UDC 51.01 : 518.5 : 519.5

MATHEMATICS

V. I. AMSTISLAVSKII

EXTENSION OF RECURSIVE HIERARCHIES AND (R)-OPERATIONS

(Presented by Academician M. A. Lavrent'ev on 21 VIII 1967)

Here we consider extensions of recursive hierarchies that arise as a result of going beyond the constructive second class of ordinal numbers. For example, the (\mathfrak{H})-hierarchy(^), whose finite levels were defined in ((^8)), was extended in ((^2)) on the basis of the constructive first, second, and third number classes and continued still further in ((^{10})) on the basis of a constructive version of higher classes of ordinal numbers. To estimate such extensions of recursive hierarchies we apply the (R)-operations(^) of A. N. Kolmogorov, which makes it possible to obtain estimates more precise than in ((^{10})) and ((^4)). Analogously to the finite levels of the classical hierarchy of (R)-sets ((^{11})), we define a sequence of classes (\mathfrak{R}_n) ((n=0,1,2,\ldots)) of sets of natural numbers and establish (where (C\mathfrak{R}_n) denotes the class of complements of sets of the class (\mathfrak{R}_n)): 1) the (\mathfrak{H})-hierarchy ((^{10})) and the (H_h)-hierarchy ((^4)) do not go beyond the class (\mathfrak{R}_2 \cap C\mathfrak{R}_2)(^{*}); 2) in each of the classes (\mathfrak{R}{n+1}\cap C\mathfrak{R})) the class (\Sigma^1_2 \cap \Pi^1_2)(^}) there is a transfinite hierarchy of sets not belonging to (\mathfrak{R}_n \cup C\mathfrak{R}_n); 3) all (\mathfrak{R}_n \subset \Sigma^1_2 \cap \Pi^1_2) (in ((^4,{}^{10) is indicated as an estimate for the sets covered by the (\mathfrak{H})- and (H_h)-hierarchies).

1. (R)-operations. (N={0,1,2,\ldots}); (a,b,i,m,n) are variables on (N); (N^N) is the set of all functions (f:N\to N); (\mathcal P N) is the set of all subsets of (N); (\xi,\eta) are variables on (\mathcal P N). Subsets of (\mathcal P N) will be called bases. The (\delta s)-operation (\Phi_M) with base (M) is the function which, to each sequence ({E_i}_{i\in N}) of sets of an arbitrary space (E) (in our case always (E=N)), assigns the set

[
\Phi_M{E_i}_i={x:(\exists \xi\in M)(\forall i\in \xi)[x\in E_i]}.
]

The completion of a base (M) is called the base

[
\widetilde M={\xi:(\exists\eta\in M)[\eta\subseteq \xi]}.
]

For equality of the (\delta s)-operations (\Phi_{M_1}) and (\Phi_{M_2}) it is necessary and sufficient that (\widetilde M_1=\widetilde M_2).

For any base (M), let

[
M^c={\xi:(N\setminus \xi)\notin M}
]

and (M'=(\widetilde M)^c); then

[
\Phi_{M'}{E_i}={x:(\forall \xi\in M)(\exists i\in \xi)(x\in E_i)}.
]

Let all tuples (i.e., finite ordered sets) of natural numbers be enumerated one-to-one by all natural numbers, with (0) the number of the empty tuple, and ([m_0,\ldots,m_i]) the number of the tuple (m_0,\ldots,m_i). Let

[
\theta(0,n)=[n]
]

and

[
\theta(m,n)=[m_0,\ldots,m_i,n]\quad\text{for }m=[m_0,\ldots,m_i];
]

we assume the enumeration of tuples is such that (\theta(m,n)) is a general recursive function (g.r.f.). Let

[
\theta_m(n)=\theta(m,n)
]

and

[
\theta_m^{-1}(\xi)={n:\theta_m(n)\in \xi}.
]

