542 | Alan Turing
(2) In §§6, 7, 8 it was supposed that the disturbances were not continuously
operative, and that the marginal reaction rates did not change with the passage of
time. These assumptions will now be dropped, though it will be necessary to
make some other, less drastic, approximations to replace them. The (statistical)
amplitude of the ‘noise’ disturbances will be assumed constant in time. Instead of
(6.6), (6.7), one then has
dx ¼ a0x þ bZ þ R1 (t ), 9
dt =>
dZ ;> (9:8)
dt
¼ cx þ d0Z þ R2(t ),
where x, Z have been written for xs, Zs since s may now be supposed Wxed. For the
4m sin2 4n sin2
same reason a À pS has been replaced by a0 and d À pS by d0. The
N N
noise )daisrteurtbwaoncneosnm-oavyebrleapsuppinpgosiendtetrovaclos ntshteitnutRett12wRh1i(tet)ndot iasen,di.eR.tt3i4fR(2t(1t, t2) and
(t3, t4 )dt are
statistically independent and each is normally distributed with variances
b1(t2 À t1) and b1(t4 À t3) respectively, b1 being a constant describing the
amplitude of the noise. Likewise for R2(t), the constant b1 being replaced by
b2. If p and p0 are the roots of (p À a0)(p À d0) ¼ bc and p is the greater (both
being real), one can make the substitution
x ¼ b(u þ v), )
Z ¼ (p À a0)u þ (p0 À a0)v, (9:9)
which transforms (9.8) into
du ¼ pu þ p0 À a0 R1(t ) À R2(t ) þ x d p0 À a0 À Z d 1 (9:11)
dt (p0 À p)b p0 À p dt (p0 À p)b dt p0 À ,
p
with a similar equation for v, of which the leading terms are dv=dt ¼ p0v. This
indicates that v will be small, or at least small in comparison with u after a lapse
of time. If it is assumed that v ¼ 0 holds (9.11) may be written
du ¼ qu þ L1(t )R1(t ) þ L2(t )R2(t ), (9:12)
dt
where
L1(t ) ¼ p0 À a0 , L2 (t ) ¼ p0 1 p , q ¼ p þ bL10 (t ) : (9:13)
(p0 À p)b À
The solution of this equation is (9:14)
Zt Zt !
u ¼ (L1(w)R1(w) þ L2(w)R2(w)) exp q(z)dz dw :
À1 w
The Chemical Basis of Morphogenesis | 543
One is, however, not so much interested in such a solution in terms of the
statistical disturbances as in the consequent statistical distribution of values of
u, x and Z at various times after instability has set in. In view of the properties of
‘white noise’ assumed above, the values of u at time t will be distributed
according to the normal error law, with the variance
Zt Zt ! (9:15)
[b1(L1(w))2 þ b2(L2(w))2] exp 2 q(z)dz dwÁ
À1 w
There are two commonly occurring cases in which one can simplify this expres-
sion considerably without great loss of accuracy. If the system is in a distinctly
estxapbÂlRewtsqta(tze),dzthÃewn ilql(bt)e, which is near to p(t), will be distinctly negative, and
small unless w is near to t. But then L1(w) and L2(w)
may be replaced by L1(t) and L2(t) in the integral, and also q(z) may be replaced
by q(t). With these approximations the variance is
( À 2q(t))À1[b1(L1(t ))2 þ b2(L2(t))2]: (9:16)
A second case where there is a convenient approximation concerns times when
athpeplsyys2teRmwt is unstable, so that q(t) > 0. For the approximation concerned to
q(z)dz must have its maximum at the last moment w ( ¼ t0) when
q(t0) ¼ 0, and it must be the maximum by a considerable margin (e.g. at least 5)
over all other local maxima. These conditions would apply for instance if q(z)
were always increasing and had negative values at a suYciently early time. One
also requires q0(t0) (the rate of increase of q at time t0) to be reasonably large; it
must at least be so large that over a period of time of length (q0(t0))À21) near to t0
the changes in L1(t) and L2(t) are small, and q0(t) itself must not appreciably
alter in this period. Under these circumstances the integrand is negligible when w
is considerably diVerent from t0, in comparison with its values at that time, and
therefore one may replace L1(w) and L2(w) by L1(t0) and L2(t0), and q0(w) by
q0(t0). This gives the value
pp(q0(t0))À21[b1(L1(t0))2 þ b2(L2(t0))2] exp Z t! (9:17)
2 q(z)dz ,
t0
for the variance of u.
The physical signiWcance of this latter approximation is that the disturbances
near the time when the instability is zero are the only ones which have any
appreciable ultimate eVect. Those which occur earlier are damped out by the
subsequent period of stability. Those which occur later have a shorter period of
instability within which to develop to greater amplitude. This principle is
familiar in radio, and is fundamental to the theory of the superregenerative
receiver.
544 | Alan Turing
Naturally one does not often wish to calculate the expression (9.17), but it is
evxapluhaRbtt0leq(azs)djuzsitiifsyienssgenaticaolmly mthoeni-nsteengsreatpedoiinntstoafbivliietywaonfd the matter. The factor
describes the extent to
which one would expect disturbances of appropriate wave-length to grow be-
tthweeefnacttiomreps tp0ba1n(qd0 t. Taking the terms in b1, b2 into consideration separately,
morphogen should (t0))À12(L1(t0))2 indicates
be regarded as lasting for athtaimt ethpe pd(iqst1u(tr0b)a)nÀc12(ebsLo1(nt0t)h)2e: Wrst
The
dimensionless quantities bL1(t0), bL2(t0) will not usually be suYciently large or
small to justify their detailed calculation.
(3) The extent to which the component for which ps is greatest may be
expected to outdistance the others will now be considered in case (d). The
greatest of the ps will be called ps0. The two closest competitors to s0 will be
s0 À 1 and s0 þ 1; it is required to determine how close the competition is. If the
variation in the chemical data is suYciently small it may be assumed that,
although the exponents ps0À1, ps0 , ps0þ1 may themselves vary appreciably in
time, the diVerences ps0 À ps0À1 and ps0 À ps0þ1 are constant. It certainly can
happen that one of these diVerences is zero or nearly zero, and there is then
‘neck and neck’ competition. The weakest competition occurs when ps0À1 ¼
ps0þ1. In this case
ps0 À ps0À1 ¼ ps0 À ps0þ1 ¼ À12(ps0þ1 À 2ps0 þ ps0À1):
But if s0 is reasonably large ps0þ1 À 2ps0 þ ps0À1 can be set equal to (d2p=ds2)s¼s0.
It may be concluded that the rate at which the most quickly growing component
grows cannot exceed the rate for its closest competitor by more than about
1 (d2 p=ds2 )s¼s0 . The formula (9.7), by which d2p=ds2 can be estimated, may be
2
rTehgearfdacetdoarspth(me 0pnr0)o=dru2cmt oafytbweodfeascctroirbse.dThine dimensionless factor never exceeds 4.
very rough terms as ‘the reciprocal of
the time for the morphogens to diVuse a length equal to a radius’. In equally
rough terms one may say that a time of this order of magnitude is required for
the most quickly growing component to get a lead, amounting to a factor whose
logarithm is of the order of unity, over its closest competitors, in the favourable
case where ps0À1 ¼ ps0þ1.
(4) Very little has yet been said about the eVect of considering non-linear
reaction rate functions when far from homogeneity. Any treatment so systematic
as that given for the linear case seems to be out of the question. It is possible,
however, to reach some qualitative conclusions about the eVects of non-linear
terms. Suppose that z1 is the amplitude of the Fourier component which is most
unstable (on a basis of the linear terms), and which may be supposed to have
wave-length l. The non-linear terms will cause components with wave-lengths
1 l, 1 l, 1 l, ... to appear as well as a space-independent component. If only
2 3 4
quadratic terms are taken into account and if these are somewhat small, then the
The Chemical Basis of Morphogenesis | 545
component of wave-length 1 l and the space-independent component will be the
2
strongest. Suppose these have amplitudes z2 and z1. The state of the system is
thus being described by the numbers z0, z1, z2. In the absence of non-linear
terms they would satisfy equations
dz0 ¼ p0z0, dz1 ¼ p1z1, dz2 ¼ p2z2,
dt dt dt
and if there is slight instability p1 would be a small positive number, but p0 and
p2 distinctly negative. The eVect of the non-linear terms is to replace these
equations by ones of the form
dz0 ¼ p0z0 þ Az12 þ Bz22,
dt
dz1 ¼ p1z1 þ Cz2z1 þ Dz0z1,
dt
dz2 ¼ p2z2 þ Ez12 þ Fz0z2:
dt
As a Wrst approximation one may put dz0=dt ¼ dz2=dt ¼ 0 and ignore z14 and
higher powers; z0 and z1 are then found to be proportional to z12, and the
equation for z1 can be written dz1=dt ¼ p0z1 À kz13. The sign of k in this
diVerential equation is of great importance. If it is positive, thenpthe eVect of
the term kz13 is to arrest the exponential growth of z1 at the value (p1 =k). The
‘instability’ is then very conWned in its eVect, for the waves can only reach a Wnite
amplitude, and this amplitude tends to zero as the instability (p1) tends to zero.
If, however, k is negative the growth becomes something even faster than expo-
nential, and, if the equation dz1=dt ¼ p1z1 À kz13 held universally, it would result
in the amplitude becoming inWnite in a Wnite time. This phenomenon may be
called ‘catastrophic instability’. In the case of two-dimensional systems cata-
strophic instability is almost universal, and the corresponding equation takes
the form dz1=dt ¼ p1z1 þ kz12. Naturally enough in the case of catastrophic
instability the amplitude does not really reach inWnity, but when it is suYciently
large some eVect previously ignored becomes large enough to halt the growth.
(5) Case (a) as described in §8 represents a most extremely featureless form of
pattern development. This may be remedied quite simply by making less drastic
simplifying assumptions, so that a less gross account of the pattern can be given by
the theory. It was assumed in § 9 that only the most unstable Fourier components
would contribute appreciably to the pattern, though it was seen above (heading
(3) of this section) that (in case (d)) this will only apply if the period of time
involved is adequate to permit the morphogens, supposed for this purpose to be
chemically inactive, to diVuse over the whole ring or organ concerned. The same
may be shown to apply for case (a). If this assumption is dropped a much more
interesting form of pattern can be accounted for. To do this it is necessary to
consider not merely the components with U ¼ 0 but some others with small
546 | Alan Turing
positive values of U. One may assume the form At À BU for p. Linearity in U
is assumed because only small values of U are concerned, and the term At is
included to represent the steady increase in instability. By measuring time from
the moment of zero instability the necessity for a constant term is avoided. The
formula (9.17) may be applied to estimate the esxtapthis2tiRctta0 lq(dzi)sdtrziibuwtiiollndoepf etnhde
amplitudes of the components. Only the factor
very much on U, and taking q(t) ¼ p(t) ¼ At À BU , t0 must be BU =A and the
factor is
exp [A(t À BU =A)2]:
The term in U 2 can be ignored if At2 is fairly large, for then either B2U 2=A2 is
small or the factor eÀBUt is. But At2 certainly is large if the factor eAt2, applying
when U ¼ 0, is large. With this approximation the variance takes the
form CeÀ21k2U , with only the two parameters C, k to distinguish the pattern
populations. By choosing appropriate units of concentration and length
these pattern populations may all be reduced to a standard one, e.g. with
C ¼ k ¼ 1. Random members of this population may be produced by consider-
ing any one of the type (a) systems to which the approximations used above
apply. They are also produced, but with only a very small amplitude scale, if a
homogeneous one-morphogen system undergoes random disturbances without
diVusion for a period, and then diVusion without disturbance. This process
is very convenient for computation, and can also be applied to two dimen-
sions. Figure 2 shows such a pattern, obtained in a few hours by a manual
computation.
Figure 2. An example of a ‘dappled’ pattern as resulting from a type (a) morphogen
system. A marker of unit length is shown. See text, §9, 11.
The Chemical Basis of Morphogenesis | 547
To be more deWnite a set of numbers ur, s was chosen, each being ± 1, and
taking the two values with equal probability. A function f (x, y) is related to these
numbers by the formula
f (x, y) ¼ P ur, s exp [ À 1 ((x À hr )2 þ (y À hs)2)]:
2
In the actual computation a somewhat crude approximation to the function
exp [ À 1 (x2 þ y2)]
2
was used and h was about 0Á7. In the Wgure the set of points where f(x, y) is positive
is shown black. The outlines of the black patches are somewhat less irregular than
they should be due to an inadequacy in the computation procedure.
10. A numerical example
The numerous approximations and assumptions that have been made in the
foregoing analysis may be rather confusing to many readers. In the present
section it is proposed to consider in detail a single example of the case of most
interest, (d). This will be made as speciWc as possible. It is unfortunately not
possible to specify actual chemical reactions with the required properties, but it
is thought that the reaction rates associated with the imagined reactions are not
unreasonable.
The detail to be speciWed includes
(i) The number and dimensions of the cells of the ring.
(ii) The diVusibilities of the morphogens.
(iii) The reactions concerned.
(iv) The rates at which the reactions occur.
(v) Information about random disturbances.
(vi) Information about the distribution, in space and time, of those morpho-
gens which are of the nature of evocators.
These will be taken in order.
(i) It will be assumed that there are twenty cells in the ring, and that they have
a diameter of 0Á1 mm each. These cells are certainly on the large rather than the
small side, but by no means impossibly so. The number of cells in the ring has
been chosen rather small in order that it should not be necessary to make the
approximation of continuous tissue.
(ii) Two morphogens are considered. They will be called X and Y, and the
same letters will be used for their concentrations. This will not lead to any real
confusion. The diVusion constant for X will be assumed to be 5 Â 10À8 cm2sÀ1
and that for Y to be 2Á5 Â 10À8 cm2sÀ1. With cells of diameter 0Á01 cm this means
that X Xows between neighbouring cells at the rate 5 Â 10À4 of the diVerence of
X-content of the two cells per second. In other words, if there is nothing altering
548 | Alan Turing
the concentrations but diVusion the diVerence of concentrations suVers an
exponential decay with time constant 1000 s, or ‘half-period’ of 700 s. These
times are doubled for Y.
If the cell membrane is regarded as the only obstacle to diVusion the permea-
bility of the membranes to the morphogen is 5 Â 10À6 cm=s or 0Á018 cm/h. Values
as large as 0Á1 cm/h have been observed (Davson & Danielli 1943, Wgure 28).
