UDC 517.11

MATHEMATICS

M. I. KANOVICH

THE COMPLEXITY OF DECIDING AN ENUMERABLE SET AS A CRITERION OF ITS UNIVERSALITY

(Presented by Academician A. A. Dorodnitsyn on 13 March 1970)

1. We shall use the terminology and concepts introduced in \((^{1-3})\). In particular, the length of the representation of a \(\Phi\)-algorithm \(\mathfrak A\) will be called its complexity and denoted by the symbol \(\mathfrak A \mathfrak S\).

Let \(\mathfrak M\) be an enumerable set, and \(n\) a natural number. A \(\Phi\)-algorithm \(\mathfrak B\) will be called \((\mathfrak M,n)\)-deciding if the algorithm \(\mathfrak B\) is applicable to all natural numbers not exceeding \(n\), and annuls precisely those of them which belong to the set \(\mathfrak M\). Let \(f\) be a general recursive function. An \((\mathfrak M,n)\)-deciding algorithm \(\mathfrak B\) will be called \((\mathfrak M,n,f)\)-deciding if, on any number \(x\) not exceeding \(n\), the algorithm \(\mathfrak B\) finishes its work in no more than \(f(x)\) steps.

A Boolean vector \(R\) is called a segment of the enumerable set \(\mathfrak M\) if
\[ \forall x\bigl(x < [B^\circ \supset (\sigma_{x+1}(R) \doteq | \equiv x \in \mathfrak M)\bigr). \]
A nonempty segment \(R\) of the set \(\mathfrak M\) will be called a \(([R^\circ - 1])\)-segment of the set \(\mathfrak M\).

Let \(\mathfrak M\) be an enumerable set, and \(n\) a natural number. A \(\Phi\)-algorithm \(\mathfrak C\) will be called an \((\mathfrak M,n)\)-characteristic if the algorithm \(\mathfrak C\) is applicable to the empty word \(\Lambda\), and the word \(\mathfrak C(\Lambda)\) is an \(n\)-segment of the set \(\mathfrak M\) (cf. \((^4)\)). Let \(t\) be a natural number. An \((\mathfrak M,n)\)-characteristic \(\mathfrak C\) will be called an \((\mathfrak M,n,t)\)-characteristic if \(\mathfrak C\) finishes its work on the empty word in no more than \(t\) steps.

2. An enumerable set \(\mathfrak M\) is called effectively nonrecursive if there exists an unbounded general recursive function \(f\) such that, whatever the natural number \(n\), every \((\mathfrak M,n)\)-deciding algorithm has complexity not less than \(f(n)\).

An enumerable set \(\mathfrak M\) will be called strictly nonrecursive if, for every partial recursive function \(\varphi\), one can indicate an unbounded general recursive function \(g\) such that, whatever the natural number \(n\), if \(\varphi\) is a strictly increasing general recursive function, then every \((\mathfrak M,n,\varphi)\)-deciding algorithm has complexity not less than \(g(n)\).

An enumerable set \(\mathfrak M\) will be called effectively complex if there exists an unbounded general recursive function \(f\) such that, whatever the number \(n\), every \((\mathfrak M,n)\)-characteristic has complexity not less than \(f(n)\).

An enumerable set \(\mathfrak M\) will be called strictly complex if, for every partial recursive function \(\varphi\), one can indicate an unbounded general recursive function \(g\) such that, whatever the number \(n\), if \(\varphi\) is a strictly increasing general recursive function, then the complexity of every \((\mathfrak M,n,\varphi(n))\)-characteristic is not less than \(g(n)\).

We shall say that an enumerable set \(\mathfrak N\) is reducible to an enumerable set \(\mathfrak M\) by weak tables if there exist a \(\Phi\)-algorithm \(\mathfrak A\) and a general recursive function \(f\) such that, whatever the natural number \(n\), the algorithm \(\mathfrak A\) is applicable to every \(f(n)\)-segment \(R\) of the set \(\mathfrak M\), and \(\mathfrak A(R) \doteq \Lambda \equiv n \in \mathfrak N\) (cf. \((^5)\)).

