UDC 51.01:518.5

MATHEMATICS

E. A. POLYAKOV

ON RECURSIVE SUBSETS OF SETS OF RECURSIVE FUNCTIONS

(Presented by Academician P. S. Novikov on 24 IV 1968)

Let (\chi) denote Kleene’s numbering of the set (A_{4p}^{(1)}) of all one-place partial recursive functions (p.r.f.), and let (K(n,x)) denote Kleene’s universal function (\left({}^{1}\right)).

Let (G \subseteq A_{4p}^{(1)}) be some family of p.r.f. We shall call a set (A \subseteq G) recursive relative to (G) (\left({}^{2}\right)) if there exists a p.r.f. (g(x)) such that

[
g(x)=
\begin{cases}
0, & \text{if } \chi x \in A,\
1, & \text{if } \chi x \in G \setminus A.
\end{cases}
]

If (G=A_{4p}^{(1)}), then Rice’s well-known theorem asserts that only two trivial subsets of (G) ((G) itself and the empty set) will be recursive.

By (A_{\mathrm{pr}}^{(1)}), (A_{\mathrm{or}}^{(1)}) we shall denote respectively the sets of all one-place primitive-recursive functions (p.r.f.) and all one-place general recursive functions (g.r.f.).

In (\left({}^{3}\right)) it was observed that if (G=A_{\mathrm{pr}}^{(1)}) or (G=A_{\mathrm{or}}^{(1)}), then the set of functions

[
A_{P}^{y_1,\ldots,y_k}={f \in G \mid P(f(y_1),\ldots,f(y_2))}
]

((P(x_1,\ldots,x_k)) is a recursive predicate; (y_1,\ldots,y_k) is a sequence of natural numbers) is a recursive subset relative to (G).

In the present note a broader family of subsets recursive relative to (G) is constructed than the one indicated above; in the case where (G) is a computable family of g.r.f., a description is obtained of all subsets recursive relative to (G). The note also gives a description of all subalgebras of the algebra of p.r.f. (\mathfrak A_{\mathrm{pr}}=\langle A_{\mathrm{pr}}^{(1)};\ +,\ ,\ i\rangle) that are recursive relative to (A_{\mathrm{pr}}^{(1)}), where the symbols (+), (), (i) denote respectively the operations of addition of two functions, superposition of two functions, and iteration of a function.

1. By a tuple we shall mean a tuple (possibly empty) consisting of natural numbers (N={0,1,2,\ldots}).

With each tuple (a=(i_0,\ldots,i_n)) and set of g.r.f. (G) we associate the set of functions

[
T_a^G={f(x)\mid f(x)\in G \wedge f(0)=i_0 \wedge \cdots \wedge f(n)=i_n}.
]

Let (G) be a family of g.r.f. such that for each tuple ((i_0,\ldots,i_n)) there is a function (g(x)\in G) for which (g(s)=i_s) ((0\leq s\leq n)).

It is not difficult to show that if ({\alpha_i}) and ({\beta_i}) are recursively enumerable sequences of tuples such that

[
A=\bigcup_{i=0}^{\infty} T_{\alpha_i}^G
\quad\text{and}\quad
G\setminus A=\bigcup_{i=0}^{\infty} T_{\beta_i}^G,
]

then the set (A) is recursive relative to (G).

The following example shows that there exist families of functions (A\subseteq G) that are recursive relative to (G) and do not have the form (A_{P}^{y_1,\ldots,y_k}).

Let us divide all tuples of odd length into two classes (\Gamma_1,\Gamma_2) as follows:

1) ((a_0,\ldots,a_{2t})\in\Gamma_1 \leftrightarrow a_0=\ldots=a_{2t}=2t);

2) ((a_0,\ldots,a_{2t})\in\Gamma_2 \leftrightarrow (\forall i)(i<2t \wedge i\ \text{even}\to (a_0,\ldots,a_i)\in\Gamma_1)\wedge(\exists i)(a_i\ \text{odd}\vee a_i<2t)).

Obviously, the sets (\Gamma_1,\Gamma_2) can be represented in the form of recursively enumerable sequences ({\alpha_i}) and ({\beta_i}), respectively. Let

[
A=\bigcup_{i=0}^{\infty}T_{\alpha_i}^{G},
]

then

[
G\setminus A=\bigcup_{i=0}^{\infty}T_{\beta_i}^{G}.
]

It is not hard to see that (A) is not of the form (A_P^{y_1,\ldots,y_k}).

1. Let (G) be some family of general recursive functions of one variable. A sequence of functions from (G), (f_1(x), f_2(x),\ldots), will be called recursively enumerable if the function (B(i,x)=f_i(x)) is a general recursive function of two variables. By (M) we shall denote some infinite recursively enumerable subset of the natural numbers.

A set of functions (A\subseteq G) is called ((B,g)_M)-closed ((g(x)) is some function from (G)), if it contains the functions

[
f_i(x)=
\begin{cases}
g(x), & \text{if } 0\le x<i,\
B(i,x), & \text{if } x\ge i,
\end{cases}
]

for all (i\in M).

