (q) from prefixes of q . Note that only sufficiently long prefixes of p are needed.

A function r : JF - JB is called computable, iff r

=

^{1jJ' (p)}

for some total recursive function p . Therefore, any computable function is continuous. For the computable operators

r :

^{JF - JB}

there is a characterization by programs (Turing-, PASCAL-, ... )

r

is computable, iff there is a program which with input p

(i.e. with input sequence p(O), p(1), p(2) , ... ) gradually yields the values r (p) (0), r (p) (1), ... (as far as this values exist).

There is a cornputable Operator

r

^:JF--B

u with r <p,q>=iJ;'(p)(q) u

whenever p E dorn (ij; ^{1 )} • (By 2

<,>:JF -JF we denote the stan- dard pairing function on JF defined by <p,q>(2i): = p(i),

<p, q> ( 2 i + 1 ) = q ( i) . ) .

The partial representation \J.i' can be extended to 1jJ : JF - [ JF - JB]

as follows: iJ;p(q): =ijJ(p) (q): = ru<p,q> . This total represen- tation of the continuous functions frorn JF to JB is the key to a theory of cornputability (and continuity) on JF which is forrnally very sirnilar to ordinary recursion theory with ~ on JN

(see chapter 1). The representation 1jJ is deterrnined uniquely (up to equivalence see chapter 4) by the following axiorns which correspond to the universal Turing-rnachine theorern and the srnn-Theorern (or translation lernrna).

( 1 )

( 2)

(3f : JF - B , cornputable) (Vp,q E JF) iJ; (q) = r <p,q> ,

u p u

(Vr: JF - B , cornputable) (31:: JF -JB , cornputable, E(JF) c JF)

(Vp,q E JF) ij;E (p) (q) = f<p,q> .

Frorn 1jJ irnmediately a representation ~ : JF - [ JF - JF ] of certain continuous functions on JF and a representation

x : ]F - [ JF - JN ] of certain continuous functionals on JF can be derived. On NC JB we assurne the discrete topology, which is the topology induced on JN by the topology on JB •

~p(q) ^{: =} {ij;p (q) if iµp (q) E ]F

div otherwise

: =

_{

ij;p (q) (O) if div otherwise

iµp (q)

The utm- and the srnn-theorern correspondingly hold for 1jJ and x •

For k 2:: 2 , on lF ^k we assume the product of Baire's topology.

Tupling functions TI (k) : lFk- JF are defined inductively by

TI (1)(q) (k+1) ) (k)

: =q ' II (p1 ' ... ,Pk+1 : =<TI (p1' ... ,pJ,Pk+1> . For any k , TI(k) is a homeomorphism, and.the projections pr. (TI (k)) - 1 =: II ~k) are computable. Let r : ]Fk - - JF be an

i i

operator. r is continuous, iff r (TI (k)) - 1 : JF - - lF is continuous.

Since II(k) is intuitively computable, we call r tobe com- putable, iff r(rr(k))- 1 is computable.

On JFJN we assume the product of Baire's topology. The mapping TI (oa) : JF JN - ]F defined by TI (oo) (p

0 ,p

1, ..• )<i,j>: :::;pi (j) is a homeomorphism, and the projections of its inverse are corn-

- . - .

Open and clopen subsets of lF correspond to provable and decidable subsets of JN • For Ac JF the following properties hold:

A is open <=> A = dom ( r) for some r E range ( x) = [ JF - - JN]

A is clopen <=>A= r -1 { o} for some total r E range ( x) = [ JF - - JN]

Therefore, bydefinition, x represents only (even exactly) con- tinuous functions JF - - JN with open domain. The "projection theorem" gives an important relation between open and clopen sets:

A is open<=> (3B<.:_JF, clopen) A= {p ¹ (:lpEJF) <p,q>EB} . Specialization to the computable case leads to the following definition.

A subset Ac ^JF is called provable, iff A = dom ( r) for some computable r : JF - - JN , i t is called decidable, iff A= r-1

{O}

/i

for some computable (total!)

r :

JF - JN • The projection theorem correspondingly holds for provable and decidable subsets of JF.

Finally the set range(~) = [JF - - JF] can be characterized.

r E [ JF - - JF] , iff r is continuous and dom ( r) is a G

O -subset (i.e. a denumerable intersection of open subsets) of JF.

