Yu. I. ZHURAVLEV

ESTIMATING THE COMPLEXITY OF LOCAL ALGORITHMS FOR SOME EXTREMAL PROBLEMS ON FINITE SETS

(Presented by Academician S. L. Sobolev, 28 IV 1964)

There exists a class of problems (the problem of constructing the shortest path between vertices of a graph, the problem of constructing minimal d.n.f.’s for functions of the algebra of logic, etc.) whose solution requires selecting, from a finite set of objects, those objects that possess certain extremal properties. Thus, in constructing minimal d.n.f.’s realizing a function \(f\), it is necessary to select, in the reduced d.n.f. \(\mathfrak{M}_f\), the conjunctions that do not occur in any minimal d.n.f.; in constructing a shortest path between two given vertices of a graph, it is necessary to select the edges of the graph that occur in at least one shortest path; and so on.

Usually the computation of such extremal properties requires algorithms of high laboriousness. However, for problems of this type the very concept of “laboriousness” of an algorithm has not been precisely formulated, and therefore there are no sufficiently good estimates of this quantity.

In note \((^1)\) a definition was given and certain properties were formulated for algorithms that compute information about elements of finite sets. At each step such an algorithm, using previously accumulated information about the neighborhood \(S(\mathfrak{A}, \mathfrak{M})\) of an element \(\mathfrak{A}\) from a set \(\mathfrak{M}\), attempts to compute the value of one property from a previously fixed set of predicates \(\{P(\mathfrak{A}, \mathfrak{M})\}\). The algorithms defined in \((^1)\) will be called local. It is natural to characterize local algorithms by two parameters: the cardinality of the set \(\{P(\mathfrak{A}, \mathfrak{M})\}\) and the size of the neighborhood \(S(\mathfrak{A}, \mathfrak{M})\). In the present note a definition of these parameters is given, and it is shown that algorithms computing certain extremal properties of conjunctions from reduced d.n.f.’s and of edges of a graph have very large values of the parameters.

I. Index of an algorithm \(A\). Let a family \(\{\mathfrak{M}\}\) of finite sets be given. To each pair \((\mathfrak{A}, \mathfrak{M})\) we assign a sequence
\[ S_1(\mathfrak{A}, \mathfrak{M}), \ldots, S_r(\mathfrak{A}, \mathfrak{M}), \ldots \]
satisfying the following conditions: \(1^\circ\). \(S_i(\mathfrak{A}, \mathfrak{M})\) is a neighborhood \((^1)\) of \(\mathfrak{A}\) in \(\mathfrak{M}\). \(2^\circ\).
\[ S_1(\mathfrak{A}, \mathfrak{M}) \subseteq S_2(\mathfrak{A}, \mathfrak{M}) \subseteq \cdots \subseteq S_r(\mathfrak{A}, \mathfrak{M}) \subseteq \cdots . \]
\(3^\circ\). For all \(i\) there exists a pair \((\mathfrak{A}_i, \mathfrak{M}_i)\), \(\mathfrak{A}_i \in \mathfrak{M}_i\), \(\mathfrak{M}_i \in \{\mathfrak{M}\}\), such that
\[ S_i(\mathfrak{A}_i, \mathfrak{M}_i) \subset S_{i+1}(\mathfrak{A}_i, \mathfrak{M}_i). \]
An algorithm \(A\) belonging to the class \(K(P_1, \ldots, P_k, P_{i_1}, \ldots, P_{i_l}, \varphi_1, \ldots, \varphi_k)\) will be called an algorithm of index \(r\) if the domain of definition of the functions \(\varphi_i\), \(i = 1,2,\ldots,k\), is
\[ \langle \mathfrak{A}^{\alpha_1\ldots\alpha_k}, S_r(\mathfrak{A}^{\alpha_1\ldots\alpha_k}, \mathfrak{M}^*) \rangle,\quad M(\mathfrak{M}^*) \in \{\mathfrak{M}\}, \]
and all marks in \(S_r(\mathfrak{A}^{\alpha_1\ldots\alpha_k}, \mathfrak{M}^*)\) are admissible \((^1)\).

