Constructivity as Continuity: An Approach to Constructive and Computable Mathematics without Constructive Logic

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Informatik-Berichte 67 – 05/1987

Constructivity as Continuity

An Approach to Constructive and Computable

Mathematics without Constructive Logic

ABSTRACT

MATHEMATICS WITHOUT (ONSTRUCTIVE LOGIC

by

Klaus Weihrauch Fernuniversität

Hagen W-Germany

A simple universal approach for investigating constructivity computability, and computational complexity in mathematics is presented. The approach is based on a general Type 2 theory of continuity and computability and can be considered as a consequent extension of the Polish recursive analysis. It does not require constructive logic but a certain kind of continuity can be

interpreted as constructivity.

This contribution outlines very shortly Type 2 theory of

continuity and computability and tries to explain the character and power of the approach by simple examples. It is claimed that for investigating constructivity and computability in in rnathematical theories constructive logic is dispensable.

1

I NTRODUCTI ON

rhe developrnent of extrernely powerful cornputers in the last decades requires a theory which closes the gap between "classical" abstract mathernatics and concrete computer science. The existence of several partly overlapping, partly cornpeting approaches to constructivity

in rnathernatics, especially in analysis, (see e.g. Beeson [2] or Troelstra [21]) shows that still there is no satisfactory generally

accepted approach.

Previous attempts to investigate aspects of constructivity can be very roughly classified as follows.

(1) Recursion theory:

Computable functions f: :N - - lN , numberings v: :N - X for studying computability on X, classical logic ( Rogers [20], Ershov [6], etc.)

(2) Restriction of logic:

0nly constructive sets, only constructive proofs, many different trends (Brouwer [5], Lorenzen [16], Bishop [3], Troelstra [21], and many others)

(3) Russian school:

A combination of (1) recursion theory and (2) constructive proofs (Markov [17], Kushner [14], Aberth [1], etc.)

(4) Polish school:

Representation of all real numbers by sequences of natural numbers, computable operators on sequences yield computable functions on JR (Grzegorczyk [7], Lacombe [15], Klaua [9], Hauck [ 8] etc.)

The approach proposed here is a consequent extension of that of the Polish school. l t is based on ordinary recursion theory (e.g. Rogers

[20] and on Type 2 theory of continuity and computability (Weihrauch [22,24]) and i t is formulated as a theory of numberings (Ershov [6]) and representations (Kreitz and Weihrauch [13), Weihrauch [24]).

l t uses naive logic for proving theorems about constructivity and cornputability. Constructivity properties appear in two versions.

The constructive specialization of ( 3 XE JN) P (x) is (

3

x E lN) P (x),

which is proved by wri ting down a number n E JN and proving P (n) . The constructive (the cornputable) specialization of (Vx) (3y)P(x,y) is:

There is a continuous (a cornputable) function which for any

x·aeterrnines y which P(x,y). Details will become clear in the examples of the next section. The fundamental role of continuity in constructive theories is well known (e.g. Nerode [19)).

The results of the approach presented here strongly suggest to interpret continuity directly as constructivity (cf. the examples

in Section 3). Of course, this kind of constructivity again differs from the previously introduced ones, but i t has the advantage that no new concepts are needed. Computability is a strengthening of continuity and, of course, depends on Church's thesis.

In a short contribution like this i t is not possible to develop the technical framework and to demonstrate the full power of the approach. Therefore in Section 2 only a coarse outline of Type 2 theory of continuity and computability is given, andin Section 3 i t is tried to explain by some simple examples the character of this approach. Detailed presentations can be found

in Weihrauch l22,24,25] and Kreitz and Weihrauch [23,11,12,13].

The ideas presented here have grown from many roots only some of which are given as references.

2 TYPE 2 THEORY OF CONTINUITY AND COMPUT~BlLITY

It is assumed that the reader is familiar with ordinary (Type 1)

recursion theory, especially with a concrete standard numbering

~=

^{:N --r p(l)} of the set of partial recursive functions from

Th! to :N satistfying the utm-theorem and the smn-theorem. The natural numbers are the concrete finite objects.

