Applications of logical computers to the construction of electrical control tables for signalling...

Zeitschr. f . math. h g i k m d Grundagen d . Math. Rd. 4, S . 222-243 (1958)

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APPLICATIONS O F LOGICAL COMPUTERS TO THE CONSTRUCTION OF ELECTRICAL CONTROL TABLES FOR SIGNALLING FRAMES

U P - AD DC -DOWN LIB I I

By ALAN ROSE in Nottingham, England

Q 1. Introduction In 1951 MCCALLUM and SMITH built a logical computer [l], [2] which could solve

problems which were reducible t o problems in the 2-valued propositional calculus. The main components of the computer were decision elements and a searching mechanism. It has since been shown that the decision elements may be replaced [3], [4], [5], [8] by a universal decision element and that, in addition to the problems considered by MCCALLUM and SMITH [l], a number of other problems are solvable [3], [6] by means of logical computers. For some problems the computer must be adapted [6] to consider many-valued propositionalcalculi. The object of the present paper is to consider the application of the 2-valued propositional calculus to certain problems connected with railways.

A railway line is normally divided into sections of track lettered AA, A B , AC, . . . BA, BB, BC, . . . , CA, CB, CC, . . . . Usually a line will be “directional”, i.e. a train may travel on the line only from right to left (a “down” line) or only from left to right (an “up” line). A short distance before the beginning (i. e. left-hand end in the case of an up line) of certain track sections signals occur. To each signal is made to correspond a different positive integer and a signal is denoted as shown below.

137 9 P 16r,

207 I -0 218

APPLICATIONS OF LOGICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 223

line the points are said to be “normal”. Otherwise they are said to be (‘reversed”. If two pairs of points are interlocked so that they must be either both normal or both reversed i t is customary to assign the same integer to both pairs of points and to follow the integer by the letters A and B in the respective cases. However, a t this stage, i t is desirable to ignore this convention.

If a signal is at “danger” no train is permitted to pass it. If a signal changes to ‘(caution)’ or “safe)) i t is said to have been “cleared”. A “route” is defined to be the track occurring between two consecutive signals and the number of a route is the number of the signal occurring a t the beginning of that route. A route is pre-selected by the energising of its route-calling relay. This relay cannot be energised if the route-calling relay of a conflicting route is energised. If a route-calling relay is energised the points whose positions thereby become of importance are set and locked in their correct positions provided it is safe for this to be done. If it is not safe for a certain pair of points these points are, as will be seen below, locked in the opposite position. The correct setting of these points then takes place auto- matically when it becomes safe and, until it so becomes, the signal cannot be cleared. The electrical control tables contain, for each route, (among other things) under the main heading “requires” the headings’) “points N ” , “points R” and “route-calling relays normal”. In this paper we shall discuss the determination of the entries in these three columns and also the determination of direct inter- lockings of points. The latter interlocking is given in the tables without direct reference to any particular routes.

Two routes are said to conflict if either (i) A pair of points is set and locked by both routes, one route requires the points

(ii) The two routes contain intersecting track sections. It is customary to use the same lettering for two intersecting track sections.

However, for the purposes of this paper, i t is necessary t o distinguish between the two sections and we shall therefore attach distinct suffixes to the letters in the two cases. Similar considerations apply to a track section containing points. Our method of lettering in illustrated is the diagrams below.

normal and the other requires them reversed, or

DA,

l3- -czl It is important to remember, in connection with (ii), that trains sometimes fail

to stop in front of a signal a t “danger’), although the distance they are likely to

1) It is customary to abbreviate “normal” and “reversed” by “N” and “R” respectively. A relay is said to be “normal” when it is not energised.

224 ALAN ROSE

I A A UP -* AB r

overrun is not very great. The division between the last track section on the overrun of a route and the next track section is marked “1”. Thus, in the diagram given below, a train on route 105 might overrun signal 107, but we do not allow for the possibility that i t might occupy track A D by so overrunning. It is not customary

AC AD I T

’05 9

207 I

APPLICATIONS OF LOGICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 225

were to overrun signal 103 when points 201 were reversed a collision might occur. The entry “101 W 201R” in1) the “route-calling relays normal” column for route 104 does not, as a result of this, become unnecessary. If the route-calling relay for route 101 were energised before the attempt to energise the relay for route 104 and points 201 were reversed from the beginning, the energising of route-calling relay 101 would lock points 201 reversed, thus making it impossible for the ener- gising of route-calling relay 104 to set them normal. I n order to prevent the later moving (after route-calling relay 101 returns to normal) of points 201 by route- calling relay 104 without conscious action by the signalman, the entry in the “route- calling relays normal’’ column must still be made.

