Reports of the Academy of Sciences of the USSR
1964. Volume 159, No. 2

CYBERNETICS AND CONTROL THEORY

Kh. A. MADATYAN

ON THE SYNTHESIS OF CIRCUITS CORRECTING THE OPENING OF CONTACTS

(Presented by Academician P. S. Novikov on 23 V 1964)

The work is devoted to questions of reliability of control systems (¹). Usually the objects of consideration are contact circuits. Suppose that in a circuit, at any moment of time, due to the action of a fault source, some faults arise. A circuit is called self-correcting with respect to the given fault source if the faulty circuit realizes the same function.

Table 1

The concept of a self-correcting circuit was introduced in (²). In that work the problem of asymptotic estimates for the Shannon function of self-correcting circuits was solved, when the fault source produces only short circuits of contacts. The following asymptotic estimates were obtained:

[
1.\quad L_3^1(n)\sim \frac{2^n}{n}.
\qquad
2.\quad L_3^m(n)\lesssim \left{\left[\frac{m}{2}\right]+1\right}\frac{2^n}{n}.
]

The case when the fault source produces only openings of contacts remained open.

In the present note a similar problem is solved for this case, and analogous asymptotic estimates are obtained.

Let (L(\Sigma)) be the number of contacts in the circuit. Denote by (L_p^m(f)=\inf L(\Sigma)), where the lower bound is taken over all self-correcting circuits for an opening in (m) contacts that realize (f). Let (L_p^m(n)=\max L_p^m(f)) (the maximum is taken over all functions (f) depending on the arguments (x_1,\ldots,x_n)).

Theorem 1. (L_p^1(n)\sim 2^n/n).

Proof. We shall indicate a method which makes it possible, for each function (f(x_1,\ldots,x_n)), to construct a self-correcting circuit for one opening, having asymptotically (2^n/n) contacts. For this we take a circuit, constructed by the method of O. B. Lupanov (³), realizing the function (f), and show how it must be transformed in order to obtain the required self-correcting circuit.

Divide the table (Table 1) specifying the function (f) into strips, each of which (except, perhaps, the last) contains exactly (s) rows, while the last contains (s') ((s'\leq s)) rows. Let (p) be the number of strips ((p\leq 2^k/s+1)). Within each strip we divide the columns into groups so that into one group

include all identical columns. Let (f_{ij}(x_1,\ldots,x_n)) be the function that coincides with (f) on the (j)-th group of the (i)-th strip and is equal to 0 on the remaining sets. Obviously,

[
f_{ij}(x_1,\ldots,x_n)=f_{ij}^{(1)}(x_1,\ldots,x_k)\ \&\ f_{ij}^{(2)}(x_{k+1},\ldots,x_n).
]

Let

[
\bigvee_j f_{ij}^{(1)}\cdot f_{ij}^{(2)}=f_i,
\qquad
\bigvee_{i=1}^{p} f_i(x_1,\ldots,x_n)=f(x_1,\ldots,x_n).
]

Take (r) equal to a power of two and less than (n-k). Then the (r)-dimensional cube with axes (x_{k+1},\ldots,x_{k+r}) can be partitioned into (2^r/r) nonintersecting spheres of radius 1 (in the sense of the Hamming metric ((4))) with centers ((\beta_{k+1}^h,\ldots,\beta_{k+r}^h)), where (h=1,\ldots,2^r/r).

The circuit (\mathfrak A'), which realizes the function (f) (without the requirement of self-correction), as is known ((^3)), is constructed by connecting in parallel circuits (\mathfrak A_i') that realize (f_i). Let us describe the structure of the circuits (\mathfrak A_i'). To this end, construct a system of such ([1,q])-multipoles (M_1',\ldots,M_6') that each subsequent one is obtained by completing the preceding one and (M_6'=\mathfrak A_i') (Fig. 1).

