«i- jffiz*^^^ Y ^^ 1 JK/^«^^--^-^K // (sC£&&£<pa~>&. F irct Drift of a Report en the FLVAC f ';' John von ! T Contract No. W-670-ORD-4926 Between the United States Army Ordnance Department and the University of Pennsylvania V r re ochool of Electrical Engineering university of Pennsylvania June 30, 1945 National Bureau of Standards Division 12 Data Processing Systems 5jW Smithsonian Institution Libraries Gift of PAUL CERUZZI c c C" DNTENTS » Sectio; ? p(SS 1 -° 3EFIII] 00IX3B i#1 Auton u Lgitalc : ,. _ ScD(ftJ3 |_ Exact :i ; :r t I i ! • . ions of such systoni i 1.3 Die " i ■■ i' Ln th> r m\ rica] rial pr iucod by such a s: ; .L Ch ck Lng ■ i :errecti ng dfund ( n r I , au1 Lbi - UAIN SUBDIVISIONS M I: E SYi 5, i ... 5 .3 5, .4 5. .: 5 .6 5 .7 weed for subdi /isions 3 First: Central ariU^ tic part: CA ;ond: Control control part: CC 3 . ... Third: '.'■ rious of memory r : (a) - i it Third: (Coat,.) Memory: M 6 2,< , CA (tcgc.1 r: C), ..' ar together Lh jiati 1 part. Af j ind jfforent pari : [npul put, mediatin 4 : i1 id .v.; i, Lh i aside. Outsid recording medium: It o 2 . 7 Fourth Input : I 7 Fifth: Dut] ut: 7 • orison of M and R, considering (a) - (h) in 2.4 7 3.0 FROCEDURE OF DISCUSSION 3.1 Program: Discussion of all divisions (specific parts) at i in 2, together with the jsscntial decisions 9 if :• "Zigzag" iiscussion of the specific parts 9 3.3 Automatic checking of errors 10 4.0 ELELENTS, SYNCHRONISM NEURON ANALOGY 4.1 Role of rel">y-like elements. Example. Rcle of synchronism LO 4.2 Ne iron; . synapses, excitatory and inhil Ltory ty; 12 .'...: Desirability of using vacuum tubes of the conventional radio tub", type 5.0 PRINCIPLES &VERNING THE A] [C. TIOM Vacuum tub* elements: Gates or triggers 14 Binary vs. I icimal system 15 Duration of binary multiplication •• 16 Tel' • perations vs. saving, equipment 1' Role of very high speed (vacuum tubes): Pri) of successive operation. Time estimal n of the princif le F irther discussion of the principle 20- Rational Bureau of Standard* Division 12 Data Processing Systems ( (" ( u a C N T E N T S Section . D E-ELEME'ITS 6.1 Re isons for the intr lucti r • thetii ment 21 6.2 1 . j rip' Lon of the s ' \ . : - - ■ . I ynchr dsm. :ating 1 ;:.•..._: r.k The r .. . E-el •. vii Lpli thresholds. 1 1 '-ys ?/, Com] ;ith vacuum tul . 7.-' CIRCUITS FOfi THE ARITHMETICAL OPERATI WS +, X 7-1 i ! of fe lin;» .. binary i s: ! i -. • iporal s ucc 05 si 7.2 E- ler at netv, id bl ck syml > <ia 7-3 The adder J-L, The multiplier: Memory r ;uirements 23 Lscussion of the lory. D< Lay, ^9 Discussion of delays 30 7.7 plier: Det died structure 31 f.8 . ■ .:_ : . Lier: Further requirements (timing, local input id output). 33 3.0 circuit: for the arithmetical operations -, -r, 3.1 Treatment of the sign 34 8.2 The subtracter 35 . : '.3 The divider: Detailed structure 36 8.4 The divider: Further requirements (cf. 7.?) 38 9.0 THE BINARY POINT 9.1 The main roi^ of the binary point: For X, f 39 7.2 Connection with the necessity of omitting 15 ;its ifter x. Decision: Only nuiribers between -1 and 1 9.3 Consequences Ln planning. Rule:-: for th< peration + , — ', x , -r A.1 /.4 Rounding off: Rule and E-element network < 42 13.0 . . FOR THE ARITHME PERATION-/. OTHER 01 1 .1 The square rootor: Detailed structure 43 • square rooter: Further "observati o List • of operations: +', -', X 1 , f . i,4 Exclusion of certain further operatic! *> , ( c CONTENTS Section 11.0 ORGANIZATION OF' CA. CQKPLETE LIST OF OPERATIONS 11.1 Input and output of CA, connections with M f>0 11.2 The operations i, j. 52 11.3 The operation s .,53 11.4 Complete list of operations: +, — , X, -r, sf , i, j,. s and conversions 55 12.0 CAPACITY OF THE MEMORY M. GENERAL PRINCIPLES 12.1 The cyclical (or delay) memory 5b 12.2 Memory capacity:- The unit. The minor cycle. Numbers and orders 57 12.3 Memory capacity:' Requirements of the types- (a) - (h) of 2.4 53 12.4 Memory capacity: Total requirements 64 12.5 The delay memory: Physical possibilities 65 12.i6 The delay memory:' Capacity of each indiv idual | dl | and the multiplication time.- The number of ] dl ]'"s needed 68 12.7 Switching vs. temporal succession 12 12.8 The iconoscope memory 73 13 .-0 ORGANIZATION OF M 13«1 \ dl / and its terminal organs A and SG 79 13.2 SG and its connections 80 13.3 The two on states of SG 81 13.4 SG and its connections: Detailed structure 82 13.5 The switching problem for the SG 83 14.-0 CC AND M 14.1 CC and the orders 84 14.2 Remarks concerning the orders (b) 86 14.3 Remarks concerning the orders (c) 86 14.4 Remarks concerning the orders (b); (Continued) 87 14.5 Waiting times. Enumeration of minor and major cycles. 88 15.0 THE CODE 15.1 The conterits of M 91 15.2 Standard numbers 91 15.3 Orders , 92 15-4 Pooling orders 96 15.5 Pooling orders. (Continued . 97 15. b Formulation of the code 98 # i ? I G U R E S Paee 1. Synchronization - clock pulses 24 2. Threshold 2 nearon by combining Threshold 1 neurons 25 3. Adder 27 h. Elementary memory. (E-element) 29 5. Line of E-elements " 29 6. Same with gate network 30 7. Simple valve - 31 3. Simple discriminator .32 9. Multiplier 32 10. Complement valve 36 11. .' Subtractor , 36 12. • ' Complete discriminator 37 13. ' Divider 37 14. Rounding off valve 42 15. Square rooter 44 16. Input and output of CA 50 17. Connections of input and output in CA 52 18. Amplification, switching and gating scheme of a I dl 69 19. Individual and s erial cycling' of a j dl | aggregate (a) , (b) 69 20. Connections of a j dl \ in detail 80 21. SG', preliminary form of SG 82 22. Supplementary connections of the L 83 .•.: ■'~^ X*0 Definitions'. ' .,■• 1.1 The considerations which follow deal with the structure of a very high speed automatic digital, computing system , and in particu- lar with its logical control . Before going "into specific 'details , so.-.e general explanatory remark* regarding these concepts may be appropriate. 1.2 An automatic computing system is a (usually highly com- posite) device, which can carry out instructions to perform calculations of a considerable order of complexity - e.g. to solve a non-linear par- tial differential equation in' 2 or 3 independent variables numerically. The instructions which govern this operation must be given to the device in absolutely exhaustive detail. They include all • -• ■ ••■■ ft ..,' ; . /,,•' •: ,.■ I-:. numerical information which is required to solve the problem under con- sideration: Initial and boundary values of the dependent variables, values of fixed parameters (constants),, tables of fixed functions which occur in the statement of the problem. These instructions must be given, in some form which the device can sense: Funchc-d into a system of punch- cards or on teletype tape, magnetically impressed on steel tape or wire, photographically impressed on motion picture film, wired into one or more, fixed or exchangeable plugboards - this list being by no means necessar- ily complete. All these procedures require the use of some code, to- express the logical and the algebraical definition of the problem under consideration, as well as the necessary numerical material (cf. above). Once these instructions are given to the device, it must be able to carry them out completely and without any need for further intelligent human intervention. At the end of the required operations the uevice must record the results again in one of the forms referred t*> c c c } above. The results are numerical data; they arc a specified part of the numerical material produced by the device in the process of carrying out the instructions Referred to above, 1.3 It is worth noting, however, that the device will in general produce essentially more numerical irate rial (in order to reach th : results) than the (final) results mentioned. Thus only a fraction of its numerical output will have to be recorded is indicated in 1.2, the winder will only circulate in the int< rior of the; device, and never be recorded for human censing. This point will receive closer consider- ation subsequently, in particular in \.U The remarks of 1*2 on the desired automatic functioning of the device must, of course, assume that it functions faultlessly. Malfunctioning of any device has, however, always a finite probability - and for a co-plicated device and a long sequence of operations it may not be possible to Keep this probability negligible. Any error may vitiate the entir output of the device. For the recognition end cor- rection of such malfunctions intelligent human intervention will in general be necessary. However, it may b possible to avoid even these pheno- mena to some extent. The device may recognize the mbst frequent mal- functions automatically, indicate their presence and location by exter- ly visible signs, and then stop. Under certain conditions it eight even carry out the necessary correction automatical!;: continue. (Cf. .) _o_ ' !► V 2.0 Main subdivision of the system ■ 2.1 In analyzing the functioning of the contemplated device, certain classificitory distinctions suggest themselves immediately. 2.2 First: Since the device is primarily a computor, it will have to perform the elementary operations of arithmetics most f re- ■ quontly. There are addition, subtraction, multiplication and division": +■ , -, x,-f . It is therefore reasonable that it should contain special- ized organs for just these operations. It must be observed, however, that while this principle as such is probably sound, the specific way in which it is realized requires close scrutiny. Even the above list of operations: +, -, x, -r f is not beyond doubt. It may be extended to include such operation as../*", , sgn, 1 1, also -^log, log, In, sin and their inverr.es, etc. One might also consider restricting it, e.g. omitting - and even x. One might also considrr more clastic arrangements. For some operations rad- ically different procedures are conceivable, e.g. using successive ap- proximation methods or function tables. These Matters will be gone into in . At any rate a central arith - metical part of the device will probably have to exist, and this consti- tutes the first specific part: CA . 2.3 Second: The logical control of the device, that is the ■ proper sequencing of its operations can be most efficiently carried out by i central, control organ. If the device is to be elastic , that is a^ nearly as possible '.11 purpose , then a distinction must be made between the specific instructions given for and defining a particular problem, and the generc-l control organs which see to it that these instructions - -3- no matter what they are^- are carried out. The former* must be stored in some way - in existing devices this is done as indicated in 1.2 - the latter are represented by definite operating parts of the device, 3y the central control we mean this latter function only, and the organs which perform it form the second specific part: CC , 2.1+ Third: Any device which is to carry out lon L ; and c plicated sequences of operations (specifically of calculations) must have a considerable memory. At least the four following phases of its operation require a memory: (a). Even in the process of carryiig out a multiplica- tion or a divisicn, a series of intermediate (partial) results must be remembered. This applies to a lesser extent even .to additions and sub- i tractions (when a carry digit nay have to be carried over several posi- .s), and to a greater extent to \j*tj 3/ , if £nesg* operations ire wanted, (Cf. .) (b) The instructions which Severn a Complicated pr lem may constitute -A considerable material, particularly so, if th> code is circumstantial (which it is in most arrangements). This material must be remembered. i (c) in many problems specific functions play an - ssen- tiai role. They are usually given in form of a table. Indeed in some ca^es this is the way in .'jhich they are given by experience (..'.g. tJ equation of state of a suostance in ;uany hydre dynamical problems), in other cases they may be given by analytical expressions, but it may nevertheless be simpler .and quicker tc bt In th ir values from a f. tabulation, than to compute them anew (on the b .sis of th... analytical ( definition) whenever as value is required. It is usually convenient to have tables of a moderate number of entries only (100-200) and to use interpolation. Linear and even quadratic interpolation -will not be sufficient in most cases, so it is- best to count on a standard of cubic cr biquadratic (or .-von higher order) interpolation, cf. . Some of the functions mentioned in the course' of 2.2 10 2 nay be handled m this way: lg, rig, In, sin and their inverses, . possibly also N / , . ■ Even the reciprocal might be treated in this manner,- thereby reducing >f -.to x. (d) For partial differential equations the initial con- ditions and the boundary -Conditions- may ..constitute an extensive numerical material, -which must, be remembered throughout :a 'given problem. . (e) For partial differential. .equations of the hyperbolic or parabolic type, integrated along a variable t, the (intermediate) re- sult's belonging to the cycle t must be remembered for the calculation of the cycle t + dt. - This material is much of the. type (d), except that it is not put into the devipe by human operators, but produced (and probably subsequently again removed and replaced by the corresponding data for t ■*- dt) by the dev ; c-- itself, -in the course of -its automatic operation. . (f) For total differential equations (d), (e) apply too, but they r--q.ire smaller memory capacities; Further memory require- - ments of the type (a) are required in problems which depend on given constants, fixed parameters, ..etcr . (g) Problems which are solved by successive approxima- > Ions (e.g. .partial differential equations of the elliptic type, treated by relaxation methods ) ■, require' a memory " of the tyj. (e); rhe (intermedial -5- * £ £ results of each approximation must be ..remembered, while those of the next -re being computed. (h) Sorting problems and certain statistical experi- ments (for which a very high speed device offers an interesting opportun- ity) require a memory for the material which is being treated; 2.5 To sum up the third remark: The device requires a con- siderable memory. While it appeared, that various part's of this memory have to perform functions which differ somewhat in their nature and con- siderably in their purpose, it is nevertheless tempting to treat the entire memory as one organ, and to have its parts even 'as interchangeable as possible for the various functions enumerated above. - This point will be considered in detail cf. At any rate the total memory -.constitutes the third- specific oart of the device: M. *■ ■■■■ — i i.--. ■— nam »»■■■■■ 2.6, The three specific parts CA, CC together C and M corre- ' spond to the associative neurons in the human nervous system. It remains to discuss the equivalents of the sensory or afferent and the motor or effe-rqnt n jurons. These are the input nnd the out put organs of the de- vice, and we shall now consider them briefly. •• In other words: A ll transfers of numerical (or other) information between the .parts C and M cf the device must be effected by the mechanisms contained in these parts. There remains, however, the necessity of getting the original definitory infermation from utside int^ the device, and also cf getting the final information, the results, from the device into, the outside. ' By. the outside we mean media cf the type described in -6- 1.2: Here information can be produced more or less directly by human action (typing, punching, photographing light impulses' produced bykeys of the same type, magnetizing metal tape or wire in some' analogous manner, etc.), it can be statically stored, and finally sensed more or less di- rectly by human' organs. The device must be endowed with the ability to maintain the input and output (sensory and motor) contact with some specific medium of this type (cf. 1.2): That medium will be' called the outside recording medium of the device: R . Mow we have: 2.7 Fourth: The device must have organs to transfer (numer- ical or other) information from R into its specific parts C and M. These organs form its input , the fourth specific part: I . It will be seen, that it is best to make all transfers' from -R (by I) into U, and never directly into C (cf. ). 2.8 Fifth: The device must have organs' to transfer (pre- sumably only numerical information) from its specific parts C and M into R. These organs form its output , the fifth specific part: . It will 'be seen that it is again best to make' -all transfers from M (by 0) into R, and never directly from C (cf. * ). 2.9 The output information, which goes into R, represents, of course, the final results of the operation of the device on the prob- lem' under consideration. These must be distinguished from the intermed- iate results, discussed. e.g. in 2.4, (e)-(g), which' remain inside M. At this point .an important question arises: Quito apart from its attribute ■ of more or less direct accessibility to human action and perception 'ft has also the properties of a memory. Indeed, it is the natural medium -7- £ # % for long time storage of all the information obtained by the automatic device on various problems. Why is it then necessary to provide for an- other type of memory, within the device M? Could not all, or at least functions of M - preferably those which involve great bulks of informa- tion - be taken over by R? Inspection of the typical functions of >),, as enumerated in 2. h, U)-(h), shows this: It would be convenient to s"hift (a) (the short-duration memory required while an. arithmetical operation is bein£ carried out) outside the device, i.e, from id into R. (Actually (a) will S be inside the device, but in CA rather than in U. Cf . -th< end of 12,2) All existing devices, even the existing desk computing machines, use the equivalent of U at this point. However (b) (logical instructions) might be sensed from outside, i.e. by I from R, and the same goes for (c) (function tables) and (e) t (g) (intermediate results). The latter may be conveyed by to R when the device, produces them, and sensed by I from R when it needs them. The same is true to some extent of (d) (ini- tial conditions 7 and parameters) and possibly even of (f) (intermediate «. results from a total differential equation). Aj to (h) (sorting and statistics 1 ), the situation is somewhat ambiguous: In many cat.es the possibility of using M' accelerates matters. decisively, but suitable blending of the use of M with a longer range use of R may be feasible without serious loss of speed and increase the amount of material that can b..: handled considerably. •Indeed, all existing (fully or partially automatic) computing, devicce uso R - as a .stack of punchcards or a length of teletype tape - for -all these ■ purposes (excepting (a), as pointed out -3- r e e above). Nevertheless it will appear that "> really high speed device would be very limited in its usefulness, unless it can rely on ..'., rather than on R, for all the purposes enumerated in 2.L, (a)-(h), with certain limitations in the case of (c), (g), (h), (cf. )• 3.0 Froceduro of Discusgie: ; 3.1 The classification of 2.0 being completed, it Ls new possible to take up the five specific parts into which the device w ls .seen tc be Subdivided, and to discuss them one by one. Such a discussion must oring out the features required for each one of these parts in it- self, as well' as in their relations to each ;thcr. It must also deter- mine the specific procedures to be used in dealing with numbers from the point of view of the device, in carrying out arithmetical operations, and providing for the general logical control. All questions of timing and of speed, and of the relative importance of various factors, must be settled within the framework of these considerations. 3.2 Tn- ideal procedure would be, to take up the five spe- cific parts in some definite order/ to treat each one of them exhaustive Lj , and go on to the next one only after the predecessor is completely dis- posed of. However, this seems hardly feasible. Two desirable fe itures of the various parts, and the decisions based .on them, emerge only after a somewhat zigzagging discussion. It is therefore necessary to take up one part first, pass after- an incomplete discussion to a second part, return after an equally incomplete : discussion of the latter with the combined results t6 the first 'part,' extend'- the discussion of the first i -9- r t part without yet concluding it, then possibly go on to a third part, etc. Furthermore, these discussions of specific parts ..ill be nixed with dis- cussions of general principles, of arithmetical procedures, of the e ments to be used, etc. In the course of such a discussion the desired features and the arrangements which seem best suited tc secure then will ;rystaili2 gradually until the device and its control issume a fairly i finite As emphasized before, this applies tc the physical device as wexl as to the arithmetical and logical arrangements which govern its functioning. 