On Natural Numberings and Representations

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deposit_hagen

Publikationsserver der Universitätsbibliothek

Mathematik und

Informatik

Informatik-Berichte 29 – 12/1982

Klaus Weihrauch

On Natural Numberings and

Representations

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On Natural Numberings and Representations

by

Klaus Weihrauch Fernuniversität

5800 Hagen W- Germany

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1. Introduction

Basic recursion theory introduces computability on sets like

*

1N IN

m , ^I: ^,2 , IF : = m explici tly. Computabili ty on other sets M like finite graphs, rational numbers, segments of ordinals, or real numbers can either be defined explicitly or be reduced to

ordinary recursion theory by a numbering v : IN---> JM or a representation 6 : JF ---> M.

In the first case a thesis like Church's thesis is needed, in the second case an argument is needed that the numbering (representation) is "effective" or "natural". The advantage in the second case is that the theory of computations and computability already exists and that only one function (numbering, representation) has tobe discussed. Usually concrete numberings are introduced without discussion or authors eloquently try to convince the reader that their numberings are "effective" whatever this means. In this paper a general concept is suggested which explains in which sense rnany concretely used numberings are effective. By the same idea "effectiveness"of representations (e.g. of real numbers by total functions on m) can be explained. Examples show the applicability of this concept. Finally the connection between certain effective numberings and representations is discussed.

This work extends and generalizes an earlier paper written by the author [ 1 ] .

2. Effective Numberings

Let M be a denumerable set. Informally a numbering of M is a relation,v s m xM, where (i,rn) E v rneans "i is a number of a".

This interpretation implies : A numberinq of M is a partial function v: IN --->M onto M.

Computability w.r.t. numberings and reducibility and equivalence of numberings can be def ined as :::ollows. Let v. : :m

--->

^{M. (}i

=

C, ... , n)

1. 1.

be numberings. A partial function f : M

1 ^{x ••• xM} ---> M is

n o

((v1, ... ,vn) ,v

0

) - computable, iff there is some computable function

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h: llin --->lli with h(dom(v

1)x . • . x dom(vk)) '=-dom(v

0

) and

f(v1(i

1), . . . ,v (i )) = v h(i , . . . , i ) for all i Edom(v)

n n o 1 n k k

(k = 1 , . . . , n) . A numbering v : lli ---> M is reducible to a numbering v ' : lli--->M', iff MS M' and id is (v,v')- computable,· v and v'

M,M'

are equi valent, iff v:::; v' and v' :::; v. Equi valent numberings imply the same kind of computability. A general theory of (total) numberings is presented in papers by Ershov [2].

Fora mere set Min general there is no creterion to distinguish between "effective" and "not effective" numberings. More additional structure is needed for this purpose. Many important sets are

defined as closures.We shall introduce "effectiveness" for numberings w.r.t. "generation systems".

Let U be a set. Let A be a subset of U and let G be a set of (partial) operations on U. The smallest subset X s U wi th A ':: X which is closed under the operations of G will be denoted [G; A].

In this section we shall start with numberings y and a of G and A and define numberings of [G; A] effective w.r.t. (y,a).

Let a : lli ---> A be a numbering of A, let y : JN ---> G be a numbering of G such that y(i): Ua(i) ___ >U for any iE dom y, where a: 1N -- 1N

is some computable function. A numbering of [G; A] which is

"effective w.r.t. (y,a)" can be defined easily as follows:

Let T , the set of a-terms, be the smalles t set T wi th lli ':: T and a

(i, (t

1, . . . , t

0 (i))) ET whenever i E JN and t

1, . . . , t

0 (i) ET. An injective mapping t : T -- lli can be defined by the conditions

a t(i) := <O,i>,t(i,(t

1, . . . , t

0 (i))) := <i+1,0> if a(i) = 0, := <i+1,1+<t(t

1) , . . . ,t(ta(i) )>> otherwise (where< > is Contor's tuple function).