II. Let \(\mathfrak{A}\) be sets and let \(\mathfrak{M} \in \{\mathfrak{M}\}\) be a collection of sets. We introduce the sequence
\[ S_1(\mathfrak{A}, \mathfrak{M}), \ldots, S_r(\mathfrak{A}, \mathfrak{M}), \ldots \]
as follows: \(S_1(\mathfrak{A}, \mathfrak{M})\) is made up of all elements \(\mathfrak{B}\) satisfying one of the conditions: \(1^\circ\). \(\mathfrak{A}\cap \mathfrak{B}\) is nonempty. \(2^\circ\). Let \(\mathfrak{A}_1,\ldots,\mathfrak{A}_q\) satisfy \(1^\circ\). Then
\[ \mathfrak{B} \subseteq \bigcup_{i=1}^{q} \mathfrak{A}_i. \]

Suppose the set \(S_{r-1}(\mathfrak{A}, \mathfrak{M})\) has been defined. We form \(S_r(\mathfrak{A}, \mathfrak{M})\) from all elements \(\mathfrak{B}\) satisfying one of the conditions: \(1^\circ\). There is an element \(\mathfrak{A}_i\), \(\mathfrak{A}_i \in S_{r-1}(\mathfrak{A}, \mathfrak{M})\), such that \(\mathfrak{A}_i\cap \mathfrak{B}\) is nonempty. \(2^\circ\). Let \(\mathfrak{B}_1,\ldots,\mathfrak{B}_q\) satisfy \(1^\circ\). Then
\[ \mathfrak{B} \subseteq \bigcup_{i=1}^{q} \mathfrak{B}_i. \]
The set \(S_r(\mathfrak{A}, \mathfrak{M})\) will be called the principal neighborhood of order \(l\) of the element \(\mathfrak{A}\) in the set \(\mathfrak{M}\).

We shall also consider the sequences
\[ S'_1(\mathfrak{A}, \mathfrak{M}), \ldots \]

..., $S'_r(\mathfrak A,\mathfrak M), \ldots$ such that
$S_{i-1}(\mathfrak A,\mathfrak M) \subseteq S'_i(\mathfrak A,\mathfrak M) \subseteq S_i(\mathfrak A,\mathfrak M)$ and there exists a pair $(\widetilde{\mathfrak A},\widetilde{\mathfrak M})$ such that
$S_{i-1}(\widetilde{\mathfrak A},\widetilde{\mathfrak M}) \subset S'_i(\widetilde{\mathfrak A},\widetilde{\mathfrak M})$.

We shall call an algorithm $A$ an index algorithm if the functions $\varphi_i$, $i=1,2,\ldots,k$, are defined on pairs
$\langle \mathfrak A^{\alpha_1\ldots\alpha_k}, S'_r(\mathfrak A^{\alpha_1\ldots\alpha_k},\mathfrak M^*)\rangle$,
$M(\mathfrak M^*)\in\{\mathfrak M\}$, and all labels in
$S'_r(\mathfrak A^{\alpha_1\ldots\alpha_k},\mathfrak M^*)$ are admissible.

III. Let $f$ be a function of the algebra of logic, $\mathfrak M_f$ the set of conjunctions entering into the reduced disjunctive normal form of the function $f$, and $N^f$ the set of maximal intervals of the function $f$. The definitions of II can obviously be applied to the pairs $(\mathfrak A,\mathfrak M_f)$, $\mathfrak A\in\mathfrak M_f$, and $(N_{\mathfrak A},N^f)$, $N_{\mathfrak A}\in N^f$ (2).

Let
\[ \Gamma=[\langle a_1,\ldots,a_s\rangle,\ \langle(a_{i_1},a_{i_2}),\ldots,(a_{i_m},a_{i_l})\rangle] \]
be an undirected graph, where $a_1,\ldots,a_s$ are the vertices of the graph and $(a_{i_1},a_{i_2}),\ldots,(a_{i_m},a_{i_l})$ are the edges of the graph. Applying the definitions of II, we obtain the definitions of the principal neighborhoods (2) of order $r$ of the vertices and edges of a graph and the definition of an index algorithm of order $r$ on the vertices and edges of a graph.

IV. Let the algorithm $A$ belong to the class
$K(P_1,\ldots,P_k,P_i,\ldots,P_{i_l},\ldots,\varphi_1,\ldots,\varphi_k)$.
The number $k$ will be called the memory size of the algorithm $A$.