The concrete infinite objects in our approach are sequences of numbers. Let JB: = N :N be Baire' s space wi th the metric

d(p,q)

=

2-k where k= 1.rn[p(n)=I= q(n)] for p*q. :~otice that JB has the cardinali ty of the continuum. The Elements p E IB

correspond to "choice sequences'' in Brouwer's constructive theory.

A function r: IBn - - -+ IB is computable iff there is a special oracle machine,called Type 2 machine, M which computes

r,

this means

r (p) =

q iff M wi th input

p

E IBn yields output

q E IB (correspondingly for L: IBn ---+JN). As a fundamental fact, computable functions r: IBn ---+ IB and L: IBn ---+ JN are

continuous (the discrete metric on JN assumed).

Let [IB - + JN] be the set of all r: IB - - - + JN such that r is continuous and dem (r) is open. There is a concrete natural representation

x : l B - [ l B - J N ]

the universal function of which is cornputable which satisfies a cornputable srnn-theorern: There is a cornputable total function I: ^lB2

- I B with x"( ) (r) = x <q,r>. The open subsets of JB

l, q,p p

forrnally correspond to the recursively enurnerable subsets of JN, and x gives rise to a theory sirnilar to that of the r.e. sets with halting problern, reducibility, productivity etc.

r:

lB - ---+ JN is cornputable iff

r

=

x

f or sorne cornputable

p

function p . Let [ lB - + lB] be the set of all continuous

functions I: ^{lB - - -+ lB} the dornain of which is a G 0-subset of ^{lB .} Then co-yrespondingly there is a concrete natural representation

1 / 1 : l B - r [ l B - t - l B )

with cornputable universal function which satisfies a cornputable srnn-theorern. A function I: lB ---+ lB is cornputable iff I = 1/1 for

p

sorne conputable function p E ^{lB .} Wi th technics frorn ordinary

recursion theory a rich Type 2 theory of continuity and cornputability can be developed (Weihrauch [22,24) ) .

The input and output sets of real world cornputers are w.l.g.

nurnbers or sequences of nurnbers, i. e. lN or lB. The elernents of other sets rnust be encoded or narned by nurnbers or sequences of nurnbers. A nurnbering of a set S is a surjective partial functions v: lN ---+ s , where v (i) = s has the rneaning ^II i is a nurnber

( or a narne) of s ^{11 •} Correspondingly, a representation of a set M is a surjective partial function o: ^lB

--~

M , where

ö (p)

=

x has the rneaning, ¹¹ p is a narne of x ^{11 •} A nurnbering

(a representation) induces cornputability (topology and cornputability) on a set. Let e. g. c5: lB ---+ M be a representation and

v: JN ----+ S be a nurnbering. X,5;. s is v - r.e. (v-recursive) iff v-1

x=

A ⁿ dorn(v) fo:c sorne r.e. (recursive) set A~ N. Y ~ M is 6-open (6-clopen, 6-provable, 6-decidable) i f f ö-¹Y =An dom(ö) for some open ( clopen, provable, decidable) subset A ~ JB • Let

f: M ----+ S be a function. Then f is called (o,v)-computable, iff there is a cornputable function ^II on narnes" r: ^lB ----+ m wi th

(Vp E dorn(fö)) fo (p) =

vr

(p) (correspondingly with ¹¹continuous¹¹ instead of "cornputable¹¹ or for f:

s

1 ^----+

s

2 or f: M

1 ^----+ M

2) .

The theory of numberings is well established (Ershov [6],

Weihrauch [24]), essential parts of a theory of representations can be found in Kreitz and Weihrauch [13] or Weihrauch [24].

