paper di A. Turing sulle macchine di Turing, Dispense di Logica I

Scarica paper di A. Turing sulle macchine di Turing e più Dispense in PDF di Logica I solo su Docsity! 230 A. M. TUKING [Nov. 12, ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM By A. M. TURING. [Received 28 May, 1936.—Read 12 November, 1936.] The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers. it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the relations of the computable numbers, functions, and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of com- putable numbers. According to my definition, a number is computable if its decimal can be written down by a machine. In §§ 9, 10 I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, the real parts of the zeros of the Bessel functions, the numbers IT, e, etc. The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable. Although the class of computable numbers is so great, and in many Avays similar to the class of real numbers, it is nevertheless enumerable. In § 81 examine certain arguments which would seem to prove the contrary. By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of Gbdelf. These results f Godel, " Uber formal unentscheidbare Satze der Principia Mathematica und ver- •vvandter Systeme, I " . Monatsheftc Math. Phys., 38 (1931), 173-198. 1936.] ON COMPUTABLE NUMBERS. 231 have valuable applications. In particular, it is shown (§11) that the Hilbertian Entscheidungsproblem can have no solution. In a recent paper Alonzo Church f has introduced an idea of "effective calculability", which is equivalent to my "computability", but is very differently defined. Church also reaches similar conclusions about the EntscheidungsproblemJ. The proof of equivalence between "computa- bility" and "effective calculability" is outlined in an appendix to the present paper. 1. Computing machines. We have said that the computable numbers are those whose decimals are calculable by finite means. This requires rather more explicit definition. No real attempt will be made to justify the definitions given until we reach § 9. For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited. We may compare a man in the process of computing a real number to ;i machine which is only capable of a finite number of conditions q1: q2. .... qI; which will be called " m-configurations ". The machine is supplied with a " t ape" (the analogue of paper) running through it, and divided into sections (called "squares") each capable of bearing a "symbol". At any moment there is just one square, say the r-th, bearing the symbol <2>(r) which is "in the machine". We may call this square the "scanned square ". The symbol on the scanned square may be called the " scanned symbol". The "scanned symbol" is the only one of which the machine is, so to speak, "directly aware". However, by altering its m-configu- ration the machine can effectively remember some of the symbols which it has "seen" (scanned) previously. The possible behaviour of the machine at any moment is determined by the ra-configuration qn and the scanned symbol <S (r). This pair qn, © (r) will be called the '' configuration'': thus the configuration determines the possible behaviour of the machine. In some of the configurations in which the scanned square is blank (i.e. bears no symbol) the machine writes down a new symbol on the scanned square: in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place to right or left. In addition to any of these operations the m-configuration may be changed. Some of the symbols written down f Alonzo Church, " An unsolvable problem, of elementary number theory ", American J. of Math., 58 (1936), 345-363. X Alonzo Church, "A note on the Entscheidungsproblem", J. of Symbolic Logic, 1 (1936), 40-41. 234 A. M. TURING [NOV. 12, If (contrary to the description in § 1) we allow the letters L, R to appear more than once in the operations column we can simplify the table considerably. m-config. symbol None 0 1 operations PO R, R, P I R, R, PO final m-config. 6 b b II. As a slightly more difficult example we can construct a machine to compute the sequence 001011011101111011111 The machine is to be capable of five ra-configurations, viz. " o ", " q ", "p ", " f ", " b " and of printing " o " , "x", " 0 " , " 1 " . The first three symbols on the tape will be " aoO " ; the other figures follow on alternate squares. On the inter- mediate squares we never print anything but "x". These letters serve to " keep the place " for us and are erased when we have finished with them. We also arrange that in the sequence of figures on alternate squares there shall be no blanks. Configuration m-config. symbol b Pa, • { ; fAny (0 or 1) rt J q i [ None 1 g ^ 1I None fAny None Behaviour operations R, Po, R, PO. R, R, PO, L, L i?, Px, L, L, L R, R PI , L E, R R L, L R,R PO, L, L final m-config. 