Reliable Circuits Using Less Reliable Relays

An investigation is made of relays whose reliability can be described in simple terms by means of probabilities. It is shown that by using a sufficiently large number of these relays in the proper manner, circuits can be built which are arbitrarily reliable, regardless of how unreliable the original relays are. Various properties of these circuits are elucidated. Part 12 INTRODUCTION In an important paper-3 von Neumann considers the problem of constructing reliable computing circuits by the redundant use of unreliable components. He studies several cases, one of which, for example, involves the construction of machines using as a basic component a “Sheffer stroke” organ.4 Von Neumann shows that under certain conditions it is possible to combine a number of unreliable Sheffer stroke organs to obtain an element which acts like a Sheffer stroke organ of higher reliability. In fact, under certain conditions one can approach perfect operation by means of a sufficiently redundant circuit. The present paper was inspired by von Neumann’s work and carries out a similar analysis for relay circuits. It appears that relays are basically more adaptable to these error-correcting circuits than the neuron-like components studied by von Neumann. At any rate, our results go further than his in several directions. In the first place, von Neumann needs to assume a certain fairly good reliability in his components in order to get started. With the Sheffer stroke organ, a probability of error less than 1/6 is absolutely necessary, and something like one in a hundred or better is required in the specific error-correcting circuits developed. The methods developed here, on the other hand, wil1 apply to arbitrarily poor relays. Secondly, the amount of redundancy required in our circuits for a ’ Murray Hill Laboratory, Bel1 Telephone Laboratories, Inc., Murray Hill, N. J. * Part 11 wil1 appear in this JOURNAL for October, 1956. 8 J. VON NEUMANN, “Probabilistic Logies,” California Institute of Technology, 1952. (Also Published in “Automata Studies,” edited by C. E. Shannon and J. McCarthy, Princeton IJniversity Press, 1956.) 4 The Sheffer stroke is the logica1 operation on two variables “nat A and not B.” It has the property that al1 logica1 functions can be generated in terms of it. 4 Sheffer stroke organ is a device with two binary inputs and one binary output which performs this logica1 operation. An unreliable component of this sort would give the proper output only with a certain probability. r9r 192 E. F. MOORE AND C. E. SHANNON [.I. F. 1. given improvement in reliability is considerably different from that required by von Neumann. For example, in one numerical case that he considers, a redundancy of about 60,000 to 1 is required to obtain a certain improvement in operating reliability. The same improvement is obtained in relay circuits with a redundancy of only 100 to 1. We also show that in a certain sense some of our circuits are not far from minimal. Thus, in the numerical case just mentioned, our results show that a redundancy of at least 67 to 1 is necessary in any circuit of the type we consider. Hence, the actual circuits which achieve this improvement with a redundancy of 100 to 1 are not too inefficient in the use of components. Another differente is that it is not necessary in the case of relays to use what von Neumann calls the “multiplexing system” in order to approach perfect operation on the final output. With his types of elements, the final output (without multiplexing) always has a definite residual unreliability. With the systems described here, this final prohability of error can approach zero. This paper is not intended for practica1 design purposes, but rather for theoretical and mathematica1 insight into the problem. There may, however, be some practica1 applications. The reliability of a commercial relay is typically very high, for example, one failure in 10; operations. However, there are cases where even this reliability is insufficient. In the first place, in large-scale computing machines an extremely large number of individual relay operations may be involved in one calculation, an error in any one of which could cause an error in the final result. Because of this, the Bel1 Telephone Laboratories’ computers have made extensive use of self-checking and error-detecting schemes. A second type of situation requiring extreme reliability occurs when human safety is dependent on correct operation of a relaycircuit, for example, railway interlocks, safety circuits on automatie elevators and in guided missiles, etc. It is possible that some of thc simpler circuits we describe may be of some use in applications such as these. However, the results of this paper wil1 not be directly applicable to actual relays which wear out with age, but only to idealized relays whose probability of failure are constant in time. IDEALIZED RELAYS We wil1 prove results only for idealized relays whose failures can be described in one specific manner by means of probabilities. Their description allows only intermittent types of failures, and allows these only under the assumption that the probability of failure remains constant as time passes. This idealization does not cover such actually possible cases as relays which wear out with age, relays whose windings burn out, or relays which have been wired into the circuit with an imperfect soldered Sept., 1956.1 CIRCUITS USING LES RELIABLE RELAYS 193 connection. It is also assumed that the circuit is not improperly designed or improperly wired