UDC 517.11

MATHEMATICS

Yu. Ya. BREITBART

ON ESSENTIAL VARIABLES OF FUNCTIONS OF THE ALGEBRA OF LOGIC

(Presented by Academician P. S. Novikov, 9 III 1966)

1°. In the present note we investigate the question of essential variables of functions of the algebra of logic (f.a.l.). In what follows, by f.a.l. we shall mean functions that depend essentially on all their variables (see (¹)).

To make the formulation of the problem more precise, we introduce the following definitions.

Definition 1. A variable \(x_i\) \((1 \le i \le n)\) of a function \(f(x_1,\ldots,\ldots,x_n)\) \((n>1)\) is called a strongly essential variable (s.e.v.) if there exists a constant \(c\) such that, upon substituting this constant in place of \(x_i\), one obtains a function depending essentially on \((n-1)\) variables.

Definition 2. A pair of variables \((x_i,x_j)\) \((1 \le i < j \le n)\) of \(f(x_1,\ldots,x_n)\) \((n>1)\) is called separable if there exists a set of \((n-2)\) constants such that, upon substituting these constants in place of all variables of \(f(x_1,\ldots,x_n)\) except \(x_i\) and \(x_j\), one obtains a function depending essentially on \(x_i\) and \(x_j\).

Definition 3. A variable \(x_i\) \((1 \le i \le n)\) of a function \(f(x_1,\ldots,\ldots,x_n)\) \((n>1)\) is called a variable of order \(k\) if there exist variables \(x_{i_1},\ldots,x_{i_k}\) \((0 \le i_1 < \cdots < i_k \le n,\ i_t \ne i\ (t=1,2,\ldots,k))\) such that each pair \((x_i,x_{i_t})\) \((t=1,2,\ldots,k)\) is separable and, whatever \(j \ne i_t\) \((t=1,2,\ldots,k)\), \(j \ne i\), the pair \((x_i,x_j)\) is not separable.

In (²–⁴) it was proved that every f.a.l. has at least one s.e.v. In (³, ⁴) it is proved that every variable of an f.a.l. is of at least first order.

We are interested in the problem of the number of s.e.v.’s of an arbitrary f.a.l. and of a lower estimate for the orders of the variables of an f.a.l.

2°. Let \(L(f)\) be the number of s.e.v.’s of the function \(f(x_1,\ldots,x_n)\), and

\[ L(n)=\min_{f(x_1,\ldots,x_n)} L(f). \]

The function \(x_1\cdot x_2\cdot \ldots \cdot x_{n-2}\cdot x_{n-1}\vee \bar{x}_1\cdot \bar{x}_2\cdot \ldots \cdot \bar{x}_{n-2}\cdot x_n\), obviously, has exactly two s.e.v.’s: \(x_{n-1}\) and \(x_n\). Thus, \(L(n)\le 2\).

Let \(f(x_1,\ldots,x_n)\) be an arbitrary f.a.l. Obviously, in order that \(f(x_1,\ldots,x_n)\) depend essentially on all its variables, it is necessary and sufficient that the representation of \(f(x_1,\ldots,x_n)\) in the form of a Zhegalkin polynomial contain an occurrence of each of the variables \(x_i\) \((1 \le i \le n)\) (see (³)).

Next we note that the number of s.e.v.’s of \(f(x_1,\ldots,x_n)\) does not change when some (possibly all) variables are replaced by their negations.

Lemma 1. If \(f(x_1,\ldots,x_n)\) is an f.a.l. whose Zhegalkin polynomial contains a conjunction of rank \(n\) (1), then every variable of \(f(x_1,\ldots,x_n)\) is an s.e.v.

Lemma 2. If there exists an \(f(x_1,\ldots,x_n)\) having exactly one s.e.v., then there will be found an \(f^*(x_1,\ldots,x_n)\) having exactly one s.e.v. and whose Zhegalkin polynomial contains a conjunction of rank \(n\).

From Lemmas 1 and 2 we obtain that \(L(n)\ge 2\). Thus, it is proved:

Theorem 1. \(L(n)=2\).