0 0 q q p q f p f 0 To illustrate the working of this machine a table is given below of the first few complete configurations. These complete configurations are described by writing down the sequence of symbols which are on the tape, 1936.] ON COMPUTABLE NUMBERS. 235 with the m-configuration written below the scanned symbol. The successive complete configurations are separated by colons. : 9 9 0 O r o o O 0 : 9 9 0 0 : 9 9 0 0 : 9 9 0 0 1 : b o q q q p 9 9 0 0 1 : 9 9 0 0 1 : 9 9 0 0 1 : 9 9 0 0 1 : P P f f 9 9 0 0 1 : 9 9 0 0 1 : o a 0 0 1 0 : f f 9 9 0 0 H-0: .... c This table could also be written in the form b :9 9 o 0 0 : 9 9 q 0 0 : ..., (C) in which a space has been made on the left of the scanned symbol and the* m-configuration written in this space. This form is less easy to follow, but we shall make use of it later for theoretical purposes. The convention of writing the figures only on alternate squares is very useful: I shall always make use of it. I shall call the one sequence of alter- nate squares JF'-squares and the other sequence ^/-squares. The symbols oi •. ^-squares will be liable to erasure. The symbols on F-squares form a continuous sequence. There are no blanks until the end is reached. There is no need to have more than one jE'-square between each pair of .F-squarcs : an apparent need of more ^/-squares can be satisfied by having a sufficiently rich variety of symbols capable of being printed on ^-squares. If a symbol /3 is on an F-square S and a symbol a is on the ^-square next on the right of S, then S and /3 will be said to be marked with a. The process of printing this a will be called marking jS (or S) with a. 4. Abbreviated tables. There are certain types of process used by nearly all machines, and. these, in some machines, are used in many connections. These processes include copying down sequences of symbols, comparing sequences, erasing all symbols of a given form, etc. Where such processes are concerned we can abbreviate the tables for the m-configurations considerably by the use of "skeleton tables". In skeleton tables there appear capital German letters and small Greek letters. These are of the nature of "variables '". By replacing each capital German letter throughout by an ^^-configuration 236 A. M. TURING [Nov. 12, and each small Greek letter by a symbol, we obtain the table for an m-configuration. The skeleton tables are to be regarded as nothing but abbreviations: they are not essential. So long as the reader understands how to obtain the complete tables from the skeleton tables, there is no need to give any exact definitions in this connection. Let us consider an example: m-config. f(e,S5,a) fi(6,93,a) Symbol Behaviour Final m-config. L f^G, 95, a) L f(<5,S3,a) f a not a R R R f2(G, From the m-configuration f(@, 93, a) the machine finds the symbol of form a which is far- thest to the left (the "first a") and the ?w-confi,guration then becomes (L If there is no a then the m-configuration be- comes 93. None R I, 93, a) 93 If we were to replace £ throughout by q (say), 93 by r, and a. by x, we should have a complete table for the m-configuration f (q, x, x). f is called an "?/i-configuration function" or "m-function". The only expressions which are admissible for substitution in an »i-function are the m-configurations and symbols of the machine. These have to be enumerated more or less explicitly: they may include expressions such as p(c, x); indeed they must if there are any m-functions used at all. If we did not insist on this explicit eaumeration, but simply stated that the machine had certain m-configurations (enumerated) and all m-configu- rations obtainable by substitution of m-configurations in certain m-func- tion.-J, we .should usually get an infinity of m-configurations; e.g., we might say that the machine was to have the m-configuration q and all m-configu- rations obtainable by substituting an m-configuration for £ in p(£). Then it would have q, p(q), pfp(q)V p(p(p(q))), ... asm-configurations. Our interpretation rule then is this. We are given the names of the ^-configurations of the machine, mostly expressed in terms of m-functions. We are also given skeleton tables. All we want is the complete table for the m-configurations of the machine. This is obtained by repeated substitution in the skeleton tables. 1936.] ON COMPUTABLE NUMBERS. 239 JAny [None JAny [None R R R not a ce2(95, a, ce3(S5,a, a). The machine finds the last symbol of form a. -> @. j8,y) R L f Any R, E, R None 3)> a ) pc2(S, a, jS). The machine prints a j8 at the end. ce(ce(255j8), a) ce3(S5,a,j8,y). The mach- ine copies down at the end ce (ce2(S5,0, y), a) £ r s t the symbols marked a, then those marked jS, and finally those marked y; it erases the symbols a, /?, y. e1((5) From e(^) the marks are ,̂ > erased from all marked sym- bols. -> @. 