Any continuous

r :

JF - - JF can be extended into a

r

^I E [ JF - - JF]

Therefore, the set [ JF - - JF] is sufficiently rich to study all the continuous functions on JF.

It is important to observe that there are a "weak" theory of continui ty and a "strong" theory of computabili ty on JF •

Studying questions of continuity helps understanding computability.

It turns out that for most problems either there exists a com- putable function or there does not exist any continuous function.

Examples will be given later.

4. Representations Continuity, Computability and Reducibility In this chapter the elements of lF , i. e. sequences of natural numbers, are used as names for ¹¹abstract¹¹ objects.

The idea is not new. The Polish school of recursive analysis essentially uses a standard representation S : lF - - JR (see e.g.

Grzegorczyk [ 9 ] ) . Also Brouwer' s [ 4 ] concept of ¹¹Wahlfolgen"

seems to base on the same idea. Here, we assume that computers as well as mathematicians can w.l.g. only operate on numbers or sequences of numbers. Other sets have tobe named by numberings or representations.

4. 1 Definition

A representation of a set M is a surjective (possibly partial) function o : lF - - M . An element x EM is called o-computable, iff x = o (p) for some computable p (i.e.

pER( 1 )) .

4.2 Examples for Standard Represenations

( 1 ) idJE' : lF -IF is the standard representation of JE' . (2) ^(II(n))-1 IF-IFn is the standard representation of lFn

( 3) (II(co))-1 IF-IFJN is the standard representation of lF

~ 4) JM : I F - 2 JN

with ^{JM (p)} : ={ i i + 1 E range (p) } is the enumeration representation of 2JN

(CO)

(5) öcf: JF--2:JN with dom(ocf): ={pE JFI range(p)

~

^{{0,1}} and}

- -1

öcf(p):=p {1} for pEdom(ocf) isthecharacteristic function representation of 2JN.

(6) The functions

t t,

and x from chapter 3 are representations.

(7) By wJF(p) : = dom(xp) a representation of the open subsets of JF is defined.

(8) By w:JE'(p) :

=

U{[vE(i)] ¹ i EM } a representation of the open p

subsets of JF is defined.

(9) By ^{y (} p ) :

=

dem ( ~ _p ) a (total) representation of the

G0-subsets of JF. is defined.

The definition of computability w.r.t. given representations corresponds to that for numberings. However, now there is also a continuous version for any definition and also the

"mixed" (o,v)-computability can be defined.

4.3 Definition: (Continuity and Computability)

Let o. : JF - - M .. (i

=

1,2) be representations and let

J. J.

v : :N - - S be a numbering. Let F : ^M₁ ^{- - M}₂ and ^{G : M}₁ ^{- - S} be functions and let A be a subset of M

1 • (1) F is weakly (o

1,o

2)-continuo~s, iff there is some continuous operator

r

E [ JE' - - JF ] wi th

Fo1 ^(p) =

o

2

r

{p) for all p E dom(Fo 1) (equivalently: Fe

1

= o

₂

r

for some continuous

r :

JF - - JF ) • (2) F is

(o

₁

,c

₂)-continuous, iff there is some continuous

operator

r

E [ JF - - JF] with Fo1(p) =o

2^r(p) for all pEdom{Fo 1) , pi dom(r) for all p E dom{o

1) \ dom{Fo 1) • (3) (Weak) (o

1,v)-continuity of G is defined accordingly

US ing

r

E [ JF - - E ) .

(4) A is 6

1-open, iff A= dom(H) for some (generally partial) ( 6

1, idJN) -continuous function H : M

1 - - JN

(5) A is

o

1-clopen (i.e.

o

1-closed and open), iff A

=

H-1

{ 0} for some total ( o 1, idJN) -continuous function

H: M1- JN.

If in ( 1) , ... , ( 5) ¹¹ continuous ^II is substi tuted by ¹¹ computable ^11, the following notions are defined:

( 1 ^{1 )} _F is weakly

(o

1

,o

2)-computable.

( 2 1 ) F is (6

1,6

2)-cornputable.

( 3 1 ) G is (weakly) (6

1,v)-computable.

( 4 1 ) A is 6

1-provable.

( 5 1 ) A is 6

1-decidable.

The following diagram expresses F6

1 (p)

=

6

2 r-(p) in ( 1) and ( 2)

JF ____ .[ ____ JF

1 1

1 1

1

1 1

ö 1 ^{1 10} 2

1 1

1 1

+ ^~

M1 ____

F ____

^M2

The function r operates on names. The additional condition in (2) expresses that r must respect dom(F). This is a re- striction on dom(F). (6

1,6

2)-continuous functions, therefore have "natural" domains. The same holds for (o

1,v)-continuous The definition of o

1-provability is based on this fact.

