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Mathematik und
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Informatik-Berichte 44 – 02/1984
Christoph Kreitz, Klaus Weihrauch
A Unified Approach to Constructive and
Recursive Analysis
In the past several attempts have been made to formulate and study constructivity and computability in ·mathematics but none of these concepts' is generally accepted. In this paper a simple and unified approach to constructive and recursive analysis is presented. Its basis is a standard theory of effectivity on IF={f:IN-lN} which is formally similar to ordinary recursion theory. lt splits in to a purely topological (11constructive11 ) version and a more special theory of computability. Effectivity an other sets M is derived by means of representations,i.e. surjective mappings 6: W --- M, where topological properties of repre- sentations play a fundamental role.
As an example representations of the real numbers are studied and it is shown that the significant differences between
<previously defined> representations are topological ones.
Finally compactness on the real numbers is studied as a further application of this theory of representations.
A Unified Approach to Constructive and Recursive Analysis by
Christoph Kreitz and Klaus Weihrauch
1. Introduction
Many mathematicians familiar with the constructivistic objections to classical mathematics concede their validity but remain unconvinced that there is any satisfactory alternative.
Among others Bishop [ 2 ] and Brigdes [ 3 ] showed that large parts of classical analysis and functional analysis can be formulated in a constructive way. But it does not seem tobe likely that their concept of constructivi.ty will be ge- nerally accepted.
The previous attempts to formulate and study effectivity in analysis can roughly be devided into three classes. The constructivists only study 11constructive11 objects and only use 11constructive11 proofs e.g. by using intuitionistic logic
(_Brouwer [ 4 ] , Lorenzen [ 15], Bi shop [ 2 ] , Bridges [ 3 ] , et a 1.). The other two attempts are based on recursion theory. The 11Russian schoo111 (Ceitin [ 5 ], Kushner [14], Aberth [ 1 ], et al.) starts with an 11effective11 partial numbering.
of the set of computable real numbers lRc by which computability on ]'~ is trans- ferred to lR . The 11Polish school11 (Grzegorczyk [ 8 ], Klaua [ 9 ], et al.) starts
C
(essentially) with an 11effective11 representation of all real numbers by
1F := lN:N, by which computability of operators an JF is transferred to computability on JR. The approach presented here is a consequent continuation of that of the Polish school. It is formulated as a theory of numberings v : 1'1-- S (Ershov( 7]) and of representations o : JF-- M (Kreitz and Weihrauch [12]) and admits to study continuity, computability and computational complexity (Ko [10]) which can be considered as different degrees of constructivity. Because of space restriction we only outline basic definitions and properties and show by examples how analysis can be developed in this context.
We shall consider the set lN of natural numbers as the (w.1.g. single) concrete set of finite objects and the set 1F = :NJII of sequences of natural numbers as the (w.1.g.
single) concrete set of infinite objects on which 11constructions11 can be performed (e.g. by a computer or by men with paper and pencil). Constructions on 1F are governed by Baire's topology, thus continuity of functions JF--- Jll or JF---- JN (JN with the discrete topology) is the weakest form of constructivity. Computability is the next additional strenger requirement for functions JF---JF, JF---- 1'I, or
™---- ™. Further conditions on the computational complexity (e.g. primitive recursive or polynomial) yield more restricted kinds of constructivity.
As a basis for the studies in Chapter 2 a unified concise Type 2 theory of continuity and computability on F is outlined, which is formally similar to ordinary Type 1 recursion theory on ™. More details can be found in a forth- coming paper (Weihrauch [23]).
For all other objects the elements of ™ or :f are used as names. Let S be a set to be named by numbers. Then any s ES must have a name and any number i s name of at most one s ES. Thus a naming of a set S by numbers is a possibly partial surjective function v : ™ --- S, which we call a numbering. The theory of numberings is
studied in detail by Ershov [7] (see also Mal 'cev [16]). Similarly a naming of a set M by elements of :f is a possibly partial surjective function o : :f - - - M which we call a representation. Constructivity on S or M is defined via constructi- vity an concrete objects, namely the names w.r.t. given numberings or representations.
