MATHEMATICS

N. A. KARPOVA

ON CONTACT CIRCUITS FOR MONOTONE FUNCTIONS

(Presented by Academician M. V. Keldysh on 25 VI 1958)

One of the basic problems of circuit theory is the synthesis of circuits that realize certain functions and contain the smallest possible number of elements. Depending on the class of circuit elements, circuits realizing one and the same function may be more or less complex in their topological structure, contain a greater or smaller number of elements, etc. It is known, for example, that a monotone function of the algebra of logic can be realized by a circuit containing only closing contacts. However, by bringing into consideration opening contacts, valves, memory elements, etc., one can obtain circuits which, in a certain sense, are simpler than the former.

Fig. 1

In the present note the complexity of realizing monotone functions in the class of contact circuits made of closing contacts is compared with their realization by circuits in which opening contacts are allowed, and it is shown that the class of circuits with arbitrary contacts is considerably simpler than the class of circuits consisting only of closing contacts.

Let us introduce an order relation between tuples of zeros and ones. Namely, we shall write
\((\alpha_1,\ldots,\alpha_n)<(\beta_1,\ldots,\beta_n)\) if, for all \(i,\ 1\leq i\leq n\), the relations \(\alpha_i\leq\beta_i\) are satisfied.

Definition. A function of the algebra of logic \(f(x_1,\ldots,x_n)\) is called monotone if for any tuples \((\alpha_1,\ldots,\alpha_n)\), \((\beta_1,\ldots,\beta_n)\) such that
\((\alpha_1,\ldots,\alpha_n)<(\beta_1,\ldots,\beta_n)\), one has
\(f(\alpha_1,\ldots,\alpha_n)\leq f(\beta_1,\ldots,\beta_n)\).

It is clear that the realization of monotone functions in the class of circuits made of closing contacts is not the best from the point of view of the number of elements in the circuit. An example is known, constructed by Yu. L. Vasil’ev, which shows that the minimal circuit made of closing contacts for the function
\[ f=x_1(x_2\vee x_3)\vee x_4(x_5\vee x_6)\vee (x_2\vee x_3)(x_5\vee x_6) \]
is not minimal in the class of all contact circuits (Fig. 1a and 1b).

Let us consider the realization of monotone functions in the class of all contact circuits (class A) and in the class of circuits made of closing contacts (class B).

For comparing the complexity of circuits from the indicated classes it is convenient to introduce the following functions characterizing complexity: \(L^{+}(f)\) — the number of contacts in a minimal circuit of class B realizing the given function \(f\); \(L(f)\) — the number of contacts in a minimal circuit of class A realizing the given function \(f\).

To characterize the relative complexity of the classes under consideration, one also introduces
\[ \lambda(f)=\frac{L^{+}(f)}{L(f)} \quad\text{and}\quad \lambda(n)=\max_{\substack{f\ \text{of } n\ \text{arguments}}}\lambda(f). \]

Below a sequence of functions \(f_n\) (\(f_n\) depends on \(n\) arguments) will be constructed for which \(\lambda(f_n)\) substantially exceeds one in the limit as \(n \to \infty\).

Lemma 1. In realizing the monotone function \(f(a,b,a_1,b_1)=aa_1 \vee bb_1 \vee a_1b_1\) by a circuit from the class B, a minimal circuit contains 5 contacts; moreover, one of the contacts \(a_1\) or \(b_1\) must be repeated twice in the circuit.

Proof. Suppose the contrary, i.e., that the relays \(a_1\) and \(b_1\) each contain one contact, and show that then the circuit must necessarily contain the chain \(ab\).

Fig. 2

To realize the given function, the circuit must contain the chain \(a_1b_1\); by assumption, it has length 2 (Fig. 2a). In addition, \(aa_1\) and \(bb_1\) must be realized. Since the contacts \(a_1\) and \(b_1\) are already contained in the circuit, \(aa_1\) can be realized in a unique way (see Fig. 2b). For the same reason, \(bb_1\) can be realized in a unique way (Fig. 2c). Consider the point of intersection of the chains \(a_1a\) and \(b_1b\) closest to the minus pole and not coinciding with the pole (denote this point by 2). There is at least one point of intersection of these chains—this is point 1. Consider the chain \(l_1l_2\), where \(l_1\) is the part of the chain \(aa_1\) from the minus pole to point 2, and \(l_2\) is the part of the chain \(bb_1\) from the plus pole to point 2. Then \(l_1\) can contain only contacts \(a\), since the chain \(l_1\) has length 1 and the contact \(a_1\) is adjacent to the plus pole. The chain \(l_2\) can contain only contacts \(b\), since the single contact \(b_1\) is adjacent to the minus pole and cannot enter \(l_2\). Thus, the chain \(l_1l_2\) realizes the conjunction \(ab\), which is not contained in the function. It remains to suppose that at least one of the contacts \(a_1\) or \(b_1\) enters the circuit more than once, and the circuit contains at least 5 contacts. A circuit with 5 contacts can be constructed as shown in Fig. 3, which completely proves Lemma 1.