3.3 In the course of this iiscussion the viewpoints of 1./+, concerned with the detection, location, and under certain co) >ns even correction, of malfunctions must also rec-jive some consideration. That is, attention must be given to facilities for checkin- errors. We will not be able to do anything like full justice to this important sub- ject, but we will try to consider it at least cursorily whenever this seem; ee5ential (cf. ). 4.0 Eleminte, Synchronism Heuron A.vilc;;y U.l W"; begin the discussion with som-i general remarks: Every digital computing device contains certain relay like elements , with discrete jquilibria. Such nn element has two or mcr distinct states in which it can exist indefinitely. These may b: perfect equilibria, in each of which the element will remain without any outsi support, while appropriate- outside stimuli will transfer it from en- equilibrium into another. Or, alternatively, there may be two states, a one of which is in equilibrium which exists when there is no outside -10- r r r 1 support, while the other depends for its existence upon the presence of an outside stimulus. The relay action manifests itself in the emission of stimuli by the element whenever it has itself received a stimulus --of the typi indicated above, The emitted stimuli must be of the same kind as the received'one, that is, they must be able tc stimulate other clc- ra nts. Fhere nust, however, be no energy relation between the receive i '■ the emitted stimuli, that is, an element which has received one stimulus, must be able to emit several of the same intensity. In other words: Being a relay the. element must receive its energy supply from another sourcu than the inc u i ig stimulus. In existing digital cemputing devices various mechan- ical, or cloctric&l devices have been used -as elements: Whoels, which can be locked into any one of ten (or more) significant positions, and which on moving fr m >ne position to anoth r ti msmit electric puis that may cause other similar wheels to move; single or combined telegraph , actuated by an electromagnet and opening or closing electric cir- cuits'; cc i i Lons of those two elements; — and finally there exists the i tempting possibility of using vacuum tubes, the grid acting as . vali i for the cathode-plate circuit. In the last mentioned case ; • j alsc be replaced by deflecting organs, i.e. the vacuum tube by a cathode ray tube — but it is likely that for some time to come the availability and various electrical advantages of the vacuum tubes proper will keep the first procedure in the foreground. Any such device may time itself autonomously, by the successive reaction times of its elements. In this case all stimuli i. ultimately originate in the input. Alternatively, they may have -11- c "their timing impressed by a fixed clock, which provides certain stimuli •that are necessary for its functioning pit definite periodically recurrent moments'. This clock may be a. rotating axis in a mechanical or a mixed, mechanico-electrical device;- and it may be an electrical, oscillator (possibly crystal controlled) in a purely electrical device. If relianc: is to be placed on synchronisms of several distinct sequences of opera- tions performed simultaneously bv the device-, the clock impressed timing is obviously preferable. We will use the term element in the . above de- fined technical sense, and call the device synchro n ous or asynchronous , according to whether its timing is impressed by a clock or autono.o ■ os , a s • do s c r ib ed abo ve. . 4.2 It is worth mentioning, that the neurons of the higher animals -u I riitely elements in the above sense.. They have all-or- ntene character, that is two states: Quiescent o.nd excited. They fulfill" the requirements c f 4.1 with an interesting variant: An excited neuron ts the standard stimulus along many lines (axons). Such a line can, however, be, connected in two different ways to the next neuron: First: In an .excitatory sy r.epsls, so that the stimulus causes the excitation of that neuron. S,cond: In an inhibitory synapsis ,, so that the stimulus absolutely prevents the excitation of that neuron by * n y stimulus on any other (excitotory) synopsis. The neuron also has a definite reaction time, between thi r.c ption of a stimulus and the emission of the stimuli caused by it, tne syro.ptic delay . Following W. Fitts and W. S. [JacCulloch ("A logical calculus of the ideas immanent in nervous activity", Bull. Math. Bio- physics, Vol. 5 (1943), pp 115-133) ws ignore the more complicated aspects -12- r i < of neuron functioning: Thresholds, temporal summation, relative inhibi- v> ticn, changes of the threshold by after effects of stimulation beyond the synaptic delay, .-tc. It is, however, convenient to consider occasion- ■ ally noiiroris with fix:dd thresholds 2 and 3» that is neurons which can be excited Drily by (simultaneous) stimuli en 2 or 3 excitatory synapses (and none on an inhibitory synapsis), Cf. It is c- asily seen, that thest -simplified neuron functions can be imitated b.\ telegraph relays or by vacuum tubes. Although the nerv- ous, system is presumably isynchronous (for the synaptic iel ys),, precise synaptic delays can Dbta-ined by using synchronous setups. Cf. 4-3 It is olear, that a very high speed computing device should - iiieally have vacuum tube elements.-.. Vacuum. tub aggregat s Lik and s :alers have been used and found reliable -at reaction times J (synaptic delays) a. l rt as a microsecond (- 10 seconds), this is a ; rf ran x which n h r i vice can approximate. Indeed;' purely mechanic 1 -device's may be entirely disregard id and practical telegraph relay re - 3tion times irt f the order cf 10 milliseconds (- 107- seconds) sr more. It Is iat r sting to note that the synaptic time of a human n ur i U I' til rd r of a milliseconds ( = 10""-' seconds). In th considerations vMhich foil w we will assume ac- •dinfily.. that the device has vacuum tubes as elements. We will also - try tc make all estimates of numbers of tubes involved, timing, etc. on the basis, that the types of tube's, used, are the conventional and com- -. -dally available ones. That is, that no tubes of unusual complexity ■ .r with fundamentally new functions are tc be used. The possibilities •I -13- f ( i v for the uso of new types of tubes will actually become clearer and mere definite after a th r ugh analysis with the conventional types (or seme equiv : . ■■ demerits, cf. ) has oeen carried out. Finf.lly it will appear that a synchronous device has censid; - / 1 i\ - s (cf . ). 5. Principles r.vorning the 'Arit hmetic"! Oper tiuns 5_-l Let us now consider certain functions of the first spe- cific part: the central arithmetical part CA. ;h- element in the sense of 4.3, ^he vacuum tube used is '. current v iv or gate, is an all-or-none device, or at least it roxi lite :. • According to whether the r rid bias is above or below cut- >i'f ; i' will i'o: currant or not. It ls tru': uhat it needs definite :■:•• - Ls on fti^ its electrodes in order to maintain either state, but there are combinations of vacuum tubes which have perfect .equilibria: J ■ " i states in each of which the combination can exist indefinitely, 3 jiy outside support, while appropriate outside stimuli (electric pulses) will transfer it from one equilibrium into another. These are the so called t rigger circuits , the basic one having two equilibria and co;. - riod or one pentode. The trigger circuits with more than two equilibria are disproportionately more involved. Thus, whether the tubes are used as gates or as -triggers, the ail-or-none, two equilibrium arrangements are the simplest ones. Since these tube arrangements are to handle numbers by means of their - , digits, it is natural to-use a system of arithmetic in which the digits are also two valued. This suggests the use of the binary system. -14- ( ~V The analogs of human neurons, discussed in 4.2 - 4.3 are equally all-or-none elements. It will appear that they are quite useful for all preliminary, orienting considerations on vacuum .tube sys- tems (cf . ). It is therefore satisfactory that here too, the natural arithmetical system to handle is the binary one. 5.2 A consistent use of the binary system is also likely to simplify the operations of multiplication and division considerably. Specifically it does away "with the decimal multiplication table, or with the alternative double procedure of building up the multiples by each multiplier or quotient digit by additions first, and then combining these (according to positional- value) by a second sequence of additions or subtractions. In other words: .Binary arithmetics has a<simpler and more one-piece logical structure than any other, particularly than the decimal one. It must be remembered, of course, that the numerical material which is directly in. human use, is likely to have to be ex- pressed in the decimal system.' Hence, the notations used in R should be decimal. But it is nevertheless preferable to use strictly binary procedures in CA, and also in whatever numerical material may enter into the central control CC. Hence d should store binary material only. This necessitates incorporating decimal-binary and binary-decimal conversion facilities into I and 6. Since these con- versions require a' good deal of arithmetical manipulating, it is most economical to use' CA, and hence for coordinating purposes also CC, in conjunction with I'-andO. The use of CA implies.^ however, .that all -15- • c ( C t -^ arithmetics used in both conversions must be-.' strictly binary. For details cf. . 5.3 At this point there arises another question of principle. In all existing devices where the element is not a vacuum tube the reaction time of the element is sufficiently long to make a .cer- tain telescoping of the steps involved in addition, subtraction, and still more in multiplication and division, desirable. To take a specific case consider binary multiplication. A reasonable precision for many differ- encial equation problems is given by carrying 3 significant decimal digits, -ft ; ; .that is by keeping the relative rounding-off errors below 10 . This , -corresponds to" 2"^ in the. binary system that is to carrying 27 signif i- ■ cant -binary digits. Hence a •.multiplication consists of pairing each 'one of 27 .multiplicand -digits with each one ti£ 27 multiplier digits, and /) forming product digits arid 1 accordingly, arid then positioning- and co.v.- bining them. These are essentially 27 'z 729 steps, and the operations of collecting and combining may about double their number. So 1000-1500 steps are essentially right. ' It is natural to observe that in the decimal system a considerably smaller number of steps obtains: 8 - 6^ steps, possibly doubled, that is about 100 steps. However, this low number is- pur- chased at the price of using a multiplication table or otherwise increas- ing or complicating the equipment. At this price the procedure can be shortened by more direct binary artifices, too, which will be considered presently. For this reason it seems not necessary to discuss the deci- mal procedure separately. . 5.4 . As pointed out before, 100.0-f-1500 successive steps per -16- c multiplication would make any -non -Vacuum tube device inacceptably slow. All such devices, excepting some of '• the latest special relays, hive reaction times of more than 10 milliseconds, and these newest relays (which may have reaction times down to 5 milliseconds) have not been in use very long. This would give an extreme minimum of 10-15 seconds per (8 decimal digit), multiplication, whereas this time is 10 seconds for fast modern ' desk.: computing cachines, and 6 seconds for the standard I«B.-M. multipliers. (For the significance of these djrations, as well as of those of possible vacuum tube devices, when applied to typical problems, of. ■ .) The logical procedure to avoid these long durations, consists of telescoping operations , that is of carrying cut simultaneously as many as possible. The complexities of carrying prevent even such sim- ple operations as addition -or subtraction to be carried out at once. In division the calculation of a digit cannot even begin unless all digits to its left are already known. Nevertheless considerable simultaneisa- tions are possible: In addition or subtraction all pairs of correspond- ing. digits can be combined at once, all first carry digits can be applied together in the next step, etc. In multiplication all the partial pro- ducts cf the form (multiplicand) x (multiplier digit) can be formed and positioned simultaneously — in the binary system such a partial product. is zero or the multiplicand, hence this is only a matter of positioning. In both addition and multiplication the above mentioned accelerated forms of addition and subtraction can be used. Also, in multiplication the partial products can be summed up quickly by adding the first pair together simultaneously with the second pair, the third pair* etc.; then 'adding; the first pair of pair sums together simultaneously with the -17- t second one, the-third one, etc.; and so on until all terras are collected. (Since 27 4 2 J , this allows to collect 27 partial sums - assuming a 27 binary digit multiplier - in 5 addition times. This scheme is due to H. Aiken.) Such accelerating, telescoping procedures are being used in all existing devices. (The use of the decimal system, with or without further telescoping artifices is also of this type, as pointed out at the end of 5.3. It is actually somewhat less efficient than purely diadic procedures. The arguments of 5.1 - 5.2 speak against con- sidering it here.) However, they save time only at exactly the rate at which they multiply the necessary equipment, that is the number of ele- ments in the device: Clearly if a duration is halved by systematically carrying out two additions at once, double adding equipment w^.11 be required (even assuming that it can be used without disproportionate control facilities and fully efficiently), etc. This way of gaining time by increasing equipment i* fully justified in non vacuum tube element devices, where gaining time is of the essence, and extensive engineering experience is available regarding the handling of involved devices containing many elements, A really all-purpose automatic digital computing system constructed along these lines must, according to all available experience, contain over 10,000 elements. 5.5 For a vacuum tube element device on the other hand, it would seem that the opposite procedure holds more- promise. As pointed out in A. 3, the reaction time of a not too complicated vacuum tube device can be made as short as one microsecond. -18- f Now at this rate even the unmanipulated duration of the multiplication, obtained in 5.3 is acceptable: 1000-1500 reaction times amount to 1-1.5 milliseconds, and this is so much faster than any conceivable non vacuum tube device, that it actually produces a serious problem of keeping the device balanced, that is to keep the necessarily human supervision beyond its input and output ends in step with its operations. (For details of this cf. .)' Regarding other arithmetical operations this can be said: Addition and subtraction are clearly much faster than multiplica- tion. On a basis of 27 binary digits (cf. 5.3), and taking carrying into consideration, each should take at most twice 2.7 steps, that is about 30-50 steps or reaction times. This amounts to .03-. 05 milliseconds. Division takes, in this scheme where shortcuts and telescoping have not been attempted in multiplying and the binary system is being used, about the same number of steps as multiplication. (cf. .) Square rooting is usually and in this scheme too, not, essentially longer than dividing. 5.6 Accelerating these arithmetical operations does there- fore not seem necessary - at least not until we have become thoroughly and practically familiar with the use of very high speed devices of this kind, and also properly understood and started to exploit the entirely new possibilities for numerical treatment of complicated prob- lems which they open up. Furthermore it seems questionable whether the method cf acceleration by telescoping processes at the price of multi- plying the number of elements required would in this situation achieve ' its purpose at all: The more complicated the vacuum tube equipment — -19- r c that is, the greater the number of elements required — the wider the tol- erances must be. Consequently any- increase in this direction will also necessitate working with longer reaction times than the above mentioned one of one microsecond. The precise quantitative effects of this factor are hard to estimate in a general way — but they are certainly much more important for vacuum tube elements than for mechanical or for telegraph relay ones. Thus it- seems worth while tc consider the following viewpoint: The device should be as simple as possible, that is, contain as few elements as possible. This can be achieved by never performing two operations simultaneously, if this would cause a significant increase in the. number of- elements required. The result will be that the device will worK more : reliably and the vacuum. tubes can be driven to shorter reaction times than otherwise. 3.7 The point to which the application of this principle can be profitably pushed will, of course, depend on the actual physical - characteristics of the available vacuum tube elements. It may be, that the optimum is not at a 10C# application of this principle and that some compromise will be found to be optimal. However, this will always de- pend on the momentary state of the vacuum tube technique, clearly the • faster the tubes are which will function reliably in this situation, the stronger the c?.se is for uncompromising application of this principle. It would seem that already with the present technical possibilities the , optimum is rather nearly at this uncompromising solution. It is also worth emphasizing that up to now all think- ing about high speed digital computing devices has tended in the opposite -20- ( t ( direction: Towards acceleration by telescoping processes at the price of multiplying the number of elements required. It would therefore seem to be more instructive to try to think out as completely as possible the opposite viewpoint: That one of absolutely refraining from the pro- cedure mentioned above, that is of carrying out consistently the principle formulated in 5.6. ( We will therefore proceed in this direction. 6.0 E-elements 6.1 The. considerations of 5.0 have defined the main princi- ples for the treatment of CA. We continue now on this basis, with some- what more. specific and technical detail. In order to. do this it is necessary to use some schematic picture for the functioning of. the standard element of the device: In- deed, the decisions regarding, the arithmetical and the logical control procedures of the device,' as well as its other functions, can only be made on the basis of some assumptions about the functioning of the ele- ments. The ideal procedure would be to treat the elements as what they are intended to be: as vacuum tubes. However, this would n necessitate. a detailed analysis of specific r^dio engineering questions at this early stage of the discussion, when too many alternatives are still open, to be treated all exhaustively and in detail. Also, the numerous alternative, .possibilities for arranging arithmetical proce- dures, logical control, etc., would superpose on the equally numerous possibilities for the choice of types and sizes of vacuum t-ubes and- other circuit elements from the point of view of practical performance,, etc. All this would produce an involved and opaque situation in which the '■' ' -21- [ r b preliminary orientation which we arc now attempting would be hardly possible. In order to avoid this we will base our considerations on a hypothetical* element, which functions essentially like a vacuum tube — e.g. like a tritde with an appropriate associated HLC-circuit — but which can be discussed as an isolated entity, without going into detailed radio frequency electromagnetic considerations. '..'e re-empha- size: This simplification is only temporary, only a transient stand- point, to make the present preiLainary discussion possible. After the conclusion's of the preliminary discussion the elements will have to be i reconsidered in ttfeir true electromagnetic nature; , But at that time the decisions of the preliminary discussion will be ■available, and. the corresponding alternatives accordingly eliminated. 6.2 ' The analogs of "human neurohs, discussed in 4.2-4.3 and again referred to at the end of 5.1, seem to- provide elements of just the kind postulated at the end of 6.1. ■ V,'e propose to use them accord- ingly for the purpose described there: as the constituent elements of the device, for the duration of the preliminary discussion. We must therefore give a precise account of the properties which we postulate for these elements. Th< element which we will discuss, to be called an £--: lei.