UDC 51.01:518.5

MATHEMATICS

I. A. LAVROV

THE USE OF ARITHMETIC PROGRESSIONS OF THE \(k\)-TH ORDER FOR CONSTRUCTING BASES OF THE ALGEBRA OF PRIMITIVE-RECURSIVE FUNCTIONS

(Presented by Academician A. I. Mal’tsev on 18 III 1966)

In this paper we study the algebra of primitive-recursive functions
\(\mathfrak A_{\mathrm{pr}}=\langle A_{\mathrm{pr}}; +, *, \vdash\rangle\), where \(A_{\mathrm{pr}}\) is the set of all unary primitive-recursive functions, and \(+, *\), and \(\vdash\) are the operations of addition, superposition, and iteration of functions. In paper \((^1)\) it was shown that the functions
\(\lambda_0(x)=x+1\) and \(q(x)=x-[\sqrt{x}]^2\) form a basis of the algebra \(\mathfrak A_{\mathrm{pr}}\). In other words, from the functions \(\lambda_0(x)\) and \(q(x)\), by means of the operations \(+, *, \vdash\), one can obtain any primitive-recursive function. In paper \((^2)\) this result is generalized as follows: \(\lambda_0(x)\) and \(Q(x)\) are a basis for the algebra \(\mathfrak A_{\mathrm{pr}}\), where \(Q(x)\) denotes the distance from \(x\) to the nearest number on the left in a certain sequence of numbers, which in that paper is called an arithmetic progression of the second order. In the present paper this result is generalized to arithmetic progressions of the \(k\)-th order, where \(k\ge 2\). In particular, instead of the function \(q(x)\) one may take the function
\(Q(x)=x[\sqrt[n]{x}]^n\). For every natural number \(k\ge 2\), consider the class of unary primitive-recursive functions \(P_k\), defined as follows. A primitive-recursive function \(A(x)\) belongs to the class \(P_k\) if: 1) \(A(0)=0\); 2) \(A(x)\) is strictly increasing; 3) \(A(x)\) is an arithmetic progression of the \(k\)-th order, i.e., the finite differences of the \(k\)-th order of the sequence

\[ A(0),\ A(1),\ A(2),\ldots,\ A(n),\ldots \tag{1} \]

are constant. In what follows, functions from \(P_k\) will be called, for short, arithmetic progressions of the \(k\)-th order.

Denote by \(Q_A(x)\) the distance from \(x\) to the nearest number on the left of the form \(A(y)\). Note that \(Q_A(x)\) is a primitive-recursive function.

Main theorem. The functions \(\lambda_0(x)=x+1\) and \(Q_A(x)\) form a basis of the algebra \(\mathfrak A_{\mathrm{pr}}\).

Concerning arithmetic progressions we shall give, without proof, the following lemma.

Lemma 1. For every arithmetic progression of the \(k\)-th order \(A(x)\) there exists a natural number \(i\) such that

\[ A(x+i)-A(i)=B(x)+x, \]

where \(B(x)\) is a certain polynomial of degree \(k\) with positive rational coefficients, taking natural values for natural \(x\).

Note that \(B(x)\) will also be an arithmetic progression of the \(k\)-th order.

For every function \(\varphi(x)\) consider the following function:

\[ \alpha_\varphi(x)= \begin{cases} 1, & \text{if } x=\varphi(y) \text{ for some } y,\\ 0 & \text{in all other cases.} \end{cases} \]

Consider a strictly increasing function \(F(x)\) such that \(F(0)=0\). For \(F(x)\) we introduce the following functions:

\[ f(x)=F(x)+x; \]

\[ T_F(0)=0,\qquad T_F(x+1)=T_F(x)+\alpha_F(x+1); \]

\[ \beta_f(x)=L[x+1+\alpha_f(x+2)]. \]

Lemma 2. \(\beta_f(x)=x+T_F(x)\).

Proof. First we prove that

\[ \alpha_f(x+T_F(x)+2)=\alpha_F(x+1). \tag{2} \]

Let \(x=F(y)+k\), where \(0\le k<F(y+1)-F(y)\). Then
\(j=x+T_F(x)+2=F(y)+k+y+2\), since \(T_F(F(y))=y\). Hence
\(F(y)+y<F(y)+y+2\le j<F(y+1)+(y+1)+1\). Therefore
\(j=F(z)+z\) if and only if \(z=y+1\), and hence
\(x+T_F(x)+2=f(z)\) if and only if \(x+1=F(z)\). Thus (2) is proved.

We now prove that \(\beta_f(x)=x+T_F(x)\). If \(x=0\), then \(\beta_f(0)=0\) and
\(0+T_F(0)=0\). Suppose it has already been proved that, for some \(x\),
\(\beta_f(x)=x+T_F(x)\). Then
\[ \beta_f(x+1)=x+T_F(x)+1+\alpha_f(x+T_F(x)+2) =x+T_F(x)+1+\alpha_F(x+1)=(x+1)+T_F(x+1). \]
Lemma 2 is proved.

