table approximations to recursive predicates

Let the function f be obtained from the functions ~, ~, J by means of the operation of restricted recursion. By Lemma 2 one can construct a function ~ from the class K n which is

~ondecreasing in each argument such that V~,~[$C~,~ ~(~, ~). By eemma I the function

belongs to the class g~ and the class ~. Then the class ~ contains the function ~,

obtained from the functions ~, ~, ~ by means of the operation of restricted recursion, where

V~,~[~(~,~)= ~ (~,~)]. The function F, constructed from the functions ~ , ~t by means of

the operation of simple minimal recursion, is contained in the class ~~. Hence the class

~ contains the function fT such that

By Lemma 3 we have V~,~[~'(~,~)=~(~,~, so V~,~[~(~,~)=~'(~, ~)]. The le~a is proved.

THEOREM i. For each n the class ~ is of the same size as the class ~

Proof. Taking into account the fact that for each n the class ~ is of the same size

as the class ~ and using Lemmas i and 4, we get that for each n the class ~g is of the

same size as the class ~. The theorem is proved.

In conclusion, the author expresses thanks to N. K. Kosovskii for posing the problem and attention to the work.

LITERATURE CITED ~

i. A. Grzegorcz~K, "Some classes of recursive functions," Rozpr. Mat., i, 1-4.6 (1953). 2. N.K. Kosovskii, "On algorithmic sequences of the initial class of the Grzegorczyk

hierarchy," Zap. Nauchn. Sem. LOMI, Akad~ Nauk SSSR, 20, 60-66 (1971).

TABLE APPROXIMATIONS TO RECURSIVE PREDICATES

R. I. Freidzon UDC 51o01:518o5

In this note the complexity of the table approximations to recursive predicates is studied. The consideration is carried out in terms of the theory of recursive functions for an axiomatic method of defining measures of complexity of notation for partial recursive functions. It is proved, in particular, that every table approximation of a given recurslve predicate can be essentially improved (from the point of view of complexity of notation for the approximating functions) at an infinite number of points.

I. Suppose fixed an admissible (in the sense of [I]) enumeration V of partially recursive functions. The single-place partial recursive function (p.r.f.) with number i will be

denoted by ~g. A recursive predicate S will be called finitary if the domain of truth of S

can be effectively given as a list; a finitary predicate S will be called n-finitary if

Every general recursive function (g.r.f.) Z, which has the property:

(A) For any natural number n, the set of the form {L: Z(L)~ ~ is finitary, will be called, following [12], ameas~e ofco~lexity ofnotation of single-placep.r.f, in the

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otde!eniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 40, pp. 131-135, 1974. Results submitted October, 12, 1972.

I This material is protec ted by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, iV. Y. 1 10011. N o part o f this publication may be rep rocluced, s tored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, w i t hou t writ ten permission o f the publisher. A copy o f this article is 1' available f rom the publisher fo r $7.50. I

l

337

enumeration ~ . In what follows, in place of Z(i) we shall write lil, if it is clear from the context which measure of complexity of notation is involved.

We shall say that a p.r.f, f is ~ ~epP~8ent~%~r~ ~uno%~on for the recursive predicate S if the function f is general recursive and one has the condition

V:~ (,~(m3= 0,== me $).

Let S be a recursive predicate, n be a natural number; by $~ we shall denote the fini-

tary predicate whose domain of truth is the set {x: ~e $ ~ ~ ~ .

A general recursive function ~ will be called a %~bZe ~Pox~m~t~on of the recursive predicate S if for any n the function q~ is a representing function for the predicate ~(~ .

THEOREM i. Let Z be a measure of complexity of notation for single-place p.r.f., ~ be be a g.r.f., S be a recursive predicate and x be a table approximation of the predicate S. If the truth domain of the predicate S is not finite, then one can construct a table approximation ~' of the predicate S such that:

(a) B~ (~,(l~'cn,)l.)<Jae(n)J);

(b)

Proof. Let n be a natural number, ~0, and we denote by fn the two-place g.r.f, whose

process of calculation reduces to the following: in order to compute ~n (m, ~), we compute

the values Ix(O)l, J~(4)l,.-. until we find an n different from the natural numbers N y, N y,

�9 .., N y, for which N~<...< N~ and one has

One finds such a number according to property (A) since the truth domain of the predicate S is not finite and hence the function ~ cannot be bounded above by a constant. The

value of ~(~,~) we set equal to ~(N~(~).