For any base (M) define the base (\mathbf R(M)) as follows:

[
\mathbf R(M)={\xi:0\in \xi \,\&\, (\forall m\in \xi)[\theta_m^{-1}(\xi)\in M]}.
]

Obviously,

[
\widetilde M_1=\widetilde M_2 \Rightarrow \widetilde{\mathbf R}(M_1)=\widetilde{\mathbf R}(M_2)
]

((\widetilde{\mathbf R}(M)) is the completion of (\mathbf R(M))). The (\delta s)-operation (\Phi_K) with base (K=\mathbf R(M)) is called the (R)-operation over (\Phi_M). (This definition corresponds to ((^{11})).)

(^*) The definition is given below.

(^ {**}) This also clarifies the connection between the (\mathfrak{H})-hierarchy (extended as in ((^{10}))) and the effective (R)-hierarchy; on this connection see ((^2)).

For any function (F:\mathscr P N\to \mathscr P N) define the function (\mathbf L_F:\mathscr P N\to \mathscr P N) as follows. First, for any (\xi\subseteq N) and ordinal (\nu<\Omega) ((\Omega) is the least uncountable ordinal) define (\xi^\nu): 1) (\xi^0=\xi); 2) (\xi^{\nu+1}=F(\xi^\nu)); 3) if (\nu) is a limit ordinal, then (\xi^\nu=\bigcap_{\mu<\nu}\xi^\mu); then put

[
\mathbf L_F(\xi)=\bigcap_{\Omega<\nu}\xi^\nu .
]

Lemma (adapted from ((^{12}))). Let (M\subseteq \mathscr P N) and
[
F(\xi)={m:m\in \xi\&\theta_m^{-1}(\xi)\in M}.
]
Then
[
\widetilde{\mathbf R}(M)={\xi:0\in \mathbf L_F(\xi)}.
]

If (X\subseteq \mathscr P N) (or (X\subseteq N)), then let, in accordance with (1), (X\in\Sigma_n^1\cap\Pi_n^1) mean that the predicate (\xi\in X) (or, respectively, (i\in X)) is expressible in both (n)-functional-quantifier Kleene forms ((^7)).

Theorem 1. For any base (M), if (\widetilde M\in\Sigma_2^1\cap\Pi_2^1), then also
[
\widetilde{\mathbf R}(M)\in \Sigma_2^1\cap\Pi_2^1 .
]

The (\Sigma_2^1)-form for (\widetilde{\mathbf R}(M)) is obtained from the definition of (\mathbf R(M)), and the (\Pi_2^1)-form from the lemma and Patnam’s method ((^{13})). (If the classes of (\Sigma_2^1)- and (\Pi_2^1)-subsets of (\mathscr P N) are numbered, for example, as in ((^7)), then one can find such general recursive functions (g_0(m,n)) and (g_1(m,n)) that, for any pair (m,n) of numbers of the two forms of the base (\widetilde M), the numbers (g_0(m,n)) and (g_1(m,n)) are numbers of the two forms of the base (\widetilde{\mathbf R}(M)).)

Define a sequence of bases (R_n) ((n=0,1,2,\ldots)) as follows: (R_0={N}) (i.e. (R_0) contains only one element, (N)), (R_{n+1}=\mathbf R(R_n)).

The (\delta s)-operation (\Phi_{R_n}) with base (R_n) is called an (R^n)-operation. Thus, an (R^0)-operation is the operation of intersection
[
\Phi_{R_0}{E_i}=\bigcap_{i\in N}E_i,
]
and an (R^{n+1})-operation is an (R)-operation over (\Phi_{R_n}'). From Theorem 1 we obtain

Corollary(^*). For any (n\in N),
[
\widetilde R_n\in\Sigma_2^1\cap\Pi_2^1
\quad\text{and}\quad
\widetilde R_n'\in\Sigma_2^1\cap\Pi_2^1 .
]

2. The classes (\mathfrak R_n). A one-place partial recursive function (p.r.f.) with Gödel number (see ((^6))) (a) is denoted by (\langle a\rangle). By (\tau) we denote the family (see ((^3))) of all recursively enumerable subsets of (N), having the following numbering: for any (a\in N), the member of the family (\tau) with number (a), (\tau a), is the domain of definition of the p.r.f. (\langle a\rangle).