5. Enumeration of computable sequences. A computable sequence y is determined by a description of a machine which computes y. Thus the sequence 001011011101111... is determined by the table on p. 234, and, in fact, any computable sequence is capable of being described in terms of such a table. It will be useful to put these tables into a kind of standard form. In the first place let us suppose that the table is given in the same form as the first table, for example, I on p. 233. That is to say, that the entry in the operations column is always of one of the forms E :E,R:E,L:Pa: Pa, R: Pa, L:R:L: or no entry at all. The table can always be put into this form by intro- ducing more m-configurations. Now let us give numbers to the w-configu- rations, calling them qx, ..., qR, as in §1. The initial m-configuration is always to be called qv We also give numbers to the symbols #]_,....., Sm 240 A. M. TUBING [Nov. 12, and, in particular, blank = 80, 0 = Slt 1 = S2. The hnes of the table are now of form Final m-config. Symbol Operations m-config. to to to Lines such as to are to be written as to and lines such as ft to be written as to s, Si Si Si Si Si s. PSk,L PSkiR PSk E, R PS0, R R PS,, R In this way we reduce each line of the table to a line of one of the forms (Nj, (N2), (i\y. From each line of form (N^ let us form an expression q( Sj]Sb L qm; from each line of form (N2) we form an expression qiSjSkRqm; and from each line of form (N3) we form an expression #,•#, SkNqm. Let us write down all expressions so formed from the table for the machine and separate them by semi-colons. In this way we obtain a complete description of the machine. In this description we shall replace q{ by the letter "D" followed by the letter "A" repeated i times, and $,- by " D " followed by "C" repeated j times. This new description of the machine may be called the standard description (S.D). It is made up entirely from the letters "A", " C", "D", "L", "R", "N", and from If finally we replace "A" by " 1 " , "C" by " 2 " , "D" by " 3 " , " L" by " 4 " , "R" by c ' 5 " , "N" by " 6 " , and "*3> by £ <7" we sh,all have a description of the machine in the form of an arabic numeral. The integer represented by this numeral may be called a description number (D.N) of the machine. The D.N determine the S.D and the structure of the 1936.] ON COMPUTABLE NUMBERS. 241 machine uniquely. The machine whose D.N is n may be described as To each computable sequence there corresponds at least one description number, while to no description number does there correspond more than one computable sequence. The computable sequences and numbers are therefore enumerable. Let us find a description number for the machine I of § 3. When we rename the m-configurations its table becomes: q-L ^ o *b1} K q2 q2 SQ P8O, R q3 q3 So PS2) R #4 ft SQ PSo>R ft Other tables could be obtained by adding irrelevant lines such as qx Sx PSVR q2 Our first standard form would be qxOQOJRq%j q%^o^o-"ft» 2*3®o^2-"ft' ft^o^oRQ\J• The standard description is DADDCRDAA ;DAADDRDAAA; I ^ ^ D D C C t f i ) ^ ^ \DAAAADDRDA; A description number is 31332531173113353111731113322531111731111335317 and so is 3133253117311335311173111332253111173111133531731323253117 A number which is a description number of a circle-free machine will be called a satisfactory number. In § 8 it is shown that there can be no general process for determining whether a given number is satisfactory or not. 6. The universal computing machine. It is possible to invent a single machine which can be used to compute any computable sequence. If this machine M is supplied with a tape on the beginning of which is written the S.D of some computing machine .At, 8KR. 2. VOL. 42. NO. 2144. B 244 A. M. TURING Subsidiary skeleton table. (Not A R, R con(£, a) [Nov. 12, con(@, a) con̂ CE, a) con2(§, a) con(@. a). Starting from an J^-square, S say, the se- A L, Pa, R con^S, a) q u e n c e Q o f s y m b o l s describ- A R,Pa,R c o n ^ a ) ing a configuration closest on the right of S is marked out R, Pa, R con2(§, a) with letters a. ->@.D G Not C R.R R, Pa, R con2(£,a) con(S, ). In the final con- figuration the machine is scanning the square which is four squares to the right of the last square of C. C is left unmarked. The table for U. hx R,R,P:,R,R,PD;R,R,PA anf anf 6. The machine prints on the .F-squares after ->anf. font not z nor R, Pz: L L,L L g(anf1} :) anf. The machine marks the configuration in the last COn (font, y) c o m p i e t e configuration with y. - !om !om con (limp, x) font. The machine finds the last semi-colon not marked with z. It marks this semi-colon with z and the configuration following it with x. Hnr,> cpe(c(fom, x, y), iim, x, y) fmp. The machine com- pares the sequences marked x and y. I t erases all letters x and y. -> Sim if they are alike. Otherwise ->• font. anf. Taking the long view, the last instruction relevant to the last configuration is found. It can be recognised afterwards as the instruction following the last semi-colon marked z. -Mim. 1936.] ON COMPUTABLE NUMBERS. 