(iii) The reactions are the most important part of the assumptions. Four
substances A, X, Y, B are involved; these are isomeric, i.e. the molecules of the
four substances are all rearrangements of the same atoms. Substances C, C0, W
will also be concerned. The thermodynamics of the problem will not be discussed
except to say that it is contemplated that of the substances A, X, Y, B the one with
the greatest free energy is A, and that with the least is B. Energy for the whole
process is obtained by the degradation of A into B. The substance C is in eVect a
catalyst for the reaction Y ! X, and may also be regarded as an evocator, the
system being unstable if there is a suYcient concentration of C.
The reactions postulated are
Y þX ! W,
W þ A ! 2Y þ B instantly,
2X ! W ,
A ! X,
Y ! B,
Y þ C ! C0 instantly,
C0 ! X þ C:
(iv) For the purpose of stating the reaction rates special units will be intro-
duced (for the purpose of this section only). They will be based on a period of
1000 s as units of time, and 10À11 mole=cm3 as concentration unit.1 There will be
little occasion to use any but these special units (s . u .). The concentration of A
will be assumed to have the large value of 1000 s . u. and the catalyst C, together
with its combined form C0 the concentration 10À3(1 þ g) s . u . , the dimension-
less quantity g being often supposed somewhat small, though values over as large
a range as from À0Á5 to 0Á5 may be considered. The rates assumed will be
Y þX !W at the rate 25 YX ,
2X ! W 16
A!X
C0 ! X þ C at the rate 7 X2,
Y !B 64
at the rate 1 Â 10À3A,
16
at the rate 55 Â 10þ3 C 0 ,
32
at the rate 1 Y:
16
1 A somewhat larger value of concentration unit (e.g. 10À9 mole=cm3) is probably more suitable. The
choice of unit only aVects the calculations through the amplitude of the random disturbances.
The Chemical Basis of Morphogenesis | 549
With the values assumed for A and C0 the net eVect of these reactions is to
convert X into Y at the rate 1 [50XY þ 7X2 À 55(1 þ g)] at the same time
32
producing X at the constant rate 116, and destroying Y at the rate Y/16. If, however,
the concentration of Y is zero and the rate of increase of Y required by these
formulae is negative, the rate of conversion of Y into X is reduced suYciently to
permit Y to remain zero.
In the special units m ¼ 1 , n ¼ 14.
2
(v) Statistical theory describes in detail what irregularities arise from the
molecular nature of matter. In a period in which, on the average, one should
expect a reaction to occur between n pairs (or other combinations) of molecules,
the actual number will diVer from the mean by an amount whose mean square is
also n, and is distributed according to the normal error law. Applying this to a
reaction proceeding at a rate F (s . u.) and taking the volume of the cell as
10À8 cm3 (assuming some elongation tangentially to the ring) it will be found
that the root mean spquare irregularity of the quantity reacting in a period t of
time (s .u .) is 0Á004 (Ft).
The diVusion of a morphogen from a cell to a neighbour may be treated as if
the passage of a molecule from one cell to another were a monomolecular
reaction; a molecule must be imagined to change its form slightly as it passes
the cell wall. If the diVusion constant for a wall is m, and quantities M1, M2 of the
relevant morphogen lie on the two sides of it, the root-mean-square irregularity
in the amount passing the wall in a period t is
p
0Á004 {(M1 þ M2)mt}:
These two sources of irregularity are the most signiWcant of those which arise
from truly statistical cause, and are the only ones which are taken into account in
the calculations whose results are given below. There may also be disturbances
due to the presence of neighbouring anatomical structures, and other similar
causes. These are of great importance, but of too great variety and complexity to
be suitable for consideration here.
(vi) The only morphogen which is being treated as an evocator is C. Changes
in the concentration of A might have similar eVects, but the change would have
to be rather great. It is preferable to assume that A is a ‘fuel substance’ (e.g.
glucose) whose concentration does not change. The concentration of C, together
with its combined form C0, will be supposed the same in all cells, but it changes
with the passage of time. Two diVerent varieties of the problem will be con-
sidered, with slightly diVerent assumptions.
The results are shown in table 1. There are eight columns, each of which gives
the concentration of a morphogen in each of the twenty cells; the circumstances to
which these concentrations refer diVer from column to column. The Wrst Wve
columns all refer to the same ‘variety’ of the imaginary organism, but there are two
specimens shown. The specimens diVer merely in the chance factors which
550 | Alan Turing
Table 1. Some stationary-wave patterns
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflWfflfflrfflfflsfflffltfflfflfflsfflfflp}e|cfflfflifflmfflfflfflfflefflfflnfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ second ‘slow
specimen: cooking’:
cell incizpfflffliffleffl}n|tfflfflpfflffl{attern fzinfflfflafflffll}|pfflfflafflfflt{tern incipient incipient four-lobed equilibrium
zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{
number X Y XY Y Y X Y
0 1Á130 0Á929 0Á741 1Á463 0Á834 1Á057 1Á747 0Á000
1 1Á123 0Á940 0Á761 1Á469 0Á833 0Á903 1Á685 0Á000
2 1Á154 0Á885 0Á954 1Á255 0Á766 0Á813 1Á445 2Á500
3 1Á215 0Á810 1Á711 0Á000 0Á836 0Á882 0Á445 2Á500
4 1Á249 0Á753 1Á707 0Á000 0Á930 1Á088 1Á685 0Á000
5 1Á158 0Á873 0Á875 1Á385 0Á898 1Á222 1Á747 0Á000
6 1Á074 1Á003 0Á700 1Á622 0Á770 1Á173 1Á685 0Á000
7 1Á078 1Á000 0Á699 1Á615 0Á740 0Á956 0Á445 2Á500
8 1Á148 0Á896 0Á885 1Á382 0Á846 0Á775 0Á445 2Á500
9 1Á231 0Á775 1Á704 0Á000 0Á937 0Á775 1Á685 0Á000
10 1Á204 0Á820 1Á708 0Á000 0Á986 0Á969 1Á747 0Á000
11 1Á149 0Á907 0Á944 1Á273 1Á019 1Á170 1Á685 0Á000
12 1Á156 0Á886 0Á766 1Á451 0Á899 1Á203 0Á445 2Á500
13 1Á170 0Á854 0Á744 1Á442 0Á431 1Á048 0Á445 2Á500
14 1Á131 0Á904 0Á756 1Á478 0Á485 0Á868 1Á685 0Á000
15 1Á090 0Á976 0Á935 1Á308 0Á919 0Á813 1Á747 0Á000
16 1Á109 0Á957 1Á711 0Á000 1Á035 0Á910 1Á685 0Á000
17 1Á201 0Á820 1Á706 0Á000 1Á003 1Á050 0Á445 2Á500
18 1Á306 0Á675 0Á927 1Á309 0Á899 1Á175 0Á445 2Á500
19 1Á217 0Á811 0Á746 1Á487 0Á820 1Á181 1Á685 0Á000
were involved. With this variety the value of g was allowed to increase at the rate
of 2À7s . u . from the value À 1 to þ 116. At this point a pattern had deWnitely begun
4
to appear, and was recorded. The parameter g was then allowed to decrease at the
same rate to zero and then remained there until there was no more appreciable
change. The pattern was then recorded again. The concentrations of Y in these
two recordings are shown in Wgure 3 as well as in table 1. For the second specimen
only one column of Wgures is given, viz. those for the Y morphogen in the
incipient pattern. At this stage the X values are closely related to the Y values,
as may be seen from the Wrst specimen (or from theory). The Wnal values can be
made almost indistinguishable from those for the Wrst specimen by renumbering
the cells and have therefore not been given. These two specimens may be said to
belong to the ‘variety with quick cooking’, because the instability is allowed to
increase so quickly that the pattern appears relatively soon. The eVect of this
haste might be regarded as rather unsatisfactory, as the incipient pattern is very
irregular. In both specimens the four-lobed component is present in considerable
The Chemical Basis of Morphogenesis | 551
1.5
1.0
0.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Figure 3. Concentrations of Y in the development of the first specimen (taken from table
1). original homogeneous equilibrium; incipient pattern; ———— final
equilibrium.
strength in the incipient pattern. It ‘beats’ with the three-lobed component
producing considerable irregularity. The relative magnitudes of the three- and
four-lobed components depend on chance and vary from specimen to specimen.
The four-lobed component may often be the stronger, and may occasionally be
so strong that the Wnal pattern is four-lobed. How often this happens is not
known, but the pattern, when it occurs, is shown in the last two columns of the
table. In this case the disturbances were supposed removed for some time before
recording, so as to give a perfectly regular pattern.
The remaining column refers to a second variety, one with ‘slow cooking’. In
this the value of g was allowed to increase only at the rate 10À5. Its initial value was
À0Á010, but is of no signiWcance. The Wnal value was 0Á003. With this pattern,
when shown graphically, the irregularities are deWnitely perceptible, but are
altogether overshadowed by the three-lobed component. The possibility of the
ultimate pattern being four-lobed is not to be taken seriously with this variety.
The set of reactions chosen is such that the instability becomes ‘catastrophic’
when the second-order terms are taken into account, i.e. the growth of the waves
tends to make the whole system more unstable than ever. This eVect is Wnally
halted when (in some cells) the concentration of Y has become zero. The
constant conversion of Y into X through the agency of the catalyst C can then
no longer continue in these cells, and the continued growth of the amplitude of
the waves is arrested. When g ¼ 0 there is of course an equilibrium with
X ¼ Y ¼ 1 in all cells, which is very slightly stable. There are, however, also
552 | Alan Turing
other stable equilibria with g ¼ 0, two of which are shown in the table. These
Wnal equilibria may, with some trouble but little diYculty, be veriWed to be
solutions of the equations (6.1) with
dX ¼ dY ¼ 0,
dt dt
and
32f (X, Y ) ¼ 57 À 50XY À 7Y 2, 32g(X, Y ) ¼ 50XY þ 7Y 2 À 2Y À 55:
The morphogen concentrations recorded at the earlier times connect more
directly with the theory given in §§6 to 9. The amplitude of the waves was then
still suYciently small for the approximation of linearity to be still appropriate,
and consequently the ‘catastrophic’ growth had not yet set in.
The functions f(X, Y) and g(X, Y) of §6 depend also on g and are
f (X, Y ) ¼ 312[ À 7X2 À 50XY þ 57 þ 55g],
g(X, Y ) ¼ 312[7X2 þ 50XY À 2Y À 55 À 55g]:
In applying the theory it will be as well to consider principally the behaviour of
the system when g remains permanently zero. Then for equilibrium
f (X, Y ) ¼ g(X, Y ) ¼ 0 which means that X ¼ Y ¼ 1, i.e. h ¼ k ¼ 1. One also
Wnds the following values for various quantities mentioned in §§ 6 to 9:
a ¼ À2, b ¼ À1Á5625, c ¼ 2, d ¼ 1Á500, s ¼ 3Á333,
I ¼ 0, a ¼ 0Á625, w ¼ 0Á500, (d À a)( Àbc)À12 ¼ 1Á980,
(m þ n)(mn)À21 ¼ 2Á121, p0 ¼ À0Á25 ± 0Á25i,
p2 ¼ À0Á0648, p3 ¼ À0Á0034, p4 ¼ À0Á 0118:
(The relation between p and U for these chemical data, and the values pn, can be
seen in Wgure 1, the values being so related as to make the curves apply to this
example as well as that in §8.) The value s ¼ 3Á333 leads one to expect a three-
lobed pattern as the commonest, and this is conWrmed by the values pn. The
four-lobed pattern is evidently the closest competitor. The closeness of the
competition may be judged from the diVerence p3 À p4 ¼ 0Á0084, which sug-
gests that the three-lobed component takes about 120 s . u. or about 33 h to gain
an advantage of a neper (i.e. about 2Á7:1) over the four-lobed one. However, the
fact that g is diVerent from 0 and is changing invalidates this calculation to some
extent.
The Wgures in table 1 were mainly obtained with the aid of the Manchester
University Computer.
Although the above example is quite adequate to illustrate the mathematical
principles involved it may be thought that the chemical reaction system is
The Chemical Basis of Morphogenesis | 553
somewhat artiWcial. The following example is perhaps less so. The same ‘special
units’ are used. The reactions assumed are
A!X at the rate 10À3A, A ¼ 103,
XþY !C at the rate 103XY ,
at the rate 106 C ,
C !XþY at the rate 62Á5C,
C!D at the rate 0Á125BC, B ¼ 103,
BþC !W instantly,
W !Y þC at the rate 0Á0625Y ,
Y !E instantly,
Y þ V ! V0 at the rate 62Á5V 0, V 0 ¼ 10À3b.
V0 ! E þ V
The eVect of the reactions X þ Y ! C is that C ¼ 10À3XY . The reaction C ! D
destroys C, and therefore in eVect both X and Y, at the rate 1 XY . The reaction
16
A ! X forms X at the constant rate 1, and the pair Y þ V ! V 0 ! E þ V
destroys Y at the constant rate 1 b. The pair B þC ! W ! Y þC forms Y at
16
1 1
the rate 8 XY , and Y ! E destroys it at the rate 16 Y . The total eVect therefore is
that X is produced at the rate f (X , Y ) ¼ 1 (16 À XY ), and Y at the rate
16
1
g(X, Y ) ¼ 16 (XY ÀY À b). However, g(X, Y ) ¼ 0 if Y < 0. The diVusion con-
stants will be supposed to be m ¼ 1 , n ¼ 116. The homogeneity condition gives
4
hk ¼ 16, k ¼ 16 À b.
It will be seen from conditions (9.4a) that case (d ) applies if and only if
4 þ k < 2Á75, i.e. if k lies between 1Á725 and p9Á325þ7.12,Cio.en. difitkiodnoe(s9.n4obt) shows that
k 4 will be instability if lie between
8 k
there in addition k þ 8 >
4Á98 and 12Á8. It will also be found that the wave-length corresponding to
k ¼ 4Á98 is 4Á86 cell diameters.
In the case of a ring of six cells with b ¼ 12 there is a stable equilibrium, as
shown in table 2.
Table 2.
cell 0 1 2 3 4 5
X 7Á5 3Á5 2Á5 2Á5 3Á5 7Á5
Y088880
It should be recognized that these equilibria are only dynamic equilibria. The
molecules which together make up the chemical waves are continually changing,
though their concentrations in any particular cell are only undergoing small
statistical Xuctuations. Moreover, in order to maintain the wave pattern a
continual supply of free energy is required. It is clear that this must be so since
there is a continual degradation of energy through diVusion. This energy is
supplied through the ‘fuel substances’ (A, B in the last example), which are
degraded into ‘waste products’ (D, E).