The class of all sets (\xi) ((\subseteq N)) representable in the form

[
\xi={m:(\exists\eta\in R_n)(\forall i\in\eta)P(m,i)},
]

where (P(m,i)) is some recursively enumerable predicate, will be called the class (\mathfrak R_n) ((n=0,1,2,\ldots)). (C\mathfrak R_n) is the class of complements (with respect to (N)) of sets of the class (\mathfrak R_n).

Obviously, the class (\mathfrak R_n) consists of all sets obtained by an (R^n)-operation over all possible recursively enumerable sequences of members of the family (\tau). We provide each of the classes (\mathfrak R_n) with a certain numbering, defining the families (\rho_n) as follows:

[
\rho_n=H_{\widetilde R_n}\tau,
]

where (H_{\widetilde R_n}) is the standard operator with base (\widetilde R_n) ((^3)). Since any (\delta s)-operation (\Phi_M) with base (M) coincides with the set-theoretic operation (\Psi_{\widetilde M}) with base (\widetilde M), then, by the definition of the standard operator, the member of the family (\rho_n) with number (a),
[
\rho_n a=\Phi_{R_n}{E_i},
]
where

[
E_i=
\begin{cases}
\tau\langle a\rangle(i), & \text{for those } i\in N \text{ for which } \langle a\rangle(i) \text{ is defined},\
\varnothing, & \text{for the remaining } i\in N .
\end{cases}
]

The families (\bar\rho_n) are defined as follows: for any (a\in N),
[
\bar\rho_n a=N\setminus \rho_n a.
]
Obviously, the totality of all members of the family (\rho_n) coincides with (\mathfrak R_n), and the totality of all members of (\bar\rho_n) with (C\mathfrak R_n).

Theorem 2. For any (n\in N),
[
\mathfrak R_n\subset \mathfrak R_{n+1},\qquad
C\mathfrak R_n\subset \mathfrak R_{n+1}
\quad\text{and}\quad
\mathfrak R_n\ne C\mathfrak R_n .
]

Proof by means of the properties of (R)-operations established in ((^5)).

(^*) Cf. ((^{11})), Corollary 2 of Theorem 12. See also our remark after Theorem 3.

Theorem 3. For all (n\in N), (\mathfrak R_n \subset \Sigma^1_2\cap \Pi^1_2).

Proof on the basis of the corollary to Theorem 1.

Remark. With the aid of the system of notations for ordinals of the first two constructive number classes (\left({}^{9}\right)), the sequence of bases (R_n) and classes (\mathfrak R_n) can be continued transfinitely, obtaining an effective analogue of the classical hierarchy of (R)-sets (\left({}^{11}\right)). As the basic space, instead of (N) one may consider the space
[
N^{m,n}=(N^N)^m\times N^n\qquad (m,n=0,1,\ldots;\; m+n>0) \left({}^{1}\right),
]
taking the family of all effectively open sets of this space instead of (\tau). The corollary of Theorem 1 and Theorems 2 and 3 can be extended to both these cases.