-ient , will be represented to be a circle 0, which receives the excitatory and inhibitory "stimuli, and emits its own stimuli along a line attached to it: 0—. This axis may branch: '0— <-, 0-^-. The emission along it follows the original stimulation by a synaptic delay , which we can assume to be a fixed time, the same for all E-elements, to be denoted by t. We propose to neglect' the other -delays ■ (due to-, conduction of the w c stimuli along the lines) aside of t. We will mark the presence of the delay t by an arrow on the line: 0-y— , > •■ - C This will also serve to identify the origin and the direction mi the line. 6.3 At this point the following observation is necessary. . In the human nervous system the conduction times along the lines (axons) can be longer than the synaptic delays, hence our above procedure of neglecting them aside of t would be unsound. In the actually intended vacuum tube interpretation, however, this procedure is justified: t is to be about a microsecond, an electromagnetic impulse travels in this time 300 meters, and as the lines are likely to be short compared to this, the conduction times may indeed be neglected. (It would take an ultra high frequency device t « 10~° seconds or less-x-to vitiate this argu- ment. ) Another point of essential divergence between the human nervous system and our intended application consists in our use of a well defined dispersionlcss synaptic delay t, common to all E-ele- ments. (The emphasis is on the exclusion of a dispersion. We will actually use E-elements with a synaptic delay 2t, cf. .) •We propose to use the delays t as absolute units of time which can be relied upon to synchronize the functions of various parts of the device. The advantages of such an arrangement are immediately plausible, specific technical reasons will appear in In order to achieve this, it is necessary to conceive the device as synchronous in the sense of i*.l. The central clock is best thought of as anElectrical oscillator, which emits in every period t a short, standard pulse of a length t' of about l/5t - l/2t. The -23- € € <L 1 FIGURE 1 > i' _ < > i — > i r C > - -i i : i <• <v 1- - t i "LOCH PUl"iii. 1 i ! FoR Ti-IEL O P £ N ! stimuli emitted nominally by an E-elemont are actually pulses of the clock,, for which the pulse acts as a gate. There is clearly a wide tolerance for the period during which the gate must be- kept open, to pass the clock-pulse without distortion. Cf. Figure 1. Thus the opening of the gate can be controlled by any electric delay device with a mean delay time t, but considerable per- missible dispersion. Nevertheless the effective synaptic delay will be t with the full precision of the clock y and the stimulus is completely renewed and synchronized after each step. For a more detailed descrip- tion in terms of vacuum tubes, cf.. 6./+ Let us now return to the description of the E-element s. An E-element receives the stimuli of its antecedents ss excitatory synapses:' 0— >-, or inhibitory synapses: 0— >-. As pointed out in /+.2, we will consider E-element s with thresholds 1, 2, 3, that is, which get excited by thesr minimum numbers of simultaneous excitatory stimuli. All inhibitory stimuli, on the other hand, will be assumed to be absolute. E-elements with the above thresholds will be denoted by 0,[2j, \3J, respectively. Since we have a strict synchronism of stimuli arriving only at times which are integer multiples of t, we may disregard pheno- mena cf tiriruj, facilitation, etc, We also disregard relative inhibition. -21+- f ( temporal summation Of stimuli, changes of threshold, changes synapses, etc. In all this we are following the procedure of W. Fitts and W. J.. MacCulloch (cf. Loc. cit. k»Z). We will also use E-elements with double synaptic delay 2tj Q-W, and mixed types: 0-d-«^>- _ The reason for our using these variants is, that they give a greater flexibility in putting together simple structures, and they can all be realized by vacuum tube circuits of the same complexity. It should be observed, that the authors quoted above -%) have shown, that most of these elements can be built up from each other. Thus 0->->- is clearly equivalent to 0-v-0->-, and in the case of [2 } -»- at least — (2j ->->- is equivalent to the network. of Figure 2, However, it -i would seem to be misleading in our application, to represent these functions as if they required 2 or 3 E-eleraents, since their complexity in a vacuum tube realization is not essentially greater than that of the simplest E— element 0->-, cf. We conclude by observing that in planning networks of E-elements, all backtracks of stimuli along the connecting lines must be avoided. Specifically: The excitatory and the inhibitory synapses and the emission points that is the three connections on ~^ )— > ' will be treated as one-way valves for stimuli from left to right in the above picture. But everywhere else the lines and their connections ^r^-v will be assumed to pass stimuli in all directions. For the delays — ^ — either assumption can be made, this last point does not happen to matter Jj in our networks. -25- * 'C 6. 5 Comparison of some typical E-element networks with their ^vacuum tube realizations indicates, that it takes usually 1-2 vacuum tubes for each E-element. In complicated networks, with many stimulating lines for each E-element, this number may become somewhat higher. On the average, however, counting 2 vacuum tubes per E-element would seem to be a reasonable estimate. This should take, care of amplification and pulse-shaping requirements too, but of course not of the power supply. For seme of the details, cf. 7.0 Circuits for the arithmetical operations 4- , X 7.1 For the device and in particular for CA a real num- ber is a sequence of binary digits. We saw in 5.3, that a standard of 27 binary digit numbers corresponds to the convention of carrying 8 signi- ^ ficant decimal digits, and is therefore satisfactory for many problems. We are not yet prepared to - make a decision on this point (cf . however, ), but we will assume for the time being, that the standard number has about 30 digits. *J | When an arithmetical operation is to be performed on such numbers, they must be present in some form in the device, and more partic- ularly in CA. Each (binary) digit is obviously representable by a stim- ulus at a certain point and time in the device, or more precisely, the value 1 for that digit can be represented by the presence and the value by the absence of that stimulus. Now the question arises, how the 30 (binary) digits of a real number are to be represented together. They could be represented simultaneously by 30 (possible) stimuli at 30 differ- ent positions in CA, or all 30 digits of one number could be represented -26- ^ © by (possible) stimuli at th same point, occurrin success periods T in time. Following the principle of 5.6 to place multiple evr ■■ in temporal succession rather than in (simultaneous) spacial juxtapositi we choose the latter alternative. Hence a number is represenl - I by a , which e.rl 1 s during % successive periods "f the stimuli correspond-. I ; inary) digits. 7.2 In the following discussions we will draw various net- works of S-elements, to perform various functions. These drawings will also be used to define block symbols . That is, after exhibiting the ucture of a particular network, a block symbol will be assigned to it, .".-hich will represent it in all its further applications including those where it enters as a constituent Into a higher order network and its bol. A bi;ck symbol shows all input and output lines of it; .. twork, but not their internal connections. The input lines will I ■■■: ! ~> , and the output lines — * . A block symbol carries the abbreviated name of its network (or its function), and the number of Lements in it as an index to the name. Cf . e.g. Figure 3. below. 7.3 v;e proceed to describe an adder n< tworks Figure 3. no addends come in on the input lines- a 1 -, ..a", and the sum- is ■ i with a delay 2 ~~ L tht addend inputs on the _in? s. (The dotted sxtra it line c is for a special pur- pose which will appear in 3.2) The carry • i by '2 . The .FlGL(-R~L 3 4g>':.'6 >- if 1 -27- a ^ corresponding digits of the two addends together with the proceeding carry digit (delay jl) excite each one of (left), (£), (T), and an out- put stimulus (that is a sura digit 1) results only when is excited with- out (2\ , or when (y\ is excited — that is when the number of l's among the three digits mentioned is odd. The carry stimulus (that is a carry digit i) results, as pointed out above, only when (2) is excited — that is when there are at least two l's among the three digits mentioned. All this consti- tutes clearly a correct procedure of binary addition. In the above we have made no provisions for handling the si^n of a number, nor for the positioning of its binary point (the analog of the decimal poin t). The^e concepts will be taken up in ., but before consid«ring them we will carry out a preliminary discussion of the multiplier and the divider,. 7 .k A multiplier network differs qualitatively from the adder in this respect: In addition every digit of each addend is used only once^, in multiplication each digit of the multiplicand is used as many times as there are digits in the multiplier. Hence the principle of 5-6.. (cf . W) also the end cf 7-1) requires, that both factors be remembered by the .multiplier network for a (relatively) considerable time: Since each number has 30 digits, the duration of the multiplication requires remem- bering for at least yp- .- 900 periods X • In other words: It is no longer possible, as in the adder, to feed in- the two factors on :.two in- put lines, and to extract in continuous operation the product on the output line — the -multiplier needs a. memory (cf. 2.4, (a)). • In discussing this memory we need not bring in M — this is a relatively small memory capacity required for immediate use in CA, v and it is best considered in CA. -28- 7«5 Tne E-elements can be used as memory devices: An element which stimulates itself, \^)ZZZ^ > will hold a stimulus indefinitely. Provided with two input lines rs, cs for receiving and for clearing (forgetting) this stimulus, and with an output line os to signalize the presence of the stimulus (during the time interval over which it is remembered), it becomes the ^ network of Figure 4. ' It should be noted that this m^ ft C,uK£ * rs c s Cfc± OS t= -X y ~\ i corresponds to the actual vacuum tube trigger circuits mentioned at the beginning of 5.1. It is worth mentioning that •1 contains one E-ele- 3 ment, while the simplest trigger circuits contain one or two vacuum' tubes (cf. loc. cit.), in agreement with the estimates of 6.5. Another observation is that m. remembers only one stimulus, that is. one binary digit. If k-fold memory capacity is wanted, then k blocks ments mi * are required, or a cyclical arrangement of k E-ele- : r- — C~\~ } >---(~~\ i F- - --y(~y^-\ •''•'Tflfi-s cycle can be provided with inputs and outputs in various ways , which can be arranged so that. when- ever a new stimulus (or rather the fact of its presence or absence, that is a binary digit) is received for remembering — say at the left end of the cycle — the old stimulus which should take its'- place — coming from the right end of the cycle — is automatically' cleared. Instead of going into these details, however, we prefer to keep the cycle open:-^\-^-(^y ... a/^Vv and provide it 'with such terminal equipment (at both ends, possibly ' ft Cum ^ ■'<■ . • — - — r ""ft. £ - fc l e ' m e n "^ S connecting them) as may be rciuirec in each particular case. This simple line is shown again in Figure 5. -29- ( I ' Terminal equipment, which will normally cycle "the output os. at ' Ik j's right and back into the input at its left end, but" upon stimulation at s suppress (clear) this returning of the., output os and connect instead. the input with the line rs, is shown in Figure 6 1A Ik. with the terminal equipment of Figure 6., is a perfect -nc, urn c rs <3S memory organ, but without it, in the form of Figure 5., it is simply a delay organ. Indeed, its sole function is to retain any stimulus fcr k periods t and then reemit i\ and to be able to do this, for successive stimuli without any interference between them. This being so, and remembering that each E-elenent represents (one or two) vacuum tubes, it would seem wasteful to use k - 2k vacuum tubes to achieve nothing more than a delay kt. There exist delay devices which can do this (in our present situation t is about a microsecond and k is about 30} "more simply. ' We do not discuss them here, but merely observe that, there arc several possible' arrangements (cf. 12.5). Accordingly, we replace the block block Ik of Figure 5 by a new dl (k) , which is to represent such. a device. It contains no E-elemont, and will itself be ! treated as a new element. We observe, that is ' dl (k) is a linear delay cir- cuit, stimuli can backtrack through it (cf. the end of 6.4)« To prevent this, it suffices to protect its ends by E -elements, that is to achieve the first and the last t delay' by — (T)^ or to use ** in some combination like Figure 6, where the E-elements of the associated network -30- r r I provide this protection. 7.7 We can now descrloe a /multiplier network. Binary multiplication consists of this: For each digital position in the multiplier (going from left to right), the multiplicand is ;hifte i by one position to the right, j.;i h then it is or is not added to the sum of parti a products already formed, according to whether the multiplier digit under consideration is 1 jr 0. Consequently the multiplier must contain an auxiliary network, which will or will not pass the multiplicand into the adder, a>?_rainr; to whether the multiplier iir.it in -uestion is 1 or 0. This can be achieved in two steps: Fir.? + , a network is required, which will emit stimuli during a certain interval of 7 periods (the interval in which the multiplicand is .vanted), provided that a certain input (con- nected to the organ which contains the aultiplier) was stimulated at a cert air: earlier moment (when the proper multiplier digit is emitted). Such a network will be called a ai^cri.minator . Second, a valve is re- hired which will pacs i stimulus only if it is also stimulate:! on a second input it possesses. Ihese two clocks together solve our problem: The discriminator mast be properly controlled, its output connected to the second input of the valve, and the multiplicand routed through the valve into the adder. Th : valve is juite simpL : Figure 7. The main stim- ulus is passed : r< ... is to os, the second input centers at s. -31- ' I i 9 ~) ) ) A . discriminator is shown on Figure 8. A stimulus at the input t defines the mo-: ment at .-Jhich the stimulus, which determines whether the lat^r emission (at os) shall take place flQi/RE 6 Q^ at all, must be r;-cr;iv.ej at the inputs. If these two stimuli coincide, the left \2) is excited. Consid< ri Its feedback, it. will remain excited until it secceeds in stimulating the middle {2) .".' The middle (k) is con- nected to (is) in such a .-manner that it can be excited by the left (2\ only at. a moment at which (is) is stimulated, but at whose predecessor (is) was not stimulated — that is at the beginning of a sequence of stimuli at (is) . The mi idle (£) then, ]uenches the left (J2) , and to- ■ t.her with (is) excites the right [2) . The ~lidle {2 ) now becomes and stays quiescent until the end of this secuence of stimuli at n.s) and beyond this, until the beginning of the next sequence. Hence the left (2) is isolated from "the two: other (2) , and thereby is ready to register the s, t stimuli for the next (is) sequence. On the other hand the feed- back of the right (?) is such, that it will stay excited for the luratic of this (is; sequence, and .emit stimuli' at cs. There is clearly a delay 2t between the input at (is) and the output at os. Now the .on multiplier network ■cah.be put together: Figury.9. .The multiplicand circulates through dl I t he- multiplier through al II F 1 CWRZ 9 a -\ cd 1 .V. L_ <M H 1 — y—M.-) at. 1 m h -32- <4 and the sum of partial products (which begins with the value and is gradually built up to the complete product) through dl III . The two inputs t, t' receive the timing stimuli required by the discriminator (they correspond to t, is in Figure 3.). 7.8 The analyri, 3 f 7.7 avoi : I the f iio\ ing essential fea- tures of the multiplier: [a) The t j y ' n • networt which controls- the in- puts t , t ' , and stir.ulat ti I ' pr ;r ," iment . . Li vi.ll clrar 1 ; have to contain j dl f--like :• r.tc ' :f. ' . (0) - 1 dl III The k (delay 1 >ngi hs ) * ' a - 1 These toi be certain' functions of synchr nizatj 1: F.ach time when the adder functions (that is in each interval it - ft r h tiplicand and the partial pro- duct sum (that is the out f md of di III ) must be brought together in such a manner, that ■ former is advanced by t (moved by one position to the right) relatively to the latter, in com- parison with their preceding encounter. Also, if th( two factors have 30 digits each, the product has 60 iigits. Hence 11 III should have about twice the k of dl I and dl II and a c Ln the former must correspond to about two cycles in the latter. (Th« timing stimuli on tt will be best regulated in phase with dl III .) On the ether hand, it is advisable to make provisions for rounding I and thereby keep the pre net off to the standard number Iigits, dl III near 30. (c) The networks required dl II to get the multiplicand and the multiplier into dl I | and (from other parts of the device), and to get the product out of dl III (d) The networKs required 'to handle the signs and the binary point positions of the factors. They are obviously dependent upon the way in which these attributes are to be dealt with arithmetically (cf. the end -33- t : 4 of 7.3 and ). Ail these points will be dealt with subsequently. The questions connected with (a) — arithmetical treatment of sign and binary point — must be taken up first, since the former is needed for subtraction, I hence for division too, and the latter is important for both m»>Ttt- plication and divisl 8.0 Circ uits for t h ;_a •• r \s ^Jc al operations -. - '3.1 til now lurr x , a sequence of (about 30) binary digits,, with no definition :' s: ;i or binary point. We must now stipulate conventions for the treatment cf these concepts. The extreme Lef* !' 1: will be reserved for the sign, so that its values 0,1 exprest the signs +, -, respectively. If the binary point is between the digital r sit 1 ind i-t-1 (from the left), then the positional valre of the sign digit is 2 1 . Hence without the ' sign convention the number x would lie in the interval ~ * < 2- ' } and with the sign convention the subinterval ~ x < Z "' is unaffected and corresponds to non negative numbers, while the interv d ?. ~, X < Z corresponds to negative numbers,. We let the latter x rej r nt a negative x*, so that the remaining digits of x are essentially the complements to the digits of -x' . V-i- precisely: 2.* ' / — (- x ) ■= ~1 , ih.t is /'* X- 2. * . To - t*~' %*' < O . ■ In other words: The digital sequences which we use A- represent, without the sign convention, the interval O = X <■ Z and with the sign convention the interval - £ =. * 2. . The second interval is correlated to the first one by subtracting 2 1 if -3U- ( ( necessary — that is their correspondence is module 2 1 . Since addition and subtraction leave relations module *: x unaffected, we can ignore these arrangements in ;arrying ,1 additions : ibtractions. The same is true fcr the :ition of '. he binary r-'nt: . If this is moved from i tc i', then each :.. . •• ' i .. " 1+iplie by " • iddition and subtraction leave this rcia' ... '. -ian1 ti these things ire, of course, the analogy c:' ' conventional _' '. pro- cedures. ) Thus we n ■ not adi any Lng tc the additj i lure of 7»3j and it will be ccr: ;t to set up i .btraction pr cedur< Li 1 :.he . ■_ .e way. The mult,. Lies ion procedure /.,', i a r, .vill have to be nsidered, and the same caution app'.i'. t; the iivision procedure to be set up. 8.2 'We now set up a suHra^ _• n twork. We can use the adder (cf. 7.3) for this pur;cse, if cne a n - - ay the first one — is fed in the negative. According to the aboy. this rieans that this addend x is .- ; Laci d by 2 X - x. That is, each ,i ;.' t . x is replaced by its comple- ment, and a unit of th< extr* ri.;hc digital position 'a then add I to this addend — or just as «el] as ±n extra addend. This last operation can be carried out by v.xil i the extra input c of the adder (cf. Figure 3.) at that I Lme. t ikes automatically care of ail carries which may be caused by thi: 'Xtra addition. The complementation of each iigi,t car. be done by a valve which does the apposite of that of Figure 7: When stimulated at s, it passes the complement of the main stimulus from is to os: Figure 11. -35- i. Bureau of Stand Division 12 Data Processing Systems Now the subtracter network is P'Cr L(7?Z /O i 5 Ol 3 V. /' / 7 :, 7? IT / / '. I I wn on Figure 11. The sub- trahend and the minuend come in n ' l e input lines s, m, and the lifference is emitted with a delay 3t against the inputs on the output line d. The two inputs t 1 , t" receive the necessary timing . .1: t 1 thrci period of subtraction, t" at its first t (corresponding to th right iigital position, cf. above). 3.3 Next ' form a divide r > >twork, in the sam sense as the multiplier network of 7.7. Binary i vision : i .< position in th< }uo1 L mt (going ; ■ ■. tract ed from the partial remain-!