A Generalisation of Productive Set

A Generalisation of Productive SetAuthor(s): R. MitchellSource: The Journal of Symbolic Logic, Vol. 31, No. 3 (Sep., 1966), pp. 455-459Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270460 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 31, Number 3, Sept. 1966

A GENERALISATION OF PRODUCTIVE SET

R. MITCHELL

Introduction. The purpose of the present paper is to introduce a generalisation of the concept of productive set introduced by Post (2) and studied by Dekker (3) and others. Throughout the paper we shall use small Latin letters to denote both non-negative integers (referred to as numbers) and functions (both partial and total) from numbers to numbers. Sets of numbers will be denoted by small Greek letters and classes of such sets by capital Latin letters. wc is the range of the nth partial recursive function.

We recall the definition of productive set: A set a is productive relative to the partial recursive function p if for all

numbers n,

0jnfz Co p(n) is defined and p(n) E a -cas.

The class of productive sets will be denoted by P. and if a is related to p by the above implication we shall say that "p is a P-function of oc", or that "o e P with P-function p".

A productive set as defined above is a set which can, in an obvious sense, be "effectively shown" to be non-recursively enumerable. We shall modify the definition in order to obtain sets which are "effectively shown" to be non-recursive. However, the classes of sets so obtained will be seen to depend on the method used to specify the recursive sets, and so we shall first examine some methods of specification and note the relations between them.

Specifications of recursive sets. There are clearly many ways in which a set may be specified as being recursive. We may, for example, make a statement that the set is recursive and then give either some enumeration of the set, some enumeration of the set and of its complement, or some enumeration of the set in increasing order. Alternatively, we may give some effective test for membership in the set. Again, the cardinal number of the set and/or of its complement may be given in addition. Each of these methods of specification can be made precise by using the theory of partial recursive functions. However, we shall here give only two explicit examples and shall point out later why we consider them to be sufficient for our purposes.

DEFINITIONS.

=o {n: wtn is recursive}, = {: W(kl(n) =(k2(0

Received June 15, 1965.

455



456 R. MITCHELL

where oc' denotes the complement of oc, and kj and k2 are the first and second "coordinate" functions of some standard effective mapping from ordered pairs of numbers onto numbers.

It is intuitively clear that knowledge of membership of a number n in al yields more available information about the encoded set wCoki(n) than does knowledge of membership of the number ki(n) in ao. We give a more formal version of this result in the theorem which follows at once.

THEOREM 1. Any partial function p which is such that for all n, if n e a0 then p(n) e or and Con = 00kip(n) is not a partial recursive.

PROOF. We use reductio ad absurdum and the fact that there is no effective method of deciding whether, for a given n, 0 & con.

Suppose then, that p is a partial recursive function with the property that for all n

n X Oo -E- p(n) is defined, p(n) & or and Coll ( Ckip(n)

and let h be a recursive function such that for all n,

X E C0h(n) '" X 0 and x E Cwn.

Then (0h(n) {0} if 0 E Wn -and (Oh(n) - if 0 0 Cw, so that h(n) E ao for all n, from which it follows that

ph(n) is defined, ph(n) & or and ('0kiph(n) =Oh(n).

Hence 0 e Wcn 4-* 0 e h(n) ? 0 E (tIklph(n) and 0 Con <> 0 0 COh(n) <"> 0 E Ctk2ph(n).

But the last two statements yield an effective test for deciding whether 0 cow, and this is known to be impossible.

The result follows by reductio ad absurdum.

Generalisations of productive set DEFINITIONS. ko is the identity function and for i 0 or 1, we shall

say that a set oc is a,-productive relative to the partial recursive function pi if for all numbers n, n E vi and Oki(n) C OC together imply that (i) Pi(d) is defined and (ii) pi(%) e oc - Cki(n).

We shall denote the class of vi-productive sets by Pi and shall say in the above circumstances that "a e Pi with Ps-function Pi."

It is clear that each member of each of the classes Po, P1 is a set which is "effectively non-recursive" in the sense mentioned in the introduction. The following two theorems establish the relationships which exist between the classes P. PO and P1. Theorem 2 being fairly obvious whilst Theorem 3 is at first a little surprising.

THEOREM 2. P C Po C P1. PROOF. Let oc e P with P-function p. Then, n E a0 and C(k.(n) C oc to-

gether imply that Con C a and hence that p(n) is defined and p(n) E a - cor Hence oc e PO with PO-function P.



GENERALISATION OF PRODUCTIVE SET 457

Next, kl is a recursive function such that

nf c- a, ki(f) C- ao

and so if a E Po with Po-function p, we have n E a and (ck(n) C c( together imply that ki(n) E ro and &kokki(T) C a and hence that pki(n) is defined and pki(n) E a - W1ki(n). Hence a E P1 with Pi-function pk1. This completes the proof of the theorem.