Chapter 3 gives an outline of a general theory of representations. An essential point is the definition of admissible representations of separable T
0-spaces.
Again in this theory topoligical (t-) and computational (c-) aspects are considered simultaneously (Kreitz and Weihrauch [12]).
In Chapter 4 as an example representations of the real numbers are studied. It is shown that the significant differences between previously defined representations are topological ones. As a further application compactness on lR will be studied in Chapter 5.
2. Type 2 Recursion Theory
As we a l ready have outl i ned computabi 1 i ty and conti nui ty on ™ and :f are the bas i s of our approach to constructive and recursive analysis. We assume the reader is fami- lier with ordinary recursion theory on ™ and with basic poperties of numberings (Mal'cev [16], Rogers [20], Ershov [ 7 ]). Let~ be a standard numbering of the
unary partial recursive functions pCl), let < > : ™n_ lN be Cantor's n-tupling func- tion. By f: A --- B (with dotted arrow) we denote a possibly partial function from A to B. Unlike to ordinary (Type 1) recursion theory for Type 2 recursion theory there is no generally accepted formalism. Below we outline a unified approach which is for- mally similar to the Type 1 formalism. More details can be found in Weihrauch1s paper [23].
Westart with some topological preliminaries. Let :f := lNlN, lB := :f u W(lN) where W(lN) is the set of words (i.e. finite sequences) over lN. On lB a partial order is defined by b~c: <=> bis a prefix of c. On lB we shall
assume the topology corresponding to the cpo (lB ,c.,e:) (Egli and Constable [6 ], Scott [21]). The topology induced on the subset :f is Baire1s topology. On lN we consider the discrete topology.
First a standard representation 1/1 of [:f- lB], the set of continuous functions from :f to lB is defined. From 1/1 we derive representations of certain continuous functions from :f to :f and from :f to lN. The construction of 1/1 rests on the following
property. Let y : W(lN)-W(lN) be isotone (w.r.t. ~). Then the function y·:
:r-
lB, defined by y(p) := sup{y(w)IWcp}, is continuous. And for any continuous function r : lF -lB , r = y for some i sotone y : W ( lN )-W ( lN) • The functi on y specifi es, how from prefixes of p elF suffiently many prefixes of r(p) = y(p) can be determined. A function r: lF-lB is called computable, iff r=y for some computable function y. The computable functi ons r : lF - lB can cas ily characteri zed by oracl e Turing machi nes whi eh on i nput p Elf from time to time read a value p(i) and from time to time write one of the values q(O), q(l), ... (in thisorder) of the result qElB. For transforming n-ary functions on lF to unary ones, the fo 11 owi ng tup 1 i ng functions II ( n): lF - lF are used:II(p,q)(i):=(p(x) if i =2x,q(x) of i =2x+l), II(l)(p):=p
( n+ 1 ) . _ ( n ) . . . _ ( n ) )
II (p 1, ... ,pn+l) .-II(II · (p1, ... ,pn~Pn+l)' notat,on. <p1, ... ,pn>·- II (p1, ... ,pn.
Alsow-ary tupling is possible: II(w\p ,p
1, ... )<i,j>:= p.(j). The functions II(n) and II(w) are homeomorphisms w.r.t. the pro~uct topologies. The1projections of their inverse are computable.
The definition of y is effective in the following sense. There is a computable (b.Y an oracle Turing machine) operator IIu: lF-lB with the following property. On input p,q it determines y(q) if y:= vNpv-~f (vN is a bijective standard numbering of W(]N)) is isotone, r(q) for some continuous r: lF- lB otherwise. Then by I/IP(q) :=w(p)(q) :=
r <p ,q> a representati on 1/J : lF - [ lF - lB ] of the conti nuous functi ons from lF to iB u
is defined, which satisfies the11universal Turing machine theorem11 and the 11smn-theorem11 Theorem:
(1) 1/J (q) = r <p,q> for some computable r E [lF- lB ]. p
u u
(2) 1/Jp<q,r>=I/JI:<p,q>(r) for some computable I:E [lF- lB] with range (I:)~ lF.