245 Sim •mt m?3 m?4 mh A not not A A . A R,Pu, L, L,Py, R, R Py con ,R ,R (stm2, Sim Sim e(mB, Sim ) 3 2 3 A C [Any [ None L, L, L, L , Pa;, j ^ , Z', con P: L, L, L ?, R, R, R •R, 22 mf2 D R, Px, L, L, L m?3 not : R, Pv, L, L, L m!3 : mL mf6 inSt, 0, : xnit S im. The machine marks out the instructions. That part of the instructions which refers to operations to be carried out is marked with u, and the final m- configuration with y. The let- ters z are erased. mi. The last complete con- figuration is marked out into four sections. The configiira- ration is left unmarked. The symbol directly preceding it is marked with x. The remainder of the complete configuration is divided into two parts, of which the first is marked with v and the last with w. A colon is printed after the whole. -> $f;. , u) Sf;. The instructions (marked u) are examined. If it is found that they involve "Print 0" or "Print 1", then 0: or 1: is printed at the end. 246 A. M. TURING [NOV. 12, in«t fl(t(in«1),tt) «**• T h e n e x t complete configuration is written down,. a R, E in^t1(a) carrying out the marked instruc- L) ce5(o»,.t>, y, x, u, w) t i o n s - T h e l e t t e r s u> v> w> x> V are erased. -^anf. i?) ce5(o», v, x, u, y, w) \nitx{N) ec5(ot>, v, x, y, u, w) co c(anf) 8. Application of the diagonal process. It may be thought that arguments which prove that the real numbers are not enumerable would also prove that the computable numbers and sequences cannot be enumerable*. It might, for instance, be thought that the limit of a sequence of computable numbers must be computable. This is clearly only true if the sequence of computable numbers is defined by some rule. Or we might apply the diagonal process. "If the computable sequences are enumerable, let a/( be the n-th computable sequence, and let </>;l(ra) be the ?n-th figure in au. Let /? be the sequence with \—<j>n(n) as its n-th. figure. Since /3 is computable, there exists a number K such that l—cf)ll(n) = <f)K(n) all n. Putting n = K, we have 1 = 2(f>K(K), i.e. 1 is even. This is impossible. The computable sequences are therefore not enumerable". The fallacy in this argument lies in the assumption that § is computable. It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps. In fact, by applying the diagonal process argument correctly, we can show that there cannot be any such general process. The simplest and most direct proof of this is by showing that, if this general process exists, then there is a machine which computes /?. This proof, although perfectly sound, has the disadvantage that it may leave the reader with a feeling that "there must be something wrong". The proof which I shall give has not this disadvantage, and gives a certain insight into the significance of the idea "circle-free". It depends not on constructing /3, but on constructing fi', whose n-th. figure is <j>n{n). * Cf. Hobson, Theory of functions of a real variable (2nd ed., 1921), 87, 88. 1936.] Otf COMPUTABLE NUMBERS. 249 9. The extent of the computable numbers. No attempt has yet been made to show that the " computable " numbers include all numbers which would naturally be regarded as computable. Al I arguments which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. The real question at issue is " What are the possible processes which can be carried out in computing a number?" The arguments which I shall use are of three kinds. (a) A direct appeal to intuition. (6) A proof of the equivalence of two definitions (in case the new definition has a greater intuitive appeal). (c) Giving examples of large classes of numbers which are computable. Once it is granted that computable numbers are all c: computable"". several other propositions of the same character follow. In particular, it follows that, if there is a general process for determining whether a formula of the Hilbert function calculus is provable, then the determination can bo carried out by a machine. I. [Type (a)]. This argument is only an elaboration of the ideas of § 1. Computing is normally done by writing certain symbols on paper. "We may suppose this paper is divided into squares like a child's arithmetic book. In elementary arithmetic the two-dimensional character of the paper is sometimes used. But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation. I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent j . The effect of this restriction of the number of symbols is not very serious. It is always possible to use sequences of symbols in the place of single symbols. Thus an Arabic numeral such as f If we regard a symbol as literally printed on a square we may suppose that the square is 0 < x < 1, 0 < y < 1. The symbol is defined as a set of points in this square, viz. the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the "distance" between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at x = 2. y = 0. With this topology the symbols form a condition- ally compact space. 