3. Hierarchies in the class (\mathfrak R_2). Introduce the one-one recursive functions (\sigma(m,n)), ((n)'), and ((n)''), establishing a one-to-one correspondence between (N^2) and (N). The function
[
\operatorname{hj}:\mathcal P N\to \mathcal P N,
]
which assigns to each (\xi\subseteq N) the set
[
\operatorname{hj}(\xi)={m:(\forall f)\,(\mathfrak R_n)T^{\xi,f}1(m,m,n)},
]
is called a hyperjump (hyperjump, (\left({}^{4,13}\right))). The hyperjump can be represented by means of the (\delta s)-operation (\Phi\right)) one obtains}); with the aid of this representation and known properties of (R)-operations (\left({}^{5,12

Theorem 4. If (\xi\in \mathfrak R_2\cap C\mathfrak R_2), then also (\operatorname{hj}(\xi)\in \mathfrak R_2\cap C\mathfrak R_2), and there exists a recursive function (g(m,n)) such that, if
[
\xi=\rho_2 m=\rho_2 n,
]
then
[
\operatorname{hj}(\xi)=\rho_2(g(m,n))'=\rho_2(g(m,n))''.
]

The field of a binary relation (P(m,n)) is the set
[
{m:(\exists n)[P(m,n)\vee P(n,m)]}.
]
An irreflexive transitive binary relation (m<{W} n) with field (W(\subseteq N)) is called a system of notations (s.o.), if: a) (1\in W) and
[
(\forall n\in W)[\,n\ne 1\Rightarrow 1<n\,];
]
b) every nonempty subset of the field (W) has a (<_{W})-minimal element.

For any function (F:\mathcal P N\to \mathcal P N) and s.o. (<{W}) with field (W), denote by (\Pi(F,<)) the class of all and only those functions (\mathfrak F:W\to \mathcal P N) for each of which there exist partial recursive functions (p(a)), (q(a,b)), and (r(a,b)) such that, if (a\in W) and (a\ne 1), then:

a) (p(a)<{W}a) and
[
{b:q(a,b)\in \mathfrak F(p(a))}={b<a};
]

b)
[
\mathfrak F(a)={m:r(a,m)\in F(U_a)},
]
where
[
U_a={\sigma(b,m):q(a,b)\in \mathfrak F(p(a))\ \&\ m\in \mathfrak F(b)}.
]

Thus, if (\mathfrak F\in \Pi(F,<{W})), then the value (\mathfrak F(a)), for (a\in W) and (a\ne 1), is determined according to the scheme given in b), through the values (\mathfrak F(b)) for (b<a).

Theorem 5. If (\mathfrak F\in \Pi(\operatorname{hj},<{W})), where (<) is an s.o. with field (W), and if
[
\mathfrak F(1)\in \mathfrak R_2\cap C\mathfrak R_2,
]
then there exists a recursive function (f(a)) such that for any (a\in W)
[
\mathfrak F(a)=\rho_2(f(a))'=\rho_2(f(a))''.
]

Proof on the basis of the recursion lemma (\left({}^{4,10}\right)) and Theorem 4.

Corollary. Let an s.o. (<_{h}) with field (D_h) and a function
[
H_h:D_h\to \mathcal P N
]
be defined by (\left({}^{4}\right)). Then
[
H_h(x)\in \mathfrak R_2\cap C\mathfrak R_2
]
for all (x\in D_h).

Now consider the transfinite iteration of the hyperjump on the basis of the constructive version of the class of all attainable ordinals. Let an s.o. (<{\widetilde C}) with field (\widetilde C), and for each (a\in \widetilde C) subsystems (I_a) ((I_a\subset <\right)). For each (a\in \widetilde C) there is in (\widetilde C) a unique element immediately following (a), (2^a). If (a\in \widetilde C) and})) with field (\operatorname{dom} I_a) (interpreted as the constructive number class of index (a)) be defined as in (\left({}^{10
[
(\forall n)[a\ne 2^n],
]
then (a=3^b5^n), where (b<{\widetilde C}a) and (n) is the Gödel number of such a partial recursive function that
[
(\forall i\in \operatorname{dom} I_b)[\,\langle n\rangle(i)<a\,].
]

Theorem 6. There exists a recursive function (g(a)) such that for any (a\in \widetilde C)
[
\sigma(I_a)=\rho_2(g(a))'=\rho_2(g(a))''.
]

Proof by means of Theorem 5.