Notice that r and I: are not only continuous but even computable. Similar to Type 1
u
recursion theory the utm-theorem and the sum-theorem characterize the representation
1jJ uniquely up to (computable) equivalence (see Chapter 3). More interesting than 1/1 itself are two representations derived from 1/J.
Definition: (1) Define a set [lF-lN] of partial functions from lF to l'l and a representation x: lF-[lF-lN] by: xp(q) :=x{p)(q) := (div if ,µp(q) =sEB,.
the first numbe~ of the sequence ,µP(q) otherwise).(2) Oefine a set [lF -lF]
of partial functions from lF fo f' and a representati on ~ : lF - [ lF - JF] by
;p(q) := ~(p)(q) := (,~P(q) if \/ip(q) E lF, div otherwise).
This definition extends well known concepts of computable operators and func- tionals to a uniform topological description, where the elements computable w.r.t. a given representation are those with computable names. The functions from [lF-lN] and from [lF-IF] have natural domains (c.f. domains of par- tial recusive functions). But the set of domains is sufficiently rich such that any continuons function is essentially considered.
Theorem: (1) [lF-N] is the set of all continuous functions z: lF----lN such that dom ( z) i s open. For any continuons function r : lF ---- JN there i s some z E [lF - l N ] which extends r. (2) A valid statement is obtained by subs-- tituting 11:N11 by "lF11 and 11open11 by 11G0-subset11 in (1).
Also the represer:tations x and; satisfy the utm- and the smn-theorem. This leads to a rich theory fo continuity and of computability which is formally similartoType 1 recursion theory. From the above theorem we conclude that by
1(:{0) a representation w' of the oper· s1,1--- - ~ is defined,
- ~ 1 ,esponds to the numberi ng i 1--- dom( q)i) - , JOSets of :N •
1✓e call a subset A~lF t-open (c-open) iff A=w'(p) for some (computable) p.
Ais t-clopen (c-clopen), iff A and lF\A are t-open (c-open). The t-open (c-open) sets are exactly the projections of the t-clopen (c-clopen) sets. The self applicability and the halting problem of x can be formulated. They are c-open, not t-clopen, c-complete and c-productive. Also effective insepera- bil ity can be defined. The sets {p
I
xp(p) = O} and {pI
xp(P) = 1} are c-effec- tively inseperable. This property can be used in the study of precomplete representations. Many other properties can be proved easily but more questions are still unsolved in this theory of continuity and computability on JF.3 Theory of representations
In order to define computability and constructivity on a set M with cardinality not greater than that of the continuum, we represent M by a surjective mapping
o : f" --- M , ca 11 ed representati on of M. Some exampl es for representations are
the enumeration representation :M: lF-P with M(p) :={i I i+lErangep}, the
w
representation ocf of Pw by characteristic functions with ocf(P) := {i!p(i) =0}, and the representations iµ: f" - [ f" - B], ~: f" - [ f" - F] ,
x : f" - [f" - l N ] , w1 : f" -{xs;.F\ x is open} introduced in chapter 2.
Effectivity properties of theorems, functions, sets, predicates etc. can be expressed by effectivity of correspondences (i.e. multivalued functions) which are triples f = (M,M' ,P) where P~M x M'.
Defintion Let o,o' be representations of M resp. M1 and let f=(M,M',P) be a correspondence. f is called weakly (0,01)-t-(c-) effective iff there is some ( comuputab l e) r E [ F - IF] such that
( oq , o I rq) E P for a 11 q E o -l dom( f) .
f is called (o,o')-t-(c-) effective, iff in addition r(q) is undefined for all qEo-1(M\domf).
(o,v)- effectivity of a correspondence f= (M,S,P) where v is a numbering of S is defined accordingly using [F -JN] instead of [F -IF]. For convenience we shall say 11continuous11 instead of 11t-effective11 and 11computable11 instead of
11c-effective11•
Since a partial function is a single valued correspondence the above definition is applicable to functions. A subset As;.::M can either be characterzed as the domain of a partial function or by its characteristic function.
A set A_SM is called o - (c-) open iff dA := (M,Jl, A x :N) is (o,id:N) t- (c-) ef- fective. Ais called o-t-{c-) clopen iff cA:=M,:N,{(x,0) !xEA} u
{(y,l) 1 yEM\A}) is (o,idlN) t- (c-) effective.