A surjective mapping h: T ---> [G; A] is defined inductively by the a

conditions: h(n) := a(n) if nE dom(a), div otherwise;

h(i, (t1, . . . , t

0 (i))) := y (i) (h(t

1), . . . ,h(t

0 (i))) if

iE dom(y) ,t. E dom(h) (j

=

1, . . . ,a(i)), and (h(t ) , . . . ,h(t (.)))

J 1 - l 0 l

E dom y(i), div otherwise. For any nE range(t) ,t (n) is a

(finite) c - term and ht-l (:::1) (if i t 2xists) is th2 -v-ä..i.ü.c in [G; A]

obtained by evaluating the term t-1(n) with number n according to

a and y. Define v := ht -1 . Obvionsly, v is a numbering of [G; A].

y,a y,a

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T is the set of all finite "Composition programs" w.r.t.

0 - -

a,dom(v _y,a) is the set of standard numbers of all meaningful _. composition programs.

THESIS: A numbering v of [G; A] is effective w.r.t. (y,a) iff i t is equivalent to v

y,a

This thesis only can be confirmed by examples. For infinite sets G and A we give an example. Let U be the set of all partial

functions over lli . Let a be a canonical numbering of all the projections, the zero, and the successor functions. Let y be a canonical numbering of all the composition operations and the operations of primitive recursion and effective minimalization.

Then v _y,a is equivalent to an "effective Gödelnumbering of the partial recursive functions".

It is easy to show that a~ a' and y ~ y' implies v ^~v .

Y, a y', a.' Let X be a finite set. Let v be an injective numbering of X. Then v is (up to equivalence) the minimurn w.r.t. ~ in the class of all numberings of X. v can be called an "absolutely effective" numbering of X. For finite sets G and A i t is reasonable to choose

absolutely effective nurnberings ^y and a. In this case our thesis gives the numberings effective w.r.t. (G; A). Seneral exarnples are given in [1].

3. Effective representations

A representation of a set M is a surjective partial rnapping 6 : IF ---> M. A function r : IF ---> IF is computable, iff there is a Turing Machine with one infinite input tape for values f(O), f(1), f(2), (in decirnal notat-ion) of the argument fE:IF, at least one work "tape and one infinite output tape onto which gradually the values g(O), g(1), ... of the result g=r(f) are written. The function fE: IF is in the dornain of r, iff for any n the machine finally writes the value q(n) onto the output tape. Cornputable operators are closely related to recursive operator in the sence of Rogers [3]. The definitions of computability and reducibility can easily be transferred frorn nurnberings to representations. For

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n-ary functions on IF we use the encoding c : IFn - IF wi th n

c (f , . . . ,f

1) (k · n+ i) := f. (k) (where i< n). An irnportant

n o n- i

observation is that also w-ary functions can be introduced via the encoding c : IFw - IF with c (f ,f

1, . . . )<i,k> .- f. (k) (where <,> is

W W Ü l

Cantor's pairing function).

The garne played with nurnberings in Section 2 can be played

accordingly with representations. Let U be a set, A be a subset of U, F be a set of operations on U which rnay be n-ary or w-ary (irnportant generalization!). Let p: JF ---> A, ^{E :}IF --->G be representations such that for sorne cornputable operator TI: IF - JN (definable by a Turing Machine) s(f): Uw--->U if n(f) = O,s(f): Un--->U if n(f) = n+ 1 for any f E dom(s). The set T now becornes a set of appropriate finite

TI'

path trees wi th nodes and leaves from IF . Using the encodings c and n c a natural injective mapping ~: T - IF can be defined by trans-

w TI'

finite induction. Accordingly the evaluation mapping h: T ---> [G; A]

TI'

can be defined by transfinite induction. The representation 6 .- h~-l is intuitively effective w.r.t. E and p.

E:, p

THESIS: A representation 6 of [G; A] is effective w.r.t. the pair of representations (s,p), iff i t is equivalent to 6

The typical w-ary operations are lirnits.