We shall call an enumerable set \(\mathfrak M\) universal with respect to weak tabular reducibility if every enumerable set is reducible to the set \(\mathfrak M\) by weak tables.

Theorem 1. Let \(\mathfrak M\) be an enumerable set. Then the following five propositions are pairwise equivalent: 1) the set \(\mathfrak M\) is universal with respect to weak tabular reducibility; 2) the set \(\mathfrak M\) is effectively nonrecursive; 3) the set \(\mathfrak M\) is strictly nonrecursive; 4) the set \(\mathfrak M\) is effectively complex; 5) the set \(\mathfrak M\) is strictly complex.

3. Let us consider questions connected with estimates of the complexity of bounded decision for enumerable sets that are universal in the sense of Turing.

3.1. First we shall give a number of definitions. By natural numbers we mean words of the form \(0P\), where \(P\) is a word in the alphabet \(|\). By the symbol \(\sigma_i(R)\), where \(i\) is a positive number not exceeding the length of \(R\), we shall denote the \(i\)-th letter of the word \(R\). For convenience we shall assume that \(\sigma_0(R) \rightleftarrows 0\).

Let \(\mathfrak F\) be a \(\Phi\)-algorithm, \(R\) a Boolean vector. We shall call a natural number \(y\) an \((\mathfrak F,R)\)-consequence of the natural number \(x\), if there exists a sequence of natural numbers
\(n_0, n_1, \ldots, n_k, n_{k+1}, y_0, y_1, \ldots, y_k, y_{k+1}\) such that:
1) \(n_0=n_{k+1}=0,\ 1\leq n_i\leq |R|\ (i=1,2,\ldots,k)\); 2) \(y_0=x,\ y_{k+1}=y\);
3) \(\forall j(0\leq j\leq k \to y_{j+1}n_{j+1}=\mathfrak F(y_j n_j\sigma_{n_j}(R)))\).

Let \(\mathfrak M\) be an enumerable set, \(\mathfrak F\) a \(\Phi\)-algorithm. We shall call a natural number \(y\) an \(\mathfrak M\)-value of the algorithm \(\mathfrak F\) on the natural number \(x\) (notation \(y=\mathfrak M\mathfrak F(x)\)), if it is false that no such segment \(R\) of the set \(\mathfrak M\) is possible for which \(y\) is an \((\mathfrak F,R)\)-consequence of the number \(x\) (cf. \((^6)\)).

Introduce the following notation:

\[ y \geqslant_{\mathfrak M\mathfrak F}(x) \rightleftarrows \neg\neg\exists z\bigl(y\geq z\ \&\ z=\mathfrak M\mathfrak F(x)\bigr), \]

\[ y \leqslant_{\mathfrak M\mathfrak F}(x) \rightleftarrows \neg\neg\exists z\bigl(y\leq z\ \&\ z=\mathfrak M\mathfrak F(x)\bigr), \]

\[ !_{\mathfrak M\mathfrak F}(x) \rightleftarrows \neg\neg\exists y\bigl(y=\mathfrak M\mathfrak F(x)\bigr). \]

We shall call a \(\Phi\)-algorithm \(\mathfrak F\) an \(\mathfrak M\)-recursive function if \(\forall x!_{\mathfrak M\mathfrak F}(x)\) (cf. \((^6)\)). An \(\mathfrak M\)-recursive function \(\mathfrak F\) will be called unbounded if \(\forall y\neg\neg\exists x(y\leqslant_{\mathfrak M\mathfrak F}(x))\). An \(\mathfrak M\)-recursive function \(\mathfrak F\) will be called nondecreasing if

\[ \forall x\neg\neg\exists y\bigl(y\geqslant_{\mathfrak M\mathfrak F}(x)\ \&\ y\leqslant_{\mathfrak M\mathfrak F}(x+1)\bigr). \]

3.2. An enumerable set \(\mathfrak M\) is called weakly effectively nonrecursive if there exists an unbounded \(\mathfrak M\)-recursive function \(\mathfrak F\) such that, whatever natural number \(n\) is taken, for every \((\mathfrak M,n)\)-deciding algorithm \(\mathfrak B\) the condition \(\mathfrak B \supset \geqslant_{\mathfrak M\mathfrak F}(n)\) is satisfied.