554 | Alan Turing
11. Restatement and biological interpretation of the results
Certain readers may have preferred to omit the detailed mathematical treatment
of §§ 6 to 10. For their beneWt the assumptions and results will be brieXy
summarized, with some change of emphasis.
The system considered was either a ring of cells each in contact with its
neighbours, or a continuous ring of tissue. The eVects are extremely similar in
the two cases. For the purposes of this summary it is not necessary to distinguish
between them. A system with two or three morphogens only was considered, but
the results apply quite generally. The system was supposed to be initially in a
stable homogeneous condition, but disturbed slightly from this state by some
inXuences unspeciWed, such as Brownian movement or the eVects of neighbour-
ing structures or slight irregularities of form. It was supposed also that slow
changes are taking place in the reaction rates (or, possibly, the diVusibilities) of
the two or three morphogens under consideration. These might, for instance, be
due to changes of concentration of other morphogens acting in the role of
catalyst or of fuel supply, or to a concurrent growth of the cells, or a change
of temperature. Such changes are supposed ultimately to bring the system out of
the stable state. The phenomena when the system is just unstable were the
particular subject of the inquiry. In order to make the problem mathematically
tractable it was necessary to assume that the system never deviated very far from
the original homogeneous condition. This assumption was called the ‘linearity
assumption’ because it permitted the replacement of the general reaction rate
functions by linear ones. This linearity assumption is a serious one. Its justiWca-
tion lies in the fact that the patterns produced in the early stages when it is valid
may be expected to have strong qualitative similarity to those prevailing in the
later stages when it is not. Other, less important, assumptions were also made at
the beginning of the mathematical theory, but the detailed eVects of these were
mostly considered in § 9, and were qualitatively unimportant.
The conclusions reached were as follows. After the lapse of a certain period of
time from the beginning of instability, a pattern of morphogen concentrations
appears which can best be described in terms of ‘waves’. There are six types of
possibility which may arise.
(a) The equilibrium concentrations and reaction rates may become such that
there would be instability for an isolated cell with the same content as any one of
the cells of the ring. If that cell drifts away from the equilibrium position, like an
upright stick falling over, then, in the ring, each cell may be expected to do
likewise. In neighbouring cells the drift may be expected to be in the same
direction, but in distant cells, e.g. at opposite ends of a diameter there is no
reason to expect this to be so.
This is the least interesting of the cases. It is possible, however, that it might
account for ‘dappled’ colour patterns, and an example of a pattern in two
The Chemical Basis of Morphogenesis | 555
dimensions produced by this type of process is shown in Wgure 2 for com-
parison with ‘dappling’. If dappled patterns are to be explained in this way they
must be laid down in a latent form when the foetus is only a few inches long.
Later the distances would be greater than the morphogens could travel by
diVusion.
(b) This case is similar to (a), except that the departure from equilibrium is
not a unidirectional drift, but is oscillatory. As in case (a) there may not be
agreement between the contents of cells at great distances.
There are probably many biological examples of this metabolic oscillation, but
no really satisfactory one is known to the author.
(c) There may be a drift from equilibrium, which is in opposite directions in
contiguous cells.
No biological examples of this are known.
(d ) There is a stationary wave pattern on the ring, with no time variation,
apart from a slow increase in amplitude, i.e. the pattern is slowly becoming more
marked. In the case of a ring of continuous tissue the pattern is sinusoidal, i.e.
the concentration of one of the morphogens plotted against position on the ring
is a sine curve. The peaks of the waves will be uniformly spaced round the ring.
The number of such peaks can be obtained approximately by dividing the so-
called ‘chemical wave-length’ of the system into the circumference of the ring.
The chemical wave-length is given for the case of two morphogens by the
formula (9.3). This formula for the number of peaks of course does not give a
whole number, but the actual number of peaks will always be one of the two
whole numbers nearest to it, and will usually be the nearest. The degree of
instability is also shown in (9.3).
The mathematical conditions under which this case applies are given in
equations (9.4a), (9.4b).
Biological examples of this case are discussed at some length below.
(e) For a two-morphogen system only the alternatives (a) to (d ) are possible,
but with three or more morphogens it is possible to have travelling waves. With a
ring there would be two sets of waves, one travelling clockwise and the other
anticlockwise. There is a natural chemical wave-length and wave frequency in
this case as well as a wave-length; no attempt was made to develop formulae for
these.
In looking for biological examples of this there is no need to consider only
rings. The waves could arise in a tissue of any anatomical form. It is important to
know what wave-lengths, velocities and frequencies would be consistent with the
theory. These quantities are determined by the rates at which the reactions occur
(more accurately by the ‘marginal reaction rates’, which have the dimensions of
the reciprocal of a time), and the diVusibilities of the morphogens. The possible
range of values of the reaction rates is so immensely wide that they do not even
give an indication of orders of magnitude. The diVusibilities are more helpful. If
556 | Alan Turing
one were to assume that all the dimensionless parameters in a system of travelling
waves were the same as in the example given in § 8, one could say that the
product of the velocity and wave-length of the waves was 3p times the diVusi-
bility of the most diVusible morphogen. But this assumption is certainly false,
and it is by no means obvious what is the true range of possible values for the
numerical constant (here 3p). The movements of the tail of a spermatozoon
suggest themselves as an example of these travelling waves. That the waves are
within one cell is no real diYculty. However, the speed of propagation seems to
be somewhat greater than can be accounted for except with a rather large
numerical constant.
(f ) Metabolic oscillation with neighbouring cells in opposite phases. No
biological examples of this are known to the author.
It is diYcult also to Wnd cases to which case (d ) applies directly, but this is
simply because isolated rings of tissue are very rare. On the other hand, systems
that have the same kind of symmetry as a ring are extremely common, and it is to
be expected that under appropriate chemical conditions, stationary waves may
develop on these bodies, and that their circular symmetry will be replaced by a
polygonal symmetry. Thus, for instance, a plant shoot may at one time have
circular symmetry, i.e. appear essentially the same when rotated through any
angle about a certain axis; this shoot may later develop a whorl of leaves, and
then it will only suVer rotation through the angle which separates the leaves, or
any multiple of it. This same example demonstrates the complexity of the
situation when more than one dimension is involved. The leaves on the shoots
may not appear in whorls, but be imbricated. This possibility is also capable of
mathematical analysis, and will be considered in detail in a later paper. The cases
which appear to the writer to come closest biologically to the ‘isolated ring of
cells’ are the tentacles of (e.g.) Hydra, and the whorls of leaves of certain plants
such as WoodruV (Asperula odorata).
Hydra is something like a sea-anemone but lives in fresh water and has from
about Wve to ten tentacles. A part of a Hydra cut oV from the rest will rearrange
itself so as to form a complete new organism. At one stage of this proceeding the
organism has reached the form of a tube open at the head end and closed at the
other end. The external diameter is somewhat greater at the head end than over
the rest of the tube. The whole still has circular symmetry. At a somewhat later
stage the symmetry has gone to the extent that an appropriate stain will bring out
a number of patches on the widened head end. These patches arise at the points
where the tentacles are subsequently to appear (Child 1941, p. 101 and Wgure 30).
According to morphogen theory it is natural to suppose that reactions, similar to
those which were considered in connection with the ring of tissue, take place in
the widened head end, leading to a similar breakdown of symmetry. The
situation is more complicated than the case of the thin isolated ring, for the
portion of the Hydra concerned is neither isolated nor very thin. It is not
The Chemical Basis of Morphogenesis | 557
unreasonable to suppose that this head region is the only one in which the
chemical conditions are such as to give instability. But substances produced in
this region are still free to diVuse through the surrounding region of lesser
activity. There is no great diYculty in extending the mathematics to cover this
point in particular cases. But if the active region is too wide the system no longer
approximates the behaviour of a thin ring and one can no longer expect the
tentacles to form a single whorl. This also cannot be considered in detail in the
present paper.
In the case of woodruV the leaves appear in whorls on the stem, the number of
leaves in a whorl varying considerably, sometimes being as few as Wve or as many
as nine. The numbers in consecutive whorls on the same stem are often equal,
but by no means invariably. It is to be presumed that the whorls originate in rings
of active tissue in the meristematic area, and that the rings arise at suYciently
great distance to have little inXuence on one another. The number of leaves in the
whorl will presumably be obtainable by the rule given above, viz. by dividing the
chemical wave-length into the circumference, though both these quantities will
have to be given some new interpretation more appropriate to woodruV than to
the ring. Another important example of a structure with polygonal symmetry is
provided by young root Wbres just breaking out from the parent root. Initially
these are almost homogeneous in cross-section, but eventually a ring of fairly
evenly spaced spots appear, and these later develop into vascular strands. In this
case again the full explanation must be in terms of a two-dimensional or even a
three-dimensional problem, although the analysis for the ring is still illuminat-
ing. When the cross-section is very large the strands may be in more than one
ring, or more or less randomly or hexagonally arranged. The two-dimensional
theory (not expounded here) also goes a long way to explain this.
Flowers might appear superWcially to provide the most obvious examples of
polygonal symmetry, and it is probable that there are many species for which this
‘waves round a ring’ theory is essentially correct. But it is certain that it does not
apply for all species. If it did it would follow that, taking Xowers as a whole, i.e.
mixing up all species, there would be no very markedly preferred petal (or
corolla, segment, stamen, etc.) numbers. For when all species are taken into
account one must expect that the diameters of the rings concerned will take on
nearly all values within a considerable range, and that neighbouring diameters
will be almost equally common. There may also be some variation in chemical
wave-length. Neighbouring values of the ratio circumferences to wave-length
should therefore be more or less equally frequent, and this must mean that
neighbouring petal numbers will have much the same frequency. But this is not
borne out by the facts. The number Wve is extremely common, and the number
seven rather rare. Such facts are, in the author’s opinion, capable of explanation
on the basis of morphogen theory, and are closely connected with the theory of
phyllotaxis. They cannot be considered in detail here.
558 | Alan Turing
The case of a Wlament of tissue calls for some comment. The equilibrium patterns
on such a Wlament will be the same as on a ring, which has been cut at a point where
the concentrations of the morphogens are a maximum or a minimum. This could
account for the segmentation of such Wlaments. It should be noticed, however, that
the theory will not apply unmodiWed for Wlaments immersed in water.
12. Chemical waves on spheres. Gastrulation
The treatment of homogeneity breakdown on the surface of a sphere is not much
more diYcult than in the case of the ring. The theory of spherical harmonics, on
which it is based, is not, however, known to many that are not mathematical
specialists. Although the essential properties of spherical harmonics that are used
are stated below, many readers will prefer to proceed directly to the last para-
graph of this section.
The anatomical structure concerned in this problem is a hollow sphere of
continuous tissue such as a blastula. It is supposed suYciently thin that one can
treat it as a ‘spherical shell’. This latter assumption is merely for the purpose of
mathematical simpliWcation; the results are almost exactly similar if it is omitted.
As in § 7 there are to be two morphogens, and a, b, c, d, m0, n0, h, k are also to
have the same meaning as they did there. The operator r2 will be used here to
mean the superWcial part of the Laplacian, i.e. r2V will be an abbreviation of
1 @2V 1 @ @V
r2 @f2 sin2 @y sin y @y
þ r2 y ,
where y and f are spherical polar co-ordinates on the surface of the sphere and r
is its radius. The equations corresponding to (7.1) may then be written
@X 9
@t >>=
¼ a(X À h) þ b(Y À k) þ m0 r2 X ,
@Y ;>> (12:1)
@t
¼ c(X À h) þ d(Y À k) þ n0r2Y :
It is well known (e.g. Jeans 1927, chapter 8) that any function on the surface of
the sphere, or at least any that is likely to arise in a physical problem, can be
‘expanded in spherical surface harmonics’. This means that it can be expressed in
the form
"#
X1 Xn
Amn Pnm(cos y) eimf :
n¼0 m¼Àn
The expression in the square bracket is described as a ‘surface harmonic of degree
n’. Its nearest analogue in the ring theory is a Fourier component. The essential
property of a spherical harmonic of degree n is that when the operator r2 is
The Chemical Basis of Morphogenesis | 559
applied to it the eVect is the same as multiplication by Àn(n þ 1)=r2. In view of
this fact it is evident that a solution of (12.1) is
X X1 Xn (Amn eiqnt þ Bnmeiqn0 t )Pnm(cos y) eimf, 9
¼hþ >=>>>
n¼0 m¼Àn (12:2)
Y X1 Xn (Cnmeiqnt þ Dnmeiqn0 t )Pnm(cos y) eif, >>;>>
¼kþ
n¼0 m¼Àn
where qn and qn0 are the two roots of
À a þ m0 n(n þ À d þ n0 n(n þ ¼ bc (12:3)
q r2 1) q r2 1)
and m0
Anm qn r2 1)
À a þ m0 n(n þ ¼ bCnm, (12:4)
À a þ r2 n(n þ ¼ cDnm:
Bnm qn0 1)
This is the most general solution, since the coeYcients Amn and Bnm can be chosen
to give any required values of X, Y when t ¼ 0, except when (12.3) has two equal
roots, in which case a treatment is required which is similar to that applied in
similar circumstances in § 7. The analogy with § 7 throughout will indeed be
obvious, though the summation with respect to m does not appear there. The
meaning of this summation is that there are a number of diVerent patterns with
the same wave-length, which can be superposed with various amplitude factors.
Then supposing that, as in § 8, one particular wave-length predominates, (12.2)
reduces to
X À h ¼ eiqn0 t Xn0 Amn0 Pnm0 (cos y) eimf, 9
>>>=
m¼Àn0
m0 >>;> (12:5)
qn0 Àaþ r2 1) (X
b(Y À k) ¼ n(n þ À h):
In other words, the concentrations of the two morphogens are proportional, and
both of them are surface harmonics of the same degree n0, viz. that which makes
the greater of the roots qn0 , qn0 0 have the greatest value.
It is probable that the forms of various nearly spherical structures, such as
radiolarian skeletons, are closely related to these spherical harmonic patterns.
