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A SIMPLIFIED UNIVERSAL T U R I N G M A C H I N E By E. F. Moore Bell Telephone Laboratories, New Jersey i. IN;RODUCTION merely as input-output media. Hence, a Turing machine c o u l d a l s o be considered In 1936 Turing (I) defined a class a mathematical model{3) of a digital of logical machines (which he called a- computer. machines, but which are now generally called Turing machines) which he used as 2. Multiple-tape Turing Machines an aid in proving certain results in mathematical logic, and which should I will modify Turing's definition of prove of interest in connection with the the machines considered by permitting theory of control and switching systems. generalized Turing machines to employ more than one tape, and permitting some Given any logical operation or arith- of the tapes to be in the form of closed loops (like a conveyor belt, or a rubber metical computation for which complete band) rather than being infinite. instructions for carrying out can be sup- plied, it is possible co design a Turing machine which can perform this operation. I will use the term "ordinary Turlng machine" to refer to what Turing called A Turing machine is defined to have an "a-machine", and the term "multiple- only a finite number of internal states tape Turing machine" to refer to a and to have an infinitely long tape on machine modified as above. which it can read, write, and erase sym- A multiple-tape Turing machine can bols, and which it can move one space at operate more like those digital computers a time in either direction. which have several tape drives, or which Turing suggested that such a machine have cyclic memories (such as mercury is an abstract mathematical model of a delay lines, magnetic drums, or loops of human being assigned to performing a com- punched paper tape). It can easily be putation, with the states of the machine shown that the class of calculations corresponding to the states of mind of which can be performed on ordinary Turing the human, and the symbols on the tape machines is exactly the same as the class corresponding to the numerical answers that multiple-tape Turing machines can and to the intermediate results on handle, although the method of handling the calculations must be different, to scratch paper (2}. permit all of the information to be Developments since the time of stored on one tape. The objectives which Turing's paper have been in the di- can be attained by having more than one rection of digital computers which show tape are to increase the similarity in a much more direct resemblance to Turing method of operationbetween this ab- machines than do human beings. In fact, straction and certain actual machines. and (in the example given later in this several present-day digital computers do report; to simplify the internal structure actually use magnetic or perfgratedpa~er and the method of ~ction of a machine tapes as auxiliary memories, insteaa ox which performs some specific task. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - . . . . . . . . . . . - - (i) Turing, A. M., On computable numbers, with an application to the Ent- scheldungs problem, Proc. Lond. Math. Soc. Series 2, Vol. ~I,-pp.~-2~,- (3) The computers a g a i n d i f f e r from T~6. these abstract mathematical machines, chiefly in making mistakes (i.e., (2) This, like most other mathematical machine breakdowns and transient models, appears to differ from the errors). Also, the Turlng machines object imitated in several auxiliary are unnecessarily general in their properties. For instance, the human properties, since no existing can make mistakes in com~utation, find physical machine can have an arbi- some of his own mistakes, ask for a trarily long tape. However, Turlng raise, as~ whether the problem is machines can be designed which behave really worth solving, and invent as if they did not have the infinite short-cut methods. tape. If no such restriction is impos- ed in the Turing machine, it corre- However, on closer analysis, it would sponds to a human attendant operating seem a reasonable conjecture that an actual computer by removing and sufficiently complicated mathematical replacing reels of tape whenever the models, somewhat like Turing machines machine indicates this should be could eventually be rigorously shown done, thus removing the limitation to have properties of the sort imposed by the mechanical properties mentioned. of the machine. 