- • but which his been shifted left by one position, preceding this subtrac- tion. If the resulting difference i r gative (that is, if its extreme left digit xc 7) then t n q '.lent digit is 1, and the next partial remainder (the oni to t ised for the following quotient iigit, before the shift left referred to above) if the differs ce 5; relative (that is, if its extreme left digit is 1) then the next |u - i '::i Iigit is 0, and the next partial remainder (in th< ;ara - nse as I is the preceding partial remainder, but in its shifted positi The alternative in division is therefor* c nr.j ir ble to that one in multiplication (cf. 7.7), with this notable difference: In multiplication it was a matter of passing or net passing an adder:,: if this: For each digital : right) , the div Lsor is ! dividend) already formed, -36- t the multiplicand, in division the question is which of two minuends to pass: the Shifted) preceding partial remainder, or this quantity minus the divisor. Hence we now need two valves where we needed one in multi- plication. Also, we need a discriminator which is somewhat more elabor- ate than that one of Figure 8.: It must not only pass a sequence of stimuli from is to os if there was a stimulus at 3 at the moment defined by the stimulation of t, but it must alternatively pass that sequence from is to another output os' if there was no stimulus at s at the mo- ment in question. Comparison of Figure 8. with Figure 12. shows, that the latter, posses- ses the desired properties. The delay between is and os or os ' is now 3t. PI $U7?£ '/z _. OS as / 1 ' i r 1 s ■ s — >— oC j C^^— y — y— — y t t' - ■^ I '-.;-. ,- ■ ..Now the divider network can be put together: Figure 13. The divisor. cir- culates; through . dl I i* while the dividend is originally in.-. , but F I (, W%£ 13 [ H our: ; Yj ^ </■- dl III is replaced, as the division pro- gresses, by the successive partial A- rf oO> JJ I— *J> & |- M *£ isr\ r L __. ... _.... - ' M" i - "> { is \ :i i < -)-j <. j- ^ -<—<t— <- A /, L-. remainders. The valve f v_^ I routes the divisor neg.itiv.ply into the -37- ( f t i v--.adder. The two valves J V]_ I immediately under it. select the partial re- mainder (cf. below) and send it from their common output line on one hand unchanged into dl II and on the other hand into the adder, from where thu timing : :' dl III the sum (actually the difference) goes into be such as to produce the required one position shift left. Thus and dl III V contain the two numbers from among which the. next partial remainder is to be selected. This selection is done by the discriminator * d, ] Which governs the two valves controlling the (second addend) input of the adder (cf. above). The sign digit of the resulting sum controls the discriminator, the timing stimulus at t must coincide with its appearance (extreme left digit of the sum), t* must be stimulated' during the period in which the two addends (actually minuend and subtrahend) are to enter the adder (advanced by 3t). t" must receive the extra stira- V ilus required in subtraction (t" in Figure 11.) coinciding with the ex- treme right digit of the difference. The quotient is assembled in , for each one of its digits the necessary stimulus is avail- dl IV able at the second output of the discriminator (os 1 in Figure 10.) it is passed into dl IV r l , timed by a through the lowest valve stimulus at t'". 6.4 The analysis of 8.3 avoided the same essential features of the divider, which 7.7 omitted for the multiplier, and which were enumerated in 7.8: (a) The timing network which controls the inputs t, t r t", t'". (b) The k (delay lengths) of the) dl I - | dl IV The details differ from those in 7.8, (b), but the problem is closely parallel. -38- ( ( f r -| (c) The networks required to get the divide] divisor into ! dl III i and '■ Ji I ' . I ti quo 1 i : I ' (d) The networks required ,\ handle ' ;ns ary p Lnt positions. As in the case ci multiplication ali ti Lll be dealt >vith subsequently. 9.0 Th-: binary poi -.t 9.1 As pointed out at the end of 3.1, the sign convention . ;' 3.1 as we'll as the binary point convention, which has not yet been determined, have no influence on addition and subtraction, but their relationship to multiplication and division is essential and requires consideration. It is clear from the definitions of multiplication and of division, as given at the beginning of 7.7 and of 3.3 respectively, that they apply only when all numbers involved art non-negative. That is, when the extreme left digit (of mu_tiplicand and multiplier, or dividend and divisor) is 0. Let us th -ref re assure this for the pr i. 1 (this subject will be taken up again in ), md coarid r the role of the binary point in multiplication and division. 9.2 As .jointed out in 7.3, (b), the product of the 30 digit numbers has 60 digits, and since the product should be a iupb< r '.vith the same standard number of significant digits as its factors, this necessi- tates omitting 30 dibits from the product. If the bin-xry point is betveen the digital positions -39- t r i and i-t- 1 - * he- -left) in on-= factor, and between j and j-fl in the other, then these -nunbers lie between 0- and 2 x ~t and between"0 and 2 J (the extreme left digit* is v 0, cf. .9-1). iience the product lies, between - and'2 1 ^~ . However,, if -it is Known to .lie. 'between and 2* c ~-'- ( / =: •■'' i / + /- / ) > then its binary point lies between k and k-t-1. Then of its 60 dibits the -first i + j-l-k (from the left) are and are ■ omitted,, and so. it is only necessary to omit the 29-i- j -fk last digits • (to the right) by some rounding-off process. ' ' This shows', that the essential effect of the positioning of the binary point is, that it determines which digits among the super- , nnmerary cnes m a product are to be omitted. > li /s'.Tj-l, then specijj.' precautions must be taken k-1 . so that no two number? are ever multiclied for which the product is >2 (it' is only limits bv <2 1+ J- 2 ). 1 his. difficulty is well known in ■ ■ planning calculations on IBLi or other automatic devices'. There is an elegant trick to get around this difficulty, .due to. G. Stibitz, but since it would complicate the structure of CA somewhat, we prefer to carry out the first discussion without -using it . We prefer instead to suppress this difficulty at this point altogether by an arrangement which produces an essentially equivalent one at another point. However, this means only -..that in planning calculations the usual care •must be exercised, and it simplifies the device and its discussion. This procedure, 'tooj' is in the spirit of the principle of 5.6. This arrangement consists in requiring k r 1 j-1, so that every multiplication can always be carried out. We also want a fixed position for the binary point, coinmcn to all numbers: 1 ■= j,= k. Hence i -' j * k = 1, that is: The binary point is always between the -40- * two 'first digital positions (from the left). In other 'words: The binary point follows always iiamediately ''after the sign "digit . . . 'Thus all non-negative "numbers will be between and 1, ana all numbers (ctf either sign) between -I and- 1. This makes it clear - once mbrS' that the multiplication can always be carried out. 9.3 . The caution formulated above is , therefore, that in planning any. calculation for the device, it is necessary to sea to it, that all numbers. which occur in the course of the calculation should always be. between -1 and 1. This can be done by multiplying the numbers- , ■ ' of the actual problem by appropriate (usually negative) powers: of 2 . (actually in many cases powers, of 10 are' appropriate, cf . • ), ^ and transforming all' formulae accordingly. From the point of view of . planning.it i's no better and no worse, than the- familiar difficulty, of positioning the decimal point in most existing automatic. 'devices. It is necessary to make certain compensatory arrangements in. I and 0, cf . '■ - Specifically the requirement that all numbers remain between -1 and 1, necessitates,. to remember these limitations in planning calculations: " ; (a) No addition or subtraction must be' performed if its result is a number not between -1 and l.(but of course between -2 and .2). * • (b) Mo division must be, performed if the. divisor is less (in absolute value) than ..the dividend. If these rulps are violated., the adder, subtracter and divider will still produce .results, .but .these will, not be the' sum diff er- jl enceand quotient respectively. -It is not difficult to include checking . organs which signalize' ail infractions of the rules (a) , (b), (cf. ). ' /■ r t I ' A 4 " Jn .-junectian with multiplication' and division some r'e- marks .about rouniin^-off are necessary. It seems reasonable to -arry both the^e cperation o-e digit beyond what is to be kept — under ths present assumptions to %ts ■ . ■ 31-st digit— and thrift omit the supernumerary digit .by sane a^iridi'hg pro- cess.'' Just plain. .ignoring that, digit would, as is v. : ell kno/ru. c.aUs.<? systematical •roundin.j-off errors biased in one. direction (towards 0), The usual Gaussian decimal procedure of rounding off to the nearest. value' of the laso digit, kept y arid in case' of a .(supernumerary digit) S to the : V even .one means in tn. binary system this:,' 'Digit pairs (3 ;, -st and 31-st) 00,10' are rounded 'to 0,1; '01 is rounded to: 00: 11 in rounded by adding 01. This requires addvt Vtiii , with carry digits and their inconveniences. In- "Stoad one .may folio* tn-; equivalent -.of 'the decimal procedure of rounding. 5's to the nearest o'l iig\i , as suggested by J, ?/■, r.auchly. In the binary system this means that digit pairs, (30-st. and 31-st) 00, 01,-10, 11 are rounded to 0, 1, 1, 1. This rounding-off rule can be stated very simply: . The 30-st digit is rounded to 1 if either the 30-'st or the 31-st digit was 1, otherwise it is rounded to 0. A rounding-off valve which does this is shown on Figure 14. A digit (stimu- lus) is passed from is .to os while s is stimulated, but when s' is also stimulated, the digit is combined with its predecessor (that is the ona to its left) according to the above, rounding-off rule, i -42- 10.0 Circuit for the ar:. t r ne tical operation V~ . Other ope rat ions 10. I A square rooter network can bo built so that it differs - little from the divider. The description which follows is prelim- inary in the same sense as those of the multiplier and the divider net- works in 7.7 and 3.3. Binary square rooting consists of this: For each digital position in the squai e root (going from left to right), the square root a f ■.: .r to ' ''" • position) is used to for; - .■. + ], and this 2a-*- i is ;■■ • ■ ; fro.'.i tr« i irti il remainder (of the v '. .: I) air ! but ..:.; . .. - :.; '' r ; left by two posit _- •. iv w dj j'i crs • ..-■ >rigi } ; ".r exhausted), before this :v. ractior. t . res j 1 - ■.: . . • - negative (that i . . L? :J ■ :::■■■■-■. left digit : .•' . ■ . r root digit is 1 , . n ■-! ,; irl mainder s to 1 e ... • I following quotie _ . 1' , before the loubJ .; '.'' . .• i f erred to above) is the different ; stion. if the •• Is negative (that is, if its extrei . f1 ligit is I) then . luare root digit is 0, and the :v .:" ; iri Lai remainder (in the as above) is the preceding partial remaindi r, but in its doubly position. This proce iure is obviously very similar to that one 1 in division (cf. 3.3), with the follow'ing differences:' First: The i pie left shifts (of the partial remainder) .ire replaced by double cn^:" (with possible additions of new digits 0). Second: The quantity which is bi ing subtracted is not one given at the start (the dividend), tut n> -.ii^- i. determined by the result obtained so far: 2a+ 1 if a is th<- squaru root up to the position und^r consideration. -U3- i ( The first difference is a rather simple matter of timing, requiring no essential additional equipment. The second difference in- Lves a change in the connection, but also no equipment. It is. tru< , that 2a + 1 must be formed from a, but this is a particularly simple oper- ation in the binary system: 2a is formed by a shift left, and since 2a +-1 is required for a subtraction, the final -+1 can be taken into ac- count by omitting the usual correction of the extreme right digit in sub- traction (cf . 8.2, it is the stimulus on t" in Figure 11. which is to be omitted) . Now the square rooter network can be put together: Figure 15 The similarity with the divider network of Figure 13. is striking. It will be noted that di" i is not needed. The f/C 7 LI7?l /£- *6£ I oU JJ eC£ ZZT oO> HP J t" v; 0T*t l^y U oL. x <■ < < t t radicand is originally in | dl III 1, but is replaced, as the square rooting progresses, by the successive partial remainders. The valve routes the square root a (as formed up to that position) negatively into the adder — the timing must be such as to produce a shift left, there- by replacing a by 2a, and the absence of the extra correcting pulse for subtraction (t" in Figures 11 and 13, cf. the discussion above) replaces it by 2a + 1. The two valves [ v ] immediately under it select th«e partial -UU- r r remainder (of. below) and senH Lt from their common output lin hand .-.changed into I dl Ii I and o.i the other hand into t e adder, fr m where tne sum (actually the difference) goes into dl III The timing must be such as to produce the required double position shift left. Thus 11 II jt and dl III contain the two numbers from among which the i i i next partial remainder is to be selected. This selection is done by the discriminator d^ which governs the two valves controlling the (second addend) input of the adder (cf. the discussion of Figure 12 in 8.3). The sign digit of the resulting sum controls the discriminator, the timing stimulus at t must coincide with its appearance (extreme left digit of the sum) t' must be stimulated during the period during which the two addends (actually minuend and subtrahend) are to enter the adder (advanced by 3t). The square root is assembled in dl IV I , for each one of its digits the necessary stimulus is available at the second output of the discriminator (os 1 in Figure 12), it is passed into dl IV through the lowest valve I V]_ , timed by a stimulus at t'" . 10.2 The concluding remarks of 3.4 concerning the divider apply essentially unchanged to the square rooter. The rules of 9.3 concerning the sizes of numbers enter- ing into various operations are easily extended t j cover square rooting: The radicand must be non negative and the square root which is produced will be non negative. Hence square rooting must only be performed if the radicand is between and 1, and the square root will also lie be- tween and 1. The other remarks in 9.3 and 9.4 apply to square room- ing too. 10.3 The networks which can add, subtract, multiply, divide -45- f e- and square root having been described, it is now possible to decide : they are to be integrated in Ca, and which operations CA should be able to perform. The first question is, whether it is necessary or worth •vhile to include all the operations enumerated above: + , ~,X, f, \f. Little need be said about +,- : These operations are so fundamental and so frequent, and the networks which execute them are I so simple (cf. Figures 3 and 11), that it is clear that they should be included. With x the need for discussion begins, and at this 'stage a certain point of principle may be brought out. Prima facie it would seem justified to provide for a multiplier, since the operation x is very important, and the multiplier of Figure 9 — while not nearly as simple as the adder of Figure 3 — is atill very simple compared with the complexity of the entire device. Also, it contains an adder and there- fore permits to carry out +,- on the same equipment as x, and it has been made very simple by following the principle formulated in 5r3 - 5.7. There are nevertheless possible doubts -about the strin- gency of these considerations. Indeed multiplication (and similarly division and square rooting) can be reduced to addition (or subtraction or halving — the latter being merely a shift to the right in the binary system) by using (preferably base 2) logarithm and antilogarithm tables. Now function tables will have to be incorporated into the complete device anyhow, and logarithm — antilogarithm tables are among the most frequently used ones — why not use them then to eliminate x (and r, v ) as special operations? The answer is, that no function table can be detailed enough to be used without interpolation (this would under the conditions contemn -46-. ( plated, require 2^ - 10 entries!), and interpolation requires multi- plication! It is true that one might use a lower precision multiplica- tion in interpolating, and gain a higher precision'' one by this procedure-.- and this could be elaborated to a complete system of multiplication by successive approximations. Simple estimates show, however, that such a procedure is actually more laborious than the ordinary arithmetical one for multiplication. Barring such procedures, one can therefore state, that function tables can be used for simplifying arithmetical (or any other) operations only after the operation x has been taken care of, not ■ before! This, then, would seem to justify the inclusion of x among the operations of CA. Finally we come to 4- and a/ . These could now certainly 1 V'' be handled by function tables: Both -f and v' with logarithm - antilogarit'hu. ones, -^ also with reciprocal tables (and x).' There arc' also well known, . 2 fast convergent iterative processes: For the reciprocal u— 2u - au m ; I (2 - au) u (two oDerations.x per stage, this converges to a), for the I , I square root u - 2 u - 2au J5 - (2 - (2au) u) u (three operations x per stace, this converges to : . hence it must be multiplied by 2a at. V -ya. the end, to give /" a). However, all these processes require more or less in- volved logical controls and they replace -f and V by not inconsiderable numbers of operations x. Now our discussions of x, r , v show, that each one of these operations lasts, with 30 (biaary) digit numbers (cf. 7.1), order of 30 t, hence it is wasteful in time to replace t , -/"by even a moderate number of x. Besides the saving in equipment is not very significant: The divider of Figure 13 exceeds the multiplier of Figure 9 -47- (( f f by above 50% in equipment, and it contains it as a part so that duplica- aions are avoidable. (Cf. ). The snuare rooter is almost iden- tical with the divider, as Figure 1$ and its discussion show. Indeed the justification of using trick methods for -f , \ , all of which amount to replacing them by several x, exists only in devices where x has been considerably abbreviated. As mentioned in 5.3 - 5.4 the duration of x and also of -$■ can be reduced to a much smaller num- .... oer of t than what we contemplate. As pointed out loc. cit., this involves telescoping and simultaneising operations, and increasing the necessary ' equipment very considerably. We saw, that such procedures are indicated in devices with elements which do not have the speed and the possibilities of vacuum tubes. Jn such devices, the, further circumstance may be impor- tant, that X: can be more efficiently abbreviated than -r (cf. 5.4), and . it may therefore be worth, while to resort to the above mentioned procedures, which replace ~ ,./ by several x. In a vacuum tube device based on the principles of 5.3 - 5.7, however, x, 4- , \/ar.e all of the same order of duration and complication and the direct arithmetical approach to all of them therefore seems to. bo justified, in preference to the trick methods discussed above. Thus all operations +, -, X, -r , Vwould seem to deserve inclusion as such in CA, more or less in. the form of the networks of Figures 3, 11, 9, 13, 15, remembering that, all these networks should actually be merged into one, which .consists essentially of the elements of the divider, Figure 13.. The whole .or, appropriate parts of this network can then be ae- lected by the action of suitably .disposed controlling E-elements, which act as valves on the necessary connections, tq make it carry out the particular one among the operations. +., - , x , v, v which is desired. (Cf. For additional rematks on specific operations and general logical control, cf- -UB- ' t 10.4 The next question is, what further operations (besides + , - , y , 4 , y/ ) would be included in CA? As pointed out in the first part of 10.3 once x is avail- able, any ether function can be obtained from function tables with interpo- lation. (For the details cf.