and that there are no bits of solder to produce short circuits between different wires. Since al1 of the above kinds of errors and failures can actually occur in practice, using real relays, the results of this paper do not strictly apply to such real relays. However, the two kinds of failures considered in this paper do actually occur in relays, so the kinds of circuits suggested are of some possible application. The first kind of failure allowed is the failure of a relay contact to close, which in actual relays is often due to a particle of dust preventing electrical closure. The second type of failure is the failure of a contact to open, which in actual relays is usually due to the welding action of the current passing through the contacts. We shall consider relay circuits in which the only causes of errors are of these two types-failure of contacts that should be closed to be actually closed and of contacts that should be open to be actually open. We wil1 assume, in fact, that there are two probabilities associated with a contact on a relay. If the relay is FIG. 1. Schematic representation of the transition probFIG. 2. One proposed way of transforming relay circuits abilities. to improve reliability. energized, the contact is closed with probability a, open with probability 1 a. If the relay is not energized, the contact is closed with probability c and open with probability 1 c. If a is greater than c, we wil1 cal1 the contact a make contact ; if a is less than c we cal1 it a break contact. We assume that different contacts are statistically independent. With actual relays this is probably not too far from the truth for contacts on diflerent relays and, indeed, this is al1 that is required for most of the results we wish to establish. In addition, we shall assume that on the successive times that a relay coil is energized its closures are statistically independent. A relay of this type governed by probabilities a and c wil1 be called a crummy6 relay. Its probability operation may be represented schematically as in Fig. 1. This wil1 be recognized as similar to diagrams used to represent a simple noisy communication Channel, and indeed such a relay can be thought of as a noisy binary Channel. The capacity of the corresponding Channel wil1 be zero if and only if a = c. We wil1 6 “Crummy = crumby, esp. lousy,” Webster’s New International Dictionary. We chose the more modern spelling universally used in comic books. 194 E. F. MOORE AND C‘. E. SH,\NNON IJ. F. 1. see later that highly reliable computers can be constructed from a sufficient number of crummy relays if and only if u =j= c. THE GENERAL METHOD OF IMPROVING RELIABILITY In a genera1 way the analysis we wil1 give depends on constructing networks of contacts which act like a single contact but with greater reliability than the contacts of which they are composed. For example, in Fig. 2A, we have a crummy relay X with a make contact .L This relay might appear as a part of a large computing circuit. In Fig. 2B we replace this by four crummy relays X1, X?, Xij, X., whose coils in parallel replace the single coil X, and whose contacts are in the series parallel combination shown, this two-terminal circuit replacing the single previous x contact. If each of these four contacts bas the probability p of being closed, it is easily seen that the probability of the four-contact circuit being closed is h(p) = 1 (1 P’)” = LP? p”_ This function is plotted in Fig. 3. It wil1 be seen that it lies above the diagonal line y = p for p greater than 0.618 and lies below the line for FIG. 3. The function describing FIG. 4. .%nother series-parallel circuit and its ;isthe behavior of Fig. 2B. sociated function. p less than 0.618. This means that if 0.618 is between the a and c of Fig. 1, Fig. 2B wil1 act like a relay with better values of a and c, that is, values nearer to zero and one. For example, if the individual relays made errors with probabilities 1 a = c = 0.01, the circuit of Fig. 2B would make errors when the coils are energized with probability 0.000396, and when the coils are not energized with probability 0.0002. Thus a large improvement in reliability, both when the coil is energized and when it is not energized, is obtained by the use of this circuit. Figure 4 shows another contact arrangement giving rise to a somewhat different function h(P) = [l (1 p)2]2 = 4p2 4p3 + pa. Sept., 1956.1 CIRCUITS USING LESS RELIABLE RELAYS 195 Here again, h(p) is the probability of the network being closed, when the individual contacts each have probability fi of being closed. The network of Fig. 4 is the dual of that in Fig. 2, and the curve is that obtained by interchanging 0 and 1 in both abscissa and ordinate in Fig. 3. The bridge network of Fig. 5 gives rise to a symmetrical curve crossing the diagonal at p = O.S. For this network we have : h(p) = 2p2 + 2p3 5fi4 + 2~5. ,411 of these networks tend to accentuate the nearness of p to its values 0 or 1 and thus tend to improve reliability. Many other networks have similar properties as we shall see. Furthermore, we wil1 show that it is possible to find a network whose curve, Fig. 6, crosses the diagonal line for a value of p between any two given numbers a and c (no matter how close together) and in fact is less than 6 at a and greater than 1 6 at c, for any positive 6. This means that an arbitrarily good relay can be made from a sufficient number of crummy relays. It may be seen that this genera1 procedure operates to improve the reliability of either make or break contacts. The only differente is the labeling of the points a and c.