CYBERNETICS AND CONTROL THEORY

E. I. NECHIPORUK

ON SELF-CORRECTING GATE CIRCUITS

(Presented by Academician P. S. Novikov on 20 I 1964)

In this note self-correcting \((^1)\) gate circuits are considered. In gate circuits two kinds of malfunctions are possible: shorting, or breakdown, and disconnection of gates. Below estimates are given for the complexity of gate circuits of depth 2 \((^2)\) that correct disconnections of gates. It is shown that the complexity of circuits correcting \(m\)-fold disconnections increases, in comparison with the complexity of ordinary circuits, by no less than \(C(m+1)\) times, where \(C\) is a certain constant, and in the case of not too “dense” matrices— asymptotically by \(m+1\) times. Thus the case considered here differs substantially from the case of correction of a single shorting in contact circuits \((^1)\), where no (asymptotic) increase of complexity is required in comparison with ordinary circuits.

\(1^\circ\). Denote by \(\chi(\mathfrak A)\) the matrix realized by the gate circuit \(\mathfrak A\). Denote by \(\mathfrak M_m(\mathfrak A)\) the set of all circuits obtained from the circuit \(\mathfrak A\) by disconnecting no more than \(m\) arbitrary gates. We shall say that the circuit \(\mathfrak A\) corrects \(m\)-fold disconnections if for every circuit \(\mathfrak C\) from \(\mathfrak M_m(\mathfrak A)\) the equality \(\chi(\mathfrak C)=\chi(\mathfrak A)\) holds.

Denote by \(B(\mathfrak A)\) the number of gates in the circuit \(\mathfrak A\). For an arbitrary matrix \(A\)* introduce the function \(B_{r,m}(A)=\min B(\mathfrak A)\), where the minimum is taken over all circuits \(\mathfrak A\) of depth no greater than \(r\), realizing the matrix \(A\) and correcting \(m\)-fold disconnections. For an arbitrary class of matrices \(\mathfrak N\) introduce the function \(B_{r,m}(\mathfrak N)=\max B_{r,m}(A)\), where the maximum is taken over all matrices \(A\) from \(\mathfrak N\). The functions \(B_{r,0}\) refer to ordinary circuits without self-correction. We shall call the number

\[ \beta_{r,m}(\mathfrak N)=\frac{B_{r,m}(\mathfrak N)}{B_{r,0}(\mathfrak N)} \]

the duplication coefficient.

Obviously, for any \(r,\mathfrak N\), \(\beta_{r,m}(\mathfrak N)\leq m+1\).

Lemma 1. In order that a circuit correct \(m\)-fold disconnections, it is necessary and sufficient that in it, between any pair of poles with nonzero conductivity, there exist no fewer than \(m+1\) distinct chains having no pairwise common gates.

Proof follows from the “transport” theorem of Ford–Fulkerson \((^3)\).

Denote by \(\|A\|\) the number of ones in the matrix \(A\). We shall call a matrix having \(s\) rows and \(t\) columns an \((s,t)\)-matrix; an \((s,t)\)-matrix will be called nondegenerate if \(s>0,\ t>0\); a nondegenerate matrix consisting only of ones will be called complete. We shall call a circuit \(\mathfrak A\) nondegenerate if \(\chi(\mathfrak A)\) is a nondegenerate nonzero matrix. Let \(a\) be a nondegenerate \((s,t)\)-matrix, \(A\) a nondegenerate nonzero matrix. Introduce the densities of matrices:

\[ d(a)=d(s,t)= \begin{cases} \dfrac{st}{s+t}, & \text{for } s>1,\ t>1,\\ 1, & \text{for } s=1 \text{ or } t=1; \end{cases} \]

\[ \delta(A)=\max d(a), \]

where the maximum is taken over all complete submatrices \(a\) of the matrix \(A\);

\[ \delta(\mathfrak N)=\min_{A\in \mathfrak N}\delta(A). \]

Theorem 1. If \(A\) is a nondegenerate nonzero matrix, then

\[ (m+1)\frac{\|A\|}{\delta(A)}\leq B_{2,m}(A). \]

* All matrices are considered Boolean.