Any continuous ( ( 6 1, 6

2) - , ( 6

1, v) -) function F is weakly continuous. The inverse holds, if F is a total function, The following Lemma shows that representation-continuous and -computable functions are closed under composition.

4.4 Theorem: (Composition of functions) Let ^6,

l be representations of M. ( i

=

1 , 2 , 3) , let

l

and G : M

2- - M

3 be functions. Then the following holds.

(F weakly ( 6 1, o

2) -continuous and G weakly (ö 2' ö

3) -continuous)

=>

GF weakly ( 6

1 , 6

1 ) -continuous.

The corresponding holds for "continuous", for "weakly com-:- putable", and for "cornputable" instead of "weakly continuous".

Proof:

By assumption there are r ,t:. E [JF - - JF] with Fö

1 (p)

=

^o₂^{r (p)}

for all pEdom(Fo

1) and Go

2 (p) =o

3t:.(p) for all pEdom(Gö 2) . Define I : = t:.r. Then I E [ JF - - JF ] and for all

pEdom(GFo

1) : GFo

1(p) =Go

2r(p) =ö

2t:.r(p) =o

2r(p) . Therefore, GF is weakly (ö

1,o

3)-continuous.

If in addition p~domr for pEdom(o

1) \dom(Fo

1) and q(t: domt:.

for q E dom ( ö

2) \ dom ( Go

2) , then for p E dom ( ö

1 ) \ dom ( GFo 1 ) : either p

$

dom(Fo

1), or p E dom(Fo

1) and Fo

1 (p) $ domG . In the first case, p

$

dom r which implies p

$

dorn E • In the second case, o

2r (p) = F6

1 (p)

q:

dom(G) , therefore r (p)

$

^dom(Go₂^),

r ( p)

$

dom ( t:. ) and p $ dom ( I ) • This proves the case "continuous".

For the "computable" cases i t suffices to observe that I

=

t:.r is computable, if t:. and r are computable.

Q.E.D.

Definition 2.3 for numberings can be viewed as a special case of definition 4.3 for representations in the following sense.

For any numbering v : JN - - S can be defined by

a representation o : JF - - S

\)

dom ( 8 ) : = { p E JF

I

p ( o) E dom ( v ) }

\)

and 8 ( p) : =v p ( o) i f p E dom ( 8 ) •

\) \)

Obviously, ids is (8 ,v)-computable.

\)

It is easy to show that v and 8 via definition 2.3 or

\)

definition 4.3 respectively induce the same kind of computability, provability and decidability for s .

Prom given representations certain new representations can be constructed. Some of these are introduced in the next definition.

4.5 Definition

Let 8 . JF - - M. be representations ( i E JN)

l. l.

( 1) The representation [ 8

1 , ••• , 8 n] of M

1 ^{x ••• x} Mn (n ~ 1.) is defined by

<p

1, ••• ,p >Edom[8

1·, •••. 8 ] :- <=> (Vi~n) p. Edom(8.),

n · n . 1. 1.

[ 81 ' · • • 18 n]<p1' · · • ,pn>: = ( 8 1 (p1) '· · • ' 8n (pn)) for <p

1, ••• ,pn> E dom[8

1, ••• ,8n].

If 8 : =8 = ••• = 8 , we wri te

1 n

(2) The representation [8

1]iEJN of the set of sequences M0 x M

1 x • • • is defined by

<p.>. EJNE dom[8.]. EJN : <=> (Vi E JN} p. E dom(8.)

l. l. l. l. . l. l.

[8.]<p.>:=(o (p

),o

1(p

1), ••• ) for <p.>Edom([8.].).

l. l. 0 0 l. l. l.