Points settings of this type are not made i f they would prevent the energising of a route-calling relay which could otherwise take place. No difficulty of this kind occurs in the above example since the only routes which, under (i), require points 201 reversed are 103B and (trivially) 101 W 20122 and these routes both conflict with route 104. Thus the conditions under which (iii) requires a pair of facing points normal (reversed) are :

(iiia) The points occur on the overrun of some route. (iiib) Every route which requires, under (i), the points reversed (normal) is

a conflicting route. I n some cases the conditions (iiia) and (iiib) might require the same pair ot

points t o be both normal and reversed. In such cases we require nothing of the points. Thus to (iiia) and (iiib) we must adjoin:

(iiic) There exists a non-conflicting route which, under (i), requires the points normal (reversed).

(iv) Wherever possible we must guard against the failure of the brakes on a stationary train. This failure might cause the train to move in the direction oppo- site to that of the track which it occupies (i. e. “set back”) and collide with another train. We do not allow for the possibility of a setting-back train travelling beyond the previous signal, or, in the case of a train standing on an overrun, the previous signal but one. Further, we do not allow for the less dangerous collision between two trains on the same track.

We must therefore impose, for trailing points (which, since we are considering setting back trains, become effectively facing points) conditions similar to (iii a), (iiib), (iiic). Clearly (iiia) has no counterpart here. On the other hand, i f two routes make the same requirement, under (ii), of the trailing points and the part of the first route traversed by a (non-accidental) train (i. e. one moving in the correct direction) before reaching the points is a proper subset of the corresponding part of the second route we should consider the second route and ignore the first, even though this may cause additional pairs of conflicting routes. Hence, before stating our conditions, we must define a route relevant to a given pair of points.

1) It is customary to abbreviate “when” by “w”.

226 ALAN ROSE

(iva) If the points do not occur on the overrun of any route then every route is relevant t o the points.

(ivp) If the points occur on the overrun of some route i then, if the points occur between the signal i and a signal immediately following signal i then route i is not relevant to the points. All other routes are relevant to the points.

Thus the conditions for a pair of trailing points to be required normal (reversed) are :

(iva) Every route relevant t o the points which requires, under (ii), the points reversed (normal) conflicts with the given route even if the given route is considered to have no overrun.

(ivb) There exists a route relevant to the points which requires, under (ii), the points normal (reversed) and which does not, when the given route is considered to have no overrun, conflict with the given route.

As will be seen above, we do not allow for the possibility of a collision between z t setting-back train and a train which has overrun its signal. As an example of the use of the conditions in (iv) we shall show that, for the track shown below, route lOlB requires points 1 normal.

UP - -u' I I I I 1 I I A

105

MAIN T I I T 1 I I I L 1

--DOWN 102 p

I I CF, UP - I I I / I

* -DOWN

We first note that points 1 are trailing points and that we must therefore consider (iv) rather than (iii). As regards (iva) we first note that the routes which, under (ii), require points 1 reversed are 102 and 104. Since points 1 occur on an overrun there will be irrelevant routes and it is easily seen that the only ones are 104 and 108. Since routes 102 and l 0 l B conflict (sections CF, and CF, intersect) even when we ignore the overrun of route 101B, condition (iva) is satisfied.

As regards (ivb) we first note that the routes which, under (ii), requirepoints 1 normal are 106 and 108, of which 108 is irrelevant. Since routes 106 and lOlB do not conflict we have, a fortior+, that they do not conflict when the overrun of route

APPLICATIONS OF LOQICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 227

lOlB is ignored. Hence condition (ivb) is satisfied. Since we have already shown that condition (iva) is satisfied y e conclude that route lOlB requires points 1 normal.

As a second example we shall show that, for the track shown below, route lOlB requires nothing of points 2. The irrelevant routes are 104 and 108. The only re- levant route which requires under (ii), the points normal (reversed) is 106 (102). Since routes lOlB and 106 conflict (even ignoring the overrun of route lOlB) condition (ivb) forbids us to require points 2 to be normal. Since routes lOlB and 102 conflict (even ignoring the overrun of route 101B) condition (ivb) forbids us to require points 2 t o be reversed. Thus route 101 B requires nothing of points 2.