Proof. Let \(B_{2,m}(A)\) be attained on a nondegenerate \((p,q)\)-circuit \(\mathfrak A\) of depth at most 2. Obviously, \(\mathfrak A\) has no “superfluous” gates. We decompose \(\mathfrak A\) into subcircuits \(\mathfrak A_k\), pairwise intersecting only at poles, so that \(\mathfrak A_0\) is a circuit of depth at most 1, while \(\mathfrak A_k\) for \(k \geqslant 1\) are circuits of depth 2 containing only one internal node (Fig. 1). We assume that every subcircuit \(\mathfrak A_k\) contains all the poles of the circuit \(\mathfrak A\). Denote by \(\varphi_k^{i,j}\) the largest number of chains containing no pairwise common gates (a flow) between the poles with numbers \(i,j\) in the subcircuit \(\mathfrak A_k\). Every chain of the circuit \(\mathfrak A\) belongs to one and only one subcircuit \(\mathfrak A_k\); therefore (Lemma 1)

Fig. 1

\[ (m+1)\|A\| \leqslant \sum_k \sum_{i,j} \varphi_k^{i,j}. \tag{1} \]

We shall show that, for nondegenerate subcircuits \(\mathfrak A_k\),

\[ \sum_{i,j} \varphi_k^{i,j} \leqslant \delta(\chi(\mathfrak A_k))\, B(\mathfrak A_k). \tag{2} \]

For \(k=0\), (2) is obvious. Let \(k \geqslant 1\). Denote by \(a_i\) and \(b_j\) (we omit the index \(k\)) the number of gates connecting the internal node of the subcircuit \(\mathfrak A_k\) with the \(i\)-th input and the \(j\)-th output, respectively. Obviously,
\(\varphi^{i,j}=\min(a_i,b_j)\). Let \(s\) be the number of inputs and \(t\) the number of outputs connected with the internal node of the subcircuit \(\mathfrak A_k\). Renumber the nonzero \(a_i,b_j\) by the numbers \(u=1,\ldots,s\), \(v=1,\ldots,t\). Since

\[ \frac{st}{s+t} \leqslant \delta(\chi(\mathfrak A_k)), \qquad B(\mathfrak A_k)=\sum_{u=1}^{s} a_u+\sum_{v=1}^{t} b_v, \]

(2) follows from the easily verified inequality

\[ \sum_{u=1}^{s}\sum_{v=1}^{t}\min(a_u,b_v) \leqslant \frac{st}{s+t}\left(\sum_{u=1}^{s} a_u+\sum_{v=1}^{t} b_v\right). \]

The assertion of the theorem follows from (1), (2), since

\[ \max \delta(\chi(\mathfrak A_k)) \leqslant \delta(A), \qquad B_{2,m}(A)=B(\mathfrak A)=\sum_k B(\mathfrak A_k). \]

\(2^\circ\). Denote by \(\mathfrak B(p,q,\alpha)\) the class of all \((p,q)\)-matrices containing \(\alpha pq\) ones (the conditions \(0\leqslant \alpha \leqslant 1\) and that \(\alpha pq\) is an integer are assumed to hold everywhere, without further mention). Put \(\alpha^*=\min(\alpha,1-\alpha)\) and

\[ H(z)=z\lg_2\frac{1}{z}+(1-z)\lg_2\frac{1}{1-z}. \tag{3} \]

Theorem 2. Suppose the following conditions are satisfied:

a) \(q_n \leqslant p_n\);

b) \(q_n \to \infty\);

c)
\[ \frac{\alpha_n \lg_2 \frac{1}{\alpha_n}\, q_n}{\lg_2 p_n}\to x; \]

d)
\[ \frac{\lg_2 p_n}{\lg_2 \frac{1}{\alpha_n^*}}\to \infty\; * . \]

* Conditions c), d) mean that the matrices from the classes \(\mathfrak B(p_n,q_n,\alpha_n)\) are not too “narrow” and not too “sparse” or “dense.”

Then

\[ \delta \bigl(\mathfrak{B}(p_n,q_n,\alpha_n)\bigr)\sim \frac{\lg_2 p_n}{\lg_2 \dfrac{1}{\alpha_n}} . \]

Proof. The lower bound follows from the existence in every matrix from \(\mathfrak{B}(p_n,q_n,\alpha_n)\) of a complete \((s_n,t_n)\)-submatrix such that

\[ t_n\sim \frac{\lg_2 p_n}{\lg_2 \dfrac{1}{\alpha_n}},\qquad \frac{s_n}{t_n}\to\infty \tag{4} \]

(the numbers \(s_n,t_n\) are the same for all matrices from \(\mathfrak{B}(p_n,q_n,\alpha_n)\)). Indeed, from (4)

\[ \delta\bigl(\mathfrak{B}(p_n,q_n,\alpha_n)\bigr)\ge \frac{s_n t_n}{s_n+t_n}\sim \frac{\lg_2 p_n}{\lg_2 \dfrac{1}{\alpha_n}} . \]

