Reports of the Academy of Sciences of the USSR
1965. Volume 160, No. 2

MATHEMATICS

Academician A. I. MAL'TSEV

POSITIVE AND NEGATIVE NUMBERINGS

A numbering \(\beta\) of a collection of certain objects \(\mathfrak B\) is a single-valued mapping of the set \(N\) of all natural numbers onto \(\mathfrak B\). The numbering equivalence \(\theta_\beta\) is the binary relation on \(N\) defined by the formula \(x\theta_\beta y \Longleftrightarrow \beta x = \beta y\). If the numbering \(\beta\) is isomorphic (for the terminology see \((^1)\)) to some computable numbering of a suitable family of recursively enumerable sets (r.e.s.), then there exists a general recursive function (g.r.f.) \(F(x,y,u,v)\) for which
\[ \beta x = \beta y \Longleftrightarrow \forall u\, \exists v\,(F(x,y,u,v)=0). \]
Thus, the numbering equivalences of computable numberings of families of r.e.s. have class \(\forall\exists\) in Kleene’s classification. Below we consider in greater detail the properties of numberings of class I, for which the equivalence \(\theta\) has either the form
\[ \beta x=\beta y \Longleftrightarrow \exists u\,(F(x,y,u)=0), \]
or the form
\[ \beta x=\beta y \Longleftrightarrow \forall u\,(F(x,y,u)=0), \]
where \(F(x,y,u)\) is a suitable g.r.f. In the first case, the set of pairs \(\langle x,y\rangle\) for which \(\beta x=\beta y\) is recursively enumerable, and the numbering \(\beta\) is called positive. In the second case, the numbering \(\beta\) is called negative. Under a negative numbering, the set of pairs \(\langle x,y\rangle\) for which \(\beta x\ne\beta y\) turns out to be recursively enumerable. A numbering \(\beta\) is decidable if it is simultaneously positive and negative.

1. Let the collection \(\mathfrak A\) have some numbering \(\alpha\). A numbering \(\beta\) of a subcollection \(\mathfrak B\subseteq\mathfrak A\) is called \(\alpha\)-computable if \(\beta n=\alpha f(n)\) for a suitable g.r.f. \(f\). A numbering \(\beta\) of the collection \(\mathfrak B\) is called uniformizable if there exists a one-to-one \(\beta\)-computable numbering of \(\mathfrak B\). A subcollection \(\mathfrak B\subseteq\mathfrak A\) is called \(\alpha\)-uniform if there exists an \(\alpha\)-computable one-to-one numbering of \(\mathfrak B\). A family \(\mathfrak B\subseteq\mathfrak A\) is called \(\alpha\)-isolated if \(\alpha\)-computable numberings of \(\mathfrak B\) exist and all of them are equivalent to one another (i.e., computable relative to one another). If the basic collection is the collection of all r.e.s. and \(\alpha\) is the usual Post \((=\) Gödel) numbering \(\pi\), then instead of \(\pi\)-computable, \(\pi\)-uniform, \(\pi\)-isolated we shall say computable, uniform, and so on.

Theorem 1. If the basic numbering \(\alpha\) of the collection of objects \(\mathfrak A\) is positive (negative), then every \(\alpha\)-computable numbering \(\beta\) of an arbitrary subcollection \(\mathfrak B\subseteq\mathfrak A\) is also positive (negative). If \(\alpha\) is positive and the subcollection \(\mathfrak B\) is \(\alpha\)-computable (i.e., possesses at least one \(\alpha\)-computable numbering), then \(\mathfrak B\) is \(\alpha\)-isolated.

The first assertion follows directly from the definitions. We prove the second. By assumption, the set \(\theta\) of pairs \(\langle x,y\rangle\) for which \(\alpha x=\alpha y\) can be arranged in a recursive sequence
\[ \langle x_0,y_0\rangle,\ \langle x_1,y_1\rangle,\ \ldots . \]
Let \(\beta,\gamma\) be arbitrary \(\alpha\)-computable numberings of \(\mathfrak B\), and let
\[ \beta n=\alpha f(n),\qquad \gamma n=\alpha g(n), \]
where \(f,g\) are suitable g.r.f. For a given \(n\), in the sequence
\[ \langle x_0,y_0\rangle,\ \langle x_1,y_1\rangle,\ \ldots \]
we look for a pair of the form \(\langle f(n),g(x)\rangle\) and put \(x=h(n)\). Then \(\beta n=\gamma h(n)\).

Corollary. Uniformizable positive numberings are decidable.

Indeed, let \(\alpha\) be positive, and let \(\beta\) be an \(\alpha\)-computable one-to-one numbering of the collection \(\mathfrak B=\mathfrak A\). According to the preceding theorem, \(\alpha\) and \(\beta\) are equivalent. But \(\beta\) is decidable, and therefore \(\alpha\) is decidable.