V. We pass to the notion of computability of a predicate in the class of local algorithms. Usually, when solving concrete problems, certain restrictions are imposed on the predicates $P_1,\ldots,P_k$ and the functions $\varphi_1,\ldots,\varphi_k$. We shall assume that certain sets $\{P(\mathfrak A,\mathfrak M)\}$ and $\{\varphi\}$ are given and that all predicates $P_1,\ldots,P_k$ and functions $\varphi_1,\ldots,\varphi_k$ participating in the definition of local algorithms are chosen respectively from $\{P(\mathfrak A,\mathfrak M)\}$ and $\{\varphi\}$.

Definition. The predicate
$P_1(\mathfrak A,\mathfrak M)\in\{P(\mathfrak A,\mathfrak M)\}$
is called $(r,k)$-computable if there exists an algorithm $A$, belonging to the class
$K(P_1,\ldots,P_k,P_1,\varphi_1,\ldots,\varphi_k)$, such that:
$1^\circ$. $P_i\in\{P(\mathfrak A,\mathfrak M)\}$, $i=1,2,\ldots,k$.
$2^\circ$. $\varphi_i\in\{\varphi\}$, $i=1,2,\ldots,k$.
$3^\circ$. For all $\mathfrak M\in\{\mathfrak M\}$, in $A(\mathfrak M)$ the label vectors of all elements have first coordinate different from $\Delta$.

VI. Let $P_2$ be the set of all functions of the algebra of logic, and $P_2(n)$ the set of functions of the algebra of logic depending on $n$ variables. Denote by $\{\mathfrak M_f\}$ the set of reduced disjunctive normal forms of all functions of the algebra of logic and by $\{\mathfrak M_f^n\}$ the set of reduced disjunctive normal forms of all functions of the algebra of logic in $n$ variables. To each DNF $\mathfrak M_f$ there corresponds one-to-one the set $\mathfrak M_f$ of all conjunctions from $\mathfrak M_f$. We shall consider the sets $\{\mathfrak M_f\}$ and $\{\mathfrak M_f^n\}$.

We introduce restrictions on the set $\{P(\mathfrak A,\mathfrak M)\}$. Consider transformations $\{\pi\}$ of the variables $x_i\to x_j^\sigma$. These transformations were studied by K. Shannon (3) and G. N. Povarov (4). The transformation $\pi$ induces a transformation on the set of conjunctions
$\pi(x_{i_1}^{\sigma_1}\cdot\ldots\cdot x_{i_k}^{\sigma_k})
=\pi(x_{i_1}^{\sigma_1})\cdot\ldots\cdot\pi(x_{i_k}^{\sigma_k})$
and on the set of DNFs
$\pi(\mathfrak A_1\vee\ldots\vee\mathfrak A_l)
=\pi(\mathfrak A_1)\vee\ldots\vee\pi(\mathfrak A_l)$.
We shall call the predicate $P(\mathfrak A,\mathfrak M)$ invariant with respect to $\{\pi\}$ if for every $\pi\in\{\pi\}$ the equality
\[ P(\mathfrak A,\mathfrak M_f)=P(\pi(\mathfrak A),\pi(\mathfrak M_f)),\qquad \mathfrak A\in\mathfrak M_f,\quad \mathfrak M_f\in\{\mathfrak M_f\} \]
holds.

The set of predicates $P(\mathfrak A,\mathfrak M_f)$ invariant with respect to $\{\pi\}$ will be denoted by $P(\pi)$.

Consider transformations $\{\overline{\pi}\}$ of reduced DNFs: to a set of conjunctions $\{\mathfrak A_i\}$ forming a reduced DNF we assign a set of conjunctions $\{\mathfrak B_i\}$,
$(\{\mathfrak B_i\}=\overline{\pi}\{\mathfrak A_i\})$, so that the following conditions are satisfied:
$1^\circ$. To each conjunction $\mathfrak A_i$ from $\{\mathfrak A_i\}$ there corresponds one-to-one a conjunction of the same rank $\mathfrak B_i$ from $\{\mathfrak B_i\}$ $(\mathfrak B_i=\overline{\pi}(\mathfrak A_i))$.
$2^\circ$. If
$N_{\mathfrak A}\subseteq \bigcup_{i=1}^{l} N_{\mathfrak A_i}$,

then

\[ N_{\bar{\pi}}(\mathfrak A) \subseteq \bigcup_{i=1}^{l} N_{\bar{\pi}}(\mathfrak A_i). \]