The most important application of the theory seems, however, to be to the
gastrulation of a blastula. Suppose that the chemical data, including the chemical
wave-length, remain constant as the radius of the blastula increases. To be quite
speciWc suppose that
560 | Alan Turing
m0 ¼ 2, n0 ¼ 1, a ¼ À4, b ¼ À8, c ¼ 4, d ¼ 7:
With these values the system is quite stable so long as the radius is less than about
2. Near this point, however, the harmonics of degree 1 begin to develop and a
pattern of form (12.5) with n0 ¼ 1 makes its appearance. Making use of the facts
that
P10(cos y) ¼ cos y, P11(cos y) ¼ P1À1(cos y) ¼ sin y,
it is seen that X À h is of the form
X À h ¼ A cos y þ B sin y cos f þ C sin y sin f, (12:6)
which may also be interpreted as
X À h ¼ A0 cos y0, (12:7)
where y0 is the angle which the rtaodBiu, sCy,, AfamndakAe0s¼wipth(tAh2eþWxBe2dþdiCre2c)t.ion having
direction cosines proportional
The outcome of the analysis therefore is quite simply this. Under certain not
very restrictive conditions (which include a requirement that the sphere be
relatively small but increasing in size) the pattern of the breakdown of homo-
geneity is axially symmetrical, not about the original axis of spherical polar co-
ordinates, but about some new axis determined by the disturbing inXuences. The
concentrations of the Wrst morphogen are given by (12.7), where y0 is measured
from this new axis; and Y À k is proportional to X À h. Supposing that the Wrst
morphogen is, or encourages the production of, a growth hormone, one must
expect the blastula to grow in an axially symmetric manner, but at a greater rate
at one end of the axis than at the other. This might under many circumstances
lead to gastrulation, though the eVects of such growth are not very easily
determinable. They depend on the elastic properties of the tissue as well as on
the growth rate at each point. This growth will certainly lead to a solid of
revolution with a marked diVerence between the two poles, unless, in addition
to the chemical instability, there is a mechanical instability causing the break-
down of axial symmetry. The direction of the axis of gastrulation will be quite
random according to this theory. It may be that it is found experimentally that
the axis is normally in some deWnite direction such as that of the animal pole.
This is not essentially contradictory to the theory, for any small asymmetry of the
zygote may be suYcient to provide the ‘disturbance’ which determines the axis.
13. Non-linear theory. Use of digital computers
The ‘wave’ theory which has been developed here depends essentially on the
assumption that the reaction rates are linear functions of the concentrations, an
assumption which is justiWable in the case of a system just beginning to leave a
The Chemical Basis of Morphogenesis | 561
homogeneous condition. Such systems certainly have a special interest as giving
the Wrst appearance of a pattern, but they are the exception rather than the rule.
Most of an organism, most of the time, is developing from one pattern into
another, rather than from homogeneity into a pattern. One would like to be able
to follow this more general process mathematically also. The diYculties are,
however, such that one cannot hope to have any very embracing theory of such
processes, beyond the statement of the equations. It might be possible, however,
to treat a few particular cases in detail with the aid of a digital computer. This
method has the advantage that it is not so necessary to make simplifying
assumptions as it is when doing a more theoretical type of analysis. It might
even be possible to take the mechanical aspects of the problem into account as
well as the chemical, when applying this type of method. The essential disadvan-
tage of the method is that one only gets results for particular cases. But this
disadvantage is probably of comparatively little importance. Even with the ring
problem, considered in this paper, for which a reasonably complete mathemat-
ical analysis was possible, the computational treatment of a particular case was
most illuminating. The morphogen theory of phyllotaxis, to be described, as
already mentioned, in a later paper, will be covered by this computational
method. Non-linear equations will be used.
It must be admitted that the biological examples which it has been possible to
give in the present paper are very limited. This can be ascribed quite simply to
the fact that biological phenomena are usually very complicated. Taking this in
combination with the relatively elementary mathematics used in this paper one
could hardly expect to Wnd that many observed biological phenomena would be
covered. It is thought, however, that the imaginary biological systems which have
been treated, and the principles which have been discussed, should be of some
help in interpreting real biological forms.
References
Child, C. M. 1941. Patterns and problems of development. University of Chicago Press.
Davson, H. & Danielli, J. F. 1943. The permeability of natural membranes. Cambridge
University Press.
Jeans, J. H. 1927. The mathematical theory of elasticity and magnetism, 5th ed. Cambridge
University Press.
Michaelis, L. & Menten, M. L. 1913. Die Kinetik der Invertinwirkung. Biochemische
Zeitschrift 49: 333–369.
Thompson, Sir D’Arcy 1942. On growth and form, 2nd ed. Cambridge University Press.
Waddington, C. H. 1940. Organisers and genes. Cambridge University Press.
CHAPTER 16
Chess (1953)
Alan Turing
Introduction
Jack Copeland
Chess and AI
Chess and some other board games are a test-bed for ideas in ArtiWcial Intelli-
gence. Donald Michie—Turing’s wartime colleague and subsequently founder of
the Department of Machine Intelligence and Perception at the University of
Edinburgh—explains the relevance of chess to AI:
Computer chess has been described as the Drosophila melanogaster of machine intelli-
gence. Just as Thomas Hunt Morgan and his colleagues were able to exploit the special
limitations and conveniences of the Drosophila fruit Xy to develop a methodology of
genetic mapping, so the game of chess holds special interest for the study of the
representation of human knowledge in machines. Its chief advantages are: (1) chess
constitutes a fully deWned and well-formalized domain; (2) the game challenges the
highest levels of human intellectual capacity; (3) the challenge extends over the full
range of cognitive functions such as logical calculation, rote learning, concept-formation,
analogical thinking, imagination, deductive and inductive reasoning; (4) a massive and
detailed corpus of chess knowledge has accumulated over the centuries in the form of
chess instructional works and commentaries; (5) a generally accepted numerical scale of
performance is available in the form of the U.S. Chess Federation and International ELO
rating system.1
History of Computer Chess
In 1945, in his paper ‘Proposed Electronic Calculator’, Turing predicted that
computers would probably play ‘very good chess’, an opinion echoed in 1949 by
Claude Shannon of Bell Telephone Laboratories, another leading early theoret-
1 D. Michie, On Machine Intelligence (2nd edn. Chichester: Ellis Horwood, 1986), 78–9.
Chess | 563
ician of computer chess.2 By 1958, Herbert Simon and Allen Newell were
predicting that within ten years the world chess champion would be a computer,
unless barred by the rules.3 Just under forty years later, on 11 May 1997,
IBM’s Deep Blue beat the reigning world champion, Gary Kasparov, in a
six-game match.
Turing was theorizing about the mechanization of chess as early as 1941.
Fellow codebreakers at GC & CS remember him experimenting with two heuris-
tics now commonly used in computer chess, minimax and best-Wrst.4 The
minimax heuristic involves assuming that one’s opponent will move in such a
way as to maximize their gains; one then makes one’s own move in such a way as
to minimize the losses caused by the opponent’s expected move.5 The best-Wrst
heuristic involves ranking the moves available to a player by means of a rule
of thumb scoring system and examining the consequences of the highest-
scoring move Wrst.
Turochamp
In 1948 Turing and the mathematical economist David Champernowne designed
a chess-playing routine known as ‘Turochamp’.
Champernowne later gave this description of Turochamp:
It was in the late summer of 1948 that Turing and I did try out a loose system of rules for
deciding on the next move in a chess game which we thought could be fairly easily
programmed for a computer. My long-suVering wife, a beginner at chess, took on the
system and lost. . . . Here is what I think I remember about the system but I may have
been inXuenced by what I have since read about other people’s systems. There was a
system for estimating the eVects of any move on White’s estimated net advantage over
Black. This allowed for:
(1) Captures, using the conventional scale of 10 for pawn, 30 for knight or bishop, 50
for rook, 100 for queen and something huge, say 5000, for king.
(2) Change in mobility; i.e., change in the number of squares to which any piece or
pawn could immediately move legitimately (1 each).
(3) Special incentives for: (a) Castling (3 points). (b) Advancing a passed pawn (1 or 2
points). (c) Getting a rook onto the seventh rank (4 points perhaps).
(I don’t think occupation of one of the 4 central squares gained any special bonus. We did
not cater to the end-game.) Most of our attention went to deciding which moves were to
2 C. E. Shannon, ‘Programming a Computer for Playing Chess’, Philosophical Magazine, 41 (1950), 256–75.
3 H. A. Simon and A. Newell, ‘Heuristic Problem Solving: The Next Advance in Operations Research’,
Operations Research, 6 (1958), 1–10.
4 Michie in interview with Copeland (Oct. 1995).
5 The foundations of the minimax approach were laid in 1928 by von Neumann in his fundamental
‘minimax theorem’ (J. von Neumann, ‘Zur Theorie der Gesellschaftsspiele’, Mathematische Annalen, 100
(1928), 295–320).
564 | Jack Copeland
be followed up. My memory about this is infuriatingly weak. Captures had to be followed
up at least to the point where no further capture was immediately possible. Checks and
forcing moves had to be followed further. We were particularly keen on the idea that
whereas certain moves would be scorned as pointless and pursued no further others would
be followed quite a long way down certain paths. In the actual experiment I suspect we
were a bit slapdash about all this and must have made a number of slips since the
arithmetic was extremely tedious with pencil and paper. Our general conclusion was
that a computer should be fairly easy to programme to play a game of chess against a
beginner and stand a fair chance of winning or least reaching a winning position.6
Turing started to code the Turochamp for the Ferranti Mark I computer at
Manchester University but he never completed the task.7
The First Working Chess Programme
Dietrich Prinz, who worked for the engineering Wrm Ferranti Ltd., wrote the Wrst
chess programme to be fully implemented. It ran in November 1951 on the
University of Manchester Ferranti Mark I.8 Prinz, like Michie, Strachey, and
others, was inXuenced by an important article published in 1950 by Donald
Davies, ‘A Theory of Chess and Noughts and Crosses’ (see Further Reading).
Prinz ‘learned all about programming the [Mark I] computer at seminars given
by Alan Turing and Cecily Popplewell’.9 (Like Turing, Prinz wrote a program-
ming manual for the Mark I.10)
Prinz’s programme was for solving simple problems of the mate-in-two
variety.11 The programme would examine every possible move until a solution
was found. On average, several thousand moves had to be examined in the
course of solving a problem, and the programme was considerably slower than
a human player. Unlike Prinz’s programme, Turochamp could in principle play a
complete game and operated not by exhaustive search but under the guidance of
heuristics.
Prinz saw in chess programming an ‘indication of the methods that might be
used to treat structural or logistic problems occurring in other Welds by means of
6 Champernowne’s account is from a letter that he wrote to Computer Chess (4 (Jan. 1980), 80–1); it is
published here by permission of W. and R. P. Champernowne.
7 D. Michie, ‘Game-Playing and Game-Learning Automata’, in L. Fox (ed.), Advances in Programming
and Non-numerical Computation (New York: Pergamon, 1966), 189.
8 B. V. Bowden (ed.), Faster Than Thought (London: Pitman, 1953), 295.
9 C. Gradwell, ‘Early Days’, reminiscences in a Newsletter ‘For those who worked on the Manchester Mk
I computers’, Apr. 1994. (I am grateful to Prinz’s daughter, Daniela Derbyshire, for sending me a copy of
Gradwell’s article.)
10 D. G. Prinz, ‘Introduction to Programming on the Manchester Electronic Digital Computer’, n.d.,
Ferranti Ltd. (a digital facsimile is in The Turing Archive for the History of Computing <www.Alan
Turing.net/prinz>).
11 Prinz’s programme is described in Bowden (ed.), Faster Than Thought, 295–7 and in D. G. Prinz,
‘Robot Chess’, Research, 5 (1952), 261–6.
Chess | 565
electronic computers’.12 Prinz used the Ferranti Mark I to solve logical problems
(as also did Audrey Bates), and in 1949, 1950, and 1951 Ferranti built three small
experimental special-purpose computers for theorem-proving and other logical
work.13 (This was several years before the Logic Theorist of Newell, Simon,
and Shaw—often incorrectly said to be the Wrst AI programme—made its
debut at the Dartmouth conference (see ‘ArtiWcial Intelligence’, above).)
Turing’s Approach in ‘Chess’
Turing says that the system of rules set out in ‘Chess’ is based on an ‘introspective
analysis’ of his own thought processes when playing (but with ‘considerable
simpliWcations’). His system anticipates much that has become standard in chess
programming: the use of heuristics to guide the search through the ‘tree’ of
possible moves and counter-moves; the use of evaluation rules which assign
numerical values, indicative of strength or weakness, to board conWgurations; the
minimax strategy; and variable look-ahead whereby, instead of the consequences
of every possible move being followed equally far, the ‘more proWtable moves
[are] considered in greater detail than the less’ (p. 571). Turing also realized the
necessity of using ‘an entirely diVerent system for the end-game’ (p. 574).
The learning procedure that Turing proposes in ‘Chess’ involves the machine
trying out variations in its method of play—e.g. varying the numerical values
that are assigned to the various pieces. The machine adopts any variation that
leads to more satisfactory results. This procedure is an early example of a genetic
algorithm or GA (see ‘ArtiWcial Life’, above).
Chess Bulldozers
Turing likened the claim that no chess programme can outplay its programmer
to the claim that no animal can swallow an animal heavier than itself. Both
claims, he said, ‘are, so far as I know, untrue’. He pointed out that a programme
might outplay its programmer simply in virtue of ‘the speed of the machine,
which might make it feasible to carry the analysis . . . farther than a man could do
in the same time’ (p. 575).
Critics question the worth of research into computer chess. Noam Chomsky has
said famously that a computer beating a grandmaster at chess is as interesting as a
bulldozer winning a weight-lifting competition.14 Deep Blue did indeed bulldoze
12 Prinz, ‘Robot Chess’, 266.
13 D. G. Prinz and J. B. Smith, ‘Machines for the Solution of Logical Problems’, in Bowden (ed.), Faster
Than Thought, ch. 15; W. Mays and D. G. Prinz, ‘A Relay Machine for the Demonstration of Symbolic Logic’,
Nature, 165/4188 (4 Feb. 1950), 197–8; M. A. Bates, ‘On the Mechanical Solution of a Problem in Church’s
Lambda Calculus’, M.Sc. thesis, University of Manchester.
14 N. Chomsky, Language and Thought (London: Moyer Bell, 1993), 93.
566 | Jack Copeland
its way to victory—256 parallel processors enabled it to examine 200 million
possible moves per second and to look ahead as many as fourteen turns of play.