We shall call an enumerable set \(\mathfrak M\) weakly effectively complex if there exists an unbounded nondecreasing \(\mathfrak M\)-recursive function \(\mathfrak F\) such that, whatever natural number \(n\) is taken, for every \((\mathfrak M,n)\)-characteristic \(\mathfrak C\) the condition \(\mathfrak C \supset \geqslant_{\mathfrak M\mathfrak F}(n)\) is satisfied.

We assume that we have a numbering of the \(\Phi\)-algorithms
\(\mathfrak A_0, \mathfrak A_1, \ldots, \mathfrak A_k, \ldots\). The algorithm with number \(k\) will be denoted by \(\langle k\rangle\).

We shall call an enumerable set \(\mathfrak M\) \(\tau\)-nonrecursive if there exists an \(\mathfrak M\)-recursive function \(\mathfrak F\) such that for any natural \(m\) and \(k\) such that \(k=\mathfrak M\mathfrak F(m)\), the following conditions are satisfied: 1) the algorithm \(\langle k\rangle\) is an unbounded \(\mathfrak M\)-recursive function; 2) if \(m\) is a Gödel number of a strictly increasing general-recursive function \(f\), then, whatever \(n\) may be, for every \((\mathfrak M,n,f)\)-deciding algorithm \(\mathfrak B\) the condition \(\mathfrak B \supset \geqslant_{\mathfrak M\langle k\rangle}(n)\) is satisfied.

We shall call an enumerable set \(\mathfrak M\) \(\tau\)-complex if there exists an \(\mathfrak M\)-recursive function \(\mathfrak F\) such that for any \(m\) and \(k\) such that \(k=\mathfrak M\mathfrak F(m)\), the following conditions are satisfied: 1) the algorithm \(\langle k\rangle\) is an unbounded nondecreasing \(\mathfrak M\)-recursive function; 2) if \(m\) is a Gödel number of a strictly increasing general-recursive function \(f\), then, whatever...

whatever $n$ may be, for every $(\mathfrak M, n, f(n))$-characteristic $\mathfrak C$ the condition
$\mathfrak C \supset \geqslant_{\mathfrak M\langle k\rangle}(n)$ is satisfied.

An enumerable set $\mathfrak M$ is called Turing universal if, for every enumerable set $\mathfrak N$, one can specify an $\mathfrak M$-recursive function $\mathfrak F$ such that
$\forall n(n \in \mathfrak N \equiv 0| = \mathfrak M\mathfrak F(n))$ (see (5)).

Theorem 2. An enumerable set is Turing universal if and only if it is $\tau$-nonrecursive.

Theorem 3. If an enumerable set is $\tau$-complex, then it is false that it is not Turing universal.

Theorem 4. An enumerable set is Turing universal if and only if it is weakly effectively nonrecursive.

Corollary 1. Every Turing-universal set is weakly effectively complex.

1. By the symbol $\overline{\mathfrak M}$ we shall denote the complement of the set $\mathfrak M$. An enumerable set $\mathfrak M$ will be called $\tau$-creative if there is a $\Phi$-algorithm $\mathfrak F$ such that, for every $n$, if $n$ is the Gödel number of an enumerable subset $\mathfrak N$ of the set $\overline{\mathfrak M}$, then $!\mathfrak M\mathfrak F(n)$, and for every number $k$ such that $k = \mathfrak M\mathfrak F(n)$ it is false that the intersection of the set $\{0,1,\ldots,k\}$ and the set $\overline{\mathfrak M}\setminus \mathfrak N$ is empty.