Usually we say "provable11 instead of 11c-open11 and 11decidable11 instead of
"c-open".
Theo- effectivity on M strongly depends on the representation o. Consider the two questions whether complementation on P is effective and whether countable
w
union an P is effective. There is no absolute answer but only one relative to
w
the considered representation: Complementation is (ocf'ocf)- computable but not even (JJl,::M) - continuous, countable union is computable w.r.t. JJI but not even weakly continuous w.r.t. ocf (use rr(w) for formalization). This difference can
be explained using the intuitive concept of finitely (or continuously) accessible (f.a.) information. Every true information n E Jvl(p) is f.a. from p, no true information n f Mp is f.a. from p. But every true infonnation nE ocf(p) or m~oef(P) is f.a. from p. Representations may be changed in a certain way without changing the induced effectivity.
For any two representations 5,o' of M resp. M' define
o:s;to' :<=> Mc;M' and nM,M' is (o,o') t.,.effective, o = - t o ' · , <=> u - tu '° < '° 1 and -" u _tu. 1 < -"
c-reducibility (:s;c) and c-equi_~alence (=c) is defined accordingly.
It is easy to show that ocf:s;c]M and that lM and ocf are not t - equival ent
Since effective functions are closed under composition two representations are t- (c-) equivalent.if and only if the define the some continuity (computa- bility) theory.
Theorem Let 5,5' be representations of M. Then (1),(2), and (3) are equivalent.
( 1) (2)
(3)
ö::;; tö 1
For any representation o1 : F --- M f (weakly) (o 1
1 ,o) t - effective For any representation a
2 : F --- M 2 g (weakly) (ö~o2) t-effective
and any correspondence f = (M,M
1 ,P) : ,=> f (weakly) (01,01) t-effective and any correspondence g = (M
2,M,P):
-=> g (weakly) ( ö ,o2) t - effecti ve.
Every representation o: F ---M induces a topology ,
0 on M by x E ,0 :<=>
o-1x=Andomö for some open subset As_F. ,
0 is called the final -topology of o and it consists exactly of all the o - open subsets of M. For example ':M, the final topology of the enumeration representation of P is determined by
w
the basis {Oe
I
ec :N, finite} where Oe:= {x~l'lI
e~x}. Clearly t- equivalent representations have the same final topologies but the c.onverse does not hold in general (a counte~exampel is presented in chapter 4). If on M already a topology -r is given then T=T0 should hold for any "reasonable" representation of M. (In some special cases there might be reasons for choosing T0 'FT,) For seperable T
0 -spaces representations equivalent to a standard-represen- tation defined as follows seem tobe the most natural ones.
Defi nt i on Let ( M, -r) be a seperab 1 e T
O - s pace and 1 et U be a numberi ng of some basis of -r. For xEM let e:u(x) := {i I xEUi}.
A standar~ - representation 5u of (M,T) is defined by
domö :=lM-le: (M) and ö (p) :=e:-1:M(p) whenever pEdomou.
u u u u
A standard-representation öu of a seperable T
0 -space has some remarkable properties
( 1)
(2)
(3)
öu is continuous and open, especially -r = 'öu For any space (M' ,-r 1) and any H : M - - M'
HO öu is continuous <=> H i s continuous ,
r,;:::; töu for any conti nuous i:; : f" ---M .
An immediate consequence is that all the standard -representations of the same space are t- equivalent and therefore the equivalence class
· {ö I o
=
töu} does not depend on the numbering U. Since every representation equivalent to öu induces the some continuity- theory we call a representation ö of a seperable T0 -space t-effective (or admissible) iff ö:töu for some standard - representation ö . The representations Ji1 and ö f of P are admissible.
U C w
The decimal - representation of the real numbers is not (see next chapter).
For admissible representations of a space (M,-r) the final topology is identical with -r. Furthermore topological continuity and continuity w.r.t. these repre- sentations are closely related.