- ). Hence it would seem that beyond x (and -t- , - which came before it), no further operations need be included is such in CA. Actually -J- , v'""were nevertheless included, and the direct arithmetical approach was used for them — but here we had the excuse that the arithmetical procedures involved had about the same- duration as those i x, md required an increase of only about 50% in equipment. . Further operations, which one might consider, will hardly jet these specifications.' Thus the cube- root differs in" its arithmetical treatment essentially from the square root, ,as the latter' requires the intermediate operation 2a-) 1 (cf. 10.1}-, which is very'-simple, particularly ^ in the binary system while the former requires at the same points 'the in- . liate speration yi'- -j- 3a -J- 1 -3a *(a+l) + 1, which is much more com- plicated, since it involves a multiplication. Other desirable operations — i the logarithm, the trigonometric functions, and their inverses — allow n hardly any properly arithmetical treatment, in these cases the direct roach involves the use of their power series, for which the general logical control facilities of the device must be adequate. On the other hand the use of function tables and interpolation, as suggested above is in most cases more effective than the direct power series approach. These considerations make the. .inclusion of further algebraic- al jr analytical operations in.CA unnecessary. There are however some quite lementiry operations, which deserve, to be included for logical or organiza- tional reasons. In order to discuss. these it is necessary to consider the ^functioning of CA somewhat more closely, although we are not yet ready to do full justice to the viewpoints brought up in 7.. 3 and at the end of 10'.3. -A3- f f 11. C Organization of CA. Complete list of operations 11. 1 As pointed out at the end of 10.2 CA will be or- ganized essentially as a divider, with suitable controls to modify its action for the requirements of the other operations. (It will, of course, also contain controls' for the purposes enumerated in ?8 . ) This imp-lies that it will in general deal with two real number variables, Ivhich ?o into the memory organs fdl* I ^ , i dl IIj of the divider network of Figure 13. [These should coincide with the \ dl I I , ■ d 1 Uj of the Multiplier, Figure 9.' The' square rooter, Figure 15, m 'is no ' d_l_ I j , but it makes the same use ol d l li ] . The adder and subtracter were not con .ectcd on Figures' 2, 11. tc such mem- ory organs, but they will have tc be 'when the organization of CA is completed.) 3o »ve must tj ink of CA as having tub input organs, i dl I and j dl II \ , .and of course one output organ. (The latter has not beer, correlated with the adder and subtract or, c'f. above. For the mul- tiplier it is idl III | , for the divider arid square rooter it is Idl IV j- . These things toe will have to le adjusted in the final or- ganization 'ci" CA.) Let us denote these two inputs of CA by I ca and J ca , and the output by C& (each of them wi'th its attached memory organ), imatically shown on Figure 15. Now the following complex of problems must bo con- sidered: As mentioned re, particularly in 2o , an extensive memory L.1 KL Li • a (Tc* r A l^± M forms an essential part of the device. Since- CA is the main internal -30-' operating unit of the device ('..: stores, CC administers, and I, main- tain the connections with the outside, cf. the analysis in 2), the connections for transfers between M and CA are very important. How arc these connections to be organised? Ic is clearly necessary to be able to transfer from any port of K to CA, i.e. to I ca , J C{ , , and conversely from CA, i.e. from , to any part of K. Direct connections between various parts J M d: therefore n:.t seem to be necessary: It is always possible tv. transfer from. one part uf M to the other via CA. (Cf., however, ) These considerations give rise tc two Questions: First: Is it nec- essary tc connect each part cf M with both I and ' J cn cr can this b? simplified? Second: How are the transfers f,r m tile part of M to an other part of M to be handled, where CA is only a through station? The first question can be answered in the light of the principle of 56.- to place multiple events in a temporal succession rather than in ( simultaneous ) spaciM juxtaposition. This means that t.. . real numbers which go frwa M into I .and J crx , will have to g>. there in two successive- steps. This being so, it is just as well to route each real number first into 1^, nnd to move it on (within CA) from 1 ctl to J Cfl when the n<-xt real number comes (trom M) into I ca . V/e restate; Every real number coming from M into CA is routed into L ca . At the same time the real number previously in I cn is moved on to >. T ca» and the real number previously is J is necessarily cleared, i.e. ^.forgotten-. It 3houdd be noted, thf.t I cn and' J can be assumed to con- tain memory organs of the type discussed in 76. (Cf. Pigure 6, there, -51- c cf. also the varioua fdl ' i in the x, '/, , ^networks in Figures* 9, 13., 15.) in .vhich the re.^l numbers they hold ire circulating. Consequently the connections of I and J ca in CA ^rt those indicated 'in Figure 17. : Tut lines - - - conduct w!.en a real number Ifrcm M)enters CA, the lines conduct at all ether times. The connections of I„ Q and J with , , OB. Cfl t | ■ ' the operating parts of CA are supposed to branch out from the twe terminals — • • Tjuj output connects with ca the outside (relatively t; CA, i.e. with M) by the line , which Conducts when a result leaves CA (irr LI) • The circulating connections of 0_„ and its connections v/ith the cperating parts of CA are not shown, nor the E-eliments which control the connections shown (nor, cf course, the operating parts of CA). (For the complete description of CA of. .) ♦ 1.2 iiuh the help of Figures 16, 17 the second question is ilso easily answered. Fur a transfer from one part cf M to another part cf M, going through CA, the portion of the route inside CA i.s cleurly a transfer from I cn Jr J c& to ca . Denoting the real numbers in I , J ca by x, y, this amounts to "combining" x, ;■ to either x or y, since tlit: "result" of any operation performed by CA (like-)-, -, x, "/'• , <if~) is supposed to uppenr at . Tr.is operation is trivial and a special case e.g. of addition: If s (or y) ij wanted it suffices to get zero in the place of y lor x) - i.„. into 1^ ( wr j ) _ . md then M>ply the -52- # € operation-*-* On the ether hand, however, it seems preferable tc intro- duce these operations as 3uch: Flr9t: "Getting eerc int& I cg | ur 3^ a ') M is u .necessarily tim$ consuming. Sec nd: The .direct transfer fr.-m I cn (cr J ca j tc O ea , which these operations require is easily effected by a small part if the CA network visualised at the beginning <£■ 11.1. Third: We propose tc; intr< duce both operations (for I ca as v;ell as fcr J cn ), because it v;ill appear that each con play ft! separate useful role in the internal administration f CA tcf. below). / introduce accordingly two new' opera* ions: i and j , corresponding tt direct transfers from I ca or J ca tc ca . These "two eperations have these further use's: It will be s^en (cf. ) that the ;utput of CA (from O ca ) can be fed back directly int^ the input of C7. [to I c - t this ru. ves the c .ntonts cf I intc J ca f.nd clears J ca , cf 11.1.). Now assume that l ctl , J ca contain the real numbers x, y, and that i or j is applied, in ccnjuncti<n with this feedback. Then the ccnttntj of I J ca are replaced by x, x cr y, k. I.e. from the point of view of .any wther two variable operations («. -i •/• . i>e « **y» *-y,3yt r ) the variables x, y have been replaced by ;:, x or y, x. Ik w the litter is an important manipulation for the V un symmetric operations (x-y, J" ), and the former is important for the symmetric -perati.ns (x+y, xy) since it lends tc doubling and squaring. Beth manipulations are frequent enough in ordinary algebra, to justify a direct treatment by means cf the operations i, j. 11.3 A further necessary operati n is ■connected with the need to be able tc sense the 3ign of a number, ^r the order relation between two numbers, and to choose accordingly between twe (suitably given) -53- t alternative courses ef acti. n. It will ?.ppe'.r later, that the ability t. ch'.cso the first „r the second en. :1 tv, . iv,n number's u, v, in depend net upen sriich an alternative, is quite adequate te mediate the ch.ice between any fcv/i ^iven alternative c ur'sea ef uctlciK (Cf. Acccrdingly, we need an operati n which can 4. "his: Given f ^ur. nur.- hers x, y, u, v, it "forms" »--if x=y. (This .-erases the order rtlati n betwetn x, y. If we put y -«-0, it senses the sign cl :■:.) In this f-.m the ..per' tUn has f>.ur variables: ;:, y, u, v. (In th'. sign form it has three variables: x, u, v.) New the scheme f. r the CA net., rk eh sen at the beginning of 11.1, which was essentially that . ne c-f the divider, Jiad rcora f . r twi variables Only, and this is equal Xjjf true f r the dieeussicn . f the inputs cf CA in 11*1. Hence four (er three) variables '.re t ■ many. Consequently it is neces- sary t.. break ^ur c perati n up int tw. variable . per'ti ns - and then we taight as well d( his .. ith the more general (four rather than three variables) fern. It is plausible to be,' in with &■ (partial! operation which merely dteid.'s whether x '- y or x<y and remembers this, but with- out taking any action yet. This is best lone by forming x - y, and then remembering its sif-n di^it only,, i.e. its first digit (from the left). {Cf. ai This digit is for x - y * o,-i.e. x^y, and it is 1 for x - y<o, i.e. x'y.j Thus this (partial) operation is essentially in the nature of a subtraction, and it can therefore present no addi- tional difliculties in a CA which can subtract. Now it seems best to arrange' things so, that once this operation has b/-.n performed, CA will simply wait until two new numbers u, v have been moved into I_„, J ca ca (thus clearing t, y out - if u, v are to' occupy I cn ,, J ca , respectively, -54- c then v must be feci in first and u second), an • n transfer (without any further instructions,) u or v into C c „ (i.e. p norm i or j) accord- ing to whethr the sign digit referred to above «■ s or 1. We introduce accordingly such nn operation: 3. It is most convenient tc arrange ~t rings so, that after x, y have occup' I-„. J„,- , a subtraction is ordered and provisions nu~do that the result x - y should remain in cr . Then x, y must be displaced frttn I ca> J C!i by u, v and s ordered. 3 will sense whether the number in C ca is ».o or <o (i.e. x^ y ,. r x<y)» clear it. from ca , n.nd "fcrm" accordingly u or v in C% . The oueitition pr . ceding s need, by the way, not b< sub- tr-.cti.n: It might be addition- 6r i c r j. Accordingly the numb-- r in c , lf which provides the criterium for • 5; yi 11 n.,t be x - y, but *i*y' .r x or y. I.e. s will for u i'r"v according, t whether the multipli- city r>n or the division, tnd the -former might indeed be sometimes useful. F r details . f those :.peratilns cf. 11.-* Combining the e< nclusicns cf 10.2, 10.4, 11.2, 11.3 a list of eight ■. per-.t i< ns' cf CA obtai-ns.s + , -, X,' ■:/. , |r-, i , j, 9. I t) '-■■' more ..ill have to be addod, because .f tho necessity f ccn- v.rting number.; between tho binary and the decimal systems, as indi- cated at tho end . f 5.2V Thus we noed n decimal - to - binary conversion and a binary - to - d< cimal' conversions • db, bd. Th-. netw rr.s which carry Cut these t,i. operations will be discussed in . This concludes for the n. ra.ent the discussion -f CA. We have enumerated tho ten operations *hich it must be at>l< r:\rnw -55- * The questions if 7.8,, the general control problems d 11.1, -ind tjj$ specific networks for db, bd still remain t, be disposed of. But it is bettor t. return tc these after V5.ri.u3 other characteristics of the device have been d. cid< d upon. .Ve postpone therefore their discus- sion and turn now tc ^ther parts ,f the d-.vic^. 12.0 Capacity cf the mem,ry M» General principles 12.1 We consider next the third specific part: the men^ry M. Memory devices were discussed in 7.5 , 7.6 , since they are needed as ports of the x, •/. , networks (of. 7.4 , 7.7 for x, 83. fur /• , 10.2 fori') and hence ^f CA itself (df. the be f inning of 11.1 ). In all these casts the devices considered hid ,a sc-qut ntial or delay ch r racter, which was in nioat cases made cyclical by suitable terminal i rgans. 'Ire precisely: The blocks li. a nd dl (k) in 7b , 76 are essentially delays , which h Id a stimulus that enters there input for a tine kt , and then emit it. Consequently they can be converted into cyclical memories, -Jhich hold a stimulus indefinitely, and make it available at the output at all times which differ from each other by h multiples of kt. It suffices for this purpose to- feed the output back into the input: •/ ;_■ |l_ *\ or >' < I ( k,, i ~ . Since the period kt contains k fundamental periods t, the capacity of such a m'-mcry device is k 3tinuli« The above schemes l&Ck the proper input, cle' ring and output facilities, but those are shown on Figure 6. It should be nrtod that In Figure 6. tho cycle around Ik goes thr ugh enc more E-ele- ment, and therefore the period of this device is actually lk + 1) t,nnd its capacity cnrrespi.nding.ly k rl stimuli. (The ! IK -f Figure &• may, of course, be replaced by a •• , -41 (k)[ , cf. 76.) -56- t Now it is by no means necessary, that neu- ry be f this cyclical (or delay) type • './e must therefore before making a decision concerning M, discuss other possible types and the advantages and dis- advantages of the cyclical type in comparison with them. 12.2 Preceding this discussion, however, we must consider the capacity which we desire in M. »Ve did already r.iontion r.bove this concept of capacity for II or a part of U.: It ia the number of stimuli which this organ can r<.uember, or itfore precise lyj the number of occasions for which it can remember whether or not a stimulus was present. The presence or absence jf a stimulus lat a ^iven occasion-, i.e. on a given line in a given moment) can be used tl express the value 1 or fir a binary digit (in a given position)-.- Hence the capacity of a memory is i the number uf Mnary digits (the values ... f) .,/hich it can retain. In >. ther words; The '(capacity), .unit of memory is (the ability to retain the value of one binary digit. vie can new express the "cost" of various types of in- fematien in these nem ry unit^. Let us consider first the memory capacity required to/ store a standard- ( real) number. As;. indicated in 71-» we shall fix the size of such a number at 30 binary digits (at least for m.,st uses, cf. ) This k-eps the relative r< unding-ef f errors below 2 , which c rresponds tu^lO- 9 , i.e. t<- carrying y significant decimal digits. Thus a standard number corresponds to 20 memory, units. To this must be \ ■' ! . . added ne unit for its 'sign ( cf. the end _rf 92.) and it is advisable t add a further unit in lieu of a symbol which characterises it as a -57- number ( K distinguish it fr o an '.rder, cf. ). In this *r.y we arrive to 32 - 2 5 units per number. ?he fact th'it a number requires 32 Memory units, makes it advis-ible to subdivide the entire nem-ry in thi3 way; First, b- vieusly, intt units , second int, fr ups i.f 32 units, to be called minor cycles . (F.r. the naj r cycles cf. ) Each standard Ireal) number accordingly • .couples precisely ine miner cycle. It simplifies the i.rgahizatUn . f the entire memory, and vari- us synchr- ni?nticn pr i.tjlens if the device rilong with it, if all other ponsfcanta of the meau.ry are also msde tc fit ints. this subdivision into mi a r cycles. Recalling the classification (a) - (h) of 24'. for the presumptive contents cf the memory M, we note: (a), iccording to our present ideas belongs to CA and not to M (it i9 handled by Jdl I to fdl l"vj , cf. the beginning of 11.1) (t ) - (g) , and probably (h) also, consist of standard numbers. (},) on the other hand consists of the operation instructions which govern the functioning of the device, to be .called standard orders . It will therefore be necessary to for- , mulate the standard orders in such a manner, that each one should also a ■ t occupy precisely one minor cycle, .'.e. 32 units. Thi9 will be done in < • 12 « 3 ,'/e are now in a position to estimate %hi Capacity re- quirements of each memory type (a) - (h) of 2.4. Au (a): Need not be discussed since it is taken care •of, in CA- (df. above). Actually, since it requires \ cl Ij to i dl__ IV] , efich of which must hold essentially a standard number, i.e. 30 units -58- ( f i (with 9r.i5.11 deviations, cf. ), this corresponds to - 120 units. Si^se this is not in M, th<. organization into minor cycles does not .apply here, but we note that - 120 units correspond - to - 4 r.; nor cycles. Of course some other parts of CA r -re memory organs too, usually with capacities of one or a few units: E.g. the discriminators oi Figures 8. and 12. The complete CA actually contains / mere j ^ I organs, corresponding to - / units, i.e. - o minor cycles ( cl'. ) • Ad (b): The capacity required for this purpose can only be estim&ted after the form oi all standard orders has. been de* cidtd upon, and several typical prblems have ■' been formulated - '.'set up" - in that terminology. This ./ill be done in . It ..ill then appear, that the capacity r'c quiT- nant's of (L) are small compared t^ those of acne of (c) - (h), particularly tc those of. (c). Ad (c): As indicated loc, cit., we c^unt on function tables of 100-2' £ entries. A function table is primarily a switch* ing pr blen, end the natural numbers of alternatives "for a switching 7 system are the powers of 2, lCf» ,) Honce 126 - 2' is a suitible number of entries. Thus' the relative precision < bt'-.ined —7 — "fi directly for the variable 19 2. , Since n relative precisi on of 2 vU is desired for the result, and (2-'~ 7 )4> 2* 20 , (2 7 )5 « 2^°, the- in- terpolate n. error must be fifth order, i.e. the interpolation bi- quadratic. (One might go tc even higher Order interpolation, and hence fewer entries in the function table. However, it ..ill appear that the capacity requirements of tc) are even for 128 •' ntri'es small conp&red e. g. tc the 3- of (c)«) With biquadratic interpolation five -59- t ( f tnble values are needed for each interpolation: Two above rmd two belc* %be rounded off vnrir.ble. Hence 128 entries allow actually the use of 124 only, and the 3c- correspond U 122 intervals, i.e. n rela- tive precision 122 • 1 fcr the variable. However even 122- <*'- 2-^° 1 (by a factor -25), Thu3 r. function table consists cf 128 numbers, i.e. it requires a capacity cf 128 minor cycles. The familiar .-mtheraati- cal prtblens hardly ever require more than five function tables (very , rarely that much), i.e. a capacity cf 640 minor cycles seen to be a . sr»fe ■. verestimate ^f the capacity required f ^ r (c). Ad (d): The3e capacities are clearly less than- or at most comparable to those required by (e). Indeed the initial values are the same thing as the intermediate values of If), except that they "belong to the first value of t. And in a partial, di-ffer- ential equation with n + 1 variables, say xi , , x n and t,- the intermediate values of a given t - to be discussed under (e) - as .well- as the initial values or the totality of all boundary values for all t correspond all three to n-dimensional manifolds (in the n + 1 - dimensional space) of x\ , , * n and t; hence they are likely to involve all about the same number of data. Another important point is, that the initial values and the boundary values are usually given - partly or wholly - by a formula - or by a moderate number of formulae. I.e., unlike the intermediate values of (e), they need not be remembc>red as individual.. numbers. Ad •(e): twr a partial differential equation with two variables, say x and t, the number of intermediate v.lues for a -60- given t is d-.termined by the number of x - lattice points used in the calculation. This is hardly ever more thin 150, and it is unlikely that more than 5 numerical quantities should be associated with each point. Ixi typical hydrodynaraical problems, ./here x iB «■ Lagrange ian label - coordinate, 50-100 points are usually a light estimate., and 2 numbers are required at each point; A f osition - co- ordinate and a velocity. Returning to the higher est imr.te of 150 points and 5 numbers at each point gives 750 numbers, i.e. it requires a capacity of 750 minor cycles. Therefore 1,000. minor cycles seen to be a safe overestimate of the capacity required for (e) in two vari- , able 9x and t) prcbler.s. For a partial differential equation with three vari- ables, say x, y and t, the estimate is harder to make. In hydro- dynamical problems, at least, important progress could be uo.de with 20 x 20 or 40 x'2C or similar numbers of x., y - lattice points - say 1,0^0 points. Interpreting x, y again in Lo<. range i an labels 3hows, that at least 4 numbers are neeaed at each point: Two position co- ordinates and two velocity components, we take 6 numbers per point to allow for possible other m.n hydrodynaraical quantities. This gives 6.CCC numbers, i.e. it requires a c u-city of 6,000 minor cycles for (e) in hydro dynamical three variable (x, y and t) problems. It will be seen (cf. ), that A memory capacity .f 6,010 min^ r cycles - i.e. of - 200, 0CC units - is still conveniently fe-siblt but that essentially higher capacities would be increa3infly difficult to control Even 200,000 units produce some- -61- ( .vh".t „f *.n unbalance - i..e. they n-.ke M bigger than the thejr parts i. f the device put t^£-ether. I- seems r.herefcre un-vise tt gt further, -;nd tc try t~ tret for variable i;:, y, z and t) pr.bler.is. It should be noted that two variable (x and t) prob- leras include all linear or circular symmetric plane or spherical sym- metric snaciul transient problems, also certain general plane or cylinder symmetric SDacial stationary problems (they must be hyper- bolic, e.g. supersonic, t is replaced by y). Three variable problems {:•:, y and t) include all spacial transient problems. Comparing this enumeration with the well known situation of fluid dynamics, elasticity, etc., shows how imoortant each on<? of these successive stages is: Complete fret dom with two variable problems^ extension to four variable Drobiems. «.s we indicated, the possibilities of the practical size for 'iA draw the natural limit for the device contemplated at present' between the second and the third alternatives. It will be seen that considerations of duration place the li:nit in the same place (cf. ) nd (fji The-memory capacities required by a total differential equation with two variables - i.e. to the lower estimate of (e). Ad (g): As pointed out in (g) in 2k., these problems ire very similar to tr.ose of (e), except that the variable t now dis- arm. Henc< '.he lower estimate of (e) (1,000 minor cycles) applies when a system of (at nost 5) one-varii.ble functions (of x) is being p^ht by successive •iDnroxii:'.