250 A. M. TUBING [NOV. 12, 17 or 999999999999999 is normally treated as a single symbol. Similarly in any European language words are treated as single symbols (Chinese, however, attempts to have an enumerable infinity of symbols). The differences from our point of view between the single and compound symbols is that the compound symbols, if they are too lengthy, cannot be observed at one glance. This is in accordance with experience. We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same. The behaviour of the computer at any moment is determined by the symbols which he is observing, and his " state of mind " at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite. The reasons for this are of the same character as those which restrict the number of symbols. If we admitted an infinity of states of mind, some of them will be '' arbitrarily close " and will be confused. Again, the restriction is not one which seriously affects computation, since the use of more compli- cated states of mind can be avoided by writing more symbols on the tape. Let us imagine the operations performed by the computer to be split up into "simple operations" which are so elementary that it is not easy to imagine them further divided. Every such operation consists of some change of the physical system consisting of the computer and his tape. We know the state of the system if we know the sequence of symbols on the tape, which of these are observed by the computer (possibly with a special order), and the state of mind of the computer. We may suppose that in a simple operation not more than one symbol is altered. Any other changes can be split up into simple changes of this kind. The situation in regard to the squares whose symbols may be altered in this way is the same as in regard to the observed squares. We may, therefore, without loss of generality, assume that the squares whose symbols are changed are always "observed" squares. Besides these changes of symbols, the simple operations must include changes of distribution of observed squares. The new observed squares must be immediately recognisable by the computer. I think it is reasonable to suppose that they can only be squares whose distance from the closest of the immediately previously observed squares does not exceed a certain fixed amount. Let us say that each of the new observed squares is within L squares of an immediately previously observed square. In connection with "immediate recognisability ", it may be thought that there are other kinds of square which are immediately recognisable. In particular, squares marked by special symbols might be taken as imme- 1936.] ON COMPUTABLE NUMBERS. 251 diately recognisable. Now if these squares are marked only by single symbols there can be only a finite number of them, and we should not upset our theory by adjoining these marked squares to the observed squares. If. on the other hand, they are marked by a sequence of symbols, we cannot regard the process of recognition as a simple process. This is a fundamental point and should be illustrated. In most mathematical papers the equations and theorems are numbered. Normally the numbers do not go beyond (say) 1000. It is, therefore, possible to recognise a theorem at a glance by its number. But if the paper was very long, we might reach Theorem 157767733443477 ; then, further on in the paper, we might find " . . . hence (applying Theorem 157767733443477) we have ... ". In order to make sure which was the relevant theorem we should have to compare the two numbers figure by figure, possibly ticking the figures off in pencil to make sure of their not being counted twice. If in spite of this it is still thought that there are other "immediately recognisable" squares, it does not upset my contention so long as these squares can be found by some process of which my type of machine is capable. This idea is developed in III below. The simple operations must therefore include: (a) Changes of the symbol on one of the observed squares. (6) Changes of one of the squares observed to another square within L squares of one of the previously observed squares. It may be that some of these changes necessarily involve a change of state of mind. The most general single operation must therefore be taken to be one of the following: (A) A possible change (a) of symbol together with a possible change of state of mind. (B) A possible change (6) of observed squares, together with a possible change of state of mind. The operation actually performed is determined, as has been suggested on p. 250, by the state of mind of the computer and the observed symbols. In particular, they determine the state of mind of the computer after the operation is carried out. We may now construct a machine to do the work of this computer. To each state of mind of the computer corresponds an " m-configuration " of the machine. The machine scans B squares corresponding to the B squares observed by the computer. In any move the machine can change a symbol on a scanned square or can change any one of the scanned squares to another square distant not more than