Theorem Let ( M;, -r i) be seperab 1 e T
O - spaces and 1 et ö i : f" ---M; be admi ss i b 1 e representat i ons ( i = 1,2) . Let F : M
1 ---M
2 , then:
(1) F (, 1,,
2)-continuous <=> F weakly (o 1,o
2)-continuous, (2) F (,
1,,
2)- continuous /\ domFEG0(,
1) => F (o 1,o
2)-continuous.
For some representations the converse of (2) also holds (e.g. for the repre- sentation p of lR by normed Cauchy - sequences - see next charter).
There are many other aspects of representations which should be studied, for example recursion-theoretic properties, co'mputable elements, the structure of equivalence degrees, closure properties etc. There are also natural representations the final topologies of which are not seperable. See Kreitz&Weihrauch [12] for further discussion.
4. Representations of the Real Numbers
An excellent recursion theoretical comparison of many representations of lR has been given by T. Deil [25] . In this chapter we show by examples that the essential differences between the representations which are mainly discussed in constructive and computable analysis already are of topological nature.
More details can be found in a paper by the authors [24].
Let lR be the set of real numbers, l et , R ~ 2 lR be the set of open subsets As;, lR , where Ais open iff it is the union of open intervals (x;y) := {zER I x<z<y}.
The space (lR,rR)is a seperable T
0 - space. From Chapter 3 we know that there are admissible representations o of lR for which especially ,
0 = 'R· The authors have shown [12] that for any complete seperable metric space an admissible representation can be defined via normed Cauchy sequences. In the case of JR a useful representation of this type is as follows:
Definition:
Let Qn := {m • 2-n
I
mE?Z}, Q0 :=UQn' let v0 be a standard numbering of Q0. Then the standard representation p of the real numbers is defined by
dom(p) := {pEF \(vk)(v
0p(k) EQk /\ \ v
0p(k) -v
0p(k+l) \ <2-k)} , p(p):=limv
0p(n) for all pEdomp.
This representation is admissible and its. final topology is 'R" As a product of admissible representations pn, defined by pn<p
1, ... ,pn> := (p(p1), ... ,P(Pn)), is admissible for any n> O. Therefore pn satisfies Rice1 s theorem which states that p-lx is not t-clopen if XERn is not trivial. Especially relations like x<y , x =y, ... are not / - decidable.
The definition of the real numbers as cuts of rational numbers induces several representations.
Defintion
Oefine a representation p< of JR as follows:
dom p < : = \ p E IF \ v
0JMP has an upper bound}
p<(p) :=supv0}1p for pEdom(p<).
A representation p> is defined correspondingly by p>(p) = infv0JMP.
The representations p< and p> are admissible w.r.t. their final topologies '< and '>' respectively, where
'<= {(x;00) 1 xElRU {-eo}} , '>= {(-co;x) \ xElRU {00} } .
It is easy to show that pEinfc(P<,P>) (c.f. [24J) and by the characterization
of the final topologies p<$ t p, p>$ t p , P<$ t p > and p>$ t p< . The charac- teristic properties of p, p< and p> are given by the finitely accessible information in each case. For any i E lN and p E lF the property v
0(i) < p<(p) can be proved in finitely many steps iff it is true, more precisely
{(d,x) EQ
0x IR I d<x} is [v
0,p<] -provable. p< is the greatest representation (w.r.t.s;c or s;t) with this property (similarly for p<). Finally p is the greatest representati on p of IR such that
{(d,e,x)EOoxOoxIRid<x<e} is [vo,vo,o]-provable.
Classical analysis suggests the following representation oc by unrestricted Cauchy sequences: dom (oc) = {p I v
0p is a Cauchy sequency}, oc(p) := limv
0 p(n) for pEdom(o ). But in this casenoprefix p[n] of p gives any information of
C
oc(p), no information is finitely accessible. This is formally expressed by the characterization of the final topology , of o : , = {0,IR} , i .e. the indiscrete
C C C
topology on IR. Therefore, oc is not usful in constructive analysis for purely topological reasons.