Defining a numbering or a representation corresponds to "giving a set" in constructive approaches. An "effective" numbering or

representation should respect the definition of the set. Numberings and representations can be characterized by the informations which can be cornputably or continuously extracted frorn the narnes. For separable T -spaces there are ''adrnissible" representations with

0

distinguished topological properties. Especially, for adrnissible representations (ö

1

,o

₂)-continuity coincides with topological continuity (for details see Kreitz and Weihrauch [13] or

Weihrauch [ 24] )

3

EXAMPLES

For demonstrating the usefulness of Type 2 theory with representations for studying constructivity three simple applications are given in this section: representations of the subsets of JN , representations of the real numbers, and the deterrnining of zeros of continuous

real functions.

( 1 ) The subsets of JN :

Although the following considerations are rather elernentary they show very well the character of the approach presented here. Let P w : = {A I A

=.

JN}, let T

=.

2Pw be the Scot t-topology on P defined by the basis {[d] 1 d~:N,d finite} where

w

[ d] = {A ~ 1'I I d ~ A} . The enumeration representation

JM: JB - • P is defined by lM := JM(p) = {n I n+1 E range(p)}

w p

Then ^JM is an adrnissible representation with final topology T.

Let ^T be Cantor's topology on P defined by the metric d(A,B)=2 , -k

C W

k: = µn[nEA <=> n $B]. Define the characteristic function representation 1'-1 cf of P w by JM.c f (p) : =p - l { O} •

Then JMcf is adrnissible with final topology ^{T •} Let A ~ ]N C

and f: X ~ A u X , g: X ,____ :N \ X . Then f is ( JM , :IM) -

continuous, f is ( lM , IM) -computable iff A is r. e. , but g

is not ( ]M , ]M) -continuous; f is ( 1-1 cf, ]Mc f) -continuous, f is ( ]M cf, ]Mc f) -computable if f A is recursi ve, but g is

( JM cf, ]Mc f) -computable. Notice that effectivi ty of a function depends on the representation. Notice that c6mplementation is not constructive w.r.t. the enumeration representation.

The interpretation of continuity as constructivity should become clear from the above examples·. More details can be found in

Weihrauch [24).

(2) The real numbers:

There are many representations of the real numbers known not tobe reasonable. Our framework shows that most of them are bad for purely topological (constructive) reasons. The naive Cauchy-representation 6 : - - - IR defined by

C

6 (p) = x: <=> ( (vQp (i)). is a Cauchy sequence and

C 1

X = 1 im \! Q p ( i) )

(where is a standard numbering of the rational numbers) has the final topölogy {~, IR} and therefore is not admissible.

The decimal representation

6 ( p) : = v p ( o) + I: { ( p ( i ) mod 1 O ) 1 O - i I i ~ 1 }

D Z

has the real line as final topology but nevertheless is not admissible. The leftcut representation, e.g.,

ÖL(p) =X<=> p-¹{0} = {i ¹ \!Q(i) < X}

is admissible but its final topology is finer than that of the real line. A standard representation r of m can be definded by

p (p) = x: <=> ( (V m > n) II vQ p(m) - vQ p(n) 1< 2-n and x=lim vQp(n)).

This representation is topologically admissible, hence especially a function f: IR ---+ :iR in continuous off i t is < p, p) -continuous.

The well-known negative decidability results on IR have

topological reasons. Define

l:

^{lB ----+ IR2 by}

l<

^{p, q>: = ( P (p) , P (q))}

then

l

is admissible. While x < y and x

*

^{y are p 2}-provable, x:;; y and x = y are not even p2-clopen (since they are not open!), hence nei ther of the properties x < y , x ~ y , x = y, and x

*

^{y on}

JR is p2

-decidable. More details can be found in Weihrauch und Kreitz [23] and Weihrauch [24).

(3) Zeros of continuous real functions:

The set C[0,1] of the continuous functions f:[0,1] - J R with the metric d(f,g):= max{lf(x)-g(x)IIO;;;,x;;:1} is a

separable metric space. Hence there is an admissible representation c5 of C[O, 1], even a "computationally"

admissible one. We summarize some properties. Let p be the above representatdon of JR.

The set {fEC[0,1] 1 (Vx)f(x)*O} is cS-provable, especially open, but not cS-closed, i.e. closed, hence not cS-decidable.