L squares from one of the other scanned 254 A. M. TURING [NOV. 12, instructions and the symbols on the tape. That is, the state of the system may be described by a single expression (sequence of symbols), consisting of the symbols on the tape followed by A (which we suppose not to appear elsewhere) and then by the note of instructions. This expression may be called the "state formula". We know that the state formula at any given stage is determined by the state formula before the last step was made, and we assume that the relation of these two formulae is expressible in the functional calculus. In other words, we assume that there is an axiom 2( which expresses the rules governing the behaviour of the computer, in terms of the relation of the state formula at any stage to the state formula at the preceding stage. If this is so, we can construct a machine to write down the successive state formulae, and hence to compute the required number. 10. Examples of large classes of numbers which are computable. It will be useful to begin with definitions of a computable function of an integral variable and of a computable variable, etc. There are many equivalent ways of defining a computable function of an integral variable. The simplest is, possibly, as follows. If y is a computable sequence in which 0 appears infinitely! often, and n is an integer, then let us define £(y, n) to be the number of figures 1 between the n-th and the (?i-\- l)-th figure 0 in y. Then <f)(n) is computable if, for all n and some y, .<f>(n) = £(y, n). An equivalent definition is this. Let H(x, y) mean <f)(x) = y. Then, if we can find a contradiction-free axiom 21̂ , such that 2^-* P, and if for each integer n there exists an integer N, such that % & and such that, if m=£<f>(n), then, for some N', % & then <j> may be said to be a computable function. We cannot define general computable functions of a real variable, since there is no general method of describing a real number, but we can define a computable function of a computable variable. If n is satisfactory, let yn be the number computed by ./U {n), and let | If *Al computes y, then the problem whether .11 prints 0 infinitely often is of the same character as the problem whether A\, is circle-free. 1936.] ON COMPUTABLE NUMBERS. 255 unless yn = 0 or yn — 1, in either of which cases an = 0. Then, as n runs through the satisfactory numbers, an runs through the computable numbersf. Now let <f)(n) be a computable function which can be shown to be such that for any satisfactory argument its value is satis- factory %. Then the function /, defined by f(an) — a^n), is a computable function and all computable functions of a computable variable are expressible in this form. Similar definitions may be given of computable functions of several variables, computable-valued functions of an integral variable, etc. I shall enunciate a number of theorems about computability, but I shall prove only (ii) and a theorem similar to (iii). (i) A computable function of a computable function of an integral or computable variable is computable. (ii) Any function of an integral variable defined recursively in terms of computable functions is computable. I.e. if 0(ra, n) is computable, and r is some integer, then rj(n) is computable, where (iii) If <f> (m, n) is a computable function of two integral variables, then <j>{n, n) is a computable function of n. (iv) If (j>(n) is a computable function whose value is always 0 or 1, then the sequence whose fi-th figure is <f>(n) is computable. Dedekind's theorem does not hold in the ordinary form if we replace *' real'' throughout by '' computable''. But it holds in the following form : (v) If G(a) is a propositional function of the computable numbers and (a) (3a)(3jB){G(a)&(-G(j8))}, (6) Q(a) and there is a general process for determining the truth value of G(a), then f A function an may be defined in many other ways so as to run through the computable numbers. J Although it is not possible to find a general process for determining whether a given number is satisfactory, it is often possible to show that certain classes of numbers are satisfactory. 256 A. M. TURING [NOV. 12r there is a computable number £ such that In other words, the theorem holds for any section of the computables such that there is a general process for determining to which class a given number belongs. Owing to this restriction of Dedekind's theorem, we cannot say that a computable bounded increasing sequence of computable numbers has a computable limit. This may possibly be understood by considering a sequence such as l ± 1 I I I J-5 2 ' 5 ' 8 ' i o j 2» • • • • On the other hand, (v) enables us to prove (vi) If a and /? are computable and a < /? and <£(a) < 0 < </>(/?), where (f>(a) is a computable increasing continuous function, then there is a unique computable number y, satisfying a < y < fi and <f>(y) = 0. Computable convergence. We shall say that a sequence fin of