Theorem 1. If there exists a function (g(x)\in G\setminus A) and a sequence ({B(i,x)}) ((B(i,x)) is a general recursive function) of functions from (G) such that, for some infinite recursively enumerable subset of the natural numbers (M), the set (A) is ((B,g)_M)-closed, then (A) is not recursive relative to (G).

Theorem 1 is a generalization of Theorem 10 (§ 11) stated in [2].

1. In what follows, by (G) we shall understand a computable family of general recursive functions, i.e.,

[
G={F(0,x),F(1,x),\ldots},
]

where (F(n,x)) is a general recursive function.

Theorem 2. If the set (A\subseteq G) is recursive relative to (G), then (A) is a computable family.

Proof. Since (A) is recursive relative to (G), there exists a partial recursive function (g(x)) such that

[
g(x)=
\begin{cases}
0, & \text{if } \chi x\in A,\
1, & \text{if } \chi x\in G\setminus A.
\end{cases}
]

For some general recursive function (\psi(x)) we have

[
F(n,x)=K(\psi(n),x).
]

By (N_g) denote the set of all solutions of the equation (g(x)=0). Clearly, the sets (N_g) and (\rho\psi) (the range of values of (\psi)) are recursively enumerable, whence

[
N_A^{\chi}=N_g\cap\rho\psi
]

is a recursively enumerable set and

[
N_A^{\chi}={\varphi(0),\varphi(1),\ldots}
]

for a suitable general recursive function (\varphi). Obviously, (K(\varphi(n),x)) is a universal function for (A).

Definition. (\beta(x)) is a limit point for (A) (\leftrightarrow \forall n\exists f\in A\ \forall x(x\le n\to \beta(x)=f(x))).

Theorem 3. If there exists a function (\beta(x)\in G\setminus A) which is a limit point for (A), then (A) is not recursive relative to (G).

Proof. Using Theorem 2, it suffices to consider the case where (A) is a computable family, i.e.,

[
A={H(0,x),H(1,x),\ldots}
]

for some general recursive function (H).

Define the function (h(n)) as follows:

[
h(n)=\mu t\bigl(H(t,0)=\beta(0)\wedge\ldots\wedge H(t,n)=\beta(n)\bigr).
]

Since (\beta) is a limit point for (A), (h(n)) is a general recursive function.

It is clear that (A) is ((B,\beta)_N)-closed, where (B(n,x)=H(h(n),x)), and (N={0,1,2,\ldots}).

Using Theorem 1, we conclude that (A) is nonrecursive relative to (G). We shall say that sequences ({\alpha_i}) and ({\beta_i}) of tuples (\alpha_i,\beta_i) are effectively separable if there exist recursively enumerable sequences of tuples (\alpha_i') and (\beta_i') such that ({\alpha_i}\subseteq{\alpha_i'}), ({\beta_i}\subseteq{\beta_i'}), and ({\alpha_i'}\cap{\beta_i'}=\varnothing) ((\varnothing) is the empty set).

Theorem 4. A set (A\subseteq G) is recursive relative to (G) if and only if
[
A=\bigcup_{i=0}^{\infty} T_{\alpha_i}^{G}
\quad\text{and}\quad
G\setminus A=\bigcup_{i=0}^{\infty} T_{\beta_i}^{G}
]
for some effectively separable sequences ({\alpha_i},{\beta_i}) of tuples.

In the proof of Theorem 4 one uses Theorem 3 of the present note and Theorem 5 (§ 10) from (2).

1. Consider the algebra of p.r.f. (\mathfrak A_{\mathrm{pr}}=\langle A_{\mathrm{pr}}^{(1)};\,+,*,i\rangle). Take a word in the alphabet (\Gamma={\alpha,\beta}) beginning with (\alpha), for example (\alpha\beta\alpha), and define the subalgebra (\mathfrak A_{\alpha\beta\alpha}) of the algebra (\mathfrak A_{\mathrm{pr}}) as follows:
   [
   \mathfrak A_{\alpha\beta\alpha}
   =
   {f(x)\mid f(x)\in A_{\mathrm{pr}}^{(1)}\land f(0)=0\land f(1)\in N\land f(2)=0}.
   ]

Theorem 5. A subalgebra of the algebra (\mathfrak A_{\mathrm{pr}}) is recursive if and only if it has the form (\mathfrak A_{a_1,\ldots,a_n}) for some word (a_1\ldots a_n) beginning with (\alpha).

Ivanovo State Pedagogical Institute
named after D. A. Furmanov

Received
15 IV 1968

CITED LITERATURE

1. A. I. Maltsev, Algorithms and Recursive Functions, 1965.
2. V. A. Uspensky, Lectures on Computable Functions, 1960.
3. E. A. Polyakov, DAN, 178, No. 2 (1968).