\(3^\circ.\) If

\[ N_{\mathfrak A} \nsubseteq \bigcup_{i=1}^{l} N_{\mathfrak A_i}, \]

then

\[ N_{\bar{\pi}}(\mathfrak A) \nsubseteq \bigcup_{i=1}^{l} N_{\bar{\pi}}(\mathfrak A_i), \]

\(\mathfrak A_i \in \{\mathfrak A_i\},\ i = 1, 2, \ldots, l.\)

We shall call a predicate \(P(\mathfrak A,\mathfrak M_f)\) invariant with respect to \(\{\bar{\pi}\}\) if, for every \(\bar{\pi} \in \{\bar{\pi}\}\), the relation

\[ P(\mathfrak A,\mathfrak M_f)=P(\bar{\pi}(\mathfrak A),\bar{\pi}(\mathfrak M_f)) \]

holds. The set of predicates \(P(\mathfrak A,\mathfrak M_f)\) invariant with respect to \(\{\bar{\pi}\}\) will be denoted by \(P(\bar{\pi})\). The transformations \(\pi\) are a special case of the transformations \(\bar{\pi}\).

VII. Consider the predicates: \(P_1(\mathfrak A,\mathfrak M_f)\) \((P'_1(\mathfrak A,\mathfrak M_f))\)—“\(\mathfrak A\) enters at least one minimal (shortest) d.n.f. of the function \(f\)”; \(P_2(\mathfrak A,\mathfrak M_f)\) \((P'_2(\mathfrak A,\mathfrak M_f))\)—“\(\mathfrak A\) enters all minimal (shortest) d.n.f.’s of the function \(f\)”; \(P_3(\mathfrak A,\mathfrak M_f)\) \((P'_3(\mathfrak A,\mathfrak M_f))\)—“\(\mathfrak A\) enters minimal (shortest) d.n.f.’s of the function \(f\), but not all of them”; \(P_4(\mathfrak A,\mathfrak M_f)\) \((P'_4(\mathfrak A,\mathfrak M_f))\)—“\(\mathfrak A\) enters at least one irredundant d.n.f. composed of the greatest number of letters (conjunctions)”; \(P_5(\mathfrak A,\mathfrak M_f)\) \((P'_5(\mathfrak A,\mathfrak M_f))\)—“\(\mathfrak A\) enters all irredundant d.n.f.’s composed of the greatest number of letters (conjunctions)”; \(P_6(\mathfrak A,\mathfrak M_f)\) \((P'_6(\mathfrak A,\mathfrak M_f))\)—“\(\mathfrak A\) enters irredundant d.n.f.’s composed of the greatest number of letters (conjunctions), but not all of them.”

Lemma. The predicates \(P_i(\mathfrak A,\mathfrak M_f)\), \(P'_i(\mathfrak A,\mathfrak M_f)\), \(i=1,2,3,4,5,6\), are invariant with respect to \(\{\pi\}\) and \(\{\bar{\pi}\}\).

VIII. Consider the set \(\{\mathfrak M_f^n\}\) (see VI) and algorithms \(A\) such that: \(1^\circ.\ A \in K(P^1,\ldots,P^k,P_\alpha,\varphi_1,\ldots,\varphi_k).\) \(2^\circ.\) The index of the algorithm \(A\) is equal to \(r\). \(3^\circ.\ P_\alpha \in \{P_i(\mathfrak A,\mathfrak M_f), P'_i(\mathfrak A,\mathfrak M_f)\},\ i=1,2,3,4,5,6\) (see VII). \(P^1,\ldots,P^k\) belong to \(P(\pi)\). \(4^\circ.\ \varphi_i=\varphi_i(\mathfrak A^{\alpha_1\ldots\alpha_k}, S(\mathfrak A^{\alpha_1\ldots\alpha_k},\mathfrak M_f^*)).\)

Theorem 1. If \(r<\infty\) and \(k<\infty\), then the predicate \(P_\alpha\) is not \((r,k)\)-computable.

IX. Consider the set \(\{\mathfrak M_f^n\}\) and algorithms \(A\) such that: \(1^\circ.\ A \in K(P^1,\ldots,P^k,P_\alpha,\varphi_1,\ldots,\varphi_k).\) \(2^\circ.\) The index of \(A\) is equal to \(r\). \(3^\circ.\ P_\alpha \in \{P_i(\mathfrak A,\mathfrak M_f), P'_i(\mathfrak A,\mathfrak M_f)\},\ i=1,2,3,4,5,6\) (see VII). \(P^1,\ldots,P^k\) belong to \(P(\bar{\pi})\).

