UDC 51.01:518.5

MATHEMATICS

Kh. Kh. Nasibullov

ON RECURSIVE FUNCTIONS OF LARGE RANGE

(Presented by Academician P. S. Novikov on 31 III 1967)

Scollem in \((^{1})\) showed that every general recursive function (g.r.f.) \(f(x)\) can be represented in the form

\[ f(x)=\psi(\mu y\{\varphi(y)=x\})=\psi(\varphi^{-1}(x)), \tag{1} \]

where \(\psi,\varphi\) are suitable primitive recursive functions (p.r.f.). Scollem further writes that, probably, a necessary and sufficient condition for \(\psi\) to be able to represent all g.r.f.’s in the form (1) with some p.r.f. \(\varphi(x)\) is again that \(\psi\) be a function of large range, but that he did not investigate this. However, as E. A. Polyakov has observed, there is no single such function \(\psi\) by means of which it would be possible to represent any one-place g.r.f. in the form (1). Nevertheless Scollem’s conjecture proved to be valid for a representation of one-place g.r.f.’s somewhat different from (1).

Lemma 1. For every p.r.f. of large range \(R(x)\) there is a p.r.f. of large range \(K(x)\) such that \(R(x)=R(K(x))\).

Let, for each \(a\in N=\{0,1,2,\ldots\}\), all solutions of the equation \(R(x)=a\) be written in the sequence

\[ x_0^{(a)}<x_1^{(a)}<x_2^{(a)}<\cdots . \tag{*} \]

Then, if \(x=x_j^{(a)}\), put \(K(x)=x_i^{(a)}\), where \(i=\mu_t\{\operatorname{rest}(j,P_t)=0\}\), \(\operatorname{rest}(x,y)\) is the remainder on division of \(x\) by \(y\), and \(P_t\) is the prime number with number \(t\). Since every \(x\in N\) belongs to one of the sequences of type (*), \(K(x)\) will be an everywhere-defined p.r.f. of large range:

\[ K(x)=\mu t_{\le x}\{\, |(L_1(t)+1)\cdot \operatorname{sg}|R(t)-R(x)|- \]

\[ -(\mu i_{\le L_1(x)}\{\operatorname{rest}(L_1(x),P_i)=0\}+1)|=0\,\}, \]

where \(L_1(x)\) is a p.r.f. of large range which, together with \(R(x)\), carries out a simple one-to-one enumeration of all pairs of natural numbers (see \((^{2})\), p. 136).

Theorem 1. If \(R(x)\) is a p.r.f. of large range, then: 1) for every partial recursive function (p.r.f.) \(f(x)\) defined at zero there exist a p.r.f. \(F(x)\) and an integer \(a\) such that

\[ f(x)=R(F^{-1}(x))+a\cdot \overline{\operatorname{sg}}\,x; \tag{2} \]

2) for every partial recursive function \(f(x)\) which is not defined at zero, there exist a p.r.f. \(F(x)\) and a partial recursive function

\[ \rho(x)= \begin{cases} \text{undefined}, & \text{if } x=0,\\ 0, & \text{if } x>0, \end{cases} \]

such that

\[ f(x)=R(F^{-1}(x))+\rho(x). \tag{3} \]

Proof. Let \(f(x)\) be an arbitrary partial recursive function. Then there is a p.r.f. \(F_1(x,y)\) such that

\[ f(x)=R(\mu y\{F_1(x,y)=0\}) \]

(see \((^{2})\), p. 137). By Lemma 1, there exists a p.r.f. of large range \(K(x)\) such that \(R(x)=R(K(x))\). Take a p.r.f. \(L_2(x)\), which together with \(K(x)\) carries out a simple one-to-one enumeration of all pairs of natural numbers, and the function

\(B(x)=\mu y\{F_1'(x,y)=0\}\) can be written in the following way:

\[ B(x)=K\left(\mu t\left\{L_2(t)\cdot \overline{\operatorname{sg}}\,F_1(L_2(t),K(t))\cdot \prod_{i=0}^{K(t)-1}\operatorname{sg} F_1(L_2(t),i)=x\right\}\right)+ \]

\[ +B(0)\cdot \overline{\operatorname{sg}}\,x, \]

i.e.

\[ B(x)=K(F^{-1}(x))+B(0)\cdot \overline{\operatorname{sg}}\,x, \]

where

\[ F(x)=L_2(x)\cdot \overline{\operatorname{sg}}\,F_1(L_2(x),K(x))\cdot \prod_{i=0}^{K(x)-1}\operatorname{sg} F_1(L_2(x),i). \]

Then

\[ f(x)=R(K(F^{-1}(x))+B(0)\cdot \overline{\operatorname{sg}}\,x)= \]

\[ =R(K(F^{-1}(x)))+R(B(0)\cdot \overline{\operatorname{sg}}\,x)-R(0), \]

\[ f(x)=R(F^{-1}(x))+(R(B(0))-R(0))\cdot \overline{\operatorname{sg}}\,x. \]

If \(f(0)\) is defined, then from (4) we obtain (2); if \(f(0)\) is not defined, then we obtain (3).