50 3. Universal Turing Machine of internal states, ql,.. .... , qn, and has a finite number of symbols(6~, So, In addition to considering Turing S1, ... Sm, which it can print on the machines which would perform various other scanned square of the tape. mathematical operations, Turing gave a description ol a universal Turing machine, The next move that the machine will which was a single ordin=ry Turing make is assumed to be determined by the machine which could perform any operation internal state of the machine and by which any oth~F~ordinary Turing machine the symbol scanned. This ordered pair could performS*J, even in case the other (an internal state qi and a symbol S~ machine was more complicated than the scanned) will be called a determinan£, universal machine. and the next action taken is to be determined only by the present determi- This ability to make a relatively nant of the machine. simple machine act like a more compli- cated machine is achieved by giving the There are only m * 3 different simple machine complicated instructions. kinds of operations which the machine can In particular, the universal Turing perform in any one step. These are t h e machine is given on one part of its tape ooeration R, in which the machine moves(7) a complete symbolic description of the to the right along the tape (i.e., it machine it is expected to imitate. Then changes from scanning its present square the universal machine stores on another to scanning the one immediately to the part of its tape (for instance, on right of this), the operation L, in alternate squsres) a copy o£ the tape which the machine moves to the left that would be on the imitated machine, along the t~pe, and the m + 1 operations and makes the changes on this part oi S k (defined for k = O, l, ..o m) which the tape which the imitated machine would erase the symbol on the square now make. The remaining part of the tape scanned, and print in its place the must be used for intermediate scratch symbol S k. work, for instance to record what state the imitated machine is in, what part of After the machine performs one of the tape it is scamning, etc. The these m + 3 operations listed above, it internal structure o£ the universal goes into a specified state q~, and Turing machine has to i n ~ u d e instructions begins the cycle again (i.e., it now to use these various kinds of data, and performs an operation based on its new to move back and forth between the dif- determinant). ferent parts of tape. The determination as to which oper- Since the method of storing all of ation the machine will perform and which this information on one tape is rather states it will go into are given by the complicated, the internal structure o£ description of the Turing machine, which the universal Turing machine which is a finite list of quadruples of the Taring described is also rather compli- form - cated, requiring a large number of states. qiSj X q& The present report describes a uni- where i = i, ... , n~ & = i, ..., n~ j = versal Turing machine which only has 0, ..., m~ and X is any Turing machine fifteen states, at the expense of being operation (i.e., X = R, X = L, or X = Sk, a multiple-tape machine. This simplifi- for k = O, ... m). The verbal interpre- cation is accomplished by putting dilTer- tation of such a quadruple is that if the ant kinds of information on different machine is in state q~ scanning symbol S-, tapes which can be moved independently, it should next perforg operation X and g~ pemnitting all the information required into state q~. at each step to be at the reading heads simultaneously. It should be noted that the first two terms of each quadruple are the determi- 4. Conventions for Ordinary Turing nant, and hence for the machine to be able Machines to act consistently according to its de- scription, we must impose the condition The following conventions concerning that no two quadruples can begin with the the description and operation o£ ordinary same determinant. This condition will be Turing machines are slightly simplified called the consistency condition, and will from those given by Turing, and a r e , ~ be taken to be part of the definition of accordance with those used by Davlst)J. an ordinary Turing machine. The machine is assumed to have a tape which has been subdivided into squares This completes the definition of the width of the tape, each square ordinary Turing machines in general, but containing exactly one symbol. The machine is able to read what is on only one square of the tape at any given time. The machine has a finite number (6) Blank tape (i.e., tape which has not yet been printed on by the machine) is assumed to originally contain the blar k symbol S o on eacn square. (4)It could be interpreted, loosely speak- ing, as a completely general-purpose digital computer. (7) The somewhat confusin~ coDvention that the machine move~ relative to the tape, instead of vice versa, (5) Martin Davis. Mimecgraphed lecture is used here only to avoid chang- notes on recursive function theory, ing the conventions established University of Illinois, 1951. by Turing and Davis. 