(M_1') is a ([1,2^r])-multipole representing a contact tree in the variables (x_{k+1},\ldots,x_{k+r}); (M_2') is a ([1,2^r/r])-multipole obtained from (M_1') by joining the (r) outputs corresponding to the points of one and the same sphere. (M_2') realizes the characteristic functions of the spheres
[
\varphi_h(x_{k+1},\ldots,x_{k+r})\qquad (1\le h\le 2^r/r).
]

Fig. 1

(M_3') is a ([1,2^{\,n-k}/r])-multipole obtained from (M_2') by connecting to each of the outputs contact trees in the variables (x_{k+r+1},\ldots,x_n). (M_3') realizes functions of the form
[
\varphi_h(x_{k+1},\ldots,x_{k+r})
x_{k+r+1}^{\sigma_{k+r+1}}\cdots x_n^{\sigma_n}.
]

(M_4') is a ([1,2^{\,n-k}])-multipole obtained from (M_3') by connecting to the outputs corresponding to the sphere with center ((\beta_{k+1}^h,\ldots,\beta_{k+r}^h)) the ([1,r])-multipole shown in Fig. 2a ((h=1,2,\ldots,2^r/r)). (M_4') realizes all conjunctions of the form
[
x_{k+1}^{\sigma_{k+1}}\cdots x_n^{\sigma_n}.
]

(M_5') is a multipole obtained from (M_4') by joining (within each sphere) certain outputs so that (M_5') realizes functions of the form
[
f_{ijh}^{(2)}(x_{k+1},\ldots,x_n)
=
f_{ij}^{(2)}(x_{k+1},\ldots,x_n)\ \&\
\varphi_h(x_{k+1},\ldots,x_{k+r})
]
[
(j=1,2,\ldots;\quad 1\le h\le 2^r/r).
]

(M_6') is a ([1,1])-multipole obtained from (M_5') by connecting to its outputs (\pi)-circuits and subsequently joining all outputs into one. Namely, to the output corresponding to the function (f_{ijh}^{(2)}) there is connected the (\pi)-circuit corresponding to the perfect disjunctive normal form of the function (f_{ij}^{(1)}(x_1,\ldots,x_n)). The multipole (M_6') realizes the function (f_i(x_1,\ldots,x_n)).

If in the given construction we take
[
r=2^{[\,\frac12\log_2 n\,]},\qquad
k=[\,2\log_2 n\,],
\qquad
s=[\,n-2\sqrt n\,],
]
then we obtain the asymptotic inequality
[
L(n)\lesssim \frac{2^n}{n}.
]

The circuit (\mathfrak A') constructed by us has one property that will be used in the next part of the proof. Suppose the circuit (\mathfrak A') is decomposed into the direct sum of three subcircuits (\mathfrak A_1'), (\mathfrak A_2'), and (\mathfrak A_3'), where (\mathfrak A_1') consists of all multipoles (M_3'), (\mathfrak A_2') consists of all multipoles (M_5'\setminus M_3'), and (\mathfrak A_3') consists of all multipoles (M_6'\setminus M_5'). Let (L(\mathfrak A_1')), (L(\mathfrak A_2')), and (L(\mathfrak A_3')) be the number of contacts in the corresponding-

in the corresponding subcircuits. It is easy to see that, for the above values of the parameters (r), (k), and (s), the relations

[
L(\mathfrak{A}'_1)=o(L(\mathfrak{A}'_2)), \qquad
L(\mathfrak{A}'_3)=o(L(\mathfrak{A}'_2)).
\tag{1}
]

hold. The latter means that (L(\mathfrak{A}'_2)\sim 2^n/n), i.e., that the subcircuit (\mathfrak{A}'_2) contains almost all the contacts of the circuit (\mathfrak{A}').

We now proceed to describe those changes in the constructed circuit which make it possible to obtain from it a self-correcting circuit satisfying the condition of the theorem. To this end, in the subcircuits (\mathfrak{A}'_1) and (\mathfrak{A}'_3), each contact (x^\sigma) should be duplicated in parallel, as indicated in Fig. 2b. We denote the resulting subcircuits by (\mathfrak{A}_1) and (\mathfrak{A}_3).