Lemma 3. \(\alpha_F(x)=\alpha_f(\beta_f(x))\).

Proof. The functions \(F(x)\) and \(\beta_f(x)\) are strictly increasing, and if \(x=F(y)\), then \(\beta_f(x)=F(y)+y=f(y)\). Lemma 3 is proved.

Consider two more series of functions

\[ g_n(x)= \begin{cases} 1, & \text{if } x=n,\\ 0 & \text{in all other cases;} \end{cases} \]

\[ \varphi_n(x)= \begin{cases} 1, & \text{if } x=ny,\\ 0 & \text{in all other cases,} \end{cases} \]

and, for a natural number \(a\ne0\), the formula

\[ \gamma_a(x)=g_2[\alpha_F(x)+\varphi_a(\beta_f(x)+(a-1)x)]. \]

Lemma 4. \(\gamma_a(x)=\alpha_K(x)\), where \(K(x)=F(ax)\).

Proof. \(\gamma_a(x)=1\) if and only if \(x=F(y)\) for some \(y\) and
\(\beta_f(x)+(a-1)x\) is divisible by \(a\); this is possible only in the case when
\(y=az\) for some \(z\). Thus \(\gamma(x)=1\) if and only if
\(x=F(az)=K(z)\) for some \(z\). Lemma 4 is proved.

Let
\[ M(x)=a_0x^k+a_1x^{k-1}+\cdots+a_{k-1}(x), \]
where all \(a_i\) are natural and \(k\ge2\). Put

\[ b=M(1), \]

\[ L(x)=\frac{M(x+2)-M(x+1)-b}{x+1}, \]

\[ S(0)=0,\qquad S(x+1)=S(x)+L(x)+\operatorname{sg}x \]

and choose the number \(d\) so that \(b+d\ne M(y)\) for any \(y\).

Consider the formula

\[ \delta(x)=L[x+T_M(x)+b\alpha_M(x)]+bg_0(x)+dg_1(x). \]

Lemma 5. \(\delta(x)=M(y+1)\) if and only if \(x=S(y)\), and also
\(\alpha_S(x)=\alpha_M(\delta(x))\).

Proof. First, \(\delta(0)=b=M(1)\). Further, we have
\(S(1)=L(0)+1=M(2)-2b+1\ne1\) and \(\delta(1)=b+d\ne M(z)\) for any

some \(z\). Continuing the computation, we shall have

\[ \delta(2)=2b+1,\ldots,\delta(S(1))=\delta(M(2)-2b+1)=M(2). \]

Suppose that for \(x=S(y+1)\) \(\delta(x)=M(y+2)\). Then

\[ \delta(S(y+2))=\delta(S(y+1)+L(y+1))= \]

\[ = M(y+2)+(y+2)L(y+1)+b=M(y+3). \]

Lemma 5 is proved.

Lemma 6. \(\alpha_\lambda(x)=\alpha_S(x+\operatorname{sg} x)\), where

\[ \lambda(0)=0,\qquad \lambda(x+1)=\lambda(x)+L(x). \]

Proof. We have \(\lambda(x)+\operatorname{sg} x=S(x)\). Note that, since \(L(x)\) is a polynomial with natural coefficients and \(L(0)\ne 0\), \(\lambda(x)\) is an arithmetic progression of order \(k-1\).

Theorem 1. If \(A(x)\) is an arithmetic progression of order \(k\) and \(k\ge 2\), then from the functions \(\lambda_0(x)=x+1,\ \alpha_A(x)\), by means of \(+\), \(*\), and \(\lfloor\ \rfloor\), one can obtain the functions \(x+[\sqrt{x}]\), \(x^2\), \(B(px+q)\), and \(A(px+q+r)\) for certain natural \(p\ne 0\), \(q\), and \(r\).

Proof. 1) Let \(A(x)\) be an arithmetic progression of second order. By Lemma 1 we find \(i\) and \(B(x)\). It is easy to note that
\(\alpha_{B(x)+x}(y)=\alpha_A(y+A(i))\). By Lemmas 2 and 3, we obtain \(\alpha_B(x)\). Let us find a number \(a\ne 0\) such that \(K(x)=B(ax)=dx^2+ex\), where \(d\) and \(e\) are natural numbers. By Lemma 4 we obtain \(\alpha_K(x)\). Then \(\alpha_M(x)=\alpha_K(x+K(1))\), where
\(M(x)=dx^2+(2d+e)x\). Applying Lemmas 2 and 3 several times, we obtain \(\alpha_{K_1}(x)\), where \(K_1(x)=dx^2+dx\). It is easy to note that \(\alpha_{K_1}(dx)=\alpha_{K_2}(x)\), where \(K_2(x)=x^2+x\). By Lemma 2 we obtain \(x+[\sqrt{x}]\) and \(\alpha_{K_3}(x)\), where \(K_3(x)=x^2\). As is known (see, for example, (3), pp. 78–79),

\[ \lfloor x+1+2\alpha_{K_3}(x+4)\rfloor=x+2[\sqrt{x}], \]

\[ \lfloor x+2[\sqrt{x}]+1\rfloor=x^2. \]