Theorem 2. For every \(n>n_0\) there exists \(\varepsilon>0\), \(\lim_{n\to\infty}\varepsilon(n)=0\), such that, if

\[ rk < \frac{2^n}{B(n)}(1-\varepsilon), \]

then the predicate \(P_\alpha\) is not \((r,k)\)-computable. (Here \(B(n)\) is a function growing more slowly than the \(k\)-fold logarithm of \(n\) for any \(k\).)

In proving Theorem 2, the following theorem, established by Yu. L. Vasil’ev \((^5,^6)\), is used essentially: in the set \(P_2(n)\) there exists a cycle of length

\[ \frac{2^n}{B(n)}(1-\varepsilon), \qquad \varepsilon \to 0 \quad \text{as } n\to\infty. \]

X. Let a bipartite graph \(\Gamma(a_i,a_j)=M_1\cup M_2\) be given, where \(M_1=\{a_1,\ldots,a_k\}\), \(M_2=\{(a_{i_1},a_{i_2}),\ldots,(a_l,a_s)\}\), the number of elements in \(M_2\) is not greater than \(n\); \(a_i,a_j\) are the poles of the graph. We shall denote by \(\Gamma_2(n)\) the set of graphs with two poles and with a number of edges not exceeding \(n\).

Graphs \(\Gamma(a_i,a_j)=M_1\cup M_2\) and \(\Gamma'(b_p,b_q)=M'_1\cup M'_2\) will be called isomorphic if there exists a one-to-one mapping \(\varphi\) of the set \(M_1\) onto the set \(M'_1\) such that: \(1^\circ.\ \langle b_p,b_q\rangle=\langle\varphi(a_i),\varphi(a_j)\rangle.\) \(2^\circ.\) If \((a_m,a_n)\) is an edge of the graph \(\Gamma\), then \((\varphi(a_m),\varphi(a_n))\) is an edge of the graph \(\Gamma'\). \(3^\circ.\) If \((a_m,a_n)\) is not an edge of \(\Gamma\), then \((\varphi(a_m),\varphi(a_n))\) is not an edge of the graph \(\Gamma'\).

In what follows we shall consider properties of vertices and edges of a graph that take identical values on the corresponding elements of isomorphic graphs. Denote by \(P(J)\) the set of predicates \(P(a,\Gamma)\), \(P((a_m,a_n),\Gamma)\) satisfying the following condition: if \(\varphi(\Gamma)=\Gamma'\), then \(P(a,\Gamma)=P(\varphi(a),\Gamma')\), \(P((\varphi(a_m),\varphi(a_n)),\Gamma')=P((a_m,a_n),\Gamma)\). In defining local algorithms on graphs, we shall consider only predicates from the set \(P(J)\).

XI. A set of edges of the graph \(\Gamma(a,b)\) forms a path between the poles if it contains a sequence \((a_1,a_i), \ldots, (a_j,b)\). A path is called a dead end if it contains no subpaths. A path is called minimal if it consists of the minimal number of edges (among all paths of the graph). The predicates \(\widetilde P_1(a_j,\Gamma)\) \(\bigl(\widetilde P_1(R,\Gamma)\bigr)\), “the vertex \(a_j\) (edge \(R\)) belongs to at least one dead-end path of the graph \(\Gamma\)”; \(\widetilde P_2(a_j,\Gamma)\) \(\bigl(\widetilde P_2(R,\Gamma)\bigr)\), “the vertex \(a_j\) (edge \(R\)) belongs to at least one minimal path of the graph \(\Gamma\),” are, obviously, invariant under isomorphic mappings of the graph \(\Gamma\), and therefore are contained in \(P(J)\). The predicate \(\widetilde P_1\) is an analogue of the predicate “the conjunction \(\mathfrak A\) belongs to at least one dead-end d.n.f. of the function \(f\).” In \({}^{2}\) it was proved (in other terms) that the latter predicate is \((2,1)\)-computable. It turns out, however, that even computing the property \(\widetilde P_1\) for vertices and edges of a graph requires algorithms with large \(r\) and \(k\).