51 there is one other modification that w i l l It should be noted that the above tape be made to simplify the construction of descriptions have the property that a the universal Turing machine given in block of N l's represents a determinant this report. This is to restrict m to be if, and only if, N is not divisible by 3. equal to l, i.e., to permit only two symbols S^, and S~ on the ordinary Turing 5. A ~ulti~le-tape Universal Turing machine c~nsidered. To simplify the Machine notation, we will let S^ = O, and S 1 = l, and note that O is the ~ymbol present on Let us consider a Turing machine blank tape. having three tapes, which will be called ~ and T~, on each of which only the This restriction does not cut down b v ~ 0 or X can occur. The descrip- the generality of the machines considered, tion of this machine is given by a list since if any operation can be performed on of sextuples below, with the first four a machine having m + 1 symbols, it can be items in each representing the determi- performed on a machine having only two nant, which in this case must consist of symbols, by replacing each symbol in the the internal state and the symbol scanned original machine by a block of binary on each of the three tapes. For instance digits, whose length is s, the least q~lO1 is the determinant which indicates integer ~ log 2 (m + 1), i.e., by coding that the sextup!e in which it occurs is the origlnal machine on a binary basis. to nave effect only if the machine is in In order to do this and remain within the internal state q~, scans 1 on T1, scans scope of the definition given for a Turing 0 on T2, and scaMs 1 on T 3. machine, it is necessary to replace each quadruple of the original machine with a As an added convention 8 is set of at most 3s - 2 quadruples (taking used in the determinant to indicate that care not to violate the consistency con- the sextuple is to have effect regardless dition) which scan through the block of of whether the corresponding tape shows binary digits to identify it and which a O or a 1. step along performing a series of oper- ations equivalent to the operation of the Yhe fifth symbol in each sextuple original machine. is one operation which is to be performed next. These operations are given as in Finally, before describing the the case of the ordinary Turing machine, universal Turing machine which can except that a subscript indicates which imitate any ordinary Turing machine tape it is to apply to. Thus R 1 indi- havin~ only two symbols, let us define cates a step to the right along T1, and precisely how the description of the 03 indicates the printing of a O on T3. ordinary Turing machine will be written on a tape, using only binary digits. Finally the sixth symbol in each sextuple gives the next state of the The tape description is obtained Turing machine. from the list of quadruples that com- pletely defines the machine, by the use The complete list of sextuples of cf a rule translating the quadruples the machine is given below, with into l's and O's in such a manner that explanations of the operations. The list will permit their use easily in the is broken up into parts A, B, and C to universal machine. permit over-all explanation of the parts. The sequenc~ of steps which this machine .The determinant q~O (or q~l) goes through in part A, part B, and then which begins each quadruple is~trans - part C will perform one step of the lated into a block of 3i + 1 (or 3i + actions of the Turing machine being 2, respectively) successive l's along imitated. the tape. In order to use this machine, The operation which is indicated the tape T1 sbould be a loop of tape, in the third term of each quadruple is containing the description of the translated into a string of O's which machine being imitated, T 2 should be an immediately follow the above described infinite tape which is blank except for block on the tape. The translation containing the determinant of the is as follows - machine being imitated (written as a string of l's, as described above), and 0 0 T~ should be a copy of the infinite tape 1 O0 which would be on the machine being L 000 imitated° R 0000 The three tapes are used for ohe The next state ell is translated purposes indicated above at each step, as a block of 35 successive l's, except that T 2 will contain the sequence immediately following the above string of all the past determinants of the of O's. machine being imitated, only the most recent of which is used at any step. For instance, the following are translations of a few quadruples - As an aid in following the written description below, Figure 1 shows most qlOlq 2 llllOOllllll of the same information in diagrammatic