UDC 517.11

MATHEMATICS

M. I. KANOVICH, N. V. PETRI

SOME THEOREMS ON THE COMPLEXITY OF NORMAL ALGORITHMS AND COMPUTATIONS

(Presented by Academician P. S. Novikov, 18 VI 1968)

The article considers two problems: a) the complexity of normal algorithms constructively generated by words of a certain alphabet; b) the complexity of deciding enumerable sets under a restriction on the running time of the deciding algorithms.

We shall use the terminology and concepts introduced in the works \((^{1,2,4})\). In particular, the length of a description of an algorithm \(\mathfrak A\) will be called the complexity of the algorithm \(\mathfrak A\) and denoted by the symbol \(\mathfrak A\mathfrak Z\).

1. Theorem 1. Let \(A\) and \(Б\) be alphabets each containing at least two letters. Let \(\mathfrak A\) be a normal algorithm over the alphabet \(Б\) such that, for any word \(Q\) in the alphabet \(Б\), from \(!\mathfrak A_{\perp}Q_{\perp}\) it follows that \(\mathfrak A_{\perp}Q_{\perp}\) is a description of a normal algorithm over the alphabet \(A\). Then one can specify a normal algorithm \(\mathfrak D\) over the alphabet \(Б\) and a natural number \(C\) such that, for any word \(Q\) in the alphabet \(Б\), the following three conditions are satisfied:

a) the algorithm \(\mathfrak D\) is applicable to the word \(Q\);

b) the word \(\mathfrak D_{\perp}Q_{\perp}\) is a description of a normal algorithm in a two-letter extension of the alphabet \(A\), of complexity not exceeding

\[ \frac{\log_{2}|Б^{\partial}|}{\log_{2}|A^{\partial}|}\cdot |Q^{\partial}|+C; \]

c) if \(!\mathfrak A_{\perp}Q_{\perp}\), then the words \(\mathfrak A_{\perp}Q_{\perp}\) and \(\mathfrak D_{\perp}Q_{\perp}\) are descriptions of normal algorithms equivalent with respect to the alphabet \(A\).

We note that in the case where one of the alphabets under consideration has one letter, analogous theorems hold.

Theorem 1 is very convenient in estimating the complexities of normal algorithms through the complexities of the algorithms participating in their construction. Thus, for example, there exists a natural number \(C\) such that, whatever the alphabet \(A\) and the normal algorithms \(\mathfrak A\) and \(\mathfrak B\) in this alphabet may be, one can construct a normal algorithm \(\mathfrak C\) in a two-letter extension of the alphabet \(A\), equivalent with respect to the alphabet \(A\) to the normal composition of the algorithms \(\mathfrak A\) and \(\mathfrak B\), whose complexity does not exceed the quantity \(2\mathfrak A\mathfrak Z+2\mathfrak B\mathfrak Z+C|A^{\partial}|\).

1. At present many results are known that are connected with estimates of the complexity of algorithms performing some work; however, the time characteristics of the computation process are usually not taken into account. We shall consider some questions connected with the computation of Boolean functions by normal algorithms under a restriction on the running time of the computing algorithms.

A Boolean function of \(n\) arguments will be called \((M,T)\)-computable if there exists a \(\Phi\)-algorithm computing it, of complexity not exceeding \(M\), and such that its running time on each \(n\)-dimensional Boolean vector is bounded by the number \(T\). If a Boolean function is not \((M,T)\)-computable, then we shall say that it is \((M,T)\)-noncomputable (here \(M\) and \(T\) are natural numbers).

Theorem 2. One can construct an algorithm \(\mathfrak A\) over the alphabet \(0|\), possessing the following properties:

a) for every natural number \(n\), if \(!\mathfrak A_{\lfloor n\rfloor}\), then \(\mathfrak A_{\lfloor n\rfloor}\) is a record of a Boolean function of \(n\) arguments;

b) for every general recursive function \(f\) and every natural number \(p\) one can specify a natural number \(m\) exceeding it such that \(!\mathfrak A_{\lfloor m\rfloor}\) and \(\mathfrak A_{\lfloor m\rfloor}\) is a record of a \((2^m/3, f(m))\)-noncomputable Boolean function.

We note that any algorithm \(\mathfrak A\) for which properties a) and b) of Theorem 2 hold is incomplete.

1. Theorem 2 makes it possible to obtain a certain estimate of the complexity of deciding enumerable sets under a restriction on the running time of the deciding algorithms.

By the symbol \(\tau(n,\mathfrak B,\mathfrak A,t)\) we shall denote the following assertion: “the algorithm \(\mathfrak B\) is applicable to all words of length \(n\) in the alphabet \(A\) and annihilates exactly those of them to which the algorithm \(\mathfrak A\) is applicable; moreover, the running time of the algorithm \(\mathfrak B\) on each word of length \(n\) in the alphabet \(A\) is bounded by the number \(t\).”

Theorem 3. Let \(A \supseteq 0|\). There exists a general recursive function \(f\) such that, for every algorithm \(\mathfrak A\) over the alphabet \(0|\) and every natural number \(n\), it is false that there is no \(\Phi\)-algorithm \(\mathfrak B\) such that \(\tau(n,\mathfrak B,\mathfrak A,f(n))\) and \(\mathfrak B \leq 2^n+64\).

The impossibility of substantially strengthening Theorem 3 follows from the following proposition.

Theorem 4. There exists an algorithm \(\mathfrak A\) over the alphabet \(0|\) such that, for every general recursive function \(f\) and every natural number \(p\), one can specify a natural number \(m\) exceeding it such that there is no \(\Phi\)-algorithm \(\mathfrak B\) for which \(\tau(m,\mathfrak B,\mathfrak A,f(m))\) and \(\mathfrak B \leq 2^m/3\).

Theorems 2 and 4 remain valid when the temporal signaling function is replaced by any other signaling function (for the definition of signaling function, see (³)).

Moscow State University
named after M. V. Lomonosov

Received
13 VI 1968

CITED LITERATURE

¹ A. A. Markov, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 42 (1954).
² A. A. Markov, Izv. AN SSSR, Ser. Mat., 31, 161 (1967).
³ B. A. Trakhtenbrot, Complexity of Algorithms and Computations. Novosibirsk, 1967.
⁴ N. A. Shanin, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 52, 226 (1958).