---tion or relazation methods, while the u.er estimate of (e) (6,000 minor cycles) applies when a system of •-■ st 6) two-variable functions (of x, y) is bein£ sought. Lany -62- c ( problems of this -type, however, deal with one. function only - this cuts the above estimates considerably (to 2>- l or 1,000 minor cycles). Problems in which only a system of individual constants is being sought by successive- aoproxinat ions, require clearly smaller capacities: ' They compare to the preceding Droblems like (f) to (e). ad (h); These problems are so manifold, that it is difficult -to plan for them systematically at this stage. In sorting problems, any device not based freely oermutable record elements (like puncheards) has certain handicaps (cf. ), besides this subject can only be adequately treated z't r an analyst of the relation of l.l and of R has been made (cf. 29 and ). It should be noted,, however, that the standard punchcard has place for 80 decimal digits.,, i.e., - 1 9-cigit decimal numbers, th ;t is - 9 numbers in our present S' nse, i.e. - 9 m nor cycles. Hence '.he 6,000 minor cycles considered in (e) correspond, to a sorting capa- city of - 700 fully used cards. In the most sorting problems the 80 columns of the cards a far from fully used - this may increase the equivalent sorting capacity of pur device proportionately above 700. This means-, that the device has a non negligible, but certainly not : ssi-ve sorting capacity. It is probably only worth using on sorting problems of more than usual mathematical complexity. In statistical experiments the memory requirements are usually small: Each individual Droblem is usually of moderate com- i plexity, each' individual nroblera is independent (or only dependent by a few data) from its •predecessors; and all that need be remembered through the entire^ sequence of individual orobi'-ms are the numbers of .--63- c 1 ( ( how many problems successively solved had their results in each one of a moderate number of p.iven distinct classes. 12. A The estimates of 12.3 can be summarized as follows: The needs of (d) - (h) are alternative, i.e. they cannot occur in the sane problem. The highest estimate reached hero was one of 6,000 rain or cycles, but already 1,000 nunor cycles would p'ermit to treat many im- portant probl-ras. (a) need not be considered in A. (b) and (c) a cumulative, i.e. they may add to (d) - (h) in the same problem. 1,000 minor cycles for each, i.e. 2,000 together, seem to be a safe over- estimate. If the higher value 6,000 is used in (d) - (h), these 2,000 may be added for (b) - (c). If the lower valuo 1,000 is used in (d) - (h), it seems, reasonable to cut the (b) - (c) capacity to 1,000 to. (This amounts to ass^imii\g fewer .function tables and somewhat less •complicated "set ups" . Actually even those estimates are generous, cf. ) Thus tot cil capacities jf 8,000 or 2,000 minor cycles obtain. It will be seen that it is desirable to have a capa- city of minor cycles which is. a power of two (cf. ). This makes the choices of 8,900 jr 2,X0 monor cycles of a convenient approxi- mate size: They lie very near to powers of, two. We consider accord- ingly" those t..o t <tal memory, capacities: 8,196 ~ 2^ or 2,0LB * 2" :..inor cyc les , i.e. 2 62,272 ■= 2 18 or 65,336 = 2 16 units. For the purposes of the discussions which follow we will use the first higher ■- a 1 1 mat e . This result deserves to be noted. It shows in a most strik- ing way whore t, he,., real difficulty, the main bottleneck of an automatic -6v < i very high st cd computing device lies! At the memory. Compared t< the relative simplicity of CA (cf. the beginning of 11.1 and ), and to the simplicity of CC and of its "-code" (cf. iftd ), !,i is somewhat impressive: The rcnuirerients formulated in 12.2, which were considerable but by no means ohantastic, necessitate a memory M - with a capacity of about a quarter -dllion units', Clearly the practicality of u device as is contemplated here depends most critically on the possibility 'of building such an ii, and on the question of how simple 8uch an ivi can be made to be. 12 .5 How can an M of a capacity of - 2 18 250*000 units be built? The necessity of introducing del-y elements of very great efficiency, as indicated in 75'., 76. \ and 12.1, becomes now obvious:' One iv-element, as shown oh Figure /„'. , has & unit memory .•■.'''•.a' ■ » capacity, hence any direct solution of the problem of construction 'd with the help of it-elements would require as many E-eloments -as the desired capacity of 1-i - indeed, because of the necessity of switch- ing and gating about four times more j (cf. )'. This is m?ni- festly impractical for the desired capacity of - 25 u ',000 - or, for that natter, for the lower alternative in 12.5', of - 65, OCX). We therefore return to the discussion of the cyclical or delay memory, which was touched upon in 12.1. (An other type will be considered in 12.6) Del fdl (k)| can. be built with great capacities k, without using any L-e laments at all. This was mentioned in 76, together with the fact that even linear electric circuits of this type -65- . exist'. Indeed, the contemplated t- of about one microsecond requires a circuit passband of 3 - 5 megacycles (remember Figure 1. '. ) and then the equipment required for delays of 1 - 3 microseconds - i.e. k = 1, 2, 3 - is simple and cheap, and that for delays up to 30 -35 microseconds - i.e. k =30, , 35 - is a-vailable and not unduly ex- pensive or complicated. Beyond this order of k, however } the linear electric circuit approach becomes impractical. This means that the delays — >*- >^ — . ???— which occur in all ^-networks of Figures 3. -15. can be easily made with linear circuits', also, that the various Jdl | of CA (cf. Figures 9, 13, 15, and the beginning of 11. 1( , which should have k values - 30, and of which onlv a moderate number will be ne-'-ded (of. (a) in 12.3), can be reasonably made with linear circuits. For k itself, however, the situation is different. ii must be made un of j dl organs, of a total capacity - 250, OuO.' If these were linear circuits, of maximum capa- city - 3 U (cf. abovo), then - 3,000 such organs would be required, which is clearly impractical. This is also true for the lower it . rna'ive of 12 .'5 , capacity - 65,000, since even then - 2,000 such organs would be n cessary. i'OW it is possible to build dl organs which have an electrical innut :ind output, but not a linear electrical circuit in between, idth k values up to several thousand. Their tv tar-.' is such, that a - U stage amplification is needed at the output, which, apart from its anpiifying character, also serves to reshape and resynchronize the output pulse." I.e.* the last stage gates the clock pulse (cf. 63.) -66- ( ( — using a non linear pert of a vacuum tube characteristic which goes across the cutoff; while all other stages effect ordinary amplification, using linear parts of vacuum tube characteristics. Thus each one of these J dl I requires - U v cuum tubes at its output, it also requires - 4 E-elements for switching and gating (cf. ). This gives probably 10 or fewer v. cuum tubes per i dl i organ. The nature of these dl organs is such, that s few hundred of them can be built and incorporated into one device without undue difficulties - although they will then certainly constitute the greater oart of the device (cf. ). Uow the a cap- city of 25U,O00 can be achieved with such dl devices, each one having a capacity 1,000 - 2,000, by using 250 - 125 of them. Such numbers are still manageable (cf.. above), and they require about 8 times more, i.e. 2,500 - 1,250 vacuum tubes. This is r.i considerable but perfectly practical number of tubes - in- deed probably considerably lower than the upper limit of practicality. Th' fact that they occur in identical groups of 10 is. also very ad- vantageous. (For details cf. ) It will be seen that the j other parts of the device- of which CA aid CC aire electrically the most complicated, renuir; together <^1,000 vacuum tubes. (CL. ) Thus the vacuum tube requirements of the device are controlled essen- tially b" *i, and 'hey ire of the order of 2,000 - 3/000. (Cf. loc. cit. -'.'rove.) This ccyifirms the conclusion of 12. U,, that the decisive cart of t'r . device,, determining more than -iny other part its feasi- bility, dimensions and cost, is the memory. -67- ■. i I We must ncW decide more accurately what the caoocity of each , dl organ should be - within the linits which were found to bo pr c-ic.l. h combination a few very simple viewpoints loads to such . decision. 12.6 v/c saw above that each I dl j organ requires about 10 .ssoci; ted vacuum tubes, essentially ind< -x-ndently of its length, (A very long i dl ] night require one more stage of amplification, i.e. il vacuum tubes.) Thus the. number of i dl i organs, and not the total capacity determines the number of vacuum tubes in M. This would justify using as few dT] organs as possible, i.e. of as high indivicu-: 1 cap:>city as nossible, 'Now it would probably be feasible to devi I'd ' ( dl i 's of the type considered with capacities considerably higher th'n the few thousand mentioned above. There are, however, r consid orations which set a limit to increases of j dl In tlit first place, the consid' rations at the end of 63. "row, that the definition of j dl 's deiav tjjae must be a frac- tion t" '. : t ( .bout 5 ~T"), so that -.a ..eh stimulus emerging from \ dl ) may gate the correct clock oulse for the output. For :. capacity !:, i.e. a delsy kt, this is relative precision 5k -2k, which is perfectly feasible for the device in question when k - 1,000, but becomes in- creasingly uncertain when k increases beyond 10,000. However, this argument is limited by the consideration that as the individual J dl | capacity increases, corrusoondinply fewer such organs are needed, -aad therefore each one can be Made <«ith corr< spondngly more attention <*nd precision. -63- £ ( K. Z i Next, there is another more sharply limiting consider- ation. If each i dl~~ ; has the capacity k, then ^0,000 of them will be needed, an ^ 0|Q P° amplifying switching and gating vacuum tube ag- gregates are necessary. Without going yet into the details of these circuits, the individual | dl ) and its associated circuits can be shown f / q ure / i "L cdt ^y schematically in Figure 18. Mote, that Figure 6. showed the block SG in detail but the block A not at all. The actual arrangement will differ from Figure 6. in some details, even regarding SG, cf. . Since \ dl \ is to be used as a memory its output must be fed back— directly or indirectly— into its input. In an aggregate of many | dl | organs— which M is going to be— we have a choice to feed each j dl \ back into itself, or to have longer cycles of i dl j's: Figure 19. (a) and (b), respectively. F I q (/ r e. *J -na >-[ eU vt | L2£j— * ^O J — Q— illh It should be noted, that (b) shows a cycle which has a capacity that is a multiple of the individual j dl | ' s capacity— i.e. this is a way to produce a cycle which is free of the individual | dl j's capacity limitations. This is r of course, dua to the reforming of the stimuli -69- c f v traversing this aggregate at each station A. The information contained in the aggregate can be observed from the outside at every station SG, and it is also here that it can be intercepted, cleared, and replaced | by other information from the outside. (For details cf . ) Both statements apply equally to both schemes (a) and (b) of Figure. 19. Thus the entire aggregates has its inputs, outputs, as well as its switching and gating controls at the stations SG — it is here that all outside connections for all these purposes must be made. ^) To omit an SG on the scheme (a) would be unreasonable: It would make the corresponding dl complete inaccessible and use- less. In the scheme (b), on the other hand, all SG. but one could be omitted (provided that all A are left in place): The aggregate would still have at least one input and output that can be switched and gated and it would therefore remain organically connected with the other parts of the device — the outside in the sense used' above. We saw in the last part of 12.5, that each A and each SG required about the sane number of vacuum tubes (4), hence the omission of an SG represents a $0% saving on tho associated equipment at that junction. Now the number of SG stations required can be estimated. (It is better to think in terms of scheme (b) of Figure 19 in general, and to turn to (a) only if all SG are known to be present, Cf. above.) Indei d: Let each dl have a capacity k, and let there be an SG after every 1 of them. Then the aggregate between any. two SG has the capacity k' = kl. (One can also use scheme (b) with aggregates of 1 dl 's each and one v 3G each.) Hence 2 ,5,° i.QPQ SG's are needed altogether, and the switching -k' -70- <c problem of M is a A » way one. On the other hand every individual K emory unit passes a position SG only at the end of each k't period. I.e. it becomes accessible to the other parts of the device only then. Hence if the information contained in it is required in any other part of the device, it becomes necessary to wait for it — this waiting time being at most k't, and averaging \ k't. This means that obtaining an item of information from M consumes an average time \ k't. This is, of course, not a time re- quirement per memory unit: Once the first unit has been obtained in ^' this way all those which follow after it (say one or more minor cycles) consume only their natural duration, t. On the other hand this variable .-. . Lting time (maximum k't, average 5 k't), must be replaced in most cases by a fixed waiting time k't, since it is usually necessary to ^_ return to the point in the process at which the information was desired, after having obtained that information — and this amounts altogether to a precise period k't. (For details cf. ). Finally, this wait k't is absent, if the part cf M in which the desired information '3? is contained follows immediately upon the point at which that information is wanted and the process continues from there. We can therefore say: The average time of transfer from a .general position in M is k't. Hence the value of k' must be obtained from the general principles of balancing the time requirements of the various operations of the device. The considerations which govern this particular case are simplt In the process of performing the calculations of mathe- matical problem a number in U will be required in the other parts of the device in order to use it in some arithmetical operations. It is excep- "/ lional if all these operations are linear, .i.e. + , - , normally''-, and -71- C'/ \c ( possibly "f , v , will also occur. It should be noted that substituting ^ a number u into a function f given by a function table, so as to form f(u), usually involves interpolation — i.e. one x if the interpolation is linear, which is usually not sufficient, and two to four x's if it is quadratic to biquadratic, which is normal. (Cf. e.g. (c) in 12.3.) A survey of several problems, which are typical for various branches of computing mathematics, shows that an average of two x (including -f , v ) per number obtained from M is certainly not too high. (For examples cf . . ) Hence every number obtained from M is used for two multiplication times or longer, therefore the waiting time required for obtaining it is not harmful as long as it is a fraction of two multipli- cation times. ^ 9 A multiplication time is of the order of 30 times t (cf. 5.3, 7.1 and. 12.2, for -f, v~cf. 5-5) say 1,000 t. Hence our condition is that k't must be a fraction of 2,000 t. Thus k'— 1,000 with k — 1,000 is perfectly feasible seems reasonable. Now a dl (cf. the second part of 12.5), hence k « k' — 1,000, 1 - 1 is a logical choice. In other words: Each dl has a capacity k — 1,000 and has an SG associated with it, as shown on Figures 18., 19. This choice implies that the number of dl j's re- quired. isr~ 3/V i ff 0° — 250 and the number of vacuum -tubes "in their asso- ciated circuits i's about 10 times more (cf. the end of 12.5.), i.e. — 2,500. 12.7 The factorization of the capacity— 250,000 into 250 dl J organs of a capacity — 1,000 each can also be interpreted in this manner: The memory capacity 250,000 presents prima facie a 250,000 — -72- C' (' or «<£ way switching problem, in order to make all parts of this memory immed- ' lately accessible to the other organs of the device. In this form the task is unmanageable for E-elements (e.g. vacuum tubes, cf. however 12.8) The above factorization replaces this by a 250 -way switching problem, and replaces for the remaining factor 1,000 the (immediate, i.e. syn- chronous) switching, by a temporal succession — i.e. by a wait of 1000 t. . This is an important general principle: A c = hk - way switching problem can be replaced by a k - way switching problem and an h-step temporal succession - i.e. a wait of ht. We had c - ' n ■■ y 250,000 and chose k - 1,000, h - 250. The size of k was determined by K. the desire to keep h down without letting the waiting time kt grow beyond one multiplication time. This gave k - 1,000, and proved to be ^ dl of capapity k. compatible with the physical possibilities of a It will be seen, that it is convenient to have k, h, and hence also c, powers of two. The above values for these quantities are near such powers, ,and accordingly we choose: Total capacity of M: c :: 262,144 = 2 XH . Capacity of a. i dl J' organ: k = 1,024 = 2 10 . Number of dl J organs in M: ha 256 = 2 8 . The two first capacities- are stated'in memory units. In terms of minor cycles of 32 - 2 memory units each:"" Total capacity, of 1! in minor cycles: c/32 - 8,192 = 2 -*. Capacity of a '._dl ; organ in minor cycles: k/32 a 32 = 2 . 12.8 The discussions up to this point were based entirely on the assumption of a delay memory. It is therefore important to note that this need not be the only practicable solution for tne" memory problem - indeed, that there exists an entirely different approach which may even -73-' (>r " «; appear prima facie to be more natural. \% The solution to which we allude must be sought- along the lines of the iconoscope . This device in its developed form remembers the state of - 400 x 500 =. 200,000 separate points, indeed it remembers for each point more than one alternative. As it is well known, it re- members whether each point has been illuminated or not, but it can dis- tinguish more than two states: Besides light and no light it can also recognize— at each point — several intermediate degrees of illumination. These memories are placed on it by a light beam, and subsequently sensed by an electron beam, but it is easy to see that small changes would make **' ...'