CYBERNETICS AND CONTROL THEORY

Yu. L. SAGALOVICH

PROOF OF THE MINIMALITY OF A CONTACT REALIZATION OF ONE CLASS OF BOOLEAN FUNCTIONS OF \(n\) VARIABLES

(Presented by Academician B. N. Petrov on 30 III 1961)

The minimality of the number of contacts in a contact circuit has been proved only for two Boolean functions of \(n\) variables. Namely, Cardo \((^1)\) showed that contact circuits realizing the functions \(x_1 \oplus x_2 \oplus \cdots \oplus x_n\) and \(\bar{x}_1 \oplus x_2 \oplus \cdots \oplus x_n\) contain \(4n - 4\) contacts, and that this number is minimal. Other proofs \((^2)\) of the minimality of a contact realization apply to circuits realizing Boolean functions of no more than 4 variables.

This note contains a proof of the minimality of a contact realization of one class of Boolean functions for arbitrary \(n\).

1. Consider the Boolean function

\[ C_n=\bar{x}_1\bar{x}_2\ldots \bar{x}_n+x_1\bar{x}_2\ldots \bar{x}_n+\cdots \]

\[ \cdots+x_1x_2\ldots x_{n-1}\bar{x}_n+x_1x_2\ldots x_n. \]

In \((^3)\) such a function is called a chain.

Lemma. In the perfect disjunctive normal form (p.d.n.f.) of the function \(C_n\), the variable \(x_i\) occurs \(i\) times with negation and \(n+1-i\) times without negation.

Theorem 1. \(C_n\) is a function that is special \((^4)\) only with respect to the variables \(x_1\) and \(x_n\).

Proof. Let

\[ C_n=x_i^{\sigma_i}f_1(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n) +f_2(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)= \]

\[ =x_i^{\sigma_i}f_1+x_i^{\sigma_i}f_2+\bar{x}_i^{\sigma_i}f_2 =x_i^{\sigma_i}(f_1+f_2)+\bar{x}_i^{\sigma_i}f_2. \]

In the p.d.n.f. of the function \(C_n\) there are only two terms adjacent with respect to \(x_i\); therefore in the p.d.n.f. of the function \(f_2\) there is only one term. Since \((^4)\) \(f_1\cdot f_2=0\), and in all there are \(n+1\) terms in the p.d.n.f. of the function \(C_n\), it follows that the p.d.n.f. of the function \(f_1+f_2\) contains exactly \(n\) terms. Consequently, in the p.d.n.f. of the function \(C_n\) there are exactly \(n\) terms containing \(x_i^{\sigma_i}\). If \(\sigma_i=1\), then in the p.d.n.f. of the function \(C_n\) there are exactly \(n\) terms containing \(x_i\) without negation. Hence, by the lemma, \(i=1\). If \(\sigma_i=0\), then in the p.d.n.f. of the function \(C_n\) there are exactly \(n\) terms containing \(x_i\) with negation. Hence, by the lemma, \(i=n\).

Theorem 2. No contact circuit realizing the function \(C_n\) can contain fewer than two contacts of the variable \(x_i\), if \(i\ne 1\) and \(i\ne n\).

Proof. If the variable \(x_i\) is represented in the circuit by only one contact, then (cf. \((^4)\)) every path between the poles either

passes, or does not pass, through \(x_i\). But then \(C_n = x_i^{\sigma_i} f_1 + f_2\), which for \(i \ne 1\) and \(i \ne n\) contradicts Theorem 1.

Theorem 3*.

\[ (x_1+\bar{x}_2)(x_2+\bar{x}_3)\cdots(x_{n-1}+\bar{x}_n)=C_n. \]

Proof. For \(n=2\) the theorem is true. Suppose

\[ (x_1+\bar{x}_2)(x_2+\bar{x}_3)\cdots(x_{n-2}+\bar{x}_{n-1})=C_{n-1}. \]

Then

\[ \begin{aligned} &(x_1+\bar{x}_2)(x_2+\bar{x}_3)\cdots(x_{n-1}+\bar{x}_n) = C_{n-1}(x_{n-1}+\bar{x}_n) \\ &\qquad = C_{n-1}\bar{x}_n + C_{n-1}x_{n-1} = C_{n-1}\bar{x}_n + x_1x_2\cdots x_{n-1} \\ &\qquad = C_{n-1}\bar{x}_n + x_1x_2\cdots x_n = C_n. \end{aligned} \]

Corollary. The minimum number of contacts in a circuit realizing the function \(C_n\) is equal to \(2n-2\). The circuit has the form:

1. In \((^5)\) it is shown that the function \(C_n\) has an inversion group \(G\) of second order. Hence it follows that the number of functions of the same type as \(C_n\) is \(2^{\,n-1}\cdot n!\), and for all of them the minimum contact realization is equal to \(2n-2\).

Laboratory of Information Transmission Systems
Academy of Sciences of the USSR

Received
27 March 1961

REFERENCES

1. C. Cardot, Ann. Telecommun., 7, 2, 75 (1952).
2. Yu. L. Vasil’ev, DAN, 127, No. 2, 242 (1959).
3. S. V. Yablonskii, Matem. sborn., 30 (72), 2, 329 (1952).
4. I. N. Povarov, Sborn. Problemy peredachi informatsii, vol. 4, 1959.
5. Yu. L. Sagalovich, Sborn. Problemy peredachi informatsii, vol. 8 (1961).

* This representation of the function \(C_n\) was proposed by V. N. Roginskii.