P '03 -- DOWN I

lo' 9 MAIN

T UP -

We note that, in view of our formulation of (iii), this condition does not cause any additional pairs of conflicting routes. As regards (iv) the situation is the same except that irrelevant routes of (iva) may now be conflicting routes even i f they were not so previously. After the completion of the work of 5 5 the computer may be used to determine the additional conflicting routes. We have only to imitate the methods of 5 4, altering the assignments of truth-values to the vj and wi in the same way as we alter the assignments t o the xi and yi in 5 6. As an example of the above type of confliction we note that, on the track used in the first example of (iv), routes lOlB and 104 require points 1 normal and reversed respectively and therefore conflict.

As an additional precaution we sometimes require that, for two integers i, i: (a) Pointsi cannot be moved from N to €2 when pointsj are N . (b) Points i cannot be moved from N to R when points j are R. (c) Points i cannot be moved from R to N when points j are R.

(d) Points i cannot be moved from R to N when points j are N , It is ?%ever necessary

since two pairs of points set a t N can only be used to cause parallel movements

to impose a condition

228 ALAN ROSE

of trains, Thus, as may be seen below, the set of all conditions (a) is not necessarily logically equivalent to the set of all conditions (c).

Corresponding to (a), (b), (c) our electrical control tables contain, for each pair of points i, the columns headed “ N - R requires points R”, “ N - R requires points N ” , “R - N requires points N” respectively. I n these columns the relevant points i are entered. If however, for t’he ordered pair of integers (i, j ) conditions (a) and (c) both apply, i t follows at once that the settings of points i and i must always be identical and the movement of the two pairs of points is controlled by the same mechanism. These points are finally renumbered iA, iB respectively.

No condition of the form (a), (b) or ( c ) is ever imposed if, by doing so, an extra pair of conflicting routes would be created. Also, as in (iii) and (iv) above, no con- dition (a) ((1))) is ever imposed if there are equal grounds for imposing (b) ((a)). It may, however, happen that (a) ((b)) applies to (k, i ) but not to ( i , j ) and that (a) and (c) both apply to (i, k ) . In this case (i, k ) is treated as ( iA, iB) and we effectively apply (a) ((b)) t o ( i , j) although this is not (and need not be) shown explicitly by the outputs of the computer. Similar considerations apply to points conditions (i), (ii), (iii), (iv).

Thus for (a) ((b)) our conditions are ( a I ) ((bI)) Whenever two routes require respectively the points i reversed and

(aII) ((HI)) There exist two non-conflicting routes which require respectively

Similarly for (c) our condition is: (cI) Whenever two routes require respectively the points i normal and the

the points j normal (reversed) the two routes conflict.

the points i reversed and the points j reversed (normal).

points j reversed the two routes conflict.

Q 2. Basic Programming Let the number of routes and the number of pairs of points under consideration

be N , M respectively. We shall denote by E , F the set of integers (possibly followed by one or more letters) which are numbers of routes and the set of integers which are numbers of points respectively. Thus E n F = 0. Let G denote the set of letter- pairs (possibly with attached suffixes) which are names of track sections. We shall use propositiond variables

pi, qa, ri ( j E F, N E G‘, i E E )

t o denote the following propositions : pi “Points j are normal.” qa “NO train is permitted (even by accident) to leave track section a and proceed

ri “Route-calling relay i is energised.” The qa are used in order that certain fictitious situations may be considered

(e. g. in connection with the absence of certain overruns in (iv)). In real situations the qa are all false.

to the next section of track.”

APPLICATIONS OF LOGICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 22‘3

If a route has more than one overrun we shall regard it as n routes i X , i Y , . . . ,

We first use decision elements to construct decision mechanisms corresponding

A , “Trackn is accessible to a train obeying a signal preceding track a, no

B, “Trackcs is accessible to a train overrunning a signal referred to in tlle

The formulae A, and B, may be constructed according to the following rules: R1. If track ix immediately follows the signal i and signal i controls the routes

iL,, iL,, . . ., iL, ( T L ~ l), where L, denotes the w t h letter of the alphabet ((0 = 1, 2 , . . ., IL), then

n being the number of distinct overruns.

t o the formulae A , , B, . The latter formulae are interpreted as follows:

other signal occurring between this signal and track a.”

previous definition, but not overrunning beyond the point marked ‘I3.*’

L ,

a

Also, if track 2 immediately pre- cedes track a then

a - dn I

23 0 ALAN ROSE

The proof that these constructions are valid is similar to that given for B, in R 1 . If B, is undefined then no train overrunning on to track a can possibly further overrun on to trackp, since no such train exists.