form. q21Lq I llllllllO00111 6. Description of the Machine To write the tape description of an entire ordinary Turing machine on a Part A tape, we write the translations of the quadruples in any order, separating each In states ql through q~ the from the next with any number of O's. multiole-tape Turing machin~ is search- 52 ink along T 1 to find the quadruple that Part B will be perginent to the next operation of the machine it is imitating. In states qA through q11 the multiple-tape Tu~ing machine-examines It is done by searching along T1 the number of O's immediately following to find a block of l's having the sale the block of l's just located on T 1 length as the block on T 2. which indicate what operation the ordinary Turing machine being imitated qlllSR2q 2 The machine steps along ia expected to perform on its tape. to the right on T1 and T 2 The corresponding operation is then per- q21~SRlqlalternately, starting in formed on T 3. state ql at the left end of the ~lock of l's on q6OOeRlq7 The multiple-tape Turing each tape. machine is scanning the first 0 following the ql108L2q3 If the machine reaches determinant block on T 2. 0 on T 2 while wire,in the In order to begin count- block of l's on T., the ing how many successive block on T 1 was snorter, O's there are, the machine and the machine begins moves to the right on T 1. preparation for comparing the next block on T1 with q710803q 8 Since the multiple-tape the same block on T 2 by Turing machine has reach- moving b a c k one space t~ ed a 1 on T1, there was return to the OAOCK on x2. only a singAe O. This indicates that the opera- q3118Rlq3 The machine moves along tion the ordinary Turing toward the right end of machine should perform the block of l's on T 1. next is to print a O, so T 3 takes this action. q30iSL2q4 When the machine has just passed the right end qTOOeRlq9 There were at least two of the block of l's on T1, successive O's on TI, so it moves back toward the the machine moves t$ the beginning of the block on right along T1, to continue T2 • counting the O's. qlOlSL2q 4 If the machine reaches q910813q8 There were exactly two a 0 on T 1 while still successive O's on Ti, so within t~e block of l's the multiple-tape m~chine on T2, the block on T 2 prints a I on T3. was shorter, and the machine moves back toward the beginning of the block q9OO~RlqlO The machine continues counting O's on TI. on T 2. The machine continues qlolOeL3q 8 There were exactly q401eL2q 4 three successive O's on moving to the left on T2, to return to the beginning T1, so the multiple- tape machine moves to of the block of l's on T 2. the left on T 3. q400eRlq 5 As soon as the machine has passed the beginning qloOOSRlqll There were exactly o~ the block of l's on T2, four successive O's on it begins moving along TI, so the multiple- t o ~ r d the next block. qlllOeR3q 8 tape machine moves to the right on T~, after having moved along Ti, qsOOSRlq5 The machine continues to the beginning of past the O's toward the the next block. next block on T I. qslOSR2q I As soon as the machine Part C has reached the begin- ning of the next block on T 1 it moves back to In states q8 and q12 through q]~, the Neginning of the the multiple-tape Turing machine prints original block on T 2 to on T^ the determinant for the next begin making the next operation, obtaining it from the symbol comparison of blocks. on T 3 and the state of the ordinary Turing machine described on T I. If qlOOeR2q6 If the comparison the length of the block of l'§ on T 1 is reaches the end of both 3¢, then the number of l's which must be blocks of l's simul- copied on T2 is 3~ + i or 3~ + 2, taneously, the blocks depending on whether T 3 is scanning a O were of equal lengths, or a i. and hence the proper . quadruple has oeen xounu. qslOO12q13 If T 3 is scanning a O, T 2 is moved to the right print a single 1 on T 2 to put a space between before going into staNe the block which has just ql3 to begin copying been used and the next from T1. block that will be printed. 53 q810iI2q12 If T~ is scanning a i, relay circuit, it would be possible to print ~wo l's on Tp build a working model of this machine q12111R2q15 before going into ~tate using perhaps twenty or twenty-five re- ql9 to begin copying lays. These figures are mentioned qlSe0el2ql3 frdm T 1. since t~.e number of relays is frequently used as a measure of the complexity of q13118R2q14 Each time this cycle a logical machine, and would enable of three steps is gone comparisons to be made between this and ql~lOSRlql 5 through, the machine actual digital computers. copies a single 1 from qlse0el2ql 3 T 1 to T2, and steps This estimated number of relays is aAong both tapes to the intended to be sufficiently generous to next position. permit the relay