§ 2. A family of r.e. sets \(\mathfrak B\) will be called finitely separable if there exists a strongly enumerable sequence of finite sets \(a_0, a_1, \ldots\) such that each of the sets of the family \(\mathfrak B\) contains at least one of the indicated finite sets, and each of these finite sets is contained in no more than one of the sets of the family \(\mathfrak B\). For example, if the family \(\mathfrak B\) is a partition (i.e., the sets \(\mathfrak B\) are nonempty, pairwise disjoint, and their union gives \(N\)), then \(\mathfrak B\) is finitely separated by the sequence \(\{0\}, \{1\}, \ldots\).

Theorem 2. Every positive numbering \(\alpha\) of an arbitrary collection of objects is isomorphic to a computable numbering of the corresponding partition \(N/\theta_\alpha\). Every finitely separable family of r.e. sets \(\mathfrak B\) that has a computable numbering \(\beta\) is an isolated positive family.

Let

\[ [a] = (x \mid ax = aa). \]

By assumption, for the occurring partial recursive function \(F(a,x,y)\) we have

\[ ax = aa \Longleftrightarrow \exists y(F(a,x,y)=0). \]

Consequently, \([a]\) is the domain of definition of the partial recursive function

\[ \Phi(a,x)=\mu_y(F(a,x,y)=0), \]

and therefore the mapping \(\alpha_1:a\to [a]\) is a computable numbering of the partition \(N/\theta_\alpha\). Since

\[ aa=ab \Longleftrightarrow \alpha_1 a=\alpha_1 b, \]

the numberings \(\alpha\) and \(\alpha_1\) are isomorphic.

To prove the second assertion, consider a strongly enumerable sequence of finite sets \(a_0,a_1,\ldots\) finitely separating \(\mathfrak B\). Let \(\beta\) be an arbitrary computable numbering of \(\mathfrak B\). Taking an arbitrary \(n\) and successively computing the elements of the sets \(n n,\ \beta i,\ a_i\) \((i,j=0,1,\ldots)\), we look for such \(i,j\) that

\[ a_i \subseteq n n,\qquad a_i \subseteq \beta j. \]

Denote by \(g(n)\) the first \(j\) encountered in this process and satisfying the indicated conditions. It is clear that \(g(n)\) is a partial recursive function and that

\[ n n\in\mathfrak B \Rightarrow g(n)\ \text{is defined and}\ n n=\beta g(n). \]

This means that an arbitrary computable numbering \(\beta\) of the family \(\mathfrak B\) is subnormal \((^1)\). But all subnormal numberings are equivalent \((^1)\), and therefore the family is isolated. It remains to show that the numbering \(\beta\) is positive. Computing successively the elements of the sets \(\beta j, a_i\) \((i,j=0,1,\ldots)\) and marking those pairs \(\langle a,b\rangle\) for which, at some moment of time, it will be

\[ a_i \subseteq \beta a \quad \text{and} \quad a_i \subseteq \beta b, \]

we enumerate all pairs for which

\[ \beta a=\beta b, \]

as required.

It follows from Theorem 2 that if some finitely separable family admits a computable unsolvable numbering, then it is not uniform. The simplest example of such a family is constructed as follows. Let \(A\) be a nonrecursive recursively enumerable set of numbers:

\[ A=\{a_0,a_1,a_2,\ldots\}. \]

We construct the sets \(A_0,A_1,\ldots\), putting into each \(A_i\), at the \(n\)-th step, the number \(i\), while into the sets \(A_{2a_n}\) and \(A_{2a_n+1}\) we put the numbers \(2a_n\) and \(2a_n+1\). The numbering

\[ \alpha:\ i\to A_i \]

is a computable positive numbering of the partition \(\{A_0,A_1,\ldots\}\). This numbering is unsolvable, since

\[ A_{2i}=A_{2i+1}\Longleftrightarrow i\in A. \]

Putting

\[ \beta n=A_0\cup A_1\cup\cdots\cup A_n, \]

we obtain a computable numbering of a nondecreasing sequence of finite sets:

\[ \beta 0\subseteq \beta 1\subseteq \cdots,\qquad \bigcup \beta n=N. \]

The numbering \(\beta\) is isomorphic to \(\alpha\) and therefore is positive and unsolvable. According to the corollary to Theorem 1, the numbering \(\beta\) is not uniformizable. At the same time, from the arguments of Friedberg \((^2)\) it follows (see \((^3)\)) that every computable family of r.e. sets containing a strongly enumerable sequence of increasing finite sets admits a one-one computable numbering. Therefore the family \(\beta N\) certainly has a one-one computable numbering and, consequently, may serve as an example of a uniform family admitting a nonuniformizable positive computable numbering.