The considered enumeration V of p.r.f, is admissible. In it the recursion theorem is

valid. Consequently, one finds a number i n such that

We introduce the notation: t~ N~.

Thus, we have constructed numbers i n and t n such that first,

and second,

[We note that in the construction of these numbers, a modification of the diagonal construc-

tion of Blum was used (cf., e.g., [2]).]

We consider the sequence of numbers t~, t2, .... This sequence is not bounded above,

since for each n, t n is the n-th number k, satisfying the condition I~(K)I~ ~(ILnl ~ . Conse-

quently, V I% (~ ~ ~-4).

We choose from the sequence of numbers tl, t2, ... a strictly increasing recursive sub- sequence t~o, t~,,.... We define a general recursive function ~' by means of the following con-

338

dition: for any natural number m, if one can realize a natural number j such that

then ~'(n~)=~, otherwise ~'(nm)= ~(n~)

The function ~' is a table approximation of the predicate S, since

Moreover, for any natural number j one has the condition

~(I~ ' ( t~ )I < l ~ ( t ~ ) I .

We have proved thus that

~'(tr.~ ) = g~.~ and hence,

3 ~ (~(I~'(~)I)< I~ (~)l).

It is also obvious that for any natural number n one has the condition

Consequently,

The theorem i s p roved.

2. It follows from Theorem 1 that if the truth domain of a given recursive predicate is not finite, then it is impossible to construct its "best" table approximation, i.e., a table approximation whose complexity at all points would not exceed the complexity of any other table approximation of the given predicate. As for recursive predicates with finite truth domain as follows from the results of [3], no algorithm is possible which for any finitary predicate would construct the number of its representing functions having the least complexity of notation among all representing functions of the given finitary predicate. The question arises: does there exist an algorithm which for any natural number n constructs the least upper bound of the least complexity of notation for representing functions of k-finitary pred-

icates for K~. It will be proved below (Theorem 2) that the validity of the assertion

about the existence of such an algorithm turns out to depend on the method of definition of the measure of complexity of notation of functions.

We introduce some definitions and notation which are needed in what follows. We shall say that the p.r.f, f is equivalent with the p.r.f, g (with write: "f ~ g"), if the functions f and g are general recursive and

V~(S(~)= O~--~g(~)= 0).

We fix a single-valued enumeration of all finitary predicates. The finitary predicates The finitary predicate with number i will be denoted by Fio A general recursive function will be called a method of computing finitary predicates if for any i, the function q~ is a representing function for the predicate F i.

We introduce the following notation:

(if for the numbers n and i one has B(n, i), then ~ is a representing function of some k- finitary predicate for K~);

where Z i s a g e n e r a l r e c u r s i v e f u n c t i o n w i t h p r o p e r t y (A) .

It is easy to see that if for the natural numbers n and t one has Az(~,t), then t is an upper bound for the complexity of notation for representing functions of k-finitary predi-

cates (for K~ ~ ) for the measure of complexity of notation Z.

339

We consider the following property of the function Z:

(B) One can realize a general recursive function L such that for each n the number L(n)

is the least natural number t for which Az(~,~).

It is obvious that for a given measure of complexity of notation of p.r.f. Z to have property (B), means that for any n one can construct the least upper bound of complexity of

notation of k-finitary predicates for 111 K~m. The next theorem establishes the"indepen-

dence" of properties (A) and (B).

THEOREM 2. One can construct an example of a measure of complexity for which property (B) holds, and an example of a measure of complexity for which property (B) does not hold.

Proof. i. We construct an example of a measure of complexity of notation for which property (B) holds. Let ~ be a method of computing finitary predicates having recursive sets of

values. We define a measure of complexity of notation of single-place p.r.f. Z~ as follows:

for any number i, if one finds a number j such that L= ~(~), then Z~(L) is equal to

rr~{~: ~e~} otherwise Z~(L)= ~.