■'■••■■■ it possible to do the placing of the memories by an electron beam also. Thus a single iconoscope has a memory capacity of the same order as our desideratum for the entire M (-250,000), and all memory units are simultaneously accessible for input and output. The situation is very much like the one described at the beginning of 12.5., and there characterized as impracticable with vacuum tube-like E-elements, The- iconoscope comes nevertheless close to achieving this: It stores 200,000 mem6ry units by means of one dielectric plate: The plate acts in this case like - 200, 000 independent memory units — indeed a condenser is a perfectly adequate memory unit, since it can hold a charge if it is properly switched and gated (and it is at this point that vacuum tubes are usually required). The 250,000-way switching and gating is done (not by about twice 250,000' vacuum tubes, which would be the obvious solution, but) by a single electron beam — the switching action proper being the steering (deflecting) this beam so as to hit the desired point on the plat e i ' e ( r e* % Nevertheless, the iconoscope in its present form is not immediately usable as a memory in our sense. The remarks which follow bring out some of the main viewpoints which will govern the use of equipment of this type for our purposes. (a) The charge deposited at. a "point" of the icono- scope plate, or rather in one of the elementary areas, influences the neighboring areas and their charger. Hence the definition of an elemen- tary area is actually not quite sharp. This is within certain limits tolerable in the present use of the iconoscope, which is the production of the visual impression of a certain image. It would, however, be sntirely unacceptable in connection with a use as a memory, as we are contemplating it, since this requires perfectly distinct and independent registration and storage of digital or logical symbols. It will prob- ably prove possible to overcome this difficulty after an adequate devel- opment — but this development may be not inconsiderable and it may neces- sitate reducing the number of elementary areas (i.e. the memory capacity ) considerably below 250,000. If this happens, a correspondingly greater number of modified iconoscope will be required in U. (b) If the iconoscope were to be used with 400 x 500 = 200,000 elementary areas (cf. above), then the necessary switching, that is the steering of the electron bean would have to be done with very . considerable precision: Since 500 elementary intervals must be distin- guished in both directions of linear deflection, a minimum relative precision of — - x t-t-t = ,1% will be necessary in each linear direction. Titis is a considerable precision, which is rarely and only with great difficulties achieved in "electrical analogy" devices, and henco a nost -75- € ( i 1 fe ; -m inopportune requirement for cur digital device. A more reasonable, but still far from trivial, linear precision of, say, ,5£ would cut the ' ' memory capacity to 10,000 (since 100 x 100 ■= 10,000, -i- x -i- =' .5%)'. There are ways to circumvent such difficulties, at least in part, but they cannot be discussed here. (c) One main virtue of the iconoscope memory is that it permits rapid switching to any desired part of the memory. It is entirely free of the octroyed temporal sequence in which adjacent memory units emerge from a delay memory. Now while this is an important advan- Y tage in some respect, the automatic temporal sequence is actually desirable in others. Indeed, when there is no such automatic temporal sequence it is necessary to state in the logical instructions which govern the prob- lem precisely at which location in the memory any particular item of information that is wanted is to be found. However, it would be unbear- ably wasteful if this statement had to be made separately for each unit of memory. Thus the digits of a number, or more generally all units of a minor cycle should follow each other automatically. Further, it is usually convenient that the minor cycles expressing the successive steps in a sequence of logical instructions should follow each other automat- ically. Thus it is probably best to have a standard sequence of the con- stituent memory units as the basis of switching, which the electron beam follows automatically, unless it receives a special instruction. Such a special instruction may then be able to interrupt this basic sequence, and to switch the electron beam to a different desired memory unit (i.e. point on the iconoscope plate). This basic temporal sequence on the iconoscope plate corresponds^ of course, to the usual method of automatic sequential scan- ning with the electron beam — i.e. to a familiar part of the standard -76- i < k i ..-^ iconoscope equipment. Only the above mentioned exceptional voluntary switches to ether points require new equipment. Tc sum .:p: It is not the presence of a basic temporal sequence of memory units which constitutes a weakness of a delay memory as compared to an iconoscope memory, but rather the inability of the former to break away from this sequence in exceptional cases (without paying the price of a waiting time, and of the additional equipment required to keep this waiting time within acceptable limits, cf. the last part of 12.6 and the conclusions of 12.7)- An iconoscope memory V should therefore conserve the basic temporal sequence by providing the usual equipment for automatic sequential scanning with the electron un, but it should at the same tine be able of a rapid switching (de- flecting) of the electron beam to any desired point under special in- ^ struction. (i) The delay organ dl contains information in the form of transient waves, and needs a feedback in order to become a (cy- clical) memory. The iconoscope on the other hand holds information in a static form (charges on a dielectric plate), and is a memory per se. Its reliable storing ability is, however, .not unlimited in time — it is a matter of seconds or minutes, './hat further measures does this neces- sitate? It I b noted that M's main function is to store information which Ls required while a problem is bring solved, since it is then that there is a .ue ■ the rapid accessibility, which the main advantage of M over outside storage (i.e. over h, cf. 2.9). Longer range storage--e.g. of certain function tables like log, sin, or -77- <r I ( r € ..-^ equations of state) or of standard logical instructions (like interpola- tion rules) b problems, or of final results until they are printed — should be definitely effected outside (i.e. in h, cf. again 2.9 and ) Hence LI should only be used for the duration of one problem and consider- ing the expected high speed of the device this will in many cases not be long '/riough tc ' .' ct th< reliability of '..'.. In some problems, however, it will be too long (cf. ), and then special measures become necessary.' The obvious solution is this: Let Nt be a time of reliable storage in the iconoscope. (Since Nt is probably a second to A Q 15 minutes, therefore t - one microsecond gives H - 10 - 10. For N,- 10' this situation will hardly ever arise.) Then two iconoscopes should be used instead of one, so that one should always bo empty while the other is in use, and after N periods t the latter should transfer its information to the former and then clear, etc. If M consists of a 'reater number of iconoscopes, say k, this scheme of renewal requires k + 1, and not, k iconoscopes. Indeed, let I Q , I,, , I, be f: ■ iconoscopes. Let -' 3 given moment I- be empty, and I, , I. ■,, I-s+is , lu in use. Aft. r -^- periods t I.., should transfer its informa- k+1 lTi tion to I- and then clear (for i = k replace i+i by 0). Thus 1^^.^ takes over the role of I-. Hence if we begin with I c , then this process goes through a complete cycle I-,, I 2 , , Ik anL! back to *o> in ^ r ^ steps of duration -JL t each i.e. of total duration Nt. Thus all I , k '-i I,, , I k are satisfactorily renewed. A more detailed [dan jf these arrangements would have to be bas- d on a knowledge of the precise orders of magnitude of N and k. ,'.' i I not do this here. We only witsh to 3 -73- { I 6 <- § •f^i emphasize this point: All these considerations bring a dynantical and cyclical element into the use of t! intrinsically static iconoscope — it forces us to treat them in a mani er soi vj I rable to t! nner ) which a delay (cyclical memory treats fc le ing] memory units. From (a) - (d) .. ' - iclude this: It is vary probable that in the end the iconoscope memorj rill prove superior t ' < a -j memory. However this may recjuire ? >n : further ic-v J . > ... nt ii se-v ral respects, and for various reasons the actual use of the iconoscope ^v ory will not be as radically lifferent from that of t i Lay ; enory, as • ight at first think. Indeed, (c) ana (d) show thai the two havi i deal in common. For these reasons it seem: reasonable to continue analysis on th.e oasis of a delay memory although the importance of iconoscope memory is fully realize i. 13»0 Or;-, :iiz ". tion ci '1 13»I We return to the discussion of a ielay m mi vy bas> d on 3?) the analysis and the conclusions of 12. t :. a 12.7. It is b ?t I :> t :.r J by considering Figure 19 again, and the alt . r . 3 which it r.hibits. a We know from 12.7 that we must think in t ,rms cf 256 - 2"' •■ ns 1 '1 1 of capacity 1.024 - 2 " each.. For ?. w] il it will nor. be neces: arj to lecide which of the two alternatives Figur* 19 ('.' ) and (b) (or >.hich combination of both) will be used. (For the decision of Consequently- we can replace Figure 19 by fcht : r Figure 13. The next taak is, then, •-■". discuss the n Lnal org A and SG. A is a 4 stagi m; . Lfier, about which more whs said .5. ^) The function of \ is solely to restore the pu] from -79-' > I ( r c to the shape and intensity with which it originally entered i dl ]. Hence it should really be considered a part off ' dl proper, and there is no occasion to analize it in terms of E-elements. SG, on the other hand, is a switching and gating. organ and we should build it up from E-elements. We therefore proceed to do this. 13.2 The purpose of SG is this: At those moments (i.e. periods ~ ) when other parts of the device (i.e.. OC, CA and perhaps I, 0) are to send information into the I dl to which this SG is attached, or when they are to receive information from it, SG must establish the . necessary connections — at such moments we say that SG is on. .At those moments when neither of these things is required, SG 'must route the output of its dl j back into the input of its (or its other) 1 dl according to the approximate alternative of Figure 19. at such moments we say that SG is off . In order to achieve this it 'is clearly necessary to take two lines from C (and 1,0) to this SG: . One, to carry the j dl output to C, and one to bring the j dl | input from C. . Since at any given time (i.e. period 7 ) only one SG will be called upon for these connections with C, i.o..beon ; (remember the principle of 5.6!) there need only be one such, pair of connecting lines, which will do for all 256 SG's. We denote these two lines, by L Q and Lj_, respectively. Now the scheme of Figure 18 can be made more detailed, as shown in Figure 20. As indicated, « is the. line connecting the outputs of all SG' s to C , and /7.-Y './ R E Z U Lj_ is the line con- necting G to the inputs o£ all SG's. When SG is off, its *jC — i ft -16. sr; TI-±_ -30- ( ' € connections o, i with L , Lj_ are interrupted, its output goes to a, this being permanently connected to the input c of the proper ) dl according to Figure 19., (a) or (b). When SG is on, its connections with a are interrupted, its output goes through o to L Q and so to C, while the pulses coming from C over Lj go into i which is now connected with a, so that these stimuli get now to a and from there to the pro- dl (input (cf. above). The line s carries the stimuli which per put SG on or off — clearly each SG must have its individual connection s (while L Q , L^ are common.) 13.3 Before' we consider the E-network of SG, one more point must be discussed. We allowed for only one state when SG is on, whereas there are actually two: First, when SG forwards information from M to C, second, when SG forwards information from C to ;U. In the first case the output of SG should be routed into L , and also into a, while no L^ connection is wanted. In the second case L^ should be connected to a (and hence to the proper dl input by the corresponding permanent connection of a). This information takes away the place of the infor- . . mat ion already in M, which would have normally gone there (i.e. the output of SG which would have gone to a if SG had remained off), hence the output of SG should go nowhere, i.e. no L connection is wanted. (This is the process of clearing . For this treatment of clearing cf . ) To sum up: Our single arrangement for the on state differs from what is needed in either of these two cases. First case: a should be connected to the output of SG, and not to L^. Second case: a should lead nowhere, not to L . • o Both maladjustments are easily corrected. In the first ~\ case it suffices to connect L not only to the organ of C which is to -81- ( ' £ t £ ^ receive its information, but also to Lj_ — in this manner the output of SG gets to a via L , the connection of L Q with L. , and L^. In the second case it suffices to connect L to nothing (except its i's) — in this manner the output of a goes into L , but then nowhere. In this way the two above supplementary connections of L and L. precise the originally unique'' on state of -SG to be the first or the second case described above. Since only one SG is on at any one time (cf . 13-2) these supplementary connections are needed only cnce. Accordingly we place them 'into C, more specifically into CC, where they clearly belong. If we had allowed for two different on states of SG itself, then the it would have been necessary to locate the E-network, which establishes the two corresponding systems of connections, into SG. Since there are 256 SG's and only one CC, it is clear that our * present arrangement saves much equipment. I3./4. 'We can now draw the E-network of SG, and also the E-net- work in CC which establishes the supplementary connections of L and h±' discussed in 13.3. Actually SG will have to be redrawn later (cf. ), we now give its preliminary form: SG-'- in Figure 21.. When s is not stim- ulated the two (2) are impassable to stimuli, while f"^ is, hence a stim- ulus entering at b goes on to a, while and i are disconnected from b and a. When s is stimulated the two (2) become passable, while f~^) is blocked, hence b is now connected to o and i to a. Hence SG" 1 " is on in the sense I . ' . of 13.2 while s is stimulated, and it is off at all other times. The triple delay on (^_J) is necessary for this reason: When SG is on, a -82- <" t i c fiQuR-E IZ t f~& Y SCL TV L. i. stimulus needs one period ? to get from b to o, i.e. to L (cf. 13.3 and the end of this section 13.4), and one to get from L^, i.e. from i (cf. ■ Figure 20), to a — that is, it takes 3 T from b.to a. It is desirable that the timing should be the same when SG-*- is off, i.e, when the stim- ulus goes via (_) from b to a — hence a triple delay is needed on \~J . The supplementary connections of L and L^ are given in ' Figure 22. When r is not stimulated the two' \^_) are passable to stimuli; while {2) is not, hence a stimulus entering at L is fed back into L^ and appears also at C^, which is supposed to- lead to C . .When r is stimulated the two f) are blocked, while (2\ becomes passable, hence a stimulus entering at C , which is supposed to come from C, goes on to Lj_, and L is isolated from all con- nections. Hence SCL produces the first state of 13.3 when r is not stim- ulated, and the second state when- r is stimulated. We also note, that in the first case a stimulus passes from L to Lj_ with a delay £• (cf . the timing question of SG^, discussed above.) 13.5 We must next give our" attention to the line s of Figure 20. and 21: As we saw in the first part of 13.4, it is the stimulation of s which turns SG on. Hence, as was emphasized at the end of 13.2, " each SG must have its own s — i.e. there must be 256 such lines s. Turn- ■■ ing a desired SG on, then, amounts to stimulating its s. Hence it is at this point that the —250-way — precisely 256-way — switching problem com- mented upon in 12.7 presents itself. 1 - More precisely: It is to be expected, that the order to turn on a certain 3G — say 'Ao, K — will appear on two lines in CC re- -83- e-j < ( t . . I for this purpose, in this manner: On- stimulus on the first' line expresses the presence of the order as such, while a sequence of ■ stimuli on the second line specifies the number k desired, k runs over 256 values, it is best to choose these as 0, 1, , 255, in which case k is the general 8-digit binary integer. Then k will be represented by a sequence of 8 (possible) stimuli on the second line, which express (by their presence or absence), in their temporal succession, k*s binary' digits (1 or 0) from right to left. The stimulus expressing the order as such must appear on the first liner, (cf. above) in some definite time relation to these stimuli on the second line— as will be seen in , it comes immediately after the last digit. Before going on, we note the difference between these 3 (binary) digit integers k and the 30 (binary) digit real numbers (lying between and 1, or, with sign,' between -1 and 1), the standard •^ real numbers of 12.2. That we consider the former as integers, i,'e* with the binary point at the right of the 8 digits, while in the latter the binary point is assumed to be to the left of the 30 digits, is mainly a matter of interpretation, (cf . ) Their difference in lengths, however, is material: A standard real number constitutes the entire content of a 32 unit minor cycle, while an 8 digit k is only part of an order which makes up such a minor cycle, (cf. ) U.O CC and M 1A.1 Our next aim is to go deeper into the analysis -'of 'CC. Such an analysis, however, is dependent upon a precise knowledge of the system of orders used in controlling 'the 'device, since the function of -84- I c e>j CC is to receive these orders', to interpret them, and then either to carry them out, or to stimulate properly those organs which will carry • them out. It is therefore our immediate task to provide a list of the . orders which control the device, i.e. to describe the code to be used in the device, and to define the mathematical and logical meaning and the operational significance of its code v.ords . Before we can formulate this code, we must go through some general considerations concerning the functions of CC and its re- lation to M. The orders which are received by C come from li, i:e. from the same place where the numerical material is stored, (cf. 2.4 and 12.3 in particular (b).) The content of M consists of minor cycles (cf. 12.2 and 12,7 )> hence by the above each minor cycle must contain a distinguishing mark, which indicates whether its is a standard number or an order. The orders which CC receives fall naturally into these four classes: (a) Orders for CC to instruct CA to carry out one of its ten specific operations (cf. 11.4). (b) Orders for CC to cause the transfer of a standard number from one place to another. (c) Orders for CC to transfer its own connection with U to a different point in M, with the purpose of getting its next order from there, (d) Orders controlling the operation of the input and the output of the device (i;e; 1 of 2.7 and of 2.3) Let us now consider these classes (a) - (d) separately; We cannot at this time add anything to the statements of 11.4 concerning (a), (cf. however ) The discussion of (d) is also better '• -85- ( c i ^ delayed (cf. ). We propose, however, to discuss (b) and ( c ) now . 14.2 Ad (b): These transfers can occur within L', or within CA, or between M and CA, The first kind ;an always be replaced by two operations of the last kind, i.e. all transfers within 11 can be routed through CA. We propose to do this, since this is in accord with the general principle of 5.6.. (cf, also the riiscussion of the second ques- tion in 11.1), and in this way we eliminate all transfers of the first kind- Transfers of the second kind are obviously handled by the oper- ating controls of CA, Hence those of the last kind alone remain. They ■ fall obviously into two classes.: Transfers from M to CA and transfers from CA t,o IL V/e may break up accordingly (b) into (b 1 ) and (b" ), corresponding to these two operations, " ILj. . 3 Ad (c): In principle CG should be instructed after e$ch order, where to find the next order that, it is to carry out. We saw, however, that this is undesirable per se, and that it should be reserved for exceptional occasions, while as a normal routine CC , should obey the orders in the temporal sequence, in which they naturally appear at the output of the DLA organ to which CC is connected, (cf. the corresponding discussion for the iconoscope memory, (c) in 12.8) There must, however, be orders available, which may be used at the exceptional occasions referred to, to instruct CC to transfer its connection to any other desired point in M. This is primarily a transfer of this connec- tion to a different DLA organ (i.e. a dl {'organ in the sense of 12.7) Since, however, the connection actually wanted must be with a definite minor cycle, the order in question must consist of two instructions: '• -86- c I i < First, the connection of CC is to be transferred to a definite DLA organ. "^ Second, CC is to wait there until a definite -period, the one in which the desired minor cycle appears at the output of this DLA, and,CC is to accept an order 1 at this time only. Apart from this,. such a transfer order' might provide, that after receiving and carrying out the order in the desired minor cycle ^ CC should return its connection to the DLA organ which contains the minor cycle tfrnt fallows upon the one containing the transfer order, wait until this minor cycle appears at the output, and then continue to accept or- ders from there on in the natural temporal sequence- Alternatively, after receiving and carrying out the order in the desired minor cycle, CC should -tinuo with that connection, and accept orders from there on in the iral temporal sequence. It is convenient to call a transfer of the ~^ first type a transient one, and one of the second type a permanent one. & '.