R3. If tracks $ and y immediately follow track a and the facing points i occur a t the junction of the three tracks and the points, when set normal, connect tracksa and B then

A, = A , (Pi 3 q A > 8, = -4 L% (Pc q d , Bp = B, & (pi 3 no) (if B, is defined),

By = B, & (p i C q,) (if Ba is defined).

B3 und B;, are undefined if B, is un- defined. I I

- Thesc constructions effectively say that, in addition to the fiction that no train

may leave track a being disregarded, a non-accidental (accidental) train may properly (accidentally) reach track p , when travelling in the correct direction, i f and only if it may reach track a: and the points are normal and a non-accidental (accidental) train may properly (accidentally) reach track y , when travelling in the correct direction, if and only if it may reach track (Y and the points are reversed (i.e. pi takes the truth-value F ) .

R4. If tracks p and y both irnme- diately precede track a and a pair of trailing points occurs a t the junction of the three tracks then

4

B, = (BP D q p ) v (BY D n,) \ P a A a = ( - 4 ~ D 48) v (A, 3 nr ) ,

“ C 3 J (if Bp and By are defined). If Bb and By are undefined1) then B, is undefined.

These constructions effectively say that, for a train to reach track a: it is neces- sary and sufficient for i t t o reach one of the tracks $, y and be not prevented froin leaving that track by a fictional prohibition. , R5. If track $ immediately follows track a and the end of track M. is the end of one or more overruns t,hen

T P - - 4 R = A , 3 q a . a

B13 is undefined. Since the end of track (Y is the end of the overruns no train can possibly overrun

onto track@. Thus Bp is undefined. The proof for AP is the same as that given in the proof of R2.

I

l) Normally the overruns on to the two routes containing tracks p and y would end st the same point (unless they both ended before points j were reached). Thus, usually, we do not need to consider cases where exactly one of BPI By is defined. Similarly the only cases of R7 will normally be (i) and (ii).

APPLICATIONS OF LOGICAL COMPUTERS TO TEE CONSTRUCTION OF CONTROL TABLES 231

Having constructed, by means of R1-5, decision mechanisms for all the A , and B, (a E G) we connect to the inputs of a If decision element (or, more probably, several 17’s combined since the number of inputs will be large) outputs of other decision elements as follows :

R6. If tracksa and f l intersect then

(i) If B,, Bp are both undefined we connect an output corresponding to the formula

A a l A p

to an input of the 17 decision element. This output is obtained by connecting the outputs of the decision mechanisms for A , and A, t o the inputs of a “/”decision element.

(ii) If B, (Bp) is defined and B, (B,) is undefined we treat similarly the formula

A , v B E I - 4 ( 4 3 v BBI4).

A , v BAA,. Aa/Bp.

(iii) If B,, BN are both defined we treat similarly both the formulae

Conditions (i), (ii), (iii) ensure that two intersecting tracks cannot be occupied simultaneously (except by two trains, both of which have overrun their signals).

- ma1 we consider, as in R6, the \ B Y

R7. If tracks a and /3 both im- mediately precede track y , the pair of trailing points number i occurs at the junction and tracks f l and y are connected when the points are nor-

formulae determined be1ow.l) The functor { , , } denotes the “conditioned incompatibility” functor whose truth- table is given by

--C3i

{P, Q, R) =T -[P, Q , RI.

Many universal decision elements may be made to correspond to this functor. Thus, for example, if

@(Pt Q, R, 8, T) = p [P b Q, R, 8 3 7’1 we may make the definition

(i) If B,, Bp are

l) See the previous

{J’, Q , R) =a. @(I7 P, Q , 1, 3).

{ A , , Pi, A@) .

both undefined we consider the formula

footnote.

232 ALAN ROSE

(ii) If B,,

R7 ensures that the trailing points are set in such a way that they can be used to deal with B train approarhing them. It also ensures that the points cannot be approached simultaneously by trains on tracks a and p. This follows at once since the formula { P , p a , &} takes the truth-value F in each of the following three cases :

arc both defined we consider the formula

{A, v B,, p t , A, v BPI.