It is also obvious that if, for some p.r.f. \(R(x)\), all one-place g.r.f.’s are representable in the form (2), then \(R(x)\) must be a function of large span.

Remark. Let \(M\) be the collection of all p.r.f.’s of large span which differ from one another only at zero. The following assertion is true: there exists such a set \(M\) that for every g.r.f. \(f(x)\) there are p.r.f.’s \(\psi(x)\in M\) and \(\varphi(x)\) such that

\[ f(x)=\psi(\varphi^{-1}(x)). \tag{5} \]

The representation (5) is a certain strengthening of the result in (1).

Next we consider the algebras \(\mathfrak A_{\text{o.r.}}=\langle A_{\text{o.r.}};+,*,-1\rangle\) and \(\mathfrak A_{\text{q.r.}}=\langle A_{\text{q.r.}};+,*,-1\rangle\) (see \((^3)\)). Let
\[ M(x)=a_0+a_1x+a_2x^2+\ldots+a_mx^m,\qquad P(x)=b_0+b_1x+b_2x^2+\ldots+b_nx^n \]
\((m,n>1,\ a_0\ne0)\) be polynomials with rational coefficients such that
\[ (\forall x)[x\in N\to M(x)\in N\ \&\ P(x)\in N]. \]

Define the function \(T_P(x)\) as the distance from \(x\) to the nearest number on the right of the form \(P(t)\).

Theorem 2. The functions \(M(x)\) and \(T_P(x)\) form a basis of the algebra \(\mathfrak A_{\text{q.r.}}\).

Proof. From the functions \(M(x)\),
\[ x\dotminus y=T_P(T_P^{-1}(x)+y)=x-y\quad\text{for }x\ge y, \]
\[ D_a(x)=\mu y\{ay=x\}=[x/a]\quad\text{for }x=a\cdot t, \]
with the aid of the operations \(+,*,-1\) one can obtain all linear functions, \(x^2\), \(x\cdot y\), and a polynomial with integer coefficients \(P_1(x)=a\cdot P(x)\), where \(a\) is a suitable number from \(N\);

\[ \overline{\operatorname{sg}}\,x= \begin{cases} 1,& x=0,\\ 0,& x>0, \end{cases} \qquad x\dotminus y= \begin{cases} x-y,& x\ge y,\\ 0,& x<y. \end{cases} \]

Let
\[ l_1(x)=\mu y\{P_1(y)=a\cdot(x+T_P(x))\}, \]
i.e.
\[ l_1(x)=P_1^{-1}(x)*a\cdot(x+T_P(x)). \]
We note that
\[ (\exists x_0)(\forall x)[x>x_0\ \&\ P(i)<x\le P(i+1)\to l_1(x)=i+1], \]
i.e. \(l_1(x)\) assumes all natural-number values greater than \(d_0\) \((P(d_0)\ge x_0)\), and

\[ (\exists t)(\forall x)[x>t\to P(x)\ge x^2]. \]

One can verify that in the sequence
\[ \langle l_1(0),T_P(0)\rangle,\ \langle l_1(1),T_P(1)\rangle,\ \langle l_1(2),T_P(2)\rangle,\ldots \]
there will occur any pair of natural numbers of the form \((a^2,a)\), where \(a>t\). Then for \(x>t\vee x=0\):

\[ [\sqrt{x}]=T_P\bigl(\mu z\{(l_1(z)+1)\cdot \operatorname{sg}(T_P^2(z)\div l_1(z))=x+1\}\bigr). \tag{6} \]

In order for formula (6) to be valid for all \(x\), it is enough to make a small transformation. Now \(q(x)=x\div [\sqrt{x}]^2\), and the functions \(x+1\) and \(q(x)\) form a basis of the algebra \(\mathfrak A_{\text{q.r.}}\) (see \((^2)\), p. 121).

Remark. Let the function \(T_{a t^n}(x)\) be equal to the distance from \(x\) to the nearest number on the right of the form \(a\cdot t^n\) \((a>0,\ n\ge2)\). Then the function \(T_{a t^n}(x)\) and an arbitrary q.r.f. \(f(x)\), where \(f(0)=d\ne0\), form a basis of the algebra \(\mathfrak A_{\text{q.r.}}\).

The proof of this assertion can be reduced to the proof of Theorem 3, after first obtaining from \(f(x)\) and
\[ T_{at^n}^{\,n}(x)=a\left(\left[\sqrt[n]{\frac{x}{a}}\right]+\operatorname{sg} x\right)^n-x \]
the function \(x^{\tilde n}\) \((\tilde n>1)\).