Fig. 3

Lemma 2. Let \(\mathfrak A=x_1 \vee x_2 \vee \cdots \vee x_n\), \(\mathfrak B=y_1 \vee \cdots \vee y_n\); then the monotone function

\[ f_{2n+2}(a,b,x_1,\ldots,x_n,y_1,\ldots,y_n)=a\mathfrak A \vee b\mathfrak B \vee \mathfrak A\mathfrak B \]

can be realized by a circuit from the class B containing no fewer than \(3n+2\) contacts.

Proof. Lemma 1 proved above allows us to assert that Lemma 2 is true for the case \(n=1\). Suppose that the assertion of Lemma 2 is valid for all \(n<n_0\), and that for \(n_0\) there is a lower estimate smaller than \(3n_0+2\). Then it is obviously impossible to establish such a correspondence that, for every \(x_i\) from \(\mathfrak A\), a relay containing one contact has a \(y_j\) from \(\mathfrak B\) corresponding to it such that the corresponding relay contains at least two contacts, and conversely; i.e., under any correspondence there will be such \(x_i\) from \(\mathfrak A\) and \(y_j\) from \(\mathfrak B\) that the indicated relays in the circuit contain one contact each. But then, substituting the constant 0 in place of all \(x_l\) from \(\mathfrak A\) and \(y_r\) from \(\mathfrak B\) different from \(x_i\) and \(y_j\), we obtain the function \(f(a,b,x_i,y_j)\), which is realized by a circuit in which the distribution of contacts is different from that asserted in Lemma 1, namely, the relays \(x_i\) and \(y_j\) contain only one contact each.

But Lemma 1 is applicable to the function \(f(a,b,x_i,y_i)\) obtained, and consequently we obtain a contradiction to the given assumption; the assertion of Lemma 2 is true for all \(n\), and a minimal realization of the given function is a circuit of the form indicated in Fig. 4a or in Fig. 4b.

Lemma 3. Let \(\mathfrak{A}=x_1\vee\cdots\vee x_n\), \(\mathfrak{B}=y_1\vee\cdots\vee y_{n-1}\); then the monotone function

\[ f_{2n+1}(a,b,x_1,\ldots,x_n,y_1,\ldots,y_{n-1}) = a\mathfrak{A}\vee a\mathfrak{B}\vee \mathfrak{A}\mathfrak{B} \]

can be realized by a circuit from class \(\mathbf{B}\) containing no fewer than \(3n\) contacts.

Proof. For the proof we shall use the result of Lemma 2. Suppose that the circuit contains fewer than \(3n\) contacts. Then, substituting the constant \(0\) for \(x_n\), we obtain a circuit realizing a function

Fig. 4

of the form indicated in Lemma 2, but for it the estimate \(3n-1\) holds. Taking into account that the function under consideration depends essentially on \(x_n\), we obtain that the circuit for the given function contains no fewer than \(3n\) contacts. On the other hand, it can be realized with this number of contacts (Fig. 4b and 4a).

Theorem. There exists a sequence of functions \(f_n\) (\(f_n\) depends on \(n\) arguments) for which \(\lambda(f_n)\ge 3/2\).

Proof. A sequence satisfying the condition of the theorem is given by the functions of the form \(f_{2n+1}\) and \(f_{2n+2}\) considered in Lemmas 2 and 3. As established above, for the realization of the indicated functions in class \(\mathbf{B}\) there is a lower bound of \(3n\) contacts.

In class \(\mathbf{A}\) one can construct a circuit for the same function with no more than \(2n+3\) contacts (Fig. 4c). Thus, in the class of circuits \(\mathbf{A}\) there is an upper bound of \(2n+3\) for the number of contacts. Directly from this, and also using the definition of \(\lambda(f_n)\), for the given sequence of functions we obtain \(\lambda(f_n)\ge 3/2\), as was required to prove.

Corollary. \(\lambda(n)\ge 3/2\).

Received
24 VI 1958