00

If (Vi) 8 = 8 . , we wri te

. J. 0 : =< 8 i

>

i E JN . (3) The representation [o

1 - o

2] of all the (81,82)- continuous functions is defined as follows.

p E dom [ o 1 - o

2 ] : <=>

$

P ( dom ( o

1 ) } <.= dom ( 6

2 ) and (Vq,q' E dom(o

1) (8

1 (q) = o

1 (q¹)=>

o ~ (q) = o ~ (q')) , 2 P 2 P

and [61 - 6

2 ] {p) {x) : =o

2^~p{q) p E doml 6

1 - o 2] .

for any qEo^-1

1 {x} i f

By 4 . 5 ( 3 ) , dom [ o 1 - o

2 ] is the set of all t-names for an operator

r

^{E [ JF} --JF] extensional w. r. t. ( 6

1 , o

2) , and [o1 -o

2 J(p) is the (o 1,o

2)-continuous function F:M

1^--M2 uniquely determined by

r

^{= ~} (Df. 4. 3 (2)). No"te that there

p

is no representation of all weakly (6 1,6

2)-continuous func- tions because this set is too large. We shall prove later that range [ ö

1 - ö

2 ] is a rather rich subclass of all the continuous functions from M

1 to M

2 , if o

1 and o 2 satisfy certain conditions.

From Definition 4.3(4)/(5) standard representation of the 6-open and 6-clopen subsets of a represented set M can be derived.

4.6

Definition

Let ö JF - -M be a represero tation.

(1) The representation w

0 of the set {ACM\ A 6-open}

is defined as follows.

p E dom(w

6) : <=> (Vq,q' E dom(o )) (6 (q) = 6 {q') =>xp(q)_=x (q')) . p

w6 (p) : = ö (dom(xp)) for p E dom(w 8) • (2) The representation ß

0 of the set {A~MI A is ö-clopen} is defined as follows.

: # ( p E dom ( w _.. ) and dom ( o ) c dom ( X ) )

u - p

:=ö(x- 1p {0}) for pEdom(ß 0) .

This means, dom(w

0) is the set of all x-names of operators

r :

^{JF - - N} extensional w. r. t.

o ,

and w

O ^(p) is the domain of the function F : M-- N determined by

r =

x p via

Def. 4.2(4). The explanation of the definition of ß

0 is similar.

The definition 4.3(2')/(4')/(5') and 4.5(3), 4.6(1)/(2) are consistentin the following sense:

F is ( o 1 , o

2) -computable

<=>

F is [ o 1 ... o

2] -computable

A is o -provable

<=>

A is ^w₈ -computable, A is o -decidable

<=>

A is ß

O -computable.

This follows immediately from Def. 4.1.

The representation and ^w

8 will be studied more detailed in connection with topological considerations in the next chapter.

The following example shows that different representations of a set may imply different kinds of computability and continuity.

4.6 Example

(a) The function u: (2JN)JN - 2JN

CX)

w i th U ( A . ) . : =

1. 1.

{xEJN ¹ (:li)xEA.} is

1. (JM , JM) -computable but not weakly

CX)

(ocf'ocf)-continuous.

(b) The function cpl: 2JN - 2JN with cpl (A) : =JN\A is ( 6 cf, 6 cf) -computable but not weakly ( JM , JM ) -continuous • ( see 4 • 2 . ( 4) / ( 5) )

The choice of a representation, therefore, depends on the func- tions which should become computable or continuous.

Proof:

( a) Suppose, p. E lF ( i E JN) and p = <p. >. Then

1. l

U(JM ⁰⁰(p))={xl (3.)xEJM(p.)}={xj (:3i,j)x+1=p.(j)}

1. 1. 1.

= {x

!

(:li,j)x+ 1= p<i,j>} = JM (p) . The identity function on lF can be used in 4.3(2) •

Now suppose, there is some continuous r : JF - - IF wi th

CX)

Uöcf{p) = öcfr (p) for be any function with

all ·p E dom ( Uo ~f) OEj:U(o~f(p)).

- .

- .

^1F ₀ ^{. Let p E 1F} ₀

Then r (p) (0) =l= 1 • Since r is continuous there is some iEJN with r[p[i)J~[r(p)[ 1 ]J=(r(p)(O)]. But there is some q E IF n [p(i]] with q<k,o> = 1 for some k > 1 ,

0

CX)

which implies o Eu (öcf (q)) = öcfr (q) , and r (q) {0)

=

1 . Since r (q) E r[p[i] J

s

[r (p) (0) J , r (q) (0) = r (p) (0) =1=1 • This is a contradiction.

(b) Define r by r (p) (i) : = 1.:. p(i) . Then r is computable and for all pEdom(ocf) we have cpl(ocf(p)) =ocfr(p) . Therefore, cpl is (ocf'ocf)-computable.