••■.. ' '■ It is clear that permanent transf ers are frequently needed, hence the second type is certainly necessary. Transient trans- fers are undoubtedly required in connection with transferring standard fc>' b" numbers (orders (c 1 ) and (c"), cf. the end of 1A.2 and in more detail in 1A..4 below). It seen:! very doubtful whether, they are over needed in true orders, particularly since such orders constitute only a small part of the contents of U (cf. (b) in 12.3), and a transient transfer order can always be expressed by two permanent transfer orders. We will therefore make all transfers permanent, except those connected with transferring standard numbers, as indicated above. 1U.L Ad (b) again: Such a transfer between CA and a defi- nite minor cycle in M (in either direction, corresponding to (b 1 ) or J (b"), cf. the end of 1/+.2) is similar to a transfer affecting CC in the -87- ( i C v sense of (o), sifice it reauires establishing a connection with the .desired DLA organ, and then waiting for the appearance of the desired minor cycLe at the output. Indeed, since only one connection t-tw; :, U and CO (actually ' or CA, i.e. C) is possible at one time, such a number transfer . : r< 3 abandoning the present 'connection cf CC with M, and t,h lg a new connection, exactly as if a transfer affecting CC in the cerise of (c) were intended, Since, however, ac- tually no such transfer of CC is desired, the connection of CC with its original DLA organ :nust be reestablished, after the number transfer has n carri- ! rat, and the waiting for the proper minpr cycle (that, one . blowing in Lhfe natural temporal sequence upon the transfer order) ,ie also necessary. I.' . this is a transient transfer, as indicated at the . of IL . . It should he noted, that during a transient transfer the place of. the minor cycle which contained the transfer order, must be remembered, since CC will have to return to its successor. I.e. CC must b& able to remember the number of the DLn organ which contains this miner cycle, and the number off periods after which the minor eye.:-: will appear at the output. (cf. for details .) la. b Some further remarks: Fir; i : Every permanent transfer involves waiting for the iesired minor cycle, i.e. in the average for half a transit through .LA organ, 512 periods J . A transient transfer involves two such , which add up exactly to one transit through a DLA organ, 1,02/* periods T . One might shorten certain transient, transfers by appropriate timing tricks, but this seems inadvisable, at least at this „ 1 -f the discussion, since the switching operation itself (i.e. -88- changing the connection of' CC) may consume a nonnegligible fraction of a minor cycle and may therefore interfere with the timing. Second: It is sometimes desirable to make a transfer from M to CA, or conversely, without any waiting time. In this case the minor cycle in M, which is involved in this transfer, should be the one immediately following (in time and in the same DLA organ) upon the one 'containing the transfer order. This obviously calls for an extra type )f immediate transfers, in addition to the two types introduced in 14.3. This type will be discussed more fully in Third: The 256 DLA organs have numbers 0, 1, , 255, i.e. all 8-digit binary numbers. -It is desirable to give the 32 minor cycles in each DLr. organ equally fixed numbers 0, 1, — -, 31, i.e. all 5-digit binary numbers. Now the DLA organs are definite physical objects, h^nce their enumeration offers no difficulties. The minor cycles in a given DLri organ, on the other hand, are merely moving loci, at which ■ rtain combinations of 32 possible stimuli may be located. Alterna- tively, looking at the situation at the output end of the DLA organ, a minor cycle is a sequence of 32 periods j , this sequence being con- sidered to be periodically returning after every 1,024 periods X . One might say that a minor cycle is a 32 r "hour" of a 1,022+ 7 "day", the "day" thus having 32 "hours'- 1 . It is now convenient to fix one of this "hours", i.e. minor cycles, as zero or ana let it be at the same time at the outputs of all 256 DLA organs of LI, We can then attribute each "hour", i.e. minor cycle, its number 0, 1, , 31, by counting from there. V.'e assum? accordingly that such a convention is established —noting that the minor cycles of any given number appear at -89- r f t \ the same time at the outputs of all 256 DLA organs of M. Thus each DLA organ has now a number jh - 0, 1, , 255 (or 8-digit binary), and each minor cycle in it has a number p = 0, 1, , 31 (or 5-digit binary). _ A minor cycle is-, completely defined within M by specifying both numbers i, p. . Due to these relationships we pro- pose to call a DLA organ a major cycle * . Fourth: As the contents of a miner cycle make their transit across a DLA organ, i.e. a major -cycle, the minor cycles number p clearly remains the same. When it. reaches the ''output and is then cycled back into the .input of a major oycle the number. p is still not changed (since it will, reach the output, again after 1,024 periods T , and we have synchronism. in all DLA organs, and- a 1,024 f' periodicity, cf. above), but /u changes to the number of the new major cycle. For individual cycling, the arrangement of Figure 19, (a), this means that /.< , too, remains unchanged. For serial cycling, the arrangement of Figure 19, (b), this means that/a. usually increases by 1, except that at the end of such a series of, say s major cycles it decreases by s-1. These observations .about the fate of a minor cycle after it has appeared at the output -of .its -major cycle apply as such when that major cycle is undisturbed, i.e. when it is off in the sense of 13.2. When it is on, in the same sense,, but in the first case of .13.3, then our observations are obviously still valid — i.e. they hold as long as the minor cycle is not being cleared-. When it is being cleared, i.e. in the second case of 13. 3, then those observations apply to the minor cycle which replaces the one that has been cleared. -90- ( 15.-0 The code 15.1 The considerations of 14. provide the ba^is for a fiamplete classi- fication of the contents of K, i.e. they enumerate a system of succe^ive disjunction which give together this classification. This classification will put us into the position to formulate the code which effects the logi- cal control of CC, and hence of the entire device. Let us therefore restate the pertinent .definitions and disjunctions. The contents of M are. the memory units, each one 'being character- ised by the presence or absence of a stimulus. It can b.e used to repre- sent accordingly the binary digit 1 or 0-, and wo will at any rate designate its contont by the binary digit i = 1 or to which it corresponds in this manner, (cf. 12.2. -and 12,5. ..with 76 ) Those units ar©-grouped together zo form 32«-unit minor cycles, and these minor , cycles, ar.o the ohtities > which will acquire direct significance in the code which wtf-will introduce. * i • • (cf. 12.2.) Wb denote tho binary digits which make up tho 32 units of a minor cycle, in their natural tomporal scquonce, by<i. , i ,. i^ , i . - ° 1 «J 31 Th<. minor cycles with those units maybe wrUton' 1= (i , i 1 , i , i_, ) i = (i >. v i.anor cycles fall into two classes: Standard number s and orders, (cf. 12.2, and 14.1,) Those two- ^Categories should be distinguished from each other by their respective first units (cf. 12.2.) i.e. by the vulue of i Q » '.Vc agree accordingly, that i = is to designate a standard number , and i = 1 an order. o ,.'.<■. 15.2. Tho remaining 31 units of a standard" number express its binary digits and its 3ign. It is in the nature of al3. arithmetical opera- tion, specif ically becauso of the role of . carry digits , that the binary J -91- f ' "^l : ! i~its of the numbers which cn.cr into them, must be fed in from right to left, i.e. those with the lowest positional value.; first. (This is so because the digits appear in a temporal succession and not simultaneously, cf, 7.1. The details arcs'- most simply evident in the discussion of the adder inT.3.) The sign plays the role of the digit farthest left, i,.e. of the highest positional valuo (cf. 8.1.) Hence it comes last, i.e. i = designates the + sign, and i_. = 1 the - sign. Finally by 9.2 the binary point follows immediately after the sign digit, ana the number 1 ? \ . this represented must '^ r moved mod 2 into the interval -1, 1. That is - Jl v - 31 = hi So ^9 --- h g T - \ 2 < mod 2) «. -*1 ;:a. 5> 15.3 Th remaining 31 units of an order, on the othor hand, must oxpresr. the nature of this order. The ordors wore classified in 14..1 into four classes (a) - (d) , and those were subdivided furthor as follows:. (a) in 11.4, (b) in 14.2, (b) and ^c) in 14..3,. 14.4, and the socond re mark in 14.5, Accordingly, the following complete list of orders obtains: (a) Orders for CC to instruct CA to carry out one of its ten specific operations enumerated in 11.4. (This is (a) in 14.1) fro desig- nate these operations by the. numbers 0, 1, 2,. — -, 9., in the order in which they occur in' 11.4, and thereby place ourselvc- into the position to refer zo any one of then by it£ number w = 0, 1, 2, _-_, 9, which is best r eiven as a 4^.digit binary (cf., hov/evor, ) Rcgardm, :) the* origin of the numbers which entor (a.", variae les) into those operations and the dis- posal of the result, this should be sai.;: According to 11.4, the formor come from I CA and o and the latter goes to , allin CA (cf., Figuros 16, -92- /' i ( - 17) J_. is fed through I,,,, ind l n . is the original input and tho CA GA CA or q^ final output of CA. Consequently thosj aro tho actual connecting links between U .. The feeding into I- . Will be described in (£), (Y) , (&) below, ooai from Op.will bo described in (</) , (fc), {.&) below. Certain oycr.tions Qre so fast (thay can be handled so as to con- sume only ■ ;.- duration of a minor cycle), that it is worth while to bypass ;,-,> when disposing of tho if result, (of. ) - provisions for cloarine I and J woro described in' 11.4. CA C » the clearing of this ought to bo seid: It would seem natural CA \ . . to cle.r Cq£ each time after its contents have been transferred into h (,cf. v) . There arc, however, cases, when it is preferable n~t to transfer out from , . nd nd to :Icar the contents of Op. Specifically: In tho f the operation s in 11.3 it turned out to bo necessary to hold in this manner in Oq^ the result of c, previous operation -, niter.- ■ natively, the previous operation might also be +, i, j, or oven x, cf. tnoro, lOthor instance: If a multiplication xy is carried out, with an which 7M contains, say, z at the beginning of tho operation, then actually z + xy will form in ; ,cf. tho discussion o.' multiplication in 7.7) It may . A therefore be occasionally desirable to hold tho rosult of an operation, which is followed . y a multiplication, in 0_ . . Formation of sums £_ xy is one nplc of this, rVb ncod ... r:f ro an additional digit c = 0, 1 to in licatc- whether should or should no*, be clcarod after the operation. 'A r c lot c = ox- press tho former t and c= 1 tho . latter. -93- c £ (3) Orders for X' to cause the transfer of a standard number 3) from a definite miner cycle in M to CA. (This is (b) in 14.1, type (b'' ) of 14.?) iho minor cycle is dcfinod,by the two indices u, p (cf. the third remark in 14.5; ?ho transfer into CA is, more precisely, one- into I (c;'. CA (a) above). (Y) Orders for CC to cause the transfer of a standard number which follo-.vs immed iately upon the order, into CA, (This is the immediate transfer of the second remurk in 14.5 in the variant which corresponds to 1 above.) It is simplest to consider a minor cycle containing a standard number (the kind ;. nalyzed in 15,2) as such an order per se, (This modifies at cttiterncnt lpc. cit, somewhat: The standard number in question is -^s* i< the minor cycle following immediately upon a minor cycle which has just given an order to CC , then the number will automatically operate as an immediate transfer order of the typo described. (cf. also the pertinent remur/.s in (£) and in (^ ) bolow.) The transfer into CA is again or.e into *CA ^ cff ^ a ^ or ^' J ^ ~bovo.) (</) Orders for CC to cause the transfor of a standard number from m CA tc a definite minor cycle in M, (this is (jb) in 14.1, type (b M ) in 14.2) The minor cycle in M is defined cy tho two indices u, p, as in (^) above. Tho transfer from CA is, more precisely, one from n — this was discussed, together with the necessary explanations and qualif ioo tions"} in (a) above. (£.) Orders for CC to cause tho transfer of a standard number from CA into tho minor cycle which follows immediately upon the one contain- ing this ordor. (This is the immediate transfer of the second remark in I -94- 1/ e 14.5, in the variant which corresponds to {(f) abovo.) The- transfer from CA is again ono from (cf. (a) or (</) above.) In this case the CC connection passes from this transfer order on to the next minor cycle, into which the standard number in question is just being sent. There would bo no point in CC now obeying (y), and £ this number sack into CA — also, there might bo timing difficulties. It is best, therefor., to except this case explicitly from the operatic, of (y). I.e.: (Y) is invalid if it follows immediately upon an (£). j (6) Orders for CC to cause the transfer of a standard number from CA into CA. (This is an operation of CA, the usefulness of whici v recognized in 11.2 cf. also' ) More precisely, from c; Lntc (cf. (a) above) {i) Orders for CC to transfer its own connection with U to ) a definite minor cycle (elsewhere), in M. (This is (c) in 14*1) The minor cycle is M is defined by the two indices u, p, as in (£ ) •'- Note, that' a (|3) could boropla ed by a {I), considering (y). tft'Ly+QQ^) s except The only difference is, that (/) is a permanent transfer, while (0) is a transient one. This nay servo to place additional emphasis on the corres- ponding considerations of 14.3 and 14.4. (y\) Orders controlling the operation of the inpat and the out- put of the device (i.e. I of 2.7 and of 2.8 ) (This is (d) in 14.1) As indicated in 14.1, the discussion of those orders is bottom delayed (cf. ). -95- J\ c ~> 15 .4 Let us nov, convDarc the nunrt rs of digits necessary to express th orders with tho number of av 1 1 i ( its in .. minor cycle _ 31, as stated at tho beginning of 15.3. Io begin .vith wc have in (t^) - (>0 3 typos of orders, to distinguish those from each other re uirjs 3 digits. ttext, tho types (a) _ (£) ( wo postpone (*)) , cf. above) have those requirements: (,a) . s1 .: ; :C:ify ^AoljJt or char Oca the number w, i.e. 4 digits, plus tho digit c - all together 5 dibits. (0), as -.veil as (d) and (£), must specify the numbers yv and p , i.e. 8 + 5 = 13 digit's, (y) is outside this ca/cegory. (£)» as well as (.6), requires no further specif icutions-; "cither o: thssc uses thc31 available dibits very efficiently. Consequently we might consider putting several such order.: into one minor cycle. On the other hand such a tendency to pool orders should bo kept | ithin very d^inite limits, for the following reasons. First, pooling several orders into one minor cycle should be . avoided, if it requires tho simultaneous performance of several operations (i.e. violates the principle of 5.6.) Secdd, it should also be avoided if it upsets the timing of the operations. Third, the entire matter is usually net important from the point of view of the total memory capacity: Indeed; it reduces the number cf those minor cycles only, which are used for "logical instructions, i.e. for -the purpose (b) in 2.1, and these represent usually only a small fraction of the total capacity of M (cf. (b) in 12.3 and ). H-ncc tho pooling of orders should rather be carried out from tho point of view of simplifying the Logical structure of tho code. • • ' . , ) -96- ( T) ...O.r. Mioso considerations discourage pooling several orders of the -re (a) - besides this would often not be logically possible either, without intervening orders of the types •,(£) - (5). Combining two orders of the typos (|3), (<7 ) , (£) is also dubious from the above points of 'ic f, bosidos it would leo •:■ only 31-3-13-13 = 2 digits free, and this (although it could be incrc .sod by various tricks to o) is uncomfortably low: It is advisable to conserve some spare capacity in the logical part of the co;c (i.'e. in tho orders), since later- on changes might bo dosirablo. (S.g. it May bocomo advisablo to increase the capacity of LI, i.e. tho number : of m jor cycles, i.e. the number Q of digits oT u. For an other reas n of. The boat chance lies inno-olinf an operation order (a) with cr- •£ controllin the transfer of its variables into Ca or the transfer of its result out of CA. Both .types may involvv 13 d ig its orders (namely (J3) or (a) ), henc<_ are c ..ount on polling (ot) with more than one such lor (cf. ti. ai ..vo estimate plus the 5 digits required by (a); !), Now . 1 u: i 11^ requires sra.isforr ing two variables into CA, honC'~ tho yst , .1 procedure consists in pooling {'a) with the disposal of i .ilt. I.e. (a) with (cf) or (£) or (6). It should be noted that every (<?) , ( &) , ( &) , -i,e. transfer frorr. CA must be preceded by an («,), and every (P), (Y),-i,o; transfer into CA, just be followed by an (a). Indeed, those transfers . are always connected with an (a) operation, the only possible exception would be an U to I', transfer, rtoutcd through (a), - even this involves '• an (a) operation (i or j in 11.4, cf. thoro and 11.2 ). C c Fitly c dors (c 1 ), (O (a) will always occur pooled with( a) , and >rdora (j3), (Y) will 1; ys occur .'-lone. ( a) , too,. may occasionally -97- c ( 1 3 occur clone: If the result of th> ■ . (a) is tc (cf. cl ---rt oh (a) in 15.3)-, r. • - '..ill ... LI; not - ., .-■■ ; • Lisposc of this rc3ul .y also (cf. th - .cit.) Wc shall ;ep be ssib-ili Los . sn: th « not' bi litioi 1 of th result, nd :ho ca.se ' - j pooled .vitb 1 order. Orders (s) urc of a sufi cic lly sxct onal logical ch ractor, r justify that they Lone, - i disra ard (y), which "is in roality c st '■or - the 7 folic types ::' oriors: (a)+(V), ( a) +(.6) ,- ( a) , (@) , (f ) , X A J |f). They re , l3=lff t S,o, r , 15, 13 di;ixs ( _■ liaucgard (*£) , -1 be discus 3d 1 tcr) plus 3 digits to distinguish the typos from c r, pluJ one ii it (i pi) to express that an crier is involved. i ^hc totals ere 22, 9, 9 V 9 ;i 17, 17 digits. This is an average jffic icy ) of - 50% in : il - the 32 digits of Lor cycle. r his offecienev pan he considered adequate, in view of -the third rein, rk of 15.4, and it it the same ti - : . vf crtablc spare c. pacity (cf. the beginni i ( of 1 . )'. 15.6 r'c arc now in the position to formulate our cede. This formulati' n will t - srooented in tho* following manner: r ■ t characterize all possible miner cycles vice. Thoac are standard numbers and order, • : ; '. cri i in 15.1 - 1?,5. In^ feho toblo which fol] ■■: pecify the f ur -following things for c c possible minor cycle: (*:) The type , i.e. its relationship to th.. cLissif ication (a) - (>•)) -98- ( < f k c of 15.3, :nd to the pooling procedures of 15.5. (II) The moaning , as "^ in l; . 1 - 15.5, (III) The shor t , symb ol, to be used in verbal or written iiscussidns of rhc code, and in particular in all further . lyss; ...' thia papor, and when sotting up pre lcir.s for the device. (Cf. ) (IV) The zodo symbol, i.e. the 32 binary digits i}i io, , I31, which correspond to the 32 units of the minor cyo3o i:. 1 tic over, there will only bo partial stctc:r.ontS on thin I . int at this time_, the precise description will be given lator(cf the numbers (binary integers) which occur in ;he£ \ jcIc, wc obsorvi this: Those numbors arc u, p. w, c, . Vc . . ill 3 n tc its (in the usual, left tc right, order) by u , u j ft »#«*. 1 .; w 3 , , w o5 c. I -99- c Table, -ji: en) - ..i:\rd er or der . for bhe . fif.be r : l ■i • i i V /2T defined b; 1-v i (.mo : ■'j\ 1 - i=%- i_, ; the sign: for +, i for -. If CC i^ : >nnected to this minor cycle, then it i ■• >rder .causing the tr :-.: - r •:' oO +(f) into I Cf .'. Ipes no ar ily however Lf t] is oi eye] ,' ] iv/s • an orier W - : . A jr .:.. . . r i'" . " t o c fc the operation ;■ in CA and to of t; result, w is fro... the list of 11. 4. T\ re the operations of 11.4, .;itr. ir current numbers .; and their symbols ;r. . .- ler (*) +(©) ~) w ->up or *Jli — >up (a) til; : the result is to be held in O ca . — >up means, • .. ,t the result is to be trans- i fen i intc : \ \ minor cycle pi.; the major cycle u; — jf, th:i U L: be transferred into the :;/:.,. : tely following upon the order — ? ., that . is to be transferred into I ; :C ->, that . . . .' - L: n nted (apart :ror.. h). wh )rder <0 & ;r to ' >r tho number in the minor .- p . yjor cycle u intc I c . o connect CO ,vith the minor cycle p in jor cycle u. ( IV) Code A«~* ut C<f— up h = l -100- ( ( 1 Remark: Orders w (or wh) — »up (or f) transfer a standard number k. from Cn into a miner cycle. If this miner cyclv: is of the type N \ (i.e. i Q - 0) , then it should clear its 31 ii representing \ , and accept the 31 digits of t, . If it is a riinnr cycle ending in up (i.e. i = 1, order' w — ?-up or wh — % up or k <-- v.o :' C <— up), then it should clear only its 13 digits represent inr up, ■'■!.. accept the last 13 digits of \ I i v «« of Standards ^ ems Hat": SMITHSONIAN INSTITUTION LIBRARIES 3 9088 01147 5779