For p.r.f. an analogous result was obtained by I. A. Lavrov \((^4)\).

It is not hard to see that a basis of the algebra \(\mathfrak A_{\mathrm{o.r.}}\) is also a basis of the algebra \(\mathfrak A_{\mathrm{p.r.}}\). The question arises: will every o.r. basis of the algebra \(\mathfrak A_{\mathrm{p.r.}}\) be a basis of the algebra \(\mathfrak A_{\mathrm{o.r.}}\) (by a basis is meant a system of generating elements of the algebra)?

Theorem 3. There exists an o.r. basis of \(\mathfrak A_{\mathrm{p.r.}}\) which is not a basis of \(\mathfrak A_{\mathrm{o.r.}}\).

Proof. From the remark to Theorem 3 it follows, for example, that \(x+1\) and \(T_{2t^2}(x)\) form a basis of the algebra \(\mathfrak A_{\mathrm{p.r.}}\). But these functions belong to a proper subalgebra of the algebra \(\mathfrak A_{\mathrm{o.r.}}\) (see \((^5)\)) and therefore cannot be a basis of \(\mathfrak A_{\mathrm{o.r.}}\).

Next we construct examples of partial recursive functions of large span, each of which together with \(x+1\) does not form a basis of the algebra \(\mathfrak A_{\mathrm{p.r.}}\); this will give a negative answer to one of A. I. Mal’tsev’s questions.

Let \(\Phi_i(x)\) be equal to the distance from \(x\) to the nearest, on the left (or on the right), number of the form \(i^t\), \(\Phi_i(0)=0\) \((i=2,3,\ldots)\). Define the function
\[ Q_i(x)= \begin{cases} \text{undefined,} & \Phi_i^{-1}(\Phi_i(x))=x,\\ \Phi_i(x) & \text{otherwise;} \end{cases} \]
\(Q_i(x)\) is undefined at those points \(x\) at which \(\Phi_i(x)\) assumes, for the first time, one of its values. It is clear that \(Q_i(x)\) is a p.r.f. of large span, and there is an interval of arbitrarily large length in which it is nowhere defined.

Theorem 4. The function \(x+1\) and the p.r.f. of large span \(Q_i(x)\) are not a basis of \(\mathfrak A_{\mathrm{p.r.}}\).

Proof. It is enough to note that the functions \(x+1\) and \(Q_i(x)\) belong to the following set \(C\), closed under \(+\), \(*\), and \(-1\):
\[ f(x)\in C \Longleftrightarrow \left\{ (\exists a)(\exists b) \left[ \begin{array}{l} a,b \text{ rational},\\ a>0, \end{array} \right. \right. \quad \&\, f(x)= \left\{ \begin{array}{ll} ax+b, & x\in D_f,\\ \text{undefined} & \\ \text{otherwise,} \end{array} \right] \vee \]
\[ \left. \vee\,(\forall m)(\forall n) \left[ m,n\in N \& m>0 \& f(mx+n) \begin{array}{c} \text{is undefined in}\\ \text{infinitely many points} \end{array} \right] \right\}; \]
where \(D_f\) is the domain of definition of \(f\).

The question of whether every o.r.f. of large span, together with the function \(x+1\), forms a basis of \(\mathfrak A_{\mathrm{p.r.}}\), remains open.

In the paper \((^6)\) R. Robinson showed that from the functions \(x+1\),
\[ \chi(x)= \begin{cases} 1, & x\ne t^2,\\ 0, & x=t^2 \end{cases} \]
with the aid of the operations \(|x-y|\), \(*\), and \(i\), one can obtain all unary p.r.f. Let us consider the analogous question for o.r.f.

Theorem 5. The functions \(1\) and \([\sqrt{x}]\) form a basis of the algebra
\[ \mathfrak A'_{\mathrm{o.r.}}=\langle A_{\mathrm{o.r.}};\ |x-y|,\ *,\ -1\rangle . \]

Let us note that if in \(\mathfrak A_{\mathrm{p.r.}}\) the operation \(+\) is replaced by the operation \(|x-y|\), then the basis is simplified.

Ivanovo State Pedagogical
Institute named after D. A. Furmanov

Received
31 III 1967

References

1. Th. Skolem, Math. Scand., 1, No. 2, 213 (1953).
2. A. I. Mal’tsev, Algorithms and recursive functions, Moscow, 1964.
3. A. I. Mal’tsev, UMM, 16, No. 3, 3 (1961).
4. I. A. Lavrov, DAN, 172, No. 2, 279 (1967).
5. E. A. Polyakov, VII All-Union Colloquium on General Algebra, Abstracts of communications and reports, Kishinev, 1965.
6. R. M. Robinson, Bull. Am. Math. Soc., 53, 925 (1947).