Let n be a natural number. Obviously, one has Am (~,r~). We shall estimate from above

the number of elements of the finitary set {i: Z4{g~n-~ ~. If the complexity of notation of

the p.r.f. ~i does not exceed n -- i, then at least one of the following conditions is sat-

fled:

(a) ~ ~-4,

(b) one can realize a natural number ~ such that g=~()) and the function q~ is a rep-

resenting function of a k-finltary predicate (for some K~ ~-4 ).

Whence it follows that the number of elements of the set {L: Z~(L)~-4} does not exceed

Z ~* ~ Since the number of distinct k-finitary predicates (for g ~ ~ ) is equal to %~*~

we get that 7A~(~,~-4). Consequently, ~=~tAz4(~,t ). Thus, as the function L from (B) one

can take the identity function.

2. We define a measure of complexity of notation of p.r.f. Z~ by the condition

VL (Z~(~)= L).

Let us assume that property (B) is satisfied for the measure Z~ we have defined. Let

L be the g.r.f, whose realizability follows from the assertion of (B), and m be a natural

number. We consider the function ~Lc~)." This function is a representing function of some

k-finitary predicate (for K~r~), and obviously

We consider the unbounded sequence of numbers L(0), L(1), .... Each of the numbers of this sequence is the number of a g.r.f, which has the least complexity of notation among all of its equivalent (with respect to the relation ~) general recursive functions. On the other hand one can show (cf., e.g., for [4]) that the set of all g.r.f, which have the least complexity of notation among all g.r.f, equivalent to them is immune. The contradiction obtained shows that the assumption that the measure Z~ property (B) is false.

The theorem is proved�9

The author expresses thanks to E. Yu. Noginaya and G. B. Marandzhyan for helpful discus- sions and comments.

LITERATURE CITED

i. H. Rogers, "G~del numberings of partial recursive functions," J, Symbolic Logic, 23, No. 3, 331-341 (1958)�9

3 4 0

,

3.

.

M. Blum, "On the siaze of machines," Inform. Control, II, No. 3, 257-265 (1967). D. Pager, "On finding programs of minimal length," Inform. Control, 14, No. 2, 550-554 (1969). A. Meyer, "Program size in restricted programming languages," Inform. Control, 21, No. 4, 382-394 (1972).

SOME PROPERTIES OF MAPPINGS OF SHEAF-SPACES

V. P. Chernov UDC 51.01

This note is a sequel to [i, 2]; the terms and notation of [I, 2] will be used without explanation.

In Sec. I some topological properties of support sets (domains of definition of constructive maps) are studied. In See. 2 the concept of neighborhood operator, which has an approximative character, is defined, and properties of such operators are considered. In Sec. 3 two theorems on normal forms of maps of strict sheaf-spaces (and by the same token of operators of finite type) are formulated. This last paragraph is independent of the pre- vious ones~

i. The term "open set" will be used as a synonym for the term "O-open set" from [3].

We shall call a set of points of a constructive topological space a Ga set (LG8 set) if it is equal to the intersection of some nonempty denumerable family of open (respectively, Lacombe) sets.

THEOREM i. Every support set of points of a sheaf-space is a G8 set.

This theorem can be proved with the aid of Theorem 3 of [I]; its proof is in [4].

THEOREM 2. Every support set of points of a sheaf-space with denumerable basis and support relation of indicial succession is a LGa space.

This theorem is a consequence of the preceding one and Theorem 6 of [i]. We note that the hypotheses of Theorem 2 are satisfied by a regular separable subspace of a complete me- tric space.

2. Let <~r,B.~> be a sheaf-space with support relation of indicia! succession; we

denote this space by ~ . Word for word as was done in [2] for sheaf-spaces, one can intro-

duce the concept of O system in the space ~. Let ~e Y . We shall call a o system t

fur~f~ental for y if

and

Let X be a p set and <~, A> be some family of its supporting subsets which is a base

for the topological space; we denote this space by ~ �9

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 40, pp. 136,1419 1974. Results submitted Decem- ber 30, 1971 and March 22, 1973.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available f rom the publisher for $ 7.50.

341

table approximations to recursive predicates

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