The huge improvement in computer chess since Turing’s day owes much more
to advances in hardware engineering than to advances in AI. Massive increases in
cpu speed and memory have meant that successive generations of machines have
been able to examine increasingly many possible moves. Turing’s expectation was
that chess-programming would contribute to the study of how human beings
think. In fact, little or nothing about human thought processes appears to have
been learned from the series of projects that culminated in Deep Blue.
The Turing Test Again
In the introductory paragraphs of his essay, Turing touches on some philosoph-
ical issues ‘unconnected with chess’. In a reference to the imitation game (Chap-
ter 11), he again states his belief that one could ‘make a machine which would
answer questions put to it, in such a way that it would not be possible to
distinguish its answers from those of a man’ (p. 569). On the question of what
justiWcation there is for this, he says that he knows of ‘no really convincing
argument to support this belief and certainly of none to disprove it.’
Consciousness
In ‘Chess’ Turing is careful to distinguish the question whether one could make a
machine that would play the imitation game successfully from the question
‘Could one make a machine which would have feelings like you and I do?’
(p. 569). Examples of ‘feelings’ are pleasure, grief, misery, and anger (Chapter 11,
p. 451). The assertion ‘No mechanism could feel’ forms the basis of the objection
to the Turing test that in Chapter 11 Turing dubs the ‘Argument from Con-
sciousness’. His remarks in the present chapter clarify the discussion that he gives
of this objection in Chapter 11.
The answer that Turing gives in ‘Chess’ to his question ‘Could one make a
machine which would have feelings’ is this (p. 569): ‘I shall never know, any more
than I shall ever be quite certain that you feel as I do.’ Thus his view appears to be
that the question ‘Can machines think?’ is independent of the question whether
machines can feel, and that an aYrmative answer may be given to the former in
the absence of our having any answer at all to the latter. If he is right—and
arguably he is15—then the assertion ‘No mechanism could feel’, even if true, goes
wide of the mark as an objection to the Turing test. As Turing sensibly says in
Chapter 11 (pp. 452–3):
15 See section 3.1 (‘Is Consciousness Necessary for Thought?’) of B. J. Copeland, ArtiWcial Intelligence: A
Philosophical Introduction (Oxford: Blackwell, 1993).
Chess | 567
I do not wish to give the impression that I think there is no mystery about conscious-
ness. . . . But I do not think these mysteries necessarily need to be solved before we can
answer the question with which we are concerned in this paper [i.e. ‘Can machines
think?’].
Consciousness and Computer Simulation
Turing asserts (p. 569) that in the case of the questions ‘Could one make a
machine to play chess, and to improve its play?’, and ‘Could one make a machine
which would answer questions put to it, in such a way that it would not be
possible to distinguish its answers from those of a man?’, the phrase ‘Could one
make a machine to . . .’ might equally well be replaced by ‘Could one programme
an electronic computer to . . .’. On pp. 569–70 Turing supports this claim
with an argument that also appears, in somewhat diVerent forms, in Chapters
11 and 13:
if some other machine had been constructed to do the job we could use an electronic
computer (of suYcient storage capacity), suitably programmed, to calculate what this
machine would do, and in particular what answer it would give.
(If no superior method presented itself, then the computer could be supplied with
a ‘look-up table’ setting out the behaviour of the other machine (see Chapter 11).)
In the case, however, of the question ‘Could one make a machine which would
have feelings like you and I do?’, Turing indicates (on p. 569) that replacing ‘Could
one make a machine to . . .’ by ‘Could one programme an electronic computer
to . . .’ might possibly result in a question that is not equivalent. Turing says in
Chapter 13 that ‘any machine of a certain very wide class’ can be replaced equiva-
lently by a digital computer, but notes that not every machine can be so replaced,
giving as examples of those that cannot bulldozers, steam-engines, and telescopes
(p. 483). Presumably, therefore, Turing considered it an open question whether a
machine capable of feeling belongs to the Wrst or second of these two classes.
The Church–Turing Thesis
‘Chess’ contains a formulation of the Church–Turing thesis that does not appear
elsewhere in Turing’s writings (p. 570):
If one can explain quite unambiguously in English, with the aid of mathematical symbols
if required, how a calculation is to be done, then it is always possible to programme any
digital computer to do that calculation, provided the storage capacity is adequate.
As elsewhere (e.g. in Chapter 10 (p. 414) and Chapter 17 (pp. 588–9)), Turing
emphasizes here that this thesis ‘is not the sort of thing that admits of clear cut
568 | Jack Copeland
proof ’, but nevertheless ‘amongst workers in the Weld it is regarded as being clear
as day’ (p. 570). (The phrase ‘how a calculation is to be done’ should be
understood as being elliptical for some such phrase as ‘how a calculation is to
be done by an obedient clerk working in accordance with a systematic method’.)
Turing applies this thesis to the question ‘Could one programme an electronic
computer to play a reasonably good game of chess?’ He points out that, in view
of the thesis, there is no need to set out an actual programme in order to show
that the answer to the question is aYrmative. It suYces to explain, ‘unambigu-
ously in English’, the rules by which the machine is to choose its move in each
position. This Turing proceeds to do.
Further reading
Davies, D. W., ‘A Theory of Chess and Noughts and Crosses’, Science News, 16 (1950),
40–64.
Newborn, M., Kasparov Versus Deep Blue: Computer Chess Comes of Age, (Springer: New
York, 1997).
Shannon, C. E. ‘Programming a Computer for Playing Chess’, Philosophical Magazine, 41
(1950), 256–75.
—— ‘A Chess-Playing Machine’, ScientiWc American, 182 (1950), 48–51.
Provenance
Turing’s essay ‘Chess’ was published in the 1953 collection Faster Than Thought,
where it formed a section of a chapter entitled ‘Digital Computers Applied To
Games’. (Edited by Vivian Bowden, Faster Than Thought is a fascinating survey of
the then state of the art in digital computing.) The chapter was co-authored by
Audrey Bates, Bowden, and Christopher Strachey. Bowden’s editorial remarks do
not make it clear which parts of ‘Digital Computers Applied To Games’ were
written by which author.16 Fortunately, Turing’s typescript of his contribution
has survived.17 It is this which is printed here.18
16 The whole article is mistakenly attributed to Turing alone in Mechanical Intelligence: Collected Works of
A. M. Turing, ed. D. C. Ince (Amsterdam: North-Holland, 1992).
17 Turing’s typescript is itself entitled ‘Digital Computers applied to Games’. In Faster Than Thought
Turing’s essay was published under the narrower and more accurate title ‘Chess’. The typescript is among the
Turing Papers in the Modern Archive Centre, King’s College, Cambridge (catalogue reference B 7).
18 Bowden’s edition of the essay diVers from Turing’s typescript in numerous respects, many of them
minor. The present edition follows Turing’s typescript. Some obvious typing errors have been corrected and
in one case a conjectured missing word has been added in square brackets. Some signiWcant diVerences
between the typescript and Bowden’s edition are mentioned in footnotes. (Bowden’s edition is reprinted in
Turing, Mechanical Intelligence, ed. Ince, 288–95.)
Chess
When one is asked ‘Could one make a machine to play chess?’, there are several
possible meanings which might be given to the words. Here are a few:
i) Could one make a machine which would obey the rules of chess, i.e. one
which would play random legal moves, or which could tell one whether a
given move is a legal one?
ii) Could one make a machine which would solve chess problems, e.g. tell
one whether, in a given position, white has a forced mate in three?
iii) Could one make a machine which would play a reasonably good game of
chess, i.e. which, confronted with an ordinary (that is, not particularly
unusual) chess position, would after two or three minutes of calculation,
indicate a passably good legal move?
iv) Could one make a machine to play chess, and to improve its play, game by
game, proWting from its experience?
To these we may add two further questions, unconnected with chess, which are
likely to be on the tip of the reader’s tongue.
v) Could one make a machine which would answer questions put to it, in
such a way that it would not be possible to distinguish its answers from
those of a man?
vi) Could one make a machine which would have feelings like you and I do?
The problem to be considered here is iii), but to put this problem into
perspective with the others I shall give the very briefest of answers to each of
them.
To i) and ii) I should say ‘This certainly can be done. If it has not been done
already it is merely because there is something better to do.’
Question iii) we are to consider in greater detail, but the short answer is ‘Yes,
but the better the standard of play required, the more complex will the machine
be, and the more ingenious perhaps the designer.’
To iv) and v) I should answer ‘I believe so. I know of no really convincing
argument to support this belief and certainly of none to disprove it.’
To vi) I should say ‘I shall never know, any more than I shall ever be quite
certain that you feel as I do.’
In each of these problems except possibly the last, the phrase ‘Could one make
a machine to . . .’ might equally well be replaced by ‘Could one programme an
electronic computer to . . .’. Clearly the electronic computer so programmed
would itself constitute a machine. And on the other hand if some other machine
Printed with the permission of Financial Times Management and the Estate of Alan Turing.
570 | Alan Turing
had been constructed to do the job we could use an electronic computer (of
suYcient storage capacity), suitably programmed, to calculate what this machine
would do, and in particular what answer it would give.
After these preliminaries let us give our minds to the problem of making a
machine, or of programming a computer, to play a tolerable game of chess.
In this short discussion it is of course out of the question to provide actual
programmes, but this does not really matter on account of the following
principle:
If one can explain quite unambiguously in English, with the aid of mathemat-
ical symbols if required, how a calculation is to be done, then it is always
possible to programme any digital computer to do that calculation, provided
the storage capacity is adequate.
This is not the sort of thing that admits of clear cut proof, but amongst
workers in the Weld it is regarded as being clear as day. Accepting this principle,
our problem is reduced to explaining ‘unambiguously in English’ the rules by
which the machine is to choose its move in each position. For deWniteness we
will suppose the machine is playing white.
If the machine could calculate at an inWnite speed, and also had unlimited
storage capacity, a comparatively simple rule would suYce, and would give a
result that in a sense could not be improved on. This rule could be stated:
Consider every possible continuation of the game from the given position.
There is only a Wnite number of them (at any rate if the Wfty-move rule makes
a draw obligatory, not merely permissive). Work back from the end of these
continuations, marking a position with white to play as ‘win’ if there is a move
which turns it into a position previously marked as ‘win’. If this does not
occur, but there is a move which leads to a position marked ‘draw’, then mark
the position ‘draw’. Failing this, mark it ‘lose’. Mark a position with black to
play by a similar rule with ‘win’ and ‘lose’ interchanged. If after this process
has been completed it is found that there are moves which lead to a position
marked ‘win’, one of these should be chosen. If there is none marked ‘win’
choose one marked ‘draw’ if such exists. If all moves lead to a position marked
‘lose’, any move may be chosen.
Such a rule is practically applicable in the game of noughts and crosses, but in
chess is of merely academic interest.
Even when the rule can be applied it is not very appropriate for use against a
weak opponent, who may make mistakes which ought to be exploited.
In spite of the impracticability of this rule it bears some resemblance to what
one really does when playing chess. One does not follow all the continuations of
play, but one follows some of them. One does not follow them until the end of
the game, but one follows them a move or two, perhaps more. Eventually a
Chess | 571
position seems, rightly or wrongly, too bad to be worth further consideration, or
(less frequently) too good to hesitate longer over. The further a position is from
the one on the board the less likely is it to occur, and therefore the shorter is the
time which can be assigned for its consideration. Following this idea we might
have a rule something like this:
Consider all continuations of the game consisting of a move by white, a reply by
black, and another move and reply. The value of the position at the end of each
of these sequences of moves is estimated according to some suitable rule. The
values at earlier positions are then calculated by working backwards move by
move as in the theoretical rule given before. The move to be chosen is that which
leads to the position with the greatest value.
It is possible to arrange that no two positions have the same value. The rule is
then unambiguous. A very simple form of values, but one not having this
property, is an ‘evaluation of material’, e.g. on the basis
P¼ 1
Kt ¼ 3
B ¼ 3 1
2
R¼5
Q ¼ 10
Checkmate ± 10001
If B is black’s total and W is white’s, then W =B is quite a good measure of
value. This is better than W ÀB as the latter does not encourage exchanges when
one has the advantage. Some small extra arbitrary function of position may be
added to ensure deWniteness in the result.
The weakness of this rule is that it follows all combinations equally far. It
would be much better if the more proWtable moves were considered in greater
detail than the less. It would also be desirable to take into account more than
mere ‘value of material’.
After this introduction I shall describe a particular set of rules, which could
without diYculty be made into a machine programme. It is understood that the
machine is white and white is next to play. The current position is called the
‘position of the board’, and the positions arising from it by later moves ‘positions
in the analysis’.2
1 Editor’s note. In the Bowden edition. Turing’s ‘ + ’ has been replaced with ‘¼’.
2 Editor’s note. In the Bowden edition, Turing’s ‘position of the board’ has been replaced by ‘position on
the board’.
572 | Alan Turing
‘Considerable’ Moves, i.e. Moves to be considered in the analysis
by the machine
Every possibility for white’s next move and for black’s reply is ‘considerable’. If a
capture is considerable then any recapture is considerable. The capture of an
undefended piece or the capture of a piece of higher value by one of lower value
is always considerable. A move giving checkmate is considerable.
Dead position
A position in the analysis is dead if there are no considerable moves in that
position, i.e. if it is more than two moves ahead of the present position, and no
capture or recapture or mate can be made in the next move.
Value of position
The value of a dead position is obtained by adding up the piece values as above, and
forming the ratio W =B of white’s total to black’s. In other positions with white to
play the value is the greatest value of: (a) the positions obtained by considerable
moves, or (b) the position itself evaluated as if a dead position, the latter alternative
to be omitted if all moves are considerable. The same process is to be undertaken
for one of black’s moves, but the machine will then choose the least value.
Position-play value
Each white piece has a certain position-play contribution and so has the black
king. These must all be added up to give the position-play value.
For a Q, R, B or Kt, count
i) The square root of the number of moves the piece can make from the
position, counting a capture as two moves, and not forgetting that the
king must not be left in check.
ii) (If not a Q) 1Á 0 if it is defended, and an additional 0Á5 if twice defended.
For a K, count
iii) For moves other than castling as i) above.
iv) It is then necessary to make some allowance for the vulnerability of the
K. This can be done by assuming it to be replaced by a friendly Q on the
same square, estimating as in i), but subtracting instead of adding.
v) Count 1Á0 for the possibility of castling later not being lost by moves of K
or rooks, a further 1Á0 if castling could take place on the next move, and
yet another 1Á0 for the actual performance of castling.