(a) p h , P both take the truth-value T . (p ) p % , Q take the truth-values F, T respectively. ( ;J) P, Q both take the truth-value T . In certain circumstances R7 causes tmo routes to conflict, even if the routes

have a common point only on their overruns. If one route has track a as part of its overrun and the other route has track p as part of its overrun then, i f the two routes were set up simultaneously, the forniulae B,, B, would both take the truth-value T. Hence the formula

{AavB.z,pa,AfivBp) would take the truth-value 1’. It seems desirable that, in these circumstances, the two routes should be regarded as conflicting routes. If we set up a route we must specify the positions of the trailing points which occur on the overrun of that route. Thus, in the above case, it would be impossible to give a satisfactory setting of points i if both routes were set up simultaneously. As an example of this type of occurrence we shall show that, with reference to the diagram for the second example of the discussion of setting back trains, routes 102 and 106 conflict. We first note that points 2 occur on the overruns of both routes. Since they are trailing points route 102 requires them reversed and route 106 requires them normal. Hence routes 103 and 106 conflict.

We shall now, tor each route i, construct a formula Ft which asserts that the route is accessible up to a certain position, this position depending on the truth-values of the q b . If route i consists of sections iy?, . . .. a2 of which sections & k + j , akf2, . . ., ( ~ 1 forin the overrun (16

Thus if Ya l , qa*. . * a , qn,e-l take the truth-value P and qz%, q, ,+ l , . . . , q., truth-value T then

1) then

Fi = Ae1 & n$ii (A ‘j+ 1 v q,,) q:; (B,,+l v Ye) . take the

Fi =TTIy=lA.l ( n = 1, 2, . . ., L), Fi = F I Z f = l (L4,j) & IIyck+i(Ba) (11. = k + 1, k + 2, . . . , 1 ) .

Hence, in this case, Fi asserts the accessibility of route i and its overrun, together with the fiction that the route (or overrun) ends after the nthsection. We note that, if we increase the value of n by 1, the number of requirements relating to points which occur on the route and overrun cannot increase by more than 1. This fact enables 11s to determine these requirements one by one.

Let us denote by C the formula corresponding to the output from the 17 decision element referred to a t the beginning of the discussion of R 6 and R7. For each

APPLICATIONS OF LOGICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 233

route i we use an “&” decision element to construct a decision mechanism for the formula

and we connect the inputs corresponding to the variables qz , q m z , . . ., qa,-l to a scanning mechanism in which 1 assignments are scanned. For the n t h assignment (n = 1, 2, . . . , I) qal, qa2, . . . , qan-l take the truth-value F and qa, z.,+~, . . ., qal - l take the truth-value T. Throughout the search ri takes the t.ruth-value T and the remaining q’s and r’s all take the truth-value F .

$j 3. Determination of the conditions to be satisfied for a route by points occurring on the route and overrun

Let the numbers of the route under consideration and of one of the routes on to which the given route overruns be i, i* respectively. We first note that, a t the n t h position ( 1 2 n s I) of the above scanning mechanism, the route is considered to

‘7 ‘*7 a k + l ak+2 011 altl

I OCk I a1 a2 I

I I I I I I I 1 1 1

(Points and intersecting tracks are not shown in this diagram)

terminate a t the end of sectiona,. For each position the machine gives outputs Ni ( R j ) corresponding to the propositions “Points number i must be normal (re- versed)” ( j € F ) . Since the set of requirements for the n t h stage is a (proper or improper) subset of the set of requirements for the (n + 1) t h stage no output which, a t the end of the n th stage, corresponds to T can, a t a later stage, correspond to F . Thus the outputs Ni and Ri may be preserved by flip-flops [7]. A flip-flop will be denoted by “F. F.” and its re-set input wire will not be shown in diagrams.

During each stage of the above search we shall carry out a second search. Let the numbers of points occurring on the whole track under consideration be a,, . . . , a M . If, at the end of the n th stage of the first search, the values of j for which N! takes the truth-value T are b, , b,, . . . , b, and the values of j for which Rj takes the truth-value T are the members of the set then, at the mth stage of the second search, pb,, p b 2 , . . ., ?)b, take the truth-value T,

pam takes the truth-value T i f and only if a, $: c, pj takes the truth-value F ( j E F ; j =+ b,, b , , . . . , b,, urn).

Thus we consider, for each value of m ( 1 5 m s M ) in turn, whether points number urn are the points (if any) whose position is of importance when the route is considered to have n + 1 sections but is of no importance when the route is considered to have only I L sections.

I n order to vary the truth-values of the pi in the above way we use a second scanning mechanism which provides inputs corresponding to variables 81, s2, . . . s x . At the mth stage s, takes the truth-value T and sl, s2, . . ., a,-1, s,+~, . . ., s~

16 Ztschr. f . math. Logik

234 ALAN ROSE

@ F. F. 1 + - C&

APPLICATIONS OF LOGICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 235

1

IJ ’ & ~ F.F. 2 Ed.