Suppose there is some continuous E : JF - JF wi th JM(I (p) = cpl JM(p) for all p E JF. Let p E JF wi th

0

t

^{JM p Then} OEJMI(p), i.e. I(p)(i-1)=1 for some iEJN.

By continui ty of I there is some k E JN wi th I[p[k]]::;[I(p)[i]]. There is some qE[p[k]] with

q(k)

=

^{1 ,} i.e. OEJM. Since I(q)E[I(p)[i]], E(q)(i-1)=1 q

i.e. OEN(E(q)) =cpl l1(q), a contradiction. Therefore cpl is not weakly ( JM , JM) -continuous.

Q.E.D.

A representation may be changed in a certain way without changing the kind of a cornputability (or continuity) defined by it.

For studying this, we define reducibility and equivalence, each of which in two versions, a continuous and a computable one.

4.7 Definition (reducibility, equivalence)

For any two representations o , : JF - - M, ( i = 1 , 2) we define:

( II Q

1

( II Q

1

l. ].

and is

is continuously (or c-) reducible to ^{Q II)}

2

and is (o

1,o

2)-computable) is reducible to ^cS ₂ ^")

It is easy to show that id is (6 1,6

2) continuous (com- M1 ,M2

putable) iff o

1 is weakly (idN ,6

2)-continuous (computable).

ßy tlefinition 4.3 a representation o of M defines a con- tinuity~theory and a computability-theory for M. Are there other representations of M defining the same continuity (com- putability) on M? The next lemma and its corollaries give defi- nite answers.

4.8 Lemma:

Let 6 and 6' be representations ~f M. Then th~ fol- lowing properties are equivalent.

( 1 ) 0 ~ 0 ¹

C

( 2) For any representation o

1.: JF - - M

1 and any F : M

1 - - M : F (weakly) (o

1,o)-continuous

=>

F (weakly) (o

1 ,o ')-con- tinuous.

(3) For any representation and any F . M--. ···2 . ^{~,r •} F (weakly) ( 6' , 6

2) -continuous

=>

F ( weakly) ( 6, o

2) -continuous.

A corresponding statement holds for (computable) reducibility and computability.

Proof:

( 1)

=> (

2) (immediate from Theorem 4.4) ( 2)

=> (

1) Choose 6 1 :

=

⁸ ! F :

=

idM .

( 1)

=> (

³⁾ (immediate frorn Theorem 4.4)

( 3) => ( 1) : Choose o

2 : =ö ' , F : =idM .

In the computable case the proof is accordingly.

Q.E.D.

4.9 Corollary:

Let

o,o'

be representations of M. Then

o

^and

c'

define the same continuity-theory (computability-theory) on M, iff o- c' (o=o')

C

Proof: (2) and (3) from Lemma 4.8 applied to

o

and to

c'

express that

c

^and

o'

define the same continuity-theory.

This is equivalent to o =eo' by Lemma 4.8. (accordingly for computabi,li ty)

Q.E.D.

4. 10 Corollary Let

o . , o

!

]. ]. be representation of M.(i=1,2), let

].

( 1) Suppose

o

₁

=

^c

o 1

^and

^o

²

⁼

^c

^o 2 ,

^then

F (weakly) (

o

₁^,

o

₂) -continuous <=> F (weakly) ( ö

1, ^o 2)

^-con-

tinuous, A o

1 -open <=> A o

1

^-open,

A

c

1-clopen <=> A

c

^1-clopen.

( 2) Suppose

o

₁

= c 1

^and

^c

₂

= c 2 ,

^then

F (weakly) (

o

₁ ^,

o

₂) -computable <=> F (weakly) ( 8

1 , ^o 2)

^-com-

putable

A

c

1-provable <=> A

c

1-provable

A

o

1 -decidable <=> A

o 1

-decidable Proof: . (immediate)

Also the operations "product", "function space" (Def. 4.5) and w,ß (Def. 4.6) behave reasonable w.r.t. reducibility and equivalence.

4.11 Lemma:

Let

o(o')

be a representation of M(M') , let o . ( o ! )

l l be

representations of perties hold.

M. (M!) (i E JN) • Then the following pro-

1 l

( 1 ) ( 2)

( 4)

( Vi , 1 :::; i :::; n) o . ::; 6 ~

l C l

(Vi)

o.::;

0~

1. C 1.

=> [ 0 1 ' ••• ' 0 ] :;;; · n c [ 0 1' ' ••• , 0 n ^{1 ] ,}

=> [

o . ] . ::; [ o

^~

] . ,

11.. C 1.1.