For k ;;; 1 let Xk: = { f E C [ O, 1 ] 1 f has exactly k zeros}

The function z

1: C[0,1] - JR with dom(Z

1)=X

1 and f Z

1 ( f) = 0 is ( c5 , p) -computable.

For k;;; 2 there is no (cS,p)-continuous function Zk: C[0,1] --~ JR with f zk (f) =O for f EXk.

For k ~ 2 there is a computable function

r:

E --+ JB with c5 (p) (pf(p)) =

o

^{for all p E cS-1xk}

There is no continuous function r: JB - - + JB such that

c5 ( p) ( p r ( p) )

=

^{o f or a 11 p E c5 -} 1 ( X

1 ^u X 2 ) .

Let Y:={fEC[0,1] 1 f(O) · f(1)<0}. Then there is

no continuous function

r:

JB --+ JB wi th c5 (p) ( p

r

(p)) =

o

for all p E cS-ly.

Let V:={f E Y I f(I)={O} for no open interval I}. Then there is a computable function r: JB - - + JB wi th

cS(p)(pf(p))=O for all pEcS- 1V.

There is no ( c5, p) - con tinuous function Z: C [O, 1] - lR with f Z (f) = 0 for all Z E V .

The above facts (or at least semantically similar ones) can also be expressed and proved in other constructive theories.

The approach presented here is particular simple. Details of proofs can be found in Weihrauch [24] .

3

CONCLUS ION

In this contribution an approach to constructivity in rnathernatics based on classical logic is presented. The general concept of onstructivity is devided into several different subproblerns.

(1) The definition of natural nurnberings and representations, ( 2) the concrete existence (

3

^nE JN) • • • ,

(3) relative continuity (interpreted as constructivity), (4) computability (as a specialization of continuity),

(5) computational complexity (which functions are easily computable?).

The means for proving statements are left in the naive metalanguage and no restriction to constructive logic is necessary. The problem to define "natural" nurnberings and representations is connected to problems of the foundations of rnathematics. Some concepts have been already suggested (e.g. "admissible" representations,see also Weihrauch [ 26]), a closed theory however does not exist. The ordinary Type 1 recursion theory can easily be completed by adding concrete existence statements, e.g. (3n) (Vx)<P (x) = x² •

n

In Type 2 theory concrete existence statements may be of the form

(3 program P) . . . Continuity enters every constructive theory which e.g.

discusses real nurnbers as an additional or derived concept.

Continuity is the tool for "spanning'' the continuum by a denumerable set. The concept is so fundamental that for nondenumerable sets i t should be required before constructivity or computability. The approach presented in this contribution suggests that continuity itself is a type of constructivity. The exarnples in Section 3 show this very clearly especially in the case of negative results which do not depend on Church's thesis but have purely topological reasons.

Physical computability on machines is of course limited by

Church's thesis. Computability is a special case of continuity. The representation concept adrnits to include naturally the study of computational complexity in analysis (N. Müller [18] ) . The results of Brent [4] and Ko and Friedrnan [10], e.g., can be easily

formulated in this frarnework. Type 2 theory of computability is the natural background for a Type 2 complexity theory. Except the exarnples

in Section 3 constructive rnetric spaces (Weihrauch [25]), cornplete partial orders (Weihrauch [24]), and cornpactness on real numbers (Kreitz and Weihrauch [11) have been successfully considered as touchstones for the approach. Functional analysis, rneasure theory, and descriptive set theory are other areas of application. As in ordinary (Type 1) recursion theory, where cornputability does not depend on the logic used for proving

staternents, in a rnathernatical theory like analysis constructivity and computability are properties inherent in the theory which do not depend on the logic used in the metalanguage. The approach presented in this contribution strongly suggests this view.

Of course i t may be interesting to investigate the formalisms (logic) which can be used for proving statements in e.g. analysis including statements about constructivity. But this is another question. The author of this paper is prepared to defend the claim that for investigating constructivity and computability in mathematical theories constructive logic is dispensable.

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