computable numbers converges computably if there is a computable integral valued function N(e) of the computable variable e, such that we can show that, if e > 0 and n > N(e) and m > N(e), then \pn—j8m| < e. We can then show that (vii) A power series whose coefficients form a computable sequence of computable numbers is computably convergent at all computable points in the interior of its interval of convergence. (viii) The limit of a computably convergent sequence is computable. And with the obvious definition of " uniformly computably convergent": (ix) The limit of a uniformly computably convergent computable sequence of computable functions is a computable function. Hence (x) The sum of a power series whose coefficients form a computable sequence is a computable function in the interior of its interval of convergence. From (viii) and TT— 4(1—i-|--i—...) we deduce that TT is computable. From e = l + l+n-j-+»-j+.. . we deduce that e is computable. 1936.] ON COMPUTABLE NUMBERS. 259 11. Application to the Entscheidungsproblem. The results of § 8 have some important applications. In particular, they can be used to show that the Hilbert Entscheidungsproblem can have no solution. For the present I shall confine myself to proving this particular theorem. For the formulation of this problem I must refer the reader to Hilbert and Ackermann's Grundziige der Theoretischen Logik (Berlin, 1931), chapter 3. I propose, therefore, to show that there can be no general process for determining whether a given formula 2( of the functional calculus K is provable, i.e. that there can be no machine which, supplied with any one 21 of these formulae, will eventually say whether 21 is provable. It should perhaps be remarked that what I shall prove is quite different from the well-known results of Godelf. G odel has shown that (in the forma- lism of Principia Mathematica) there are propositions 21 such that neither '21 nor — 21 is provable. As a consequence of this, it is shown that no proof •of consistency of Principia Mathematica (or of K) can be given within that formalism. On the other hand, I shall show that there is no general method which tells whether a given formula % is provable in K, or, what comes to the same, whether the system consisting of K with —21 adjoined as an cextra axiom is consistent. If the negation of what Godel has shown had been proved, i.e. if, for each 21, either 21 or — 21 is provable, then we should have an immediate solution of the Entscheidungsproblem. For we can invent a machine JC which will prove consecutively all provable formulae. Sooner or later JC will reach either 21 or —21. If it reaches 21, then we know that 2( is provable. If it reaches — 21, then, since K is consistent (Hilbert and Ackermann, p. 65), we know that 21 is not provable. Owing to the absence of integers in K the proofs appear somewhat lengthy. The underlying ideas are quite straightforward. Corresponding to each computing machine i t we construct a formula Un (it) and we show that, if there is a general method for determining whether Un (.11) is provable, then there is a general method for deter- mining whether i t ever prints 0. The interpretations of the propositional functions involved are as follows : Rst( x> V) is to be interpreted as "in the complete configuration x (of J/l) the symbol on the square y is S". t Loc. cit. S2 260 A. M. TURING [NOV. 12, I(x, y) is to be interpreted as "in the complete configuration x the square y is scanned". KQm(x) is to be interpreted as "in the complete configuration x the m-configuration is qm. F(x, y) is to be interpreted as sty is the immediate successor of x ". Inst {qt Sj 8k L 37} is to be an abbreviation for (x, y, x', y') I (BSj(x, y) k I(x, y) k K8i(x) k F(x, x') k F(y', y)) fI{x'iy')kBSk{x',y)kKqi{x') k (z) \_F{y', z)v(RSj(x, z) + Rak(x', z) Inst {q{ 8, Sk R qt} and Inst {qt 8j Sk N q{] are to be abbreviations for other similarly constructed expressions. Let us put the description of .11 into the first standard form of § 6. This description consists of a number of expressions such as "q{ 8i Sk Lqt" (or with ROT N substituted for L). Let us form all the corresponding expres- sions such as Inst {qt $3- Sk L qt} and take their logical sum. This we call Des(.U). The formula Un(.U) is to be {3u)[N{u) &, (x)(N{x)->{3x')F(x, X')) &. (y, z)(F(y, z)->N(y) k N(z)) & (y) R>%(% y), & I(u, u) & Kqi{u) & Des(..U)l ->(35) (30 [N(s) & N(t) & RSl(s, t)). [K{u)&... &Des(.U)] may be abbreviated to A(M). When we substitute the meanings suggested on p. 259-60 we find that Un(.U) has the interpretation "in some complete configuration of M, S-^ (i.e. 0) appears on the tape ". Corresponding to this I prove that (a) If Sx appears on the tape in some complete configuration of • U, then Un(U) is provable. (b) If Un (• U) is provable, then 8X appears on the tape in some complete configuration of • 11. When this has been done, the remainder of the theorem is trivial. 1936.] ON COMPUTABLE NUMBERS. 