, and

(1), (3), and (4) hold correspondingly for the computable case.

( 2 1 ) Q ::; Q 1 => Q ^{CO ::; (} Q 1 ) CO

(An effective version of (2) would require reducibility uni- form in i E JN • )

Proof:

( 1) and (2) By assumption there are continuous operators with

o.(p)=o~ ^r1..(p) for pEdom(o.). Define for (1)

l 1. 1.

r.

l

r<p1 ' · • · ,pn>: =<r 1 (p1),. · • ,rn (pn)>

and r<p.>: =<r. (p.)>. for (2) •

.1 .1 .1 .1

(3) Suppose, o

1

^{(p) =o1}r(p) (pEdomo

1)

^{and o2(q) =o}

2

^t.(q)

( q E dom ( o

2) ) where r , 6. E [ JF - - JF ] . By the utm-theorem and the smn-theorem for 1)J there is some total I: E [ JF - - - JF ] with ~I: (p) (q) = t,,~Pr (q) for all p,q E JF. Then. for all pE dom[o

1 - o

2] , xEM

1 , qE (o

1)-

^{1 {x}:}

[o 1 - o2 ] ^{(p) (x)}

=

_{o 2~Pr} ^(q)

=

⁰

2

^{6.~Pr (q)}

⁼

^o

2

^{~r (p) (q)}

⁼

[o 1 ^-o 2

](t:(p))(x).

,. This implies [

o

₁ - o

2] S [

o 1 -

^o

2] .

(4) Suppose, ö (p) = o 'r (p) for p E dom(ö) , where r E [JF - - JF]

By the utm- and srr~-theorem for x there is some continuous

I: : JF - F w i th

xt:

^{(p) (q) = xpr (q)} for all p,qEJF • Then for p E dom(w

0,) : w 6 , ( p) = o ' dom ( x P) = o ' r r

=

⁸ dom XI: ( p)

=

^w _O ^{( I: ( p) ) ,}

and f or p E dom ( ß O , ) :

-1 domx = o dom ( x r)

p p

s8, (p)

=

ö' (x~1

{o})

=

^o'rr-1x~1{o}

=

oxI:(;){o}

=

^ß₀^(I:(p))

In the computable case, the functions under considerations are computable.

Q.E.D.

The class of·representations with the relation Sc(S) is a preorder, i. e. cS ~ cS and cS ~ cS ' " cS '

s

cS" => cS :;; cS ^{11 •}

For any preorder the equivalence classes under the induced or- der are a partial order, i.e. in addition the law

x ~ y " y ~ x ::;>x = y holds.

In a partial order the infimum (supremum) of a subset is well- defined if i t exists, in a preorder i t is defined up to equi- valence (if i t exists).

Let (A,s) be a transitive reflexive order, i.e. a preorder.

For Y c A we define

Inf (Y) ={a E A \ (Vy E Y) a s y /\ (Vb E A) [ (Vy E Y)b s; y => b s a]}

Sup(Y) ={aEA\ (VyEY)ysa/\ (VbEA)[(VyEY)ysb=>asb]}

Each of the sets Inf(Y) and Sup(Y) is either empty or con- sists exactly of a single equivalence class, where

a

=

^{b :} <=> ( a

s

^{b /\ b}

s

^{a) .}

For the class of representations Inf and Sup denotes the above sets w. r. t. the preorder s , Int and Sufc denotes these sets w. r. t. the preorder ~ •

4.12 Theorem Let o

1,ö

2 be representations. Representations ö and ö are defined as follows:

dom(~_): ={<p 1,p

2> p

1 Edom(ö

1) Ap

2 Edom(ö

2) /\ o₁(p

1) =ö 2(p

2)} , .§_<p

1,p

2>:=ö 1(p

1) for <p 1,p

2>Edom(ö), and

dom(6) : ={2p \ p E dom(o

1)} u {2p + 1 1 p E dom(o

2)} , 6(2p) : =o

1 (p) if 2p E dom(6) , 6(2p+ 1): =o

2 (p) if 2p+ 1 Edom(6) .

The .representations ö and o have the following pro-·

perties.

( 1 ) §_ E Inf { o 1 , o 2} S Int . { o 1 , o 2} (2) 8ESup{ö

1,ö

2} SSu!?:: {ö1,o 2}