236 ALAN ROSE

MAIN

I CG I Ch,

At the end of each search by the second scanning mechanism the two first flip- flops and the third flip-flop are, of course, re-set. It would seem desirable for de- cision mechanisms for all C & Fi (i € E ) to be set up initially and for an automatic process to connect these decision mechanisms successively to the Ni and Ri decision mechanisms. This automatic mechanism must also connect the correct variables qg to the first scanning mechanism and re-set the two second flip-flops. Thus if i < i f , i f € E and there is no member if’ of E such that i < i f t < it then the decision mechanism for C & li’c is connected to the Ni and Ri decision mechanisms as soon as the machine process for route i is completed. At the same time the qa relevant to route i are disconnected from the first scanning mechanism and the qa relevant to route if are connected. Thus, for the track shown below, we should use the circuit shown in the appended diagram. (In practice a much larger area of track would be considered.) As we ore not considering routes occurring to theleft of routes 101 and 107 or to the right of route 102 no trains are considered liable to overrun on t o routes 101, 102 and 107. Routes lOlX and 107X are considered to overrun on to route 103,4 and routes 101 Y and 107 Yare considered to overrun on to route 103B.

B UP MAIN M A I N

, CKl Ch, CL I

4 RELIEF log 4

I -DOWN BF, \

APPLICATIONS OF LOQICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 237

1 Nl 1 R1 N2 I R2 1 N , 1 R, 1 N.4

Route101X T F F I T F F F Route 101 Y ~ T i F F I T F F T Route1012 I T ~ F T ! F i T F 13’

R4 1 Ns I R5 T I F F F F F F 1 F F

Requires

Points N I Points R Route No.

~

101 -~ ~

1, 2* 2* ( 3 w 2 N ) (4 w2R)* (4 w 2 R ) *

8 4. Determination of those routes which conflict with a given route (other than routes caused by points condition (iv))

We again carry out a scanning process with respect to the formula C & Fi , i being the number (possibly followed by one or more letters) of the given route. We set all the qa a t F and carry out a scanning process such that if the numbers of the routes on the track are b , , b , , . . .) blv then, a t the kth ( 1 5 k l N ) assignment of truth-values considered,

r i , rb , take the truth-value T, rq takes the truth-value F (7 E E ,

For this assignment the truth-values of the pj are determined as follows: (i) If one of the routes i, bk gave N j the truth-value T and the other gave Mj

the truth-value T then p i takes the truth-value T if and only if route bk gave N , the truth-value T.

+ i, bk) .

238 ALAN ROSE

(ii) If the hypotheses of (i) do not both apply then p j takes the truth-value T if and only if a t least one of the routes i, bk gave Ni the truth-value T.

Thus the settings of the points are correct for the simultaneous preselection of routesi and bk except that, i f the two routes require incompatible settings of a pair of points, the setting for route bk is provided.

The scanning process is carried out by using a third scanning mechanism which provides inputs for variables uh,, ub2, . . . , t.6bN, val, v, , . . . , v U M , wU1, wU2, . . . , toadl.

At the kth assignment ‘1cb, takes the truth-value T and %bl, . . , % b k - l , %hk+l, . . , ub, all take the truth-value F (k = 1, 2, . . ., N ) . The variable w j (wj) takes the truth-value T if and only if route bk gave N j (Rj) the truth-value 1’. The asfiignments for the vj and wj may be recorded magnetically during the process for 0 3 and the third scanning mechanism automatically prepared and brought into use when the whole process of $ 3 is completed. It is clear that the inputs for the variables r b . , rb, may be provided by decision mechanisms for the formulae ub, v Xb , . . . , ub, v zb, respectively, where zi is preset a t T and the variables z9 (4 + i, bq E E ) are preset a t E’. Similarly the inputs for pu , . . ., pax may be provided by decision mechanisms for the formulae vU1 v (Nal 3 wUl), . . . , vaM v (NaM 3 wax) respectively. The inputs Zb, , . . . , z b , , N U 1 , . . . , NaM , E l a 1 , . . . , RaM may be obtained by duplicating the third scanning mechanism and, during the process for route i, setting the duplicate mechanism a t the 5th assignment (i = b,) and using the variables u, v, w as inputs for the corresponding z, N , R respectively.