For all (a\in \widetilde C), the sets (\mathfrak H_a(\subseteq N)) are defined as follows (\left({}^{10}\right)): 1) (\mathfrak H_1=N); 2) (\mathfrak H_{2^a}=\operatorname{hj}(\mathfrak H_a)), where (a\in \widetilde C); 3) if (a\in \widetilde C) and (a=3^b5^n), then
[
\mathfrak H_a={m:(m)1\in \operatorname{dom} I_b\ \&\ (m)_0\in \mathfrak H}.*
]

[
\text{* For } m>0,\ (m)_i \text{ is the exponent of the } i\text{-th prime number in the representation of } m
]
[
\text{as a product of powers of primes; } (0)_i=0\ \text{(see }({}^{6})\text{).}
]

From Theorem 6, with the aid of Theorem 4, we obtain

Corollary. For all (a \in \widehat{C}), (\mathfrak{H}_a \in \mathfrak{R}_2 \cap C\mathfrak{R}_2).

4. Hierarchies in the classes (\mathfrak{R}_{n+1}). Theorem 5 can be generalized as follows.

Theorem 7. For any (n=0,1,2,\ldots): if (\mathfrak{F} \in \Pi(F,<{W})), where (<N) is representable in the form}) is a c.o. with field (W), and the function (F:\mathfrak{P}N \to \mathfrak{P

[
F(\xi)={m:(\forall \eta \in R_n)(\exists i \in \eta)P^{\xi,\eta}(m,i)}
]

with some general-recursive (P^{\xi,\eta}), and if (\mathfrak{F}(1)\in \mathfrak{R}{n+1}\cap C\mathfrak{R}), then there exists an o.r.f. (f(a)) such that, for every (a\in W),

[
\mathfrak{F}(a)=\rho_{n+1}(f(a))'=\overline{\rho}_{n+1}(f(a))''.
]

On the basis of this theorem, in each of the classes (\mathfrak{R}{n+1}\cap C\mathfrak{R}(m,m,i)) and, as the c.o. (<}) it is easy to indicate (taking, for example, (P^{\xi,\eta}(m,i)\Longleftrightarrow T_1^{\xi,\eta{W}), the Kleene system (<}) ((^{9}))) a transfinite sequence of sets that do not belong to (\mathfrak{R{n}\cup C\mathfrak{R}) and have strictly increasing degrees of recursive unsolvability.

Institute of Cybernetics
Academy of Sciences of the Azerbaijan SSR

Received
19 VII 1967

CITED LITERATURE

(^{1}) J. W. Addison, Fund. math., 46, 123 (1958).
(^{2}) J. W. Addison, S. C. Kleene, Proc. Am. Math. Soc., 8, 1002 (1957).
(^{3}) V. I. Amstislavskii, DAN, 169, No. 5, 995 (1966).
(^{4}) H. B. Enderton, Trans. Am. Math. Soc., 111, 457 (1964).
(^{5}) L. Kantorovitch, E. Livenson, Fund. math., 20, 54 (1933).
(^{6}) S. C. Kleene, Introduction to Metamathematics, Moscow, 1957.
(^{7}) S. C. Kleene, Trans. Am. Math. Soc., 79, 312 (1955).
(^{8}) S. C. Kleene, Bull. Am. Math. Soc., 61, 193 (1955).
(^{9}) S. C. Kleene, Am. J. Math., 77, 405 (1955).
(^{10}) D. L. Kreider, H. Rogers Jr., Trans. Am. Math. Soc., 100, 325 (1961).
(^{11}) A. A. Lyapunov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 40, 1 (1953).
(^{12}) A. A. Lyapunov, Tr. Mosk. matem. obshch., 6, 195 (1957).
(^{13}) H. Putnam, Proc. Am. Math. Soc., 15, 44 (1964).