261 LEMMA 1. / / S± appears on the tape in some complete configuration of .At, then Un(.At) is provable. We have to show how to prove Un (it). Let us suppose that in the n-th complete configuration the sequence of symbols on the tape is &r(n,o)> *̂ r(n,i)5 •••> $i<n,nh followed by nothing but blanks, and that the scanned symbol is the i(n)-th, and that the m-configuration is q^n). Then we may form the proposition , u) & RSrluJvF>, u') & ... & RSr{H,Mn\ which we may abbreviate to CCn. As before, F{u, u') & F{u', u") & ... & F{u^\ w(r)) is abbreviated to F<r). I shall show that all formulae of the form A{-W) & F™^- CCn (abbre- viated to CFn) are provable. The meaning of CFn is " The n-th. complete configuration of i t is so and so ", where "so and so " stands for the actual n-th. complete configuration of i t . That CFn should be provable is therefore to be expected. CF0 is certainly provable, for in the complete configuration the symbols are all blanks, the m-configuration is qx, and the scanned square is u, i.e. CC0 is (y) RSo{u, y) & I(u, u) & KQl(u). A(o\i)->CC0 is then trivial. We next show that CFn^-CFn+1 is provable for each n. There are three cases to consider, according as in the move from the n-th to the (n-j-l)-th configuration the machine moves to left or to right or remains stationary. We suppose that the first case applies, i.e. the machine moves to the left. A similar argument applies in the other cases. If r[n,i(n)}=a, r(n-\-l, i(n-\-l)} = c, k(i(n)j =b, and k(i(n-\-l)) =d, then Des (it) must include Inst {qa 8b Sd L q^ as one of its terms, i.e. Hence A(.AV) & Fin+n^1nat{qa8b8dLqc} & But Inst{qa Sb 8dLqc} & ^ n + w^(CC n - is provable, and so therefore is A (• It) & F(n+»-> (CCn -» C(L ., 264 A. M. TURING [NOV. 12, into which M is convertible (cf. foot-note p. 252). The machine £> includes ^2 a s a Par^. The motion of the machine X when supplied with the formula My is divided into sections of which the n-th. is devoted to finding the n-th figure of y. The first stage in this n-th. section is the formation of {My} {Nn). This formula is then supplied to the machine £2, which converts it successively into various other formulae. Each formula into which it is convertible eventually appears, and each, as it is found, is compared with and with Aa:|Aa;'[{a;}(a;')] |, i.e. Nv If it is identical with the first of these, then the machine prints the figure 1 and the n-th section is finished. If it is identical with the second, then 0 is printed and the section is finished. If it is different from both, then the work of .!!2 is resumed. By hypothesis, {My}(Nn) is convertible into one of the formulae N2 or Nx; consequently the n-th section will eventually be finished, i.e. the n-th. figure of y will eventually be written down. To prove that every computable sequence y is A-defUiable, we must show how to find a formula My such that, for all integers n, {My}(Nn)c(mvN1+<j)y{n). Let .11 be a machine which computes y and let us take some description of the complete configurations of -U by means of numbers, e.g. we may take the D.N of the complete configuration as described in §6. Let £(n) be the D.N of the w-th complete configuration of M. The table for the machine ..U gives us a relation between £(n-\-l) and £(n) of the form where py is a function of very restricted, although not usually very simple, form : it is determined by the table for. U. py is A-defmable (I omit the proof of this), i.e. there is a W.F.F. Ay such that, for all integers n, Let U stand for Xu[{{u}(Ay))(Nr)], where r=£(0); then, for all integers n, {Uy}(NJ conv N,{n). 1936.] ON COMPUTABLE NUMBERS. It may be proved that there is a formula V such that 265 conv Nx if, in going from the n-th to the (n-\- l)-th complete configuration, the figure 0 is printed. conv JV2 if the figure 1 is printed, conv N3 otherwise. Let Wy stand for so that, for each integer n, conv {Wy} (Nn), and let Q be a formula such that \{Q}(Wy)UNs) convNr(s), where r(s) is the 5-th integer q for which {Wy} (NQ) is convertible into either N-L or JVa. Then, if j|f7 stands for it will have the required property f. The Graduate College, Princeton University, New Jersey, U.S.A. t In a complete proof of the A-definability of computable sequences it would be best to modify this method by replacing the numerical description of the complete configurations by a description which can be handled more easily with our apparatus. Let us choose certain integers to represent the symbols and the m-configurations of the machine. Suppose that in a certain complete configuration the numbers representing the successive symbols on the tape are s1s2... sn, that the m-th symbol is scanned, and that the ?n.-configur- ationhas the number t; then we may represent this complete configuration by the formula where etc. „ N» ..., # , „ , _ , ] , [Nt, NaJ, [NSM+V ..., NSlt]], [a, 6] stands for \u f" -{ {u} (a) )(&)]» [a, 6, c] stands for AM P I \ {u} (a)}(b) J (c)l,