The route bk will conflict u-ith route i if and only i f , a t the kth assignment, the formula C & B’< takes the truth-value F . (Our construction autonlatically gives Pbk the truth-value T.) Thus the formula C b k , denoting the proposition “route bk

is a conflicting route”, will take the truth-value T i f and only if, a t some stage (the kth is the only possible one) the formula

1 ’ . .

rb, 3 & Fi

takes the truth-value T. Hence outputs corresponding to Cbk ‘ >yHT]- ‘s, (bk E E ) may be obtained from

C& F; the decision mechanisms shown on the left.

We may then make our entries in the “route-calling relays normal” column as follows :

(i) If routes i , bk each have only one overrun and CbE takes the truth-value T then route bk is entered.

(ii) If route i is treated as routes iX, i Y , . . . , route bk has only one overrun and all these routes give Gbk the truth-value T then route bk is entered.

(iii) If route bk is treated as routes b k X , bkY, . . . , route i has only one overrun and route i gives all of C b k X , Cb, y , . . . the truth-value I’ then route bh is entered.

(iv) If route i is treated as routes i X , i Y , . . . , route bk has only one overrun and some, but not all, of these routes give the truth-value T then route bk is entered

APPLICATIONS OF LOGICAL COMPUTERS TO TEE CONSTRUCTION OF CONTROL TABLES 239

conditionally. The condition is obtained by a method similar to that used in the previous example. Thus i f route i is the route 101 of the previous example and routes lOlX, 101 Y , 1012 give the formula C,,, the truth-valuesP, F , T respectively we enter in the “route-calling relays normal” column “102 w 2”’. Similarly, by (ii), if routes 101X, 101 Y , 1012 all give C,,, the truth-value T then “104” is entered.

(v) If route i has only one overrun, route bk is treated as routes b k X , bkY, . . . and route i gives some, but not all, of C b k X , C b k Y , . . . the truth-value T then route bk

is entered conditionally. As in (iv) the condition will relate to the positions of points which differ for different routes b k X , bkY, . . . . Thus, for example, if route 112 gives C, , ,x , C,,, p, Clolzthe truth-values F , F, T respectively we enter in the “route- calling relays normal” column for route 112 “101 w 2”’.

If the conditions stated in (ii) or (iv) apply except that route be, has more than one overrun and i f , further, all propositions Cb, have the same truth-value then the entry is the same as in (ii) or (iv) respectively. Similar considerations apply to (iii) and (v) if route i has more than one overrun. We shall refer to the four cases derived above from cases (ii), (iii), (iv), (v) respectively as cases (vi), (vii), (viii), (is). Since two routes whose overruns intersect are not regarded as confliding, cases (i)-(ix) deal with almost all entries. Similar methods apply to ot’her cases.

As an example of the use of (ix) let us suppose that routes 139, 143 are treated as routes 139X, 139 Y , 1392,143X, 143 Y and that routes 143X, 143 Y give N,, , R,, respectively the truth-value p. Let us further suppose that routes 139X, 139Y, 1392 all give C,, ,x , C,,, y the truth-values F , T respectively. We then enter in the “route-calling relays normal” column of route 139 “143 W 47R”.

When the computer indicates a confliction between two routes whose numbers are (for the sume value of i) of the forms iX, iY we do not, of course, require a corresponding entry in the “route-calling relays normal” column for route i. How- ever, i f routes iX and iY overrun onto routes i*A and i*B respectively and the facing points j occur on the overruns of routes i and are normal (reversed) for

z T

J: RELIEF P

240 ALAN ROSE

route iX (iY) then the computer will indicate, among other things, that entries a A w jR” should be made in the "route-calling relays normal” columns

of routes i*A, i respectively. These entries will be indicated by outputs corresponding to T for theforniulae Cip, C i e ~ when the computer considers routes i*A, iY respec- tively. These entries should not be disregarded.

If we wish to set up route i*A we must be able to set the points j normal but, if route i is already set up with the points j reversed, the points j are locked in the “reversed” position and cannot be set normal. If, in these circumstances, we did not require the entry “i W jR”route-calling relay i*A might later (after the deenergising of route-calling relay i) move points j without conscious action by the signalman.