For a P, count
vi) 0Á2 for each rank advanced.
vii) 0Á3 for being defended by at least one piece (not P).
For the black K, count
Chess | 573
viii) 1Á0 for the threat of checkmate.
ix) 0Á5 for check.
We can now state the rule for play as follows.
The move chosen must have the greatest possible value, and, consistent with
this, the greatest possible position-play value. If this condition admits of
several solutions a choice may be made at random, or according to an
arbitrary additional condition.
Note that no ‘analysis’ is involved in position-play evaluation. This is in order
to reduce the amount of work done on deciding the move.
The game below was played between this machine and a weak player who did
not know the system. To simplify the calculations the square roots were rounded
oV to one decimal place (i.e. the table below was used). No ‘random choices’
actually arose in this game. The increase of position-play value is given after
white’s move if relevant. An asterisk indicates that every other move had a lower
position-play value. 0—0 indicates castling.3
Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Square Root 0 1 1Á4 1Á7 2Á0 2Á2 2Á4 2Á6 2Á8 3Á0 3Á2 3Á3 3Á5 3Á6
White (Machine) Black
1. P—K4 4Á2* P—K4
2. Kt—QB3 3Á1* Kt—KB3
3. P—Q4 3Á1* B—QKt5
4. Kt—KB3(1) 2Á0 P—Q3
5. B—Q2 3Á6* Kt—QB3
6. P—Q5 0Á2 Kt—Q5
7. P—KR4(2) 1Á1* B—Kt5
8. P—QR4(2) 1Á0* Kt  Kt ch.
9. P Â Kt B—KR4
2Á4* P—QB3
10. B—Kt5 ch. 0—0
11. P Â P À1Á5 R—Kt1
0Á6 Q—R4
12. P Â P 1Á2* Kt—Q2
Kt—B4(4)
13. B—R6 0Á4 B—Kt3
14. Q—K2 3Á0* Kt  KtP
15. KR—Kt1(3) Kt—B4
16. R—Kt5(5) KR—QB1
17. B—Kt5
18. 0—0 Continued
19. B—B6
3 Editor’s note. In the Bowden edition a diVerent game is to shown, diverging from the present game at
the 21st move.
574 | Alan Turing
White (Machine) Black
20. B—Q5 À0Á7 B Â Kt
21. P Â B(2) QÂP
Q—R6 ch.
22. B—K3(6) Kt—R5
R—Kt7
23. K—Q2 Q—B6 ch.
24. B Â RP(7) R—R7
BÂB
25. P—B4 KÂR
R—R8 mate
26. K—B1
27. B Â BP ch.
28. R Â KtP ch.(5)
29. B—K3(8)
Notes:
1. If B—Q2 3Á6* then P Â P is foreseen.
2. Most inappropriate moves from a positional point of
view.
3. If 0—0 then B Â Kt, B Â B, Q Â P.
4. The fork is unforeseen.
5. Heads in the sand!
6. Only this or B—K1 can prevent Q—R8 mate.
7. Fiddling while Rome burns.
8. Mate is foreseen, but ‘business as usual’.
Numerous criticisms of the machine’s play may be made. It is quite defenceless
against ‘forks’, although it may be able to see certain other kinds of combination.
It is of course not diYcult to devise improvements of the programme [so] that
these simple forks are foreseen. The reader may be able to think of some such
improvements for himself. Since no claim is made that the above rule is particu-
larly good, I have been content to leave this Xaw without remedy; clearly a line
has to be drawn between the Xaws which one will attempt to eliminate and those
which must be accepted as a risk. Another criticism is that the scheme proposed,
although reasonable in the middle game, is futile in the end game. The change-
over from the middle game to the end-game is usually suYciently clean cut for it
to be possible to have an entirely diVerent system for the end-game. This should
of course include quite deWnite programmes for the standard situations, such as
mate with rook and king, or king and pawn against king. There is no intention to
discuss the end-game further here.
If I were to sum up the weakness of the above system in a few words I would
describe it as a caricature of my own play. It was in fact based on an introspective
analysis of my thought processes when playing, with considerable simpliWca-
tions. It makes oversights which are very similar to those which I make myself,
and which may in both cases be ascribed to the considerable moves being rather
inappropriately chosen. This fact might be regarded as supporting the rather glib
view which is often expressed, to the eVect that ‘one cannot programme a
machine to play a better game than one plays oneself ’. This statement should I
Chess | 575
think be compared with another of rather similar form. ‘No animal can swallow
an animal heavier than himself.’ Both statements are, so far as I know, untrue.
They are also both of a kind that one is rather easily bluVed into accepting, partly
because one thinks that there ought to be some rather slick way of demonstrating
them, and one does not like to admit that one does not see what this argument is.
They are also both supported by normal experience, and need rather exceptional
cases to falsify them. The statement about chess programming may be falsiWed
quite simply by the speed of the machine, which might make it feasible to carry
the analysis a move farther than a man could do in the same time. This eVect is
rather less than might be supposed. Although electronic computers are very fast
where conventional computing is concerned, their advantage is much reduced
where enumeration of cases, etc., is involved on a large scale. Take for instance
the problem of counting the possible moves from a given position in chess. If the
number is 30 a man might do it in 45 seconds and the machine in 1 second. The
machine has still an advantage, but it is much less overwhelming than it would
be for instance where calculating cosines.
In connection with the question, numbered iv) above, as to the ability of a
chess-machine to proWt from experience, one can see that it would be quite
possible to programme the machine to try out variations in its method of play
(e.g. variations in piece value) and adopt the one giving the most satisfactory
results. This could certainly be described as ‘learning’, though it is not quite
representative of learning as we know it. It might also be possible to programme
the machine to search for new types of combination in chess. If this product
produced results which were quite new, and also interesting to the programmer,
who should have the credit? Compare this with the situation where the Defence
Minister has given orders for research to be done to Wnd a counter to the bow
and arrow. Should the inventor of the shield have the credit, or should the
Defence Minister?
CHAPTER 17
Solvable and Unsolvable Problems (1954)
Alan Turing
Introduction
Jack Copeland
Unsolvable Problems
In Chapter 1 Turing proves the existence of mathematical problems that cannot
be solved by the universal Turing machine. There he also advances the thesis,
now called the Church–Turing thesis, that any systematic method for solving
mathematical problems can be carried out by the universal Turing machine.
Combining these two propositions yields the result that there are mathematical
problems which cannot be solved by any systematic method—cannot, in other
words, be solved by any algorithm.
Substitution Puzzles
In ‘Solvable and Unsolvable Problems’ Turing sets out to explain this result to a
lay audience. The article Wrst appeared in Science News, a popular science journal
of the time. Starting from concrete examples of problems that do admit of
algorithmic solution, Turing works his way towards an example of a problem
that is not solvable by any systematic method. Loosely put, this is the problem of
sorting puzzles into those that will ‘come out’ and those that will not. Turing
gives an elegant argument showing that a sharpened form of this problem is not
solvable by means of a systematic method (pp. 591–2).
The sharpened form of the problem involves what Turing calls ‘the substitu-
tion type of puzzle’. An typical example of a substitution puzzle is this. Starting
with the word BOB, is it possible to produce BOOOB by replacing selected
occurrences of the pair OB by BOOB and selected occurences of the triple BOB
by O? The answer is yes:
BOB ! BBOOB ! BBOBOOB ! BOOOB:
Solvable and Unsolvable Problems | 577
Turing suggests that any puzzle can be re-expressed as a substitution puzzle.
Some row of letters can always be used to represent the ‘starting position’
envisaged in a particular puzzle, e.g. in the case of a chess problem, the pieces
on the board and their positions. Desired outcomes, for example board positions
that count as wins, can be described by further rows of letters, and the rules of
the puzzle, whatever they are, are to be represented in terms of permissible
substitutions of groups of letters for other groups of letters.
As Turing points out, it is not only ‘toy’ puzzles that can be re-expressed as
substitution puzzles, but also mathematical problems, for instance the problem
of Wnding a proof of a given mathematical theorem within an axiom system
(which Turing describes as ‘a very good example of a puzzle’). The axioms—
which are simply strings of mathematical symbols—form the starting position.
The theorem—another string of symbols—is the winning position. The rules of
the puzzle are substitutions that enable strings of mathematical symbols to be
transformed into other strings, much as in the case of the transition from BOB to
BBOOB in the earlier example.
Turing calls the substitution formulation of any puzzle its ‘normal form’ and
states the following normal form principle (p. 588):
Given any puzzle, we can Wnd a corresponding substitution puzzle which is equivalent to
it in the sense that given a solution of the one we can easily use it to Wnd a solution of the
other.
Normal Forms and the Church–Turing Thesis
The normal form principle for puzzles closely parallels the Church–Turing thesis,
which says that given any systematic method, we can Wnd a corresponding
Turing machine that is equivalent to it.
Neither the normal form principle for puzzles nor the Church–Turing thesis is
susceptible to deWnite proof (see ‘Computable Numbers: A Guide’). While few
doubt that the Church–Turing thesis is in fact true, the very nature of the thesis
has always been a matter for debate. Church, for example, described the thesis as
a deWnition.1 Post, on the other hand, described it as a ‘working hypothesis’ that
is in need of ‘continual veriWcation’, and he criticized Church for masking this
hypothesis as a deWnition.2 Turing’s remarks in ‘Solvable and Unsolvable
Problems’ about the status of the normal form principle for puzzles are of
outstanding interest for the light that they may cast on his view concerning
1 A. Church, ‘An Unsolvable Problem of Elementary Number Theory’, American Journal of Mathematics,
58 (1936), 345–63 (356).
2 E. L. Post, ‘Finite Combinatory Processes - Formulation 1’, Journal of Symbolic Logic, 1 (1936), 103–5
(105).
578 | Jack Copeland
the status of the Church–Turing thesis. In this connection, see also the
material from Turing’s draft typescript quoted in n. 9 on p. 590.
Turing says of the normal form principle (pp. 588–9):
The statement is . . . one which one does not attempt to prove. Propaganda is more
appropriate to it than proof, for its status is something between a theorem and a
deWnition. In so far as we know a priori what is a puzzle and what is not, the statement
is a theorem. In so far as we do not know what puzzles are, the statement is a deWnition
which tells us something about what they are. One can of course deWne a puzzle by some
phrase beginning, for instance, ‘A set of deWnite rules . . .’, but this just throws us back on
the deWnition of ‘deWnite rules’. Equally one can reduce it to the deWnition of ‘computable
function’ or ‘systematic procedure’. A deWnition of any one of these would deWne all the
rest.
Turing would perhaps have said much the same concerning not only the
Church–Turing thesis but also the thesis introduced in Chapter 13:
A digital computer will replace any rival design of calculating machine.
In so far as we do not know what calculating machines are, the statement is a
deWnition which tells us something about what they are.
Proof of Unsolvability
Having introduced the normal form principle for puzzles, Turing turns to his
central project of establishing that ‘there cannot be any systematic procedure for
determining whether a puzzle be solvable or not’ (p. 590). In particular, there
cannot be a systematic procedure for determining whether substitution puzzles
are or are not solvable. Turing argues by reductio ad absurdum. He shows that the
supposition that there is a systematic procedure for determining whether substi-
tution puzzles are or are not solvable leads to an outright contradiction, and on
that basis concludes that there can be no such procedure. The argument turns on
the impossibility of applying a certain procedure to itself.
Any systematic procedure is in eVect a puzzle, since in following the procedure
one applies rules to some ‘starting position’ until one or another result is
achieved. So if there were a systematic procedure for determining whether each
puzzle is or is not solvable, then by the normal form principle, there is a
substitution puzzle—call it K—that is equivalent to this procedure. When
applied to any substitution puzzle, K—if it exists—must ‘come out’ either with
the result s o l va b l e or with the result n ot s o l v a bl e. Since K is applicable
to any substitution puzzle, K can be applied to itself in order to determine
whether it itself is or is not solvable. Turing shows (p. 592) that this supposed
ability of K to pronounce on its own solvability leads to outright contradiction,
and so concludes that K cannot exist.
Solvable and Unsolvable Problems | 579
The Meaning of ‘Unsolvable’
Turing points out that the result he has established, namely that there is no
systematic method for deciding whether or not substitution puzzles come out, is
often expressed by saying that there is no decision procedure for puzzles of this
type, and that the decision problem for this type of puzzle is unsolvable. He
continues (p. 592): ‘so one comes to speak (as in the title of this article) about
‘‘unsolvable problems’’ meaning in eVect puzzles for which there is no decision
procedure. This is the technical meaning which the words are now given by
mathematical logicians.’
As Turing says, this terminology is potentially confusing. It is natural to use
the words ‘unsolvable problem’ to mean a problem for which no solution can
possibly be found. It would be a confusion to think that Turing has shown that
the problem of deciding whether or not substitution puzzles come out is an
unsolvable problem in this natural sense. Indeed, with suYcient time, inventive-
ness, and patience, mathematicians may always be able to establish whether or
not any given substitution puzzle comes out. If that is so, then the problem of
deciding whether or not substitution puzzles come out is solvable, in the natural
sense of the word.
What Turing has shown is that there is no systematic method for deciding
whether or not substitution puzzles come out, i.e. there is no general procedure,
applicable by rote, that one can employ in order to decide whether or not each
substitution puzzle comes out. The ‘decision problem’ for substitution puzzles is
the problem of Wnding such a rote procedure (a ‘decision procedure’); in showing
that there is no such procedure, Turing has shown that the decision problem for
substitution puzzles is unsolvable in the natural sense.
Turing therefore recommends that, in order to ‘minimize confusion’, one
should ‘always speak of ‘‘unsolvable decision problems’’, rather than just ‘‘un-
solvable problems’’ ’ (p. 592).
Significance of Turing’s Result
Turing ends the chapter with a comment on the signiWcance of what he has
shown. His result concerning the decision problem for substitution puzzles ‘may
be regarded as going some way towards a demonstration, within mathematics
itself, of the inadequacy of ‘‘reason’’ unsupported by common sense’. For he has,
he says, set ‘certain bounds to what we can hope to achieve purely by reasoning’.