As in (4), it is easy to show that for every \(t_n\), not greater than \([\alpha_n q_n]\), in every matrix from \(\mathfrak{B}(p_n,q_n,\alpha_n)\) there exists a complete \((s_n,t_n)\)-submatrix, where

\[ s_n=\left]\,p_n\frac{C^{t_n}_{[\alpha_n q_n]}}{C^{t_n}_{q_n}}\,\right[\, * . \tag{5} \]

We shall show that for a certain choice of the number \(t_n\) the pair \((s_n,t_n)\) satisfies the conditions (4). Put

\[ t_n\to\infty,\qquad \frac{t_n}{\alpha_n q_n}\to 0 . \tag{6} \]

In view of c), d) these conditions are compatible and follow from (4). From (5)

\[ \lg_2 s_n \ge \lg_2 p_n+ H\!\left(\frac{t_n}{\alpha_n q_n}\right)\alpha_n q_n - H\!\left(\frac{t_n}{q_n}\right)q_n+o(1) = \lg_2 p_n-t_n\lg_2\frac{1}{\alpha_n}- \]

\[ -(\alpha_n q_n-t_n)\lg_2\!\left(1-\frac{t_n}{\alpha_n q_n}\right) +(q_n-t_n)\lg_2\!\left(1-\frac{t_n}{q_n}\right)+o(1) = \]

\[ = \lg_2 p_n-t_n\lg_2\frac{1}{\alpha_n} -\frac{1-\alpha_n}{\alpha_n}\frac{t_n^2}{q_n}(1+o(1))\lg_2 e+o(1)\ ** . \]

Introduce a parameter \(\varepsilon_n\) and put

\[ t_n=\left[(1-\varepsilon_n) \frac{\lg_2 p_n}{\lg_2 \dfrac{1}{\alpha_n}}\right]. \]

Then

\[ \lg_2 s_n \ge \varepsilon_n\lg_2 p_n - \frac{1-\alpha_n}{\alpha_n}\frac{t_n^2}{q_n}(1+o(1))\lg_2 e +o(1). \]

Put

\[ \lambda_n=\frac{\lg_2 p_n}{\lg_2 \dfrac{1}{\alpha_n}},\qquad \omega_n= \frac{\lg_2 \dfrac{1}{\alpha_n}\,q_n}{\lg_2 p_n},\qquad \varphi_n=\frac{\lg_2 p_n}{\lg_2 \lambda_n}. \]

In view of d), \(\varphi_n\to\infty\). The conditions (4) follow from the conditions
\(\varepsilon_n\to 0,\ \varepsilon_n\omega_n\to\infty\ *** ,\ \varepsilon_n\varphi_n\to\infty\).
Put

\[ \varepsilon_n=(\min(\omega_n,\varphi_n))^{-1/2}. \]

Then all the conditions are fulfilled.

\[ \text{* } [a]\ (]a[)\text{ denotes the integer nearest to }a\text{ from below (from above).} \]

\[ \text{** The following facts are used: Stirling’s formula, (6), }H(z)\text{ increases monotonically} \]
\[ \text{in a neighborhood of zero, }H(0)=0,\text{ (3), } \lg_2(1-z)=-z(1+o(1))\lg_2 e\text{ as }z\to 0. \]

\[ \text{*** }\ \varepsilon_n\lg_2 p_n\, \frac{\alpha_n q_n}{(1-\alpha_n)t_n^2} \asymp \varepsilon_n\omega_n \frac{\lg_2 \dfrac{1}{\alpha_n}}{1-\alpha_n} \to\infty,\quad \text{since } \frac{\lg_2 \dfrac{1}{\alpha_n}}{1-\alpha_n}\ge 1. \]

Upper estimate. Denote by \(S(p,q,\alpha,s,t)\) the number of matrices from \(\mathfrak B(p,q,\alpha)\) having at least one complete \((s,t)\)-submatrix. We have

\[ S(p,q,\alpha,s,t)\le C_p^s C_q^t C_{pq-st}^{\alpha pq-st}. \]

Denote by \(\varkappa_n\) the cardinality of the class \(\mathfrak B(p_n,q_n,\alpha_n)\). Put \(\nu_n=\dfrac{\lg_2 q_n}{\lg_2 p_n}\). Under conditions a), b) and \(s_nt_n=o(\alpha_n p_n q_n)\), we have

\[ \lg_2\frac{S(p_n,q_n,\alpha_n,s_n,t_n)}{\varkappa_n} \le (s_n+\nu_n t_n)\lg_2 p_n-s_nt_n\lg_2\frac{1}{\alpha_n}.\!^{*} \tag{7} \]