XII. We shall assume that, before the algorithm begins to operate, on all vertices and edges of the graph the predicate \(P_1(\mathfrak A,\Gamma)\), “\(\mathfrak A\) is a pole of the graph,” has been computed.

\[ P_1(\mathfrak A,\Gamma)(\mathfrak A_i,a_j)=1, \quad \text{if } \mathfrak A \text{ is a vertex of the graph and } \mathfrak A\in\{a_i,a_j\}. \]

\(P_1(\mathfrak A,\Gamma)=0\) on all the remaining vertices of the graph \(\Gamma(a_i,a_j)\) and all edges of the graph \(\Gamma(a_i,a_j)\).

XIII. Some of the predicates \(P\in P(J)\) have meaning simultaneously for vertices and edges of the graph, others only for vertices or only for edges. If a predicate \(P\) has meaning only for edges, then we shall assume that \(P(a,\Gamma)\equiv 0\). We extend similarly predicates that have meaning only for vertices of the graph.

XIV. Let \(\mathfrak A,\mathfrak B\) be elements of the graphs \(\Gamma\) and \(\Gamma'\) (edges or vertices). We shall call the neighborhoods \(S(\mathfrak A^{\alpha_1\ldots\alpha_k},\Gamma^*)=S\) and \(S'=S(\mathfrak B^{\alpha_1\ldots\alpha_k},\Gamma'^*)\) isomorphic if there exists a mapping \(\varphi\) that establishes a one-to-one correspondence between the vertices belonging to \(S\) and \(S'\), the edges belonging to \(S\) and \(S'\), with
\[ \varphi(\mathfrak A)=\mathfrak B \quad \text{and} \quad \varphi(\mathfrak A^{\gamma_1\ldots\gamma_k})=\mathfrak B^{\gamma_1\ldots\gamma_k}. \]

In defining local algorithms over graphs we shall consider monotone functions \(\varphi_i\) satisfying the following condition: if \(S(\mathfrak A^{\alpha_1\ldots\alpha_k},\Gamma^*)\) and \(S(\mathfrak B^{\alpha_1\ldots\alpha_k},\Gamma'^*)\) are isomorphic neighborhoods, then
\[ \varphi_i\bigl(\mathfrak A^{\alpha_1\ldots\alpha_k}, S(\mathfrak A^{\alpha_1\ldots\alpha_k},\Gamma^*)\bigr) = \varphi_i\bigl(\mathfrak B^{\alpha_1\ldots\alpha_k}, S(\mathfrak B^{\alpha_1\ldots\alpha_k},\Gamma'^*)\bigr). \]

XV. Consider the set \(\Gamma_2(n)\) and an algorithm \(A\) whose predicates satisfy the restrictions of X (are contained in \(P(J)\)) and whose functions \(\varphi_i\) satisfy the restriction of XIV.

Theorem 3. There exists a constant \(\beta\) such that, if
\[ r<\beta n(1-\varepsilon), \quad \varepsilon\to 0 \text{ as } n\to\infty,\quad k<\infty, \]
then the predicate \(\widetilde P_1(a,\Gamma)\), \(\bigl(\widetilde P_1(R,\Gamma)\bigr)\), “the vertex \(a\) (edge \(R\)) belongs to at least one dead-end path between the poles of the graph \(\Gamma\),” is not \((r,k)\)-computable.

Theorem 4. There exists a constant \(\beta'\), \(\beta'>\beta\), such that, if
\[ r<\beta' n(1-\varepsilon), \quad \varepsilon\to 0 \text{ as } n\to\infty,\quad k<\infty, \]
then the predicate \(\widetilde P_2(a,\Gamma)\) \(\bigl(\widetilde P_2(R,\Gamma)\bigr)\), “the vertex \(a\) (edge \(R\)) belongs to at least one minimal path between the poles of the graph \(\Gamma\),” is not \((r,k)\)-computable.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
16 IV 1964

CITED LITERATURE

\({}^{1}\) Yu. I. Zhuravlev, DAN, 151, No. 5, 1025 (1963).
\({}^{2}\) Yu. I. Zhuravlev, Collection Problems of Cybernetics, No. 8, 5 (1962).
\({}^{3}\) C. Shannon, Bell Syst. Techn. J., 28, No. 1, 59 (1949).
\({}^{4}\) G. N. Povarov, DAN, 100, No. 5, 909 (1955).
\({}^{5}\) Yu. L. Vasil’ev, Collection Problems of Cybernetics, No. 10, 5 (1963).
\({}^{6}\) Yu. L. Vasil’ev, Fifth All-Union Colloquium on General Algebra, Abstracts of Reports, 1963, p. 14.