The most commonly used representations are the r - adic representations (r=2,8,10, ... ). The finite r-adic fractions are dense in IR, the infinite r - adic fractions, however, are not appropriate for representing the real
numbers. Again this has topological reasons. For simplicity we consider the case r = 10. Define oDEZ by dom(ooEz) = {p E lF 1 (vn > 0) p(n) < 10} ,
ooEz(P) := (TT1P(O) -TT2p(O)) L {p(i) .10-i I i ElN}. It can be shown that ODEZ has the final topology 'R' especially oDEZs;cP• But oDEZ is not admissible since
p $t oDEZ . The names w. r. t. oDEZ contain more finitely accessible information than those w.r.t. p. A very undesirable consequence is that many trivial functions on IR are not even (oDEZ'oDEZ) - continuous.
Example: The function x t--3x is not (oDEZ'oDEZ) - continuous: Consider the
sequences (Pn) and (qn) on lF defined by Pn = (0 0 3 ... 3 0 ... ), qn = (0 O 3 ... 3 4 0 ... ) (each with n- times 11311) . Suppose 3 • oDEz(r) = oDEZr(r) (rEdom(oDEz)) for some r: lF ---- JF. Then l im Pn = l im qn = (0 0 3 3 •.. ) , but l im r p(n)
*
l im r q(n), hence r is not continuous. Therefore the decimal representation is not very useful for analysis, again for topological reasons.Finally we consider the representations by characteristic functions of left cuts (or right cuts) which is also often considered. Let v be a total bijective num- bering of some dense(w.r.t. 'R)subset of IR. Define olv by
dom(olv):={pElFI (3xER)(vi)(p(i)=0 <=> v(i)<x}, oL)P) :=supvp-1
{0}
for p E dom (oL). Then olv is admissible and its final topology 'Lv is gene- rated by the basis {(x;y] 1 x,yES, x<y}. Especially, one easily shows ol ::; p
V C
and 'R~'Lv. Notice that the final topology 'Lv depends on the dense subset S S. lR whi eh i s numbered by v. Therefore no l eft cut representat ion olv of lR be called 11natural11 for topological reasons. Right cut representations oRv are defined accordingly. It should be noted that the representation defined by continued fractions is the infimum of olv and oRv if v is some standard num- bering of Q.
Among all the representations of lR we have considered p is the only one which has the appropriate topological properties for the study of analysis. The
special definition of p guarantees some positive computability properties which become important in complexity theory (Ko [10], Kreitz and Weihrauch [11]).
5 Compactness on lR
As an application of the theory of representations to constructive analysis we consider compactness on lR. We show that (at least) two different reasonable kinds of compactness can be formulated in our theory. Let I, defined by
I<j,k>:=(v
0(j)-2-k;v
0(j)+2-k), be a standard numbering of a basis of 'R·
A standard representation w of 'R is defined by w(p) := u {Ik I k E :Mp}. The
corresponding representation a. of the closed subsets is a.(p) := lR\w(p). Another way specifying a closed set Aist to list all the open sets Ik such that
A11Ik=1=0. Let dom(a.c):={pl (3A,closed):Mp={klA11Ik=1=0}}
a.c(p) := ~ ~ {I<j,k> 1 <j,k>ElMP}. The representations a. and a.c are incompa- rable w.r.t. topological reducibility because the finitely accessible informa- tions are too different.Both representations are admissible w.r.t. their final topologies. Let a.1 be the standard infimum of a. and a.c. This representation corresponds to the concept of locatedness in constructive analysis (Bishop [2]).
For a cl osed set A,A = {x
I
d(x,A) = O}, therefore the cl osed sets can be charac- terized by distance functions which are (p,p)-continuous. The ~-names of the corresponding operators lF yield a representation a.1 of the nonempty closed subsets of lR which directly expresses locatedness. The representation a.1(restricted to the nonempty sets) is equivalent to a.1• A subset of lR is compact, iff it is closed and bounded. Therefore, the restriction a (ac,al) of a. (a.c,a.l) to the bounded subsets yields a representation of the compact subsets K(lR).