Corollary 2. An enumerable set is Turing universal if and only if it is $\tau$-creative (cf. (7) and Theorem 1 from (8)).

1. Let $f$ and $g$ be general recursive functions. By the symbol $f \triangleleft g$ we shall denote the following assertion: “the function $f$ is nondecreasing and $\forall n(f(n)\leqslant g(n))$.” By the symbol $\chi(f)$ we shall denote the formula
   $\exists m\forall n(f(n)=0\vee f(n)=m)$. By the symbol $\mathfrak M_g$ we shall denote the set defined by the condition
   $n \in \mathfrak M_g \Leftrightarrow \exists x(g(x)=n)$.

The function $g$ is called a re-enumeration of the enumerable set $\mathfrak M$ if
$\forall n(n\in\mathfrak M \equiv n\in\mathfrak M_g)$. Let $g$ be a general recursive function, and let $\mathfrak M$ be an enumerable set. We shall call the function $g$ $\mathfrak M$-regular if there exists an $\mathfrak M$-recursive function $\mathfrak F$ such that
$\forall nm(g(n)=g(m)\supset m\leqslant \mathfrak M\mathfrak F(n))$.

By the symbol $\eta(g)$ we shall denote the assertion: “there is a $\Phi$-algorithm $\mathfrak C$ such that, for every number $m$ which is the Gödel number of a general recursive function $f$ such that $f\triangleleft g$ and $\chi(f)$, the condition
$!_{\mathfrak M_g}\mathfrak C(m)\ \&\ \forall n(f(n)\leqslant \mathfrak M_g\mathfrak C(m))$ is satisfied.”

Theorem 5. For every re-enumeration $g$ of a Turing-universal set $\mathfrak M$ one can specify an $\mathfrak M$-recursive function $\mathfrak F$ such that, for every number $m$ which is the Gödel number of a general recursive function $f$ such that $f\triangleleft g$, the condition $\forall n(f(n)\leqslant \mathfrak M\mathfrak F(m))$ is satisfied.

Theorem 6. If a general recursive function $g$ is a re-enumeration of a $\tau$-complex set $\mathfrak M$, then $\eta(g)$ holds.

Theorem 7. Let $g$ be a re-enumeration of an enumerable set $\mathfrak M$. Let $g$ be $\mathfrak M$-regular and let $\eta(g)$ hold. Then the set $\mathfrak M$ is weakly effectively complex.

1. An enumerable set $\mathfrak M$ will be called strongly nonrecursive if, for every partial recursive function $\varphi$, one can specify such a nondecreasing general recursive function $g$ that, if $\varphi$ is a general recursive function, then: 1) the function $g$ is unbounded; 2) whatever the natural number $n$ may be, the complexity of every $(\mathfrak M,n,\varphi)$-deciding algorithm is not less than $g(n)$.

Let $f$ be a general recursive function. We shall say that the enumerable set $\mathfrak M$ $f$-reduces to the enumerable set $\mathfrak N$ if there exists a $\Phi$-algorithm $\mathfrak A$ such that, whatever the natural number $m$ may be, the following conditions are satisfied: 1) if $f(m)\in\mathfrak N$, then $!\mathfrak A(m0|)$ and $\mathfrak A(m0|)\doteq \Lambda \equiv m\in\mathfrak M$; 2) if $f(m)\notin\mathfrak N$, then $!\mathfrak A(m0|)$ and $\mathfrak A(m0|)\doteq \Lambda \equiv m\in\mathfrak M$.

By the letter \(\mathfrak L\) denote the general recursive function defined by the equality \(\mathfrak L(n) = [\log_2(n+1)]\).