Exarnple: Let U :=IR, A= \O,G= {l}, l : IRw--->IR,

E: , p

d orn ( 1 ) : = { ( x . ) . ¹ ⁽V i , j ) j > i => ¹ x . -x . ¹

<

2 -i } , 1 ( ( x . ) . ) : = 1 im x . ,

l lEW l J l lEW i l

p any standard representation of (D, (Vf E IF) s(f) := l.

Then 6 is a representation of IR equivalent to those represent-

E: , p

ations which are used in recursive analysis [4]. More exarnples will be given in the next section.

4. Relations between effective nurnberings and representations

Let 6:= JF--->M be a reoresentation. M := {mEMI 6(f) =rn for _~ _C some cornputable f} is the set of (6-) cornputable elernents in M.

The partial nurnbering n with n (i) = 6<P., where <P. is an "adrnissible"

l l

Gödelnurnbering of the unary partial recursive functions, is called the nurnbering derived from 6. On the set M there are two kinds of

C

computability: (n,n)-cornputability and the restriction of (6,6)- cornputability.

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THEOREM:

Let n: JN--->M be the numbering derived from the representation

C

6 : IF ---> M. Let p : M ---> M be a function. Let

C C

C, :<=> p has a (6,6)-computable extension p: M--->M,

-

0

C :<=> p is

Tl

Then c

0 => C

11 •

(n,n)-computable.

The proof is not difficult. An interesting question is under which conditions C implies C,. Farnous theorems give partial answers (see

n o

below). We conclude with exarnples.

Example 1:

Let v be a (y,a)-effective nurnbering of [G; A] (see section 2).

y,a

From y and a canonical representations s : IF ---> G and 6: IF ---> A can be deduced. We have the representation 6 (see section 3)

E: 'p

and the nurnbering n derived from it. In this case:

v

=

n and C, <=> C. This shows the consistency of the two

y , a o ·n

definitions of effective nurnberings and effective representations.

Exarnple 2: (Theorem of Myhill/Shepherdson)

Let U be an effective cpo [5], let a be an effective nurnbering of a basis A of U, p a canonical associated representation of A, let G : = [

LJ} ,

^where

LJ

is w-ary wi th dom (

LJ ) :

= { ( x. ) . 1 {x. 1 i E w} ,:;:; D

J. J.EW J.

is directed} and

LJ

^{(x. ) .} ^{: =} ^{lub {x.}¹i E w}, let s be a canonical

J. 1.EW J.

representation of G. Then "C

11 ::::;> C

0" is the (generalized) Myhill/

Shepherdson theorem for effective cpo's (provided the function p is total) .

Exarnple 3:

Let (U,d) be a complete metric space, let a be a nurnbering a: JN-A of a dense subset of U such that {(i,j ,p,q) E JN x JN x 02 x CQI

p< d(a(i) ,a(j)) < q} is r.e . . Then a construction similar to that form (see end of section 3) yields a natural representation 6 of U. Let n be the nurnbering derived from 6. Then under certain

conditions (n,n)-computability implies (6,6)-computybility (Theorem of Ceitin, Moschovakis, Kreisel/Lacornbe/Shoenfield).

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Example 4:

Let U := ® = (section of countable ordinals), A := {O},

G := {successor, sup}, where sup is the least upper bound for

< <

fundamental sequences. Let 6: IF--""'1/ ® be the representation effective w.r.t. (G, A). Let n be the numbering derived from 6.

Then n is the numbering of a universal system of notation for the computable ordinals in the sense of Kleene (see Rogers [3]). It is unknown under which conditions C~ implies C

0.

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[1] A. Reiser, K. Weihrauch, Natural numberings and generalized computability, EIK 16 (1980) 1 - 3, 11 - 20.

[2] J.L.Ersov, Theorie der Numerierungen I, Zeitschr. f. math.

Logik u. Grundlagen der Mathematik 19 (1973), 289 - 388.

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

[4] A. Grzegorczyk, Computable functionals, Fundamenta mathematicae 42 (1955), 168 - 202.

[5] K. Weihrauch/G. Schäfer, Admissible representations of

effective cpo's, Informatik Berichte 16 (1982), Fernuniversität Hagen, ( to appear in TCS 19 83) .