Unfortunately sup: K(lR) - J R is not (a,p) - ((ac•P) - , (a. 1,p)-) continuous, that is, a p-name for sup(S) cannot be continuously obtained from an ä-name of S (etc.). However the following holds:
Theorem
There are computable functions E,r,t.: lF ----lF such that:
p< E (p) = sup a.c(P) if a.c(P) is bounded,
p> r <i ,p> = sup a.(p) if a.(p) s;. [-i ; i]
p t.<i,p>=supa.1(p) ifa.1(p)s;.[-i;i]
Therefore, for determining the supremum (continuously) it is not sufficient to know that a.1(p) is bounded but a bound must be known (c.f. Bishop1s concept of constructive compactness). The information of a bound can be inserted into the name. Define representations a.b and a~ of the compact subsets of lR by
dom(ab) =f<i,p> 1 a.(p)~[-i; i]}, a.b<i,p>:=a(p), and a.~ correspondingly with a.
1. Then sup is (a.b,P>)-computable and (a.~,p)-computable (but not (ab,p)-continuous).
The compact subsets of lR can also be characterized by their Heine - Borel
property Which leads to (at least) two different characterizations. Let C p : = { I j I j E Jv1 p } , C p , n : = {I j 1 ( :l i E D n ) j + 1 = p ( i ) } . Let u s s ay II Q : f" - - - - fi proves compactness of S ~ lR11 i ff
(vp)[S~ ucp <=> pEdom(Q) and s~ucp => s~ucp,Q(p)].
This induces a representation i<: w of the compact sets by dom . w (K. ) = {p
IX·
p proves compactness of some S~R}, Kw(p) = the set S~lR for which xp proves compactness.Ourfirst constructive version of the Heine-Borel theorem for lR is as follows:
Theorem (Heine - Borel, weak)
b
a. =c Kw
Since sup, therefore, is not (Kw,P) -continuous, Kw is called the weak Heine- Borel representation. As we already know~ the names p w.r.t. a.~ contain more finitely accessible information. It can be shown that from such a name p for any covering Cq a minimal covering can be obiained. Let . K be the restriction of ~ w to those p E f" such that cq,x (q) is a minimal covering of K.w(p), i .e.
K (p)
q:.
u E for any proper sub~et E of C ( ) . Then the (strong) Heine -Borelw q,xp q
theorem can be formulated as follows:
Theorem (Heine-Borel, strong) a1 =c b K
The proofs and more details will be presented in a forthcoming paper [13].
6. Conclusion
In this contribution we have presented an approach to constructive analysis
which is based on Type 1 and Type 2 recursion theory and on the theory of numberings and of representations. For the type 2 theory, topology plays an important role.
The approach extends the approach of the Polish school. Since for any representa- tion o there is a canonical numbering v
0:=o~ of the computable elements it includes the approach of the Russian school, and-also essential ideas of the approaches
like Bishop's can be expressed and studied without change of logic. We claim that this is the adequate way for investigating constructivity,computability and computational complexity in analysis (and other areas of mathematics).
The properties 11continuous11, 11computable11, 11easily (e.g. polynomially) compu- table" yield a basic hierarchy of constructivity. It has turned out that in most cases a property which is not 11effective11 is not even continuous and a pro- perty which is "effective" is easily computable. This means that most of the ne- gative results have topological reasons and are independent of Church's thesis.
The fundamental role of topology in recursive analysis has already been pointed out by Nero de [ 18] and others.
The approach presented here does not depend on specific representations but diffe- rent representations may be chosen from case to case. First of all a representation has tobe topologicallYsound depending on the intended- application. Then computabi-
lity and computational complexity have tobe considered. For any representation o the corresponding numbering v
6 of the o-computable elements brings up the
question how v0-computability and o-computability are related. Complete or partial answers are known only for very few cases (Myhill and Shepharson [17], Ceitin [51, Spreen [22]). Seemingly the clear topological properties of o are concealed by the composition with the numbering ~ the topological properties of which are not easily to understand.
In this contribution we only gave some examples of application. It should be mentioned that the kind of computability introduced by Pour-El arid Richards [19] an the LP-spaces can very naturally be defined in our framework by use of representations. A general theory of constructive normed spaces can be developed.
Even measure theory can easily be approached. By d(X,Y) :=µ(Xt:,, Y) a metric on the _ _ open sets Jk := u{In In E Dk} can be defined. The completion of this space is (essentially) the set of Borel sets factorized by the null sets.
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