realization of this machine to include the solution of q130ieL2q4 When the Turing machine se,reral circuit problems. First, the reaches the end of the complexity of the relay sec~encing block of l's on T I the required will probably prohibit many of copying is completed. the possible relay combinations from The tapes must now be being useable. Second, the digits being prepared for the opera- read on each tape must be used in many tions in Part A, by parts of the circuit, sr~d hence extra moving back toward the relays to follow these digits may be beginning of the block required, and finally, since while on T2, and entering the the tape is in motion the information as cycle of Part A at to the present-determinant is unreadable, the appropriate point. it may be necessary to insert extra states (not in the logical description 7. Physical Rea!izability given) to record each determinant. If it were desired to build a work- ing model of the machine just described, standard teletype tape equipment could 8. Conclusion not be used, since it does not have the properties assumed. A punch which would be able to operate on tape moving in This machine indicates that very either direction, and to read from the complicated logical processes can be same line that it is in a position to done using a fairly small number of punch in would be required. mechanical or electrical components, provided large amounts of a memory But since holes in punched tape medium are available. With present cannot be erased once they are punched, speeds and costs of components, it in order to make a machine using punched would not be economically feasible to tape capable of imitating the behavior use such a machine to perform compli- of an ordinary Turing machine which has cated operations because of the extreme this erasing property the coding of slowness and fairly large amount of the description of the machine would memory required. have to be in a more complicated fashion. This has been shown by Shannon (8) to be However, with present trends toward always possible, although his method of the development of high-speed, elect- doing it is somewhat complicated. ronic logical components, this result suggests that it may be possible to It sh-,u!d be mentioned, however, reduce the number of components required that the properties of the tapes assumed for logical control purposes, particular- in Turing machines are very much like ly if any cheap memory devices are the properties a~tained by magnetic tapes, developed. It would be impossible to which have erasability, reversibility, draw any precise conclusions from this and the ability to use the same reading extreme example as to the extent to head for either reading or writing. which memory and speed can be traded for complexity since no precise quantitative If a tape mechanism were available relations between them have been obtain- which had the properties assumed and ed. which could be connected directly to (8) Unpublished work. 54 15-STATE UNIVERSAL TURING MACHINE I~ 00eR - L, _ _ . . . . . /COMPARE BLOCKS ! , 2~_" IDENTICAL BLOCK H A S ' xN~.~.~...~e Tel ~ ! j ~ O N T! ~. r 2 (,~ , I "~,,,e "2 ~ , ~ - ~ slOeRz~ , ~8~K-O~ ~T/ W A S S H O R T E R A ! ZERO SHORTER / 1-/-),,~30,eL2~,~Jl~~, L , ____EJ, I l THER~'~'~E \ / I MOVETOmGNT Y j l k Io,.O~".o,.l K\ ~ z ZEROS , l E N D OF 8LOCK l~'luu°"'~sl \ (~9~9'0e '3~ 8 ~ ~ ~ . O N TI l I \ y ' - \.~ ~-........~ ~.ROCEEDp A S T ZEROS h , '~'-, r~- . ~ - ~1 TO N E X T B L O C K O N T! ~ I 1~900eR'~'oJ LaY I ~sOOeR,~s~ I~'z°le Lm~'l'J I THERE WERE ~ I ' " I COPY BLOCK A ,~ , ~ zERos ,/fl\ i ' ,'Ro,~ ~, TO T~ ~__~ l- + _ -- EXAMIHESYMBOL , I~'o00,e""~"}TH.REW.,~. i .~,o,,=,~,=, o..3 L~,@°2'=~,,L!~,,"e",~,*l Figure 1 SIMPLE LEARNING BY A DIGITAL COMPUTER The Computation Laboratory Harvard University 1. Introduction Techniques of this type may have Digital comouters can readily be some value for those who, like psy- programmed to exhibit modes of behavior chologists and neurophysiologists, are which are usually associated only wlth interested In the potentialities of the ne2vous systems of living organisms. existing digital computers as models of Thls paper describes a concrete example the structure and of the functions of of one practical technique by which the animal nervous systems. The description Electronic Delay Storage Automatic will be given In two stages. In the Calculator (EDSAC) of the University first stage (Section 2) the behavior of ~athematlcal Laboratory, Cambridge, was the EDSAC under the control of a re- made capable of modifying its behavior s o_~.~ learning program will be presented on the basis of experience acquired In from the point of view of an experlmenter the course of operation. who can control the input of the machine 55