The phrase ‘purely by reasoning’ here presumably means ‘purely by algorith-
mic methods’. Some mathematical problems require for their solution not only
‘reason’, in this sense, but also what Turing refers to in Chapter 3 as ‘intuition’
(see also Chapter 4). There he says (pp. 192–3):
580 | Jack Copeland
The activity of the intuition consists in making spontaneous judgements which are not the
result of conscious trains of reasoning. . . . Often it is possible to Wnd some other way of
verifying the correctness of an intuitive judgement. We may, for instance, judge that all
positive integers are uniquely factorizable into primes; a detailed mathematical argument
leads to the same result. This argument will also involve intuitive judgements, but they
will be less open to criticism than the original judgement about factorization. . . . The
necessity for using the intuition is . . . greatly reduced by setting down formal rules for
carrying out inferences which are always intuitively valid. . . . In pre-Go¨del times it was
thought by some that it would probably be possible to carry this programme to such a
point that all the intuitive judgements of mathematics could be replaced by a Wnite
number of these rules. The necessity for intuition would then be entirely eliminated.
The argument of ‘Solvable and Unsolvable Problems’ illustrates why it is that
the need for intuition cannot always be eliminated in favour of formal rules.
Go¨del’s Theorem
Turing notes that the unsolvability of the decision problem for substitution
puzzles aVords an elegant proof of the following rather general statement
(p. 593):
no systematic method of proving mathematical theorems is suYciently com-
plete to settle every mathematical question, yes or no.
The proof Turing gives is as follows. Each statement of the form ‘such-and-such
substitution puzzle comes out’ can be expressed in the form of a mathematical
statement. So if there were a systematic method of settling every question that
can be posed in mathematical form, this method would serve as a decision
procedure for substitution puzzles. Given that there is no such decision proced-
ure, it follows that no systematic method is able to settle every mathematical
question.
Turing remarks that the above statement follows ‘by a famous theorem of
Go¨del’ and describes himself as providing ‘an independent proof ’ of the state-
ment (p. 593). Turing might also have pointed out that his own ‘On Computable
Numbers’ yields a proof of this statement.
Go¨del’s famous incompleteness theorem of 1931 is, however, importantly less
general than the above statement, since it concerns only one particular systematic
method of proving mathematical theorems, the system set out by Whitehead and
Russell in Principia Mathematica3 (as explained in ‘Computable Numbers: A
Guide’). Go¨del did later generalize his result of 1931 to all formal systems
(containing a certain amount of arithmetic), but emphasized the importance
that Turing’s work played in this generalization. Go¨del said in 1964:
3 A. N. Whitehead and B. Russell, Principia Mathematica, vols. i–iii (Cambridge: Cambridge University
Press, 1910–13).
Solvable and Unsolvable Problems | 581
[D]ue to A. M. Turing’s work, a precise and unquestionably adequate deWnition of the
general concept of formal system can now be given . . . Turing’s work gives an analysis of
the concept of ‘mechanical procedure’ (alias ‘algorithm’ or ‘computation procedure’ or
‘Wnite combinatorial procedure’). . . . A formal system can simply be deWned to be any
mechanical procedure for producing formulas, called provable formulas.4
In his references to Go¨del’s work, Turing hides his own light under a bushel.
Further reading
Boone, W. W., review of Turing’s ‘The Word Problem in Semi-Groups with Cancellation’,
Journal of Symbolic Logic, 17 (1952), 74–6.
Turing, A. M., ‘The Word Problem in Semi-Groups with Cancellation’, Annals of Math-
ematics, 52 (1950), 491–505. Reprinted in Pure Mathematics: Collected Works of A. M.
Turing, ed. J. L. Britton (Amsterdam: North-Holland, 1992).
Provenance
What follows is the text of the original printing of ‘Solvable and Unsolvable
Problems’ in Science News.5 Unfortunately Turing’s own typescript appears to
have been lost. However, a sizeable fragment of a draft typescript, with additions
in Turing’s handwriting, has been preserved.6 (Turing recycled the draft pages,
covering the reverse sides with handwritten notes concerning morphogen-
esis.) The fragment corresponds to pp. 584–9. For the most part the pub-
lished version follows the draft pages closely (except for punctuation and
occasional changes of word and word-order). SigniWcant diVerences between
the draft and the published version are mentioned in footnotes.
4 K. Go¨del, ‘Postscriptum’, in M. Davis (ed.), The Undecidable (New York: Raven, 1965), 71–3 (71–2); the
Postscriptum, dated 1964, is to Go¨del’s 1934 paper ‘On Undecidable Propositions of Formal Mathematical
Systems’ (ibid. 41–71).
5 Footnotes have been renumbered consecutively. Footnotes not marked ‘Editor’s note’ appeared in
Science News. A page reference to Science News been replaced by the number (in square brackets) of the
corresponding page of this volume.
6 The fragment is among the Turing Papers in the Modern Archive Centre, King’s College, Cambridge; at
the time of writing it is uncatalogued.
Solvable and Unsolvable Problems
If one is given a puzzle to solve one will usually, if it proves to be diYcult, ask the
owner whether it can be done. Such a question should have a quite deWnite
answer, yes or no, at any rate provided the rules describing what you are allowed
to do are perfectly clear. Of course the owner of the puzzle may not know the
answer. One might equally ask, ‘How can one tell whether a puzzle is solvable?’,
but this cannot be answered so straightforwardly. The fact of the matter is that
there is no systematic method of testing puzzles to see whether they are solvable
or not. If by this one meant merely that nobody had ever yet found a test which
could be applied to any puzzle, there would be nothing at all remarkable in the
statement. It would have been a great achievement to have invented such a test,
so we can hardly be surprised that it has never been done. But it is not merely
that the test has never been found. It has been proved that no such test ever can
be found.
Let us get away from generalities a little and consider a particular puzzle. One
which has been on sale during the last few years and has probably been seen by
most of the readers of this article illustrates a number of the points involved
quite well. The puzzle consists of a large square within which are some smaller
movable squares numbered 1 to 15, and one empty space, into which any of the
neighbouring squares can be slid leaving a new empty space behind it. One may
be asked to transform a given arrangement of the squares into another by a
succession of such movements of a square into an empty space. For this puzzle
there is a fairly simple and quite practicable rule by which one can tell whether
the transformation required is possible or not. One Wrst imagines the transform-
ation carried out according to a diVerent set of rules. As well as sliding the
squares into the empty space one is allowed to make moves each consisting of
two interchanges, each of one pair of squares. One would, for instance, be
allowed as one move to interchange the squares numbered 4 and 7, and also
the squares numbered 3 and 5. One is permitted to use the same number in both
pairs. Thus one may replace 1 by 2, 2 by 3, and 3 by 1 as a move because this is
the same as interchanging Wrst (1, 2) and then (1, 3). The original puzzle is
solvable by sliding if it is solvable according to the new rules. It is not solvable by
sliding if the required position can be reached by the new rules, together with a
‘cheat’ consisting of one single interchange of a pair of squares.1 Suppose, for
instance, that one is asked to get back to the standard position—
This article Wrst appeared in Science News, 31 (1954), 7–23, published by the Penguin Press, and is printed
by permission of the Estate of Alan Turing.
1 It would take us too far from our main purpose to give the proof of this rule: the reader should have
little diYculty in proving it by making use of the fact that an odd number of interchanges can never bring a
set of objects back to the position it started from.
Solvable and Unsolvable Problems | 583
1 234 10 1 4 5
5 678 9 268
from the position
9 10 11 12 11 3 15
13 14 15 13 14 7 12
One may, according to the modiWed rules, Wrst get the empty square into the
correct position by moving the squares 15 and 12, and then get the squares 1, 2,
3, . . . successively into their correct positions by the interchanges (1, 10), (2, 10),
(3, 4), (4, 5), (5, 9), (6, 10), (7, 10), (9, 11), (10, 11), (11, 15). The squares 8, 12,
13, 14, 15 are found to be already in their correct positions when their turns are
reached. Since the number of interchanges required is even, this transformation
is possible by sliding.2 If one were required after this to interchange say square 14
and 15 it could not be done.
This explanation of the theory of the puzzle can be regarded as entirely satisfac-
tory. It gives one a simple rule for determining for any two positions whether one
can get from one to the other or not. That the rule is so satisfactory depends very
largely on the fact that it does not take very long to apply. No mathematical
method can be useful for any problem if it involves much calculation. It is
nevertheless sometimes interesting to consider whether something is possible at
all or not, without worrying whether, in case it is possible, the amount of labour or
calculation is economically prohibitive. These investigations that are not con-
cerned with the amount of work involved are in some ways easier to carry out,
and they certainly have a greater aesthetic appeal. The results are not altogether
without value, for if one has proved that there is no method of doing something it
follows a fortiori that there is no practicable method. On the other hand, if one
method has been proved to exist by which the decision can be made, it gives some
encouragement to anyone who wishes to Wnd a workable method.
From this point of view, in which one is only interested in the question, ‘Is
there a systematic way of deciding whether puzzles of this kind are solvable?’, the
rules which have been described for the sliding-squares puzzle are much more
special and detailed than is really necessary. It would be quite enough to say:
‘Certainly one can Wnd out whether one position can be reached from another by
2 It can in fact be done by sliding successively the squares numbered 7, 14, 13, 11, 9, 10, 1, 2, 3, 7, 15, 8, 5,
4, 6, 3, 10, 1, 2, 6, 3, 10, 6, 2, 1, 6, 7, 15, 8, 5, 10, 8, 5, 10, 8, 7, 6, 9, 15, 5, 10, 8, 7, 6, 5, 15, 9, 5, 6, 7, 8, 12, 14,
13, 15, 10, 13, 15, 11, 9, 10, 11, 15, 13, 12, 14, 13, 15, 9, 10, 11, 12, 14, 13, 15, 14, 13, 15, 14, 13, 12, 11, 10, 9,
13, 14, 15, 12, 11, 10, 9, 13, 14, 15.
584 | Alan Turing
a systematic procedure. There are only a Wnite number of positions in which the
numbered squares can be arranged (viz. 20922789888000) and only a Wnite
number (2, 3, or 4) of moves in each position. By making a list of all the
positions and working through all the moves, one can divide the positions into
classes, such that sliding the squares allows one to get to any position which is in
the same class as the one started from. By looking up which classes the two
positions belong to one can tell whether one can get from one to the other
or not.’ This is all, of course, perfectly true, but one would hardly Wnd such
remarks helpful if they were made in reply to a request for an explanation of
how the puzzle should be done. In fact they are so obvious that under such
circumstances one might Wnd them somehow rather insulting. But the fact of the
matter is, that if one is interested in the question as put, ‘Can one tell by a
systematic method in which cases the puzzle is solvable?’, this answer is entirely
appropriate, because one wants to know if there is a systematic method, rather
than to know of a good one.
The same kind of argument will apply for any puzzle where one is allowed to
move certain ‘pieces’ around in a speciWed manner, provided that the total
number of essentially diVerent positions which the pieces can take up is Wnite.
A slight variation on the argument is necessary in general to allow for the fact
that in many puzzles some moves are allowed which one is not permitted to
reverse. But one can still make a list of the positions, and list against these Wrst
the positions which can be reached from them in one move. One then adds the
positions which are reached by two moves and so on until an increase in the
number of moves does not give rise to any further entries. For instance, we can
say at once that there is a method of deciding whether a patience can be got out
with a given order of the cards in the pack: it is to be understood that there is
only a Wnite number of places in which a card is ever to be placed on the table. It
may be argued that one is permitted to put the cards down in a manner which is
not perfectly regular, but one can still say that there is only a Wnite number of
‘essentially diVerent’ positions. A more interesting example is provided by those
puzzles made (apparently at least) of two or more pieces of very thick twisted
wire which one is required to separate. It is understood that one is not allowed to
bend the wires at all, and when one makes the right movement there is always
plenty of room to get the pieces apart without them ever touching, if one wishes
to do so. One may describe the positions of the pieces by saying where some
three deWnite points of each piece are. Because of the spare space it is not
necessary to give these positions quite exactly. It would be enough to give
them to, say, a tenth of a millimetre. One does not need to take any notice of
movements of the puzzle as a whole: in fact one could suppose one of the pieces
quite Wxed. The second piece can be supposed to be not very far away, for, if it is,
the puzzle is already solved. These considerations enable us to reduce the number
of ‘essentially diVerent’ positions to a Wnite number, probably a few hundred
Solvable and Unsolvable Problems | 585
millions, and the usual argument will then apply. There are some further compli-
cations, which we will not consider in detail, if we do not know how much
clearance to allow for. It is necessary to repeat the process again and again
allowing successively smaller and smaller clearances. Eventually one will Wnd
that either it can be solved, allowing a small clearance margin, or else it cannot be
solved even allowing a small margin of ‘cheating’ (i.e. of ‘forcing’, or having the
pieces slightly overlapping in space). It will, of course, be understood that this
process of trying out the possible positions is not to be done with the physical
puzzle itself, but on paper, with mathematical descriptions of the positions, and
mathematical criteria for deciding whether in a given position the pieces overlap,
etc.
These puzzles where one is asked to separate rigid bodies are in a way like the
‘puzzle’ of trying to undo a tangle, or more generally of trying to turn one knot
into another without cutting the string. The diVerence is that one is allowed to
bend the string, but not the wire forming the rigid bodies. In either case, if
one wants to treat the problem seriously and systematically one has to replace
the physical puzzle by a mathematical equivalent. The knot puzzle lends itself
quite conveniently to this. A knot is just a closed curve in three dimensions
nowhere crossing itself; but, for the purpose we are interested in, any knot can
be given accurately enough as a series of segments in the directions of the three
coordinate axes. Thus, for instance, the trefoil knot (Figure 1a) may be regarded
as consisting of a number of segments joining the points given, in the usual (x, y,
z) system of coordinates, as (1, 1, 1), (4, 1, 1,), (4, 2, 1), (4, 2, À1), (2, 2, À1), (2,
2, 2), (2, 0, 2), (3, 0, 2), (3, 0, 0), (3, 3, 0), (1, 3, 0), (1, 3, 1), and returning again
with a twelfth segment to the starting point (1, 1, 1).3 This representation of the
knot is shown in perspective in Figure 1b. There is no special virtue in the
representation which has been chosen. If it is desired to follow the original curve
more closely a greater number of segments must be used. Now let a and d
represent unit steps in the positive and negative X-directions respectively, b and e
in the Y-directions, and c and f in the Z-directions: then this knot may be
described as aaabffddccceeaffbbbddcee.4 One can then, if one wishes, deal
entirely with such sequences of letters. In order that such a sequence
should represent a knot it is necessary and suYcient that the numbers of
a’s and d’s should be equal, and likewise the number of b’s equal to the
number of e’s and the number of c’s equal to the number of f ’s, and it must
not be possible to obtain another sequence of letters with these properties
by omitting a number of consecutive letters at the beginning
3 Editor’s note. In place of this sentence Turing’s draft has: ‘Thus for instance the trefoil knot may be
regarded as consisting of a number of segments joining the points (0, 0, 0), (0, 2, 0), (1, 2, 0), (1, 2, 2),
(1, À1, 2), (1, À1, 1), (À1, À1, 1), (À1, 1, 1), (2, 1, 1), (2, 0, 1), (2, 0, 3), (0, 0, 3), (0, 0, 0).’