Theorem 8. For every enumerable set \(\mathfrak N\) one can specify an enumerable set \(\mathfrak M\) such that: 1) the set \(\mathfrak M\) is \(\mathfrak L\)-reducible to the set \(\mathfrak N\); 2) if the set \(\mathfrak N\) is nonrecursive, then the set \(\mathfrak M\) is strongly nonrecursive.

Theorem 8 points to the impossibility of any substantial strengthening of Theorem 1. Theorem 8 also makes it possible to detect a discrepancy between table reducibility (see the definition in \({}^{9}\)) and weak table reducibility, using a method different from that of the paper \({}^{10}\).

Theorem 9. Let a strongly nonrecursive set \(\mathfrak M\) be table reducible to a set \(\mathfrak N\). Then the set \(\mathfrak N\) is strongly nonrecursive.

By the letter \(E\) denote the general recursive function defined by the equality \(E(n)=n\).

Theorem 10. Whatever the hyperimmune set \(\mathfrak M\), the unbounded general recursive function \(f\), and the natural number \(m\), one can specify a natural number \(n\), exceeding \(m\), such that it is not true that an \((\mathfrak M,n,E)\)-deciding algorithm of complexity not exceeding \(f(n)\) is impossible.

Theorems 8–10 give the following corollary (cf. § 1.4 of the paper \({}^{10}\)):

Corollary 3. For every hyperimmune set \(\mathfrak N\) one can specify an enumerable set \(\mathfrak M\) such that: 1) the set \(\mathfrak M\) is \(\mathfrak L\)-reducible to the set \(\mathfrak N\); 2) it is not true that \(\mathfrak M\) is table reducible to the set \(\mathfrak N\).

1. In order to show the impossibility of any substantial strengthening of Theorem 2, we note that an enumerable set \(\mathfrak M\) is nonrecursive if and only if for any partial recursive function \(\varphi\) one can specify a nondecreasing \(\mathfrak M\)-recursive function \(\mathfrak F\) such that, if \(\varphi\) is a general recursive function, then: 1) \(\mathfrak F\) is an unbounded \(\mathfrak M\)-recursive function; 2) for every natural number \(n\) and every \((\mathfrak M,n,\varphi)\)-deciding algorithm \(\mathfrak B\), the condition \(\mathfrak B \supseteq_{\mathfrak M} \mathfrak F(n)\) holds.

In connection with Theorems 1 and 3, let us also note that, for any enumerable set \(\mathfrak M\) and any general recursive function \(\varphi\), one can specify an unbounded general recursive function \(g\) such that, whatever \(n\) may be, every \((\mathfrak M,n,\varphi(n))\)-characteristic has complexity not less than \(g(n)\).

The author expresses profound gratitude to A. A. Markov for his attention and advice in writing this paper.

Moscow State University
named after M. V. Lomonosov

Received
4 III 1970

CITED LITERATURE

\({}^{1}\) A. A. Markov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 42 (1954).
\({}^{2}\) A. A. Markov, Izv. AN SSSR, ser. matem., 31, 161 (1967).
\({}^{3}\) S. K. Kleene, Introduction to Metamathematics, 1954.
\({}^{4}\) A. N. Kolmogorov, “Problems of Information Transmission,” 1, no. 1, 3 (1965).
\({}^{5}\) R. M. Friedberg, H. Rogers, Jr., Zs. Math. Logik u. Grundlagen der Math., 5, 117 (1959).
\({}^{6}\) M. Davis, Computability and Unsolvability, N. Y., 1958.
\({}^{7}\) A. H. Lachlan, Proc. Am. Math. Soc., 19, no. 1, 99 (1968).
\({}^{8}\) M. I. Kanovich, DAN, 186, no. 5, 1008 (1969).
\({}^{9}\) A. I. Mal’tsev, Algorithms and Recursive Functions, M., 1965.
\({}^{10}\) A. H. Lachlan, Zs. Math. Logik u. Grundlagen der Math., 11, 17 (1965).
\({}^{11}\) N. A. Shanin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 52, 226 (1958).