Suppose the upper estimate is false. Then there exist a number \(\omega\), greater than one, and a subsequence of \(\{n\}\) (for it we retain the notation \(\{n\}\)) such that for every matrix \(A_n\) from \(\mathfrak B(p_n,q_n,\alpha_n)\), \(\omega\lambda_n\le \delta(A_n)\), i.e., every matrix \(A_n\) contains a complete \((u_n,v_n)\)-submatrix \(a_n\) (\(u_n,v_n\) depend on \(A_n\)) for which \(\omega\lambda_n\le d(a_n)\). Every matrix \(a_n\) contains a complete \((s_n,t_n)\)-submatrix \(b_n\) (\(s_n,t_n\) depend on \(a_n\)) such that \(s_nt_n=o(\alpha_n p_nq_n)\) and \(\omega\lambda_n\le d(b_n)\). Indeed, without loss of generality assume that \(v_n\le u_n\). If \(v_n\le 2\omega\lambda_n\), put \((s_n,t_n)=(u_n,v_n)\); if \(2\omega\lambda_n<v_n\), put \(s_n=t_n=]2\omega\lambda_n[\). Condition \(s_nt_n=o(\alpha_np_nq_n)\) follows from c). But when \(\omega\lambda_n\le d(b_n)\), the right-hand side of (7) tends to \(-\infty\), i.e., as \(n\) grows, almost all matrices from \(\mathfrak B(p_n,q_n,\alpha_n)\) contain no complete \((s_n,t_n)\)-submatrices. The contradiction obtained completes the proof of the theorem.

Corollary. Under the conditions of Theorem 2,

\[ \frac{\alpha_n\lg_2\dfrac{1}{\alpha_n}}{H(\alpha_n)}(m+1) \le \beta_{2,m}\bigl(\mathfrak B(p_n,q_n,\alpha_n)\bigr)\le m+1. \]

Indeed, from (5),

\[ B_{2,0}\bigl(\mathfrak B(p_n,q_n,\alpha_n)\bigr)\sim H(\alpha_n)\frac{p_nq_n}{\lg_2 p_n}. \]

Theorem 3. Under the conditions of Theorem 2 and \(\alpha_n\to 0\),

\[ \beta_{2,m}\bigl(\mathfrak B(p_n,q_n,\alpha_n)\bigr)\sim m+1. \]

Theorem 4. If \(\min(p_n,q_n)\to\infty\), then for the class of all \((p_n,q_n)\)-matrices

\[ \frac{m+1}{e\ln 2}\le \beta_{2,m}\le m+1. \]

Proof. From the corollary,

\[ (m+1)\max\left(\alpha,\lg_2\frac{1}{\alpha}\right)\le \beta_{2,m}. \]

The maximum is attained at \(\alpha=\dfrac{1}{e}\).

Leningrad State University
named after A. A. Zhdanov

Received
15 I 1964

CITED LITERATURE

\(^{1}\) Yu. G. Potopov, S. V. Yablonskii, DAN, 134, No. 3 (1960).
\(^{2}\) O. B. Lupanov, DAN, 111, No. 6 (1956).
\(^{3}\) C. Berge, Graph Theory and Its Applications, IL, 1962, p. 87.
\(^{4}\) E. I. Nechiporuk, DAN, 148, No. 1 (1963).
\(^{5}\) E. I. Nechiporuk, in: Problems of Cybernetics, vol. 9, 1963.

\[ {}^{*}\ x_n\leqslant\!\cdot\, y_n \text{ means that } x_n\leqslant y_n \text{ for sufficiently large } n. \]
The following facts are used: Stirling’s formula,

\[ H'(z)=\lg_2\frac{1-z}{z},\qquad H''(z)<0 \quad \text{for } 0<z<1, \]

\[ H\left(\frac{\alpha_np_nq_n-s_nt_n}{p_nq_n-s_nt_n}\right) = H\left(\alpha_n-\frac{(1-\alpha_n)s_nt_n/p_nq_n}{1-s_nt_n/p_nq_n}\right) \leqslant \]

\[ \leqslant H(\alpha_n) -\frac{(1-\alpha_n)s_nt_n/p_nq_n}{1-s_nt_n/p_nq_n} \lg_2\frac{1-\alpha_n}{\alpha_n}. \]