\(2^\circ\). Let \(B_k\) be the set of all sequences of length \(k\) of 0’s and 1’s. If \(\widetilde{\sigma}=(\sigma_1,\ldots,\sigma_k)\), then denote

\[ |\widetilde{\sigma}|=\sum_{i=0}^{k}\sigma_i\cdot 2^i . \]

Lemma 3. In order that the variable \(x_i\) \((1\le i\le n)\) of the function \(f(x_1,\ldots,x_n)\) be a variable of order \(k\), it is necessary and sufficient that \(f(x_1,\ldots,x_n)\) can be represented in the form

\[ \begin{aligned} f(x_1,\ldots,x_n)=&\ x_i\cdot \bigvee_{\widetilde{\sigma}=(\sigma_1,\ldots,\sigma_k)\subset A} x_{i_1}^{\sigma_1}\cdots x_{i_k}^{\sigma_k} \ \vee \\ &\ \vee\ \bar{x}_i\cdot \bigvee_{\widetilde{\delta}=(\delta_1,\ldots,\delta_k)\subset B} x_{i_1}^{\delta_1}\cdots x_{i_k}^{\delta_k} \ \vee \\ &\ \vee_{\widetilde{\gamma}=(\gamma_1,\ldots,\gamma_k)\subset C} \bigvee x_{i_1}^{\gamma_1}\cdots x_{i_k}^{\gamma_k}\cdot f_{|\widetilde{\gamma}|}(x_1,\ldots,x_{i_1-1},x_{i_1+1},\ldots,x_{i-1},x_{i+1},\ldots,x_{i_k-1}, \\ &\hspace{9.5cm} x_{i_k+1},\ldots,x_n), \end{aligned} \tag{1} \]

where \(A\cup B\cup C=B_k\), pairwise disjoint; \(A\cup B\) is nonempty, and \(f_{|\widetilde{\gamma}|}\) are certain Boolean-algebra functions essentially depending on no more than \((n-k-1)\) variables.

Let us outline the proof. If \(f(x_1,\ldots,x_n)\) is representable in the form (1), then every pair \((x_i,x_{i_t})\) \((t=1,2,\ldots,k)\) is distinguished, and there are no other distinguished pairs of the form \((x_i,x_j)\) \((j\ne i,\ j=i_t)\) \((t=1,2,\ldots,k)\).

Now suppose that \(f(x_1,\ldots,x_n)\) has a variable of \(k\)-th order. Without loss of generality this variable may be assumed to be \(x_1\) (renaming the variables if necessary). Let \(x_2,\ldots,x_{k+1}\) be such variables of \(f(x_1,\ldots,x_n)\) that each pair \((x_1,x_i)\) \((i=2,\ldots,k+1)\) is distinguished and there are no other distinguished pairs of the form \((x_1,x_j)\) \((j>k+1)\); this can also be achieved by renaming the variables. There exist \(\delta_2,\ldots,\delta_n\) such that

\[ f(0,\delta_2,\ldots,\delta_n)\ne f(1,\delta_2,\ldots,\delta_n). \]

Then, whatever \(j>k+1\) may be,

\[ f(0,\delta_2,\ldots,\delta_{k+1},\ldots,\delta_j,\ldots,\delta_n) = f(0,\delta_2,\ldots,\delta_{k+1},\ldots,\bar{\delta}_j,\ldots,\delta_n). \]

Consequently,

\[ \begin{aligned} f(0,\delta_2,\ldots,\delta_{k+1},x_{k+2},\ldots,x_n)&=a,\\ f(1,\delta_2,\ldots,\delta_{k+1},x_{k+2},\ldots,x_n)&=\bar a. \end{aligned} \]

From this the assertion of the lemma is easily obtained.

Corollary 1. If a Boolean-algebra function \(f(x_1,\ldots,x_n)\) has a variable of order \(k\), then there is at least one variable of order

\[ p\ge \frac{(n-k-1)}{(2^k-1)}. \]

Corollary 2. Every Boolean-algebra function of \(n\) variables \((n>3)\) has no more than one variable of order 1.

Theorem 2. Whatever the Boolean-algebra function of \(n\) variables \((n>2)\) may be, it has at least one variable of order

\[ k\ge \left[\left(1-\frac{1}{\log_2\log_2 n}\right)\log_2 n\right]. \]

Proof. Let \(k\) be the greatest of the orders of the variables of \(f(x_1,\ldots,x_n)\). Then, by Corollary 1, we have

\[ \frac{n-k-1}{2^k-1}<k. \]

Hence we obtain that

\[ k\ge \left[\left(1-\frac{1}{\log_2\log_2 n}\right)\log_2 n\right]. \]

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
8 III 1966

CITED LITERATURE

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3. N. A. Solov’ev, in: Problems of Cybernetics, vol. 9, 333 (1963).
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