§ 3. Let \(\beta\) be a computable numbering of some family of partial recursive functions of one variable. This means that, for a suitable partial recursive function \(U(n,x)\), we have the identity

\[ (\beta n)(x)=U(n,x). \]

Computing successively \(U(0,0),\)

\(U(0,1), U(1,0), \ldots\) and marking those pairs \(\langle a,b\rangle\) for which, for some \(x\), \(U(a,x) \ne U(b,x)\), we enumerate in general all pairs \(\langle a,b\rangle\) for which \(\beta a \ne \beta b\). Therefore all computable numberings of families of general recursive functions are negative. The usual numbering of the family of all primitive recursive functions (3) may serve as a typical example of an undecidable negative numbering.

Theorem 3. Every negative numbering \(\alpha\) of an arbitrary family of objects is isomorphic to a suitable computable numbering of a suitable family of general recursive functions taking only the values \(0,1\).

By hypothesis, the aggregate \(\neg \theta\) of all pairs \(\langle a,b\rangle\) satisfying the relation \(aa \ne ab\) can be arranged in a recursive sequence
\[ \neg \theta = \{\langle a_0,b_0\rangle,\langle a_1,b_1\rangle,\ldots\}. \]
We construct a function \(U(n,x)\) and auxiliary sets \(S^{m}_{i0}, S^{m}_{i1}\) by means of the following algorithm. For an arbitrary \(m\) put
\[ U(a_m,m)=0,\quad U(b_m,m)=1,\quad S^{m}_{00}=\{a_m\},\quad S^{m}_{01}=\{b_m\}. \]
Suppose that, for some \(s\), finite sets \(S^{m}_{s0}, S^{m}_{s1}\) have already been constructed so that
\[ i\in S^{m}_{s0}\Rightarrow U(i,m)=0 \]
and
\[ i\in S^{m}_{s1}\Rightarrow U(i,m)=1, \]
and moreover, if \(i\in S^{m}_{s0}, j\in S^{m}_{s1}\), then \(\langle i,j\rangle\in\neg\theta\). Let \(x\) be the least natural number not contained in
\[ S^{m}_{s0}\cup S^{m}_{s1}. \]
Take some pair \(\langle a,b\rangle\), where \(a\in S^{m}_{s0}, b\in S^{m}_{s1}\). Since \(aa\ne ab\), we have \(aa\ne ax\) or \(ab\ne ax\), and consequently, by running through the pairs in the sequence \(\neg\theta\), we shall find in this sequence either a pair of the form \(\langle a,x\rangle\), or a pair of the form \(\langle b,x\rangle\). For each pair \(\langle a,b\rangle\), \(a\in S^{m}_{s0}, b\in S^{m}_{s1}\), select a pair \(\langle u_{a,b},x\rangle\in\neg\theta\), where \(u_{a,b}=a,b\). Among the selected pairs there necessarily occur either all pairs of the form \(\langle a,x\rangle\), \(a\in S^{m}_{s0}\), or all pairs of the form \(\langle b,x\rangle\), \(b\in S^{m}_{s1}\). In the first case put
\[ U(x,m)=1,\quad S^{m}_{s+1\,1}=S^{m}_{s1}\cup\{x\},\quad S^{m}_{s+1\,0}=S^{m}_{s0}, \]
and in the second
\[ U(x,m)=0,\quad S^{m}_{s+1\,0}=S^{m}_{s0}\cup\{x\},\quad S^{m}_{s+1\,1}=S^{m}_{s1}. \]
The induction assumptions are thereby preserved and the function \(U(x,m)\) is constructed. It is clear that
\[ aa=ab \Longleftrightarrow \forall x\,(U(a,x)=U(b,x)), \]
and therefore the computable numbering \(n\mapsto U(n,x)\) is isomorphic to the numbering \(\alpha\).

Among the various numberings of a family, the precomplete ones (3) are usually of interest. However, for negative numberings the following holds.

Remark. No negative numbering \(\alpha\) of an aggregate \(\mathfrak A\) consisting of more than one object can be precomplete.

Indeed, let \(a,b\) be distinct objects of \(\mathfrak A\). The aggregate of all \(\alpha\)-numbers of all objects distinct from \(a\) is recursively enumerable. Similarly, the aggregate of all \(\alpha\)-numbers of all objects distinct from \(b\) is recursively enumerable. In other words, the families \(\mathfrak A-\!a\) and \(\mathfrak A-\!b\) are \(\alpha\)-completely enumerable. They are distinct from \(\mathfrak A\) and their union coincides with \(\mathfrak A\); but this, according to (1), cannot occur for precomplete numberings.

Received
19 X 1964

CITED LITERATURE

1. A. I. Mal’tsev, Algebra and Logic, 3, No. 4, 3 (1964).
2. R. M. Friedberg, J. Symbolic Logic, 23, 309 (1958).
3. A. I. Mal’tsev, Algorithms and Recursive Functions, “Nauka”, 1965.