If we wish to set up route i and route calling relay i*A is already energised then points j would normally have been set normal automatically. However this may not have been possible (e. g. if points j and j* were interlocked by (cI)) and, if the entry “i*A w jR” were not made, a serious delay in the setting up of the already preselected route i*A might result.

w jR”, ii ‘*

D D *j D 4 ( ~ j b 4)

APPLICATIONS OF LOGICAL COMPUTERS TO THB CONSTRUCTION OF CONTROL TABLES 241

fi (Vi) takes the truth-value T if and only i f route bARR gives Ni (Rj) the truth-value T.

only if route bA gives C k R the truth- P J

i

Since route bL is relevant t o the points j and makes, under (ii), a requirement of the points if and only if, for the iith assignment, one of the formulae xi D zj, yi ) yj takes the truth-value T we have that condition (iva), in the N r case, is equivalent to the assertion that, for each of the N assignments considered, the formula

takes the truth-value F. Similarly condition (ivb) is, in this case, equivalent to the assertion that, for at least one of the N assignments considered, the formula

( ~ j D B j ) 3 &

(Xj D Z j ) 3 di

242 ALAN ROSE

takes the truth-value T. Similar considerations apply to the RY case and t,he final outputs therefore correspond to NY and RY as required.

Additional entries in the “route-calling relays normal” column may then be determined by a repetition of the method of 94.

Q 6. Determination of the requirements for the interlocking of points In this section of the work we use a fifth scanning mechanism. This is identical

with the first form of the fourth except that, in the definition of the mechanism, “N;’ and “R;’ are replaced by “at least one of N j , N i , NY” and “at least one of R j , R:, RY” respectively and we use the form of Ck determined in the last para- graph of $5. We also use variables r: , T ; . . . . , r i 7 . At the Ath assignment (1 s As N ) ri takes the truth-value T and the remaining rf ’s take the truth-value F.

For each pair of points i our first step is to set up, for each route b, , the mechanism shown below and carry out a search with the fifth scanning mechanism.

Route b, will require the points k normal (reversed) i f and only if, for some assignment considered, the formula xi & r:, (yi & r;) takes the truth-value T. Thus the outputs xb (Ya ) correspond to the pro- positions “points i are required normal (reversed) by route b;’.

Thus our points condition (aI) is equivalent to the assertion that, for all of the N assignments considered, the formula

P P

“j 83 z;==l( y b , 3 d b v )

takes the truth-value F . Similarly condition (aII) is equivalent t o the

assertion that, for a t least one of the N assignments considered, t he formula

yj 83 zL1 ( y b , db,)

takes the truth-value T . Clearly the treatment of (bI) and (bII) is the same as the above except that the roles of the two formulae are interchanged and the treatment of (cI) is the same as that of (bI) except thateachinput ya, is replaced by the corresponding input X b v . Thus we may use the circuit shown below, the outputs Rj”, N,!”, N T being interpreted as follows :

Rj” If points i are to be moved from N to R points j must be R. N Y If points i are to be moved form N to R pointsj must be N . Ny If points i are t o be moved from R t o N points j must be N .

APPLICATIONS OF LOGICAL COMPUTERS TO THE CONSTRUCTION OF CONTROL TABLES 243

The outputs Xb Y b obtained above are preserved and used as inputs for decision mechanisms for the formulae

l L P

cf=l (yb, 3 dby)> cf==l (xby 3 d b y ) *

x* 2 :-I cy., Pd+I yi 2 != 1 Ix8vp %,

References 111 D. B. MCCALLUM and J. B. SMITH, Mechanised reasoning - logical computers and their

design. Electronic Engineering 23, 126-133 (1951). [2] D. B. MCCALLUM and J. B. SMITH, Feedback logical computers. Electronic Engineering

23, 4 5 8 4 6 1 (1951). [3] J. M. PUGMIRE and A. ROSE, Formulae corresponding to universal decision elements.

This Zeitschr. 4, 1-9 (1958). [4] A. ROSE and J. E. PARTON, Improvements in or relating to decision element circuits.

British Patent Application No. 28685/56. (This patent has been assigned to the National ResearchDevelopment Corporation who have also made aPatent Application in theU. S. A.).

[5] ALAN ROSE, Sur les BlBments universels de dbcision. C. r. Acad. Sci. Paris 244, 2343-2345 (1957).

[6] ALAN ROSE, Many-valued logical machines. Proc. Cambridge Phil. SOC. 54, 307-321 (1958).

[7] ALAN ROSE, The use of universal decision elements as flip-flops. This Zeitschr. 4, 169-174 (1958).

[8] BOLESEAW SOBOCINSKI, On a universal decision element. J. Computing Systems I, 71-80 (1953).

(Eingegangen am 23. April 1958)