It remains for us to obtain the functions \(B(px+q)\) and \(A(px+q+r)\). Having obtained \(x^2\), one can obtain \(K(x)\), i.e. \(B(ax)\). But

\[ A(ax+i)=B(ax)+ax+A(i). \]

2) Let \(A(x)\) be an arithmetic progression of order \(k+1\), and suppose that for all arithmetic progressions of order \(k\) the theorem has been proved. As in the first case, we find \(\alpha_B(x)\). Let us find a number \(a\ne 0\) such that the function \(K(x)=B(ax)\) is a polynomial with natural coefficients. Then \(\alpha_{M(x)+x}(y)=\alpha_K(x+K(1))\), where \(M(x)\) is some polynomial with natural coefficients. Applying Lemmas 2–6, we obtain \(\alpha_\lambda(x)\), where \(\lambda(x)\) is an arithmetic progression of order \(k\). By the induction hypothesis one can obtain \(x+[\sqrt{x}]\), \(x^2\), and \(\lambda(px+q+r)\) for certain \(p\ne 0\), \(q\), and \(r\). We have

\[ \lambda(px+q+r)+\operatorname{sg}(px+q+r)=S(px+q+r), \]

\[ \delta(S(px+q+r))=M(px+q_1), \]

\[ M(px+q_1)+(px+q_1)+K(1)=K(px+q_2), \]

\[ K(px+q_2)=B(p_1x+q_3), \]

\[ B(p_1x+q_3)+(p_1x+q_3)+A(i)=A(p_1x+q_3+i). \]

Theorem 1 is proved.

Remark. In the proof of the theorem the functions \(g_n(x)\), \(\varphi_n(x)\), which are easily obtained from the functions \(x+1\) and \(\operatorname{sg} x\), are used. The function \(\operatorname{sg} x=d_A(a+\operatorname{sg} x)\), where \(a\) is such a number that \(a=A(y)\) for some \(y\), but \(a+1\ne A(z)\) for any \(z\). The functions \(mx+n\) and \(\operatorname{sg} x\) are easily obtained from \(x+1\).

Now we shall prove the main theorem.

Theorem 2. The functions \(\lambda_0(x)=x+1\) and \(Q_A(x)\) form a basis of the algebra \(\mathfrak A_{\mathrm{pr}}\).

Proof. Let

\[ P(x,y)=A(px+py+q+i)+(x-y). \]

Then

\[ P(x,y)=B(px+py)+q)+(p+1)x+(p-1)y+q+A(i). \]

It is not difficult to note that \(P(x,y)<A(px+py+q+i+1)\). We have

\[ \overline{\operatorname{sg}}\,x=\operatorname{sg} Q_A(a+\operatorname{sg}x),\qquad a_A(x)=\overline{\operatorname{sg}}\,Q_A(x), \]

where \(a\) is a number such that \(a\ne A(y)\) for any \(y\), but \(a+1=A(z)\) for some \(z\).

By Theorem 1 we obtain the function \(B(px+q)\), and hence also \(P(x,y)\). Then

\[ \varepsilon(x,y)=Q_A(P(x,y))=x-y,\qquad \text{if } x\ge y. \]

By Theorem 1 we also obtain the functions \(x+[\sqrt{x}]\) and \(x^2\). Then

\[ \varepsilon(x+[\sqrt{x}],x)=[\sqrt{x}],\qquad \varepsilon(x,[\sqrt{x}]^2)=q(x). \]

The proof of Theorem 2 is complete, since, by R. Robinson’s theorem, \(x+1\) and \(q(x)\) are a basis of the algebra \(\mathfrak A_{\mathrm{pr}}\).

Theorem 3. The functions \(\lambda_0(x)=x+1\) and \(nQ_A(x)\) \((n\ne 0)\) form a basis of the algebra \(\mathfrak A_{\mathrm{pr}}\).

Proof. \(a_A(x)\) is obtained as in Theorem 2. Instead of the function \(\varepsilon(x,y)\) one can obtain \(n\varepsilon(x,y)\). The rest of the proof repeats Theorem 2 verbatim from \((2)\).

The author takes this opportunity to express gratitude to D. A. Zakharov for a number of very valuable remarks.

Received
12 III 1966

REFERENCES

\(^{1}\) R. M. Robinson, Bull. Am. Math. Soc., 53, 925 (1947).
\(^{2}\) I. A. Lavrov, E. A. Palyutin, Seventh All-Union Colloquium on General Algebra, Summaries of Communications and Reports, Kishinev, 1965.
\(^{3}\) R. Peter, Recursive Functions, Moscow, 1954.