4 Editor’s note. Turing’s draft has ‘bbacceeefddbbaaaeccddfff ’.
586 | Alan Turing
(a)
(b)
Y
(1,3,1)
(1,3,0) dd (3,3,0)
3c
(2,2,2) bbb
ee (4,2,1)
ccc ff
b
2
(4,2,−1)
ee (2,2,−1)
dd X
5
Z (1,1,1) aaa (4,1,1)
a (3,0,2) (3,0,0)
4 1
3 ff
2 (2,0,2)
1 23 4
O
1
Figure 1. (a) The trefoil knot (b) a possible representation of this knot as a number of
segments joining points.
or the end or both. One can turn a knot into an equivalent one by operations of
the following kinds—
(i) One may move a letter from one end of the row to the other.
(ii) One may interchange two consecutive letters provided this still gives a
knot.
(iii) One may introduce a letter a in one place in the row, and d somewhere else,
or b and e, or c and f, or take such pairs out, provided it still gives a knot.
(iv) One may replace a everywhere by aa and d by dd or replace each b and e
by bb and ee or each c and f by cc and V. One may also reverse any such
operation.
—and these are all the moves that are necessary.
Solvable and Unsolvable Problems | 587
It is also possible to give a similar symbolic equivalent for the problem of
separating rigid bodies, but it is less straightforward than in the case of knots.
These knots provide an example of a puzzle where one cannot tell in advance
how many arrangements of pieces may be involved (in this case the pieces
are the letters a, b, c, d, e, f ), so that the usual method of determining whether
the puzzle is solvable cannot be applied. Because of rules (iii) and (iv) the lengths
of the sequences describing the knots may become indeWnitely great. No system-
atic method is yet known by which one can tell whether two knots are the same.
Another type of puzzle which we shall Wnd very important is the ‘substitution
puzzle’. In such a puzzle one is supposed to be supplied with a Wnite number of
diVerent kinds of counters, perhaps just black (B) and white (W ). Each kind is in
unlimited supply. Initially a number of counters are arranged in a row and one is
asked to transform it into another pattern by substitutions. A Wnite list of the
substitutions allowed is given. Thus, for instance, one might be allowed the
substitutions5
(i) WBW ! B
(ii) BW ! WBBW
and be asked to transform WBW into WBBBW, which could be done as follows
WBW WWBBW WWBWBBW WBBBW
(ii) (i)
(ii)
Here the substitutions used are indicated by the numbers below the arrows, and
their eVects by underlinings. On the other hand if one were asked to transform
WBB into BW it could not be done, for there are no admissible steps which
reduce the number of B’s.
It will be seen that with this puzzle, and with the majority of substitution
puzzles, one cannot set any bound to the number of positions that the original
position might give rise to.
It will have been realized by now that a puzzle can be something rather more
important than just a toy. For instance the task of proving a given mathematical
theorem within an axiomatic system is a very good example of a puzzle.
It would be helpful if one had some kind of ‘normal form’ or ‘standard form’
for describing puzzles. There is, in fact, quite a reasonably simple one which I
5 Editor’s note. Turing’s draft has: ‘Thus for instance one might be allowed the substitutions
WBW ! B
BWWW ! WB
BWB ! WWWB
WWB ! W
and be asked to transform WBWWBWBBB into WBB, and this one could do Wrst by substituting W for
WWB and getting WBWBBBB and then successively WWWWBBBB, WWBBB, WBB.’
588 | Alan Turing
shall attempt to describe. It will be necessary for reasons of space to take a good
deal for granted, but this need not obscure the main ideas. First of all we may
suppose that the puzzle is somehow reduced to a mathematical form in the sort
of way that was used in the case of the knots. The position6 of the puzzle may be
described, as was done in that case, by sequences of symbols in a row. There is
usually very little diYculty in reducing other arrangements of symbols (e.g. the
squares in the sliding squares puzzle) to this form. The question which remains
to be answered is, ‘What sort of rules should one be allowed to have for
rearranging the symbols or counters?’ In order to answer this one needs to
think about what kinds of processes ever do occur in such rules, and, in order
to reduce their number, to break them up into simpler processes. Typical of such
processes are counting, copying, comparing, substituting. When one is doing
such processes, it is necessary, especially if there are many symbols involved, and
if one wishes to avoid carrying too much information in one’s head, either to
make a number of jottings elsewhere or to use a number of marker objects as well
as the pieces of the puzzle itself. For instance, if one were making a copy of a row
of counters concerned in the puzzle it would be as well to have a marker which
divided the pieces which have been copied from those which have not and
another showing the end of the portion to be copied. Now there is no reason
why the rules of the puzzle itself should not be expressed in such a way as to take
account of these markers. If one does express the rules in this way they can be
made to be just substitutions. This means to say that the normal form for puzzles
is the substitution type of puzzle. More deWnitely we can say:
Given any puzzle we can Wnd a corresponding substitution puzzle which is
equivalent to it in the sense that given a solution of the one we can easily use it to
Wnd a solution of the other. If the original puzzle is concerned with rows of pieces of a
Wnite number of diVerent kinds, then the substitutions may be applied as an
alternative set of rules to the pieces of the original puzzle. A transformation can
be carried out by the rules of the original puzzle if and only if it can be carried out by
the substitutions and leads to a Wnal position from which all marker symbols have
disappeared.
This statement is still somewhat lacking in deWniteness, and will remain so. I
do not propose, for instance, to enter here into the question as to what I mean by
the word ‘easily’. The statement is moreover one which one does not attempt to
prove. Propaganda is more appropriate to it than proof, for its status is some-
thing between a theorem and a deWnition. In so far as we know a priori what is a
puzzle and what is not, the statement is a theorem. In so far as we do not know
what puzzles are, the statement is a deWnition which tells us something about
what they are. One can of course deWne a puzzle by some phrase beginning, for
instance, ‘A set of deWnite rules . . .’, but this just throws us back on the deWnition
6 Editor’s note. Turing’s draft has ‘positions’.
Solvable and Unsolvable Problems | 589
of ‘deWnite rules’. Equally one can reduce it to the deWnition of ‘computable
function’ or ‘systematic procedure’. A deWnition of any one of these would deWne
all the rest. Since 1935 a number of deWnitions have been given, explaining in
detail the meaning of one or other of these terms, and these have all been proved
equivalent to one another and also equivalent to the above statement. In eVect
there is no opposition to the view that every puzzle is equivalent to a substitution
puzzle.7
After these preliminaries let us think again about puzzles as a whole. First let
us recapitulate. There are a number of questions to which a puzzle may give rise.
When given a particular task one may ask quite simply
(a) Can this be done?
Such a straightforward question admits only the straightforward answers, ‘Yes’
or ‘No’, or perhaps ‘I don’t know’. In the case that the answer is ‘Yes’ the answerer
need only have done the puzzle himself beforehand to be sure. If the answer is to
be ‘No’, some rather more subtle kind of argument, more or less mathematical, is
necessary. For instance, in the case of the sliding squares one can state that the
impossible cases are impossible because of the mathematical fact that an odd
number of simple interchanges of a number of objects can never bring one back
to where one started. One may also be asked
(b) What is the best way of doing this?
Such a question does not admit of a straightforward answer. It depends partly
on individual diVerences in people’s ideas as to what they Wnd easy. If it is put in
the form, ‘What is the solution which involves the smallest number of steps?’, we
again have a straightforward question, but now it is one which is somehow
of remarkably little interest. In any particular case where the answer to (a) is
‘Yes’ one can Wnd the smallest possible number of steps by a tedious and
usually impracticable process of enumeration, but the result hardly justiWes the
labour.
When one has been asked a number of times whether a number of diVerent
puzzles of similar nature can be solved one is naturally led to ask oneself
(c) Is there a systematic procedure8 by which I can answer these questions, for
puzzles of this type?
If one were feeling rather more ambitious one might even ask
(d) Is there a systematic procedure8 by which one can tell whether a puzzle is
solvable?
I hope to show that the answer to this last question is ‘No’.
There are in fact certain types of puzzle for which the answer to (c) is ‘No’.
7 Editor’s note. At this point Turing’s draft contains the following: ‘Some of these other deWnitions will be
found in Refs (1), (3), (11), (13), and (16) vol II. Some equivalence theorems are proved in (4) and (14), and
some propaganda on the matter will be found in (13). A very satisfactory account of all these problems will
be found in (5).’ Tantalizingly, the list of references is omitted.
8 Editor’s note. Turing’s draft has ‘systematic method’.
590 | Alan Turing
Before we can consider this question properly we shall need to be quite
clear what we mean by a ‘systematic procedure’8 for deciding a question.9 But
this need not now give us any particular diYculty. A ‘systematic procedure’
was one of the phrases which we mentioned as being equivalent to the idea of
a puzzle, because either could be reduced to the other. If we are now clear as
to what a puzzle is, then we should be equally clear about ‘systematic
procedures’. In fact a systematic procedure is just a puzzle in which there is
never more than one possible move in any of the positions which arise and in which
some signiWcance is attached to the Wnal result.
Now that we have explained the meaning both of the term ‘puzzle’ and
of ‘systematic procedure’, we are in a position to prove the assertion made
in the Wrst paragraph of this article, that there cannot be any systematic pro-
cedure for determining whether a puzzle be solvable or not. The proof does
not really require the detailed deWnition of either of the terms, but only
the relation between them which we have just explained. Any systematic pro-
cedure for deciding whether a puzzle were solvable could certainly be
put in the form of a puzzle, with unambiguous moves (i.e. only one move
from any one position), and having for its starting position a combination
of the rules, the starting position and the Wnal position of the puzzle under
investigation.
The puzzle under investigation is also to be described by its rules and starting
position. Each of these is to be just a row of symbols. As we are only considering
substitution puzzles, the rules need only be a list of all the substitution pairs
appropriately punctuated. One possible form of punctuation would be to separ-
ate the Wrst member of a pair from the second by an arrow, and to separate the
diVerent substitution pairs with colons. In this case the rules
B may be replaced by BC
WBW may be deleted
would be represented by ‘: B ! BC : WBW ! :’. For the purposes of the argu-
ment which follows, however, these arrows and colons are an embarrassment. We
9 Editor’s note. At this point Turing’s draft contains the following material, which is crossed out. ‘It is a
phrase which, like many others e.g. ‘‘vegetable’’ one understands well enough in the ordinary way. But one
can have diYculties when speaking to greengrocers or microbiologists or when playing ‘‘twenty questions’’.
Are rhubarb and tomatoes vegetables or fruits? Is coal vegetable or mineral? What about coal gas, marrow,
fossilised trees, streptococci, viruses? Has the lettuce I ate at lunch yet become animal? The fact of the matter
is that when one is applying a word, say an adjective, to something deWnite, one chooses the word itself so
that it describes what one wants to describe fairly and squarely. If it doesn’t one had better look for another
word. But if one is playing twenty questions this just can’t be done. The questions are about ‘‘the object’’,
and one doesn’t know what it is. The same sort of diYculty arises about question c) above. An ordinary sort
of acquaintance with the meaning of the phrase ‘‘systematic method’’ won’t do, because one has got to be
able to say quite clearly about any kind of method that might be proposed whether it is allowable or not.
Fortunately a number of satisfactory deWnitions were found in the late thirties, and they have . . .’ [the
fragment ends at this point].
Solvable and Unsolvable Problems | 591
shall need the rules to be expressed without the use of any symbols which are
barred from appearing in the starting positions. This can be achieved by the
following simple, though slightly artiWcial trick. We Wrst double all the symbols
other than the punctuation symbols, thus ‘: BB ! BBCC : WWBBWW ! :’. We
then replace each arrow by a single symbol, which must be diVerent from those on
either side of it, and each colon by three similar symbols, also chosen to avoid
clashes. This can always be done if we have at least three symbols available, and the
rules above could then be represented as, for instance, ‘CCCBBWBBCC
BBBWWBBWWBWWW ’. Of course according to these conventions a great variety
of diVerent rows of symbols will describe essentially the same puzzle. Quite apart
from the arbitrary choice of the punctuating symbols the substitution pairs can be
given in any order, and the same pair can be repeated again and again.
Now let P(R,S) stand for ‘the puzzle whose rules are described by the row of
symbols R and whose starting position is described by S’. Owing to the special
form in which we have chosen to describe the rules of puzzles, there is no reason
why we should not consider P(R,R) for which the ‘rules’ also serve as starting
position: in fact the success of the argument which follows depends on our doing
so. The argument will also be mainly concerned with puzzles in which there is at
most one possible move in any position; these may be called ‘puzzles with
unambiguous moves’. Such a puzzle may be said to have ‘come out’ if one
reaches either the position B or the position W, and the rules do not permit
any further moves. Clearly if a puzzle has unambiguous moves it cannot both
come out with the end result B and with the end result W.
We now consider the problem of classifying rules R of puzzles into two classes,
I and II, as follows:
Class I is to consist of sets R of rules, which represent puzzles with unambigu-
ous moves, and such that P(R,R) comes out with the end result W.
Class II is to include all other cases, i.e. either P(R,R) does not come out, or
comes out with the end result B, or else R does not represent a puzzle with
unambiguous moves. We may also, if we wish, include in this class sequences of
symbols such as BBBBB which do not represent a set of rules at all.
Now suppose that, contrary to the theorem that we wish to prove, we have a
systematic procedure for deciding whether puzzles come out or not. Then with
the aid of this procedure we shall be able to distinguish rules of class I from those
of class II. There is no diYculty in deciding whether R really represents a set of
rules, and whether they are unambiguous. If there is any diYculty it lies in
Wnding the end result in the cases where the puzzle is known to come out: but
this can be decided by actually working the puzzle through. By a principle which
has already been explained, this systematic procedure for distinguishing the two
classes can itself be put into the form of a substitution puzzle (with rules K, say).
When applying these rules K, the rules R of the puzzle under investigation form
the starting position, and the end result of the puzzle gives the result of the test.
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