THEOREM 3. Po C P (and hence by Th. 2, Po = P). PROOF. The method of proof is similar to that used by Myhill (4) in

showing that any productive set is completely productive. We shall first establish that if a E Po, then a has a recursive (totally defined) Po-function.

Suppose, then, that acE Po with Po-function p. Now there is a recursive function h, say, with the property that for all n, x,

X E (0h(n) - p(n) is defined and x E con.

If p(n) is undefined, then W0h(n) is the empty set so that we have coh(n) C and h(n) e qo. Since a e Po with PO-function p, this implies that ph(n) is defined and ph(,n) E a.

Hence, for all n, at least one of p(n), ph(n) is defined and we can define a recursive function / by /(n) = P(n) or ph(n) depending on which of two .fixed Turing machines for p and Ph yields a value (from argument n) in the least number of steps.

We now show that / is a Po-function of a. If n X oo and Wn C a, then p(n) is defined and p(n) E -co,,. But, by the definition of A, this implies that Oh(n) = (on and h(n) e oo and hence also that ph(n) is defined and ph(n) E oc- On. So if n Ea and a),,, C: o, it follows that /(n) E oc con.

Hence / is a recursive Po-function of a as required. We conclude the proof by showing that if oce Po with recursive Po-

function /, then a is productive. Let a E Po with recursive PO-function I. By the "fixed point theorem"

of Myhill (4) there is a recursive function g, say, such that for all n, x, X G og(n) A X oen and x = /g(n). Now, for any n, either Co(n) = or og(n) {/g(n)}. So wg(n) is certainly recursive and g(n) E ao. But also (Og(n) Con and so

C0jz _C O g(n) E qo and (ogwn) a

/g(n) a - Wg(n)

/g(n) a - n

Hence a E P with P-function /g. This completes the proof. REMARK. Theorem 3 is possible only because of the fact whilst a given

encoding of a set in ao tells us that the set is recursive, we have no effective



458 R. MITCHELL

membership test available - nor by Theorem 1 can we effectively find such a test - and consequently the only usable information we have is the recursive enumerability of the set.

We show next that P C Pi. THEOREM 4. If a is a recursively enumerable non-recursive set, then

c E P1. PROOF. Let ac satisfy the conditions of the theorem, and suppose an

index of oc is given. Define a partial recursive function p as follows.

= f The first number to appear in some standard enumeration of both

a ) and W k,,(), if there is such a number; undefined otherwise.

Now, if n E al and cok,(,,) C (X, it follows that the intersection of a and Wk,(n)

is not empty, so that we must have

p(n) is defined and p(n) E -a

Hence a E Pi with Pi-function p. REMARKS. Theorem 4 states that any recursively enumerable non-

recursive set is "demonstrably" non-recursive in the sense that it belongs to the class P1. Indeed, if we are given an index of the set we can actually produce an index of a P,-function of it. The result is in contrast with the fact that no recursively enumerable set is productive, so we see that the class P1 properly contains the class P of productive sets. We shall call sets in the class P1 almost productive (abbreviated a.p.).

Concluding remarks. As mentioned earlier, we could have given numerous other specifications of the recursive sets. In (1), five types of specification were studied and these led to five generalizations of productive set. It was found that these fell into two types, the first being the trivial generalization yielding exactly the productive sets, and the second yielding the a.p. sets. It appears that any specification of recursive sets which does not contain an available effective test for membership will lead only to the productive sets, whilst any specification which contains an available membership test will lead to the a.p. sets. Such things as knowledge of cardinality appear to be irrelevant as far as our technique of generalisation is concerned.

The structure of the set of a.p. sets is in many respects similar to that of the productive sets, and further, concepts such as "almost contra- productive", "almost semiproductive" etc. are readily defined.

It is hoped to discuss some of the properties of a.p. and related sets in a later paper.



GENERALISATION OF PRODUCTIVE SET 459

REFERENCES

[1] R. MITCHELL, Some properties o/ non-recursive sets, Ph. D. Thesis, University of Newcastle upon Tyne, May 1962.

[2] E. L. POST, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc., Vol. 50 (1944), pp. 284-316.

[3] J. DEKKER, Productive sets, Trans. Amer. Math. Soc., Vol. 78 (1955), pp. 129-149.

[4] J. MYHILL, Creative sets, Zeit. fur Math. Log. und Grund. der Math., Vol. 1

(1955), pp. 97-108.

BRISTOL COLLEGE OF SCIENCE AND TECHNOLOGY.



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A Generalisation of Productive Set