Fig. 2

For the subcircuit (\mathfrak{A}'2) we shall give a special construction. Recall that in synthesizing the multipoles (M'_5) we performed, in a certain way, unions of the outputs of the multipoles (M'_4); moreover, unions of this sort were carried out only within individual spheres of each multipole. We also note that in the ([1,r])-pole (Fig. 3a), which was connected to the outputs of (M'_3), all contacts are controlled by different relays. Consider one of the unions performed for the outputs of the sphere with center ((\beta^h)).},\ldots,\beta^h_{k+r

Number all those outputs of the multipole (M'4) which participated in the union under consideration and which are controlled by one and the same relay (x). Assign these numbers to those outputs of the multipole (M'3) from which the contact (X^{\bar{\beta}^h}{k+1}) emerged, and then delete this contact. Now connect, by contacts (X^{\bar{\beta}^h_2).}}_{k+1}), consecutively output (M'_3) with number 1 to output 2, output 2 to output 3, and so on, to the end. Connect the first and the last outputs with the corresponding (\pi)-circuit (Fig. 3a). We carry out this construction for all distinct relays in the given union, and then for all unions. Denote the subcircuit thus obtained by (\mathfrak{A

Fig. 3

Thus, we have constructed a circuit (\mathfrak{A}=\mathfrak{A}_1\cup \mathfrak{A}_2\cup \mathfrak{A}_3). It is easy to see that it realizes the same function (f(x_1,\ldots,x_n)). The fact that this circuit is self-correcting follows from the fact that every chain of the circuit (\mathfrak{A}) having nonzero conductivity has been duplicated.

Let us estimate the complexity (L(\mathfrak{A})) of the circuit (\mathfrak{A}). From (1) and the relations

[
L(\mathfrak{A}_1)=2L(\mathfrak{A}'_1), \qquad
L(\mathfrak{A}_3)=2L(\mathfrak{A}'_3), \qquad
L(\mathfrak{A}_2)\leq L(\mathfrak{A}'_2)+\sqrt{n}\,2^{\,n-2\sqrt{n}}
]

it follows that (L(\mathfrak{A})\lesssim 2^n/n). Thus, (L^1_p\leq L(\mathfrak{A})\lesssim 2^n/n). The lower estimate for (L^1_p(n)) follows from (5). The theorem is proved.

By a slight complication of this method of synthesis it is not difficult to obtain the following result:

Theorem 2.

[
L^m_p(n)\lesssim \left(\frac{m+1}{2}\right)\frac{2^n}{n}.
]

For this it is necessary:

1) In the circuits (\mathfrak A'_1) and (\mathfrak A'_3), duplicate each contact (x^\sigma) ((m+1)) times.

2) If (m) is odd, then each contact in Fig. 3a must be duplicated ((m+1)/2) times.

3) If (m) is even, then each contact in Fig. 3a must be duplicated (m/2) times and, in addition, output number 1 must be connected with number 4, output number 3 with number 6, ..., (i) with (i+3), etc., (q+1) with 2 (Fig. 3b). Then we obtain

[
L(\mathfrak A_1)=(m+1)L(\mathfrak A'_1),\qquad
L(\mathfrak A_3)=(m+1)L(\mathfrak A'_3),
]

[
L(\mathfrak A_2)\leq
\left(\frac{m+1}{2}\right)L(\mathfrak A'_2)+
\frac{m+1}{2}\sqrt{n2^{\,n-2}\sqrt[n]{n}}.
\tag{2}
]

From (1) and (2) it follows that

[
L_p^m(n)\lesssim
\left(\frac{m+1}{2}\right)\frac{2^n}{n}.
]

Theorem 2 gives, for almost all functions, self-correcting circuits that are twice as simple as those obtained by the trivial method.

I take this opportunity to express my gratitude to my scientific adviser, Prof. S. V. Yablonskii, for his guidance of the present work.

Moscow State University
named after M. V. Lomonosov

Received
18 V 1964

References

¹ S. V. Yablonskii, Problems of Cybernetics, vol. 2, 1959.
² Yu. G. Potapov, S. V. Yablonskii, DAN, 134, No. 3 (1960).
³ O. B. Lupanov, DAN, 119, No. 1 (1958).
⁴ R. W. Hamming, Bell Syst. Techn. J., 29, No. 1 (1950).
⁵ C. E. Shannon, Bell Syst. Techn. J., 28, No. 1 (1949).