Life Games and Statistical Models

Life

Games and Statistical Models

(evolution equations/pattern/order/stochastic elements/competition)

M. DRESDEN AND D. WONG

Institute for Theoretical Physics,StateUniversity of New York at Stony Brook, StonyBrook,NewYork, 11794 CommunicatedbyMark Kac, December 19, 1974

ABSTRACT Aset.ofequations isobtained, which de-scribes the rules of a class of games (life games). These games simulate the processes ofgrowth, death, survival, and competition. The equations are nonlinear difference equations, where the degree of nonlinearity is directly relatedto the number of interacting neighbors. The time evolution andthe development ofgeometricpatterns can be studied starting from these equations. Extensions and generalizations, such as the introduction of stochastic elements, can easily be accommodated in the formalism. Some significant unsolved problems are noted.

1. Background, the connection between life games and statisticalmechanicalmodels

Eversince vonNeumann's basic studies (1) there has been a continued interest in games simulating births, growth, and death, as models for life processes. An especially interesting game ofthis type was invented by J. H. Conway (2, 3); he called it "life" because of its suggestive similarities to the time evolutionof asocietyof-or organized group of-living organisms. In this game the universe considered is

two-dimensional,

divided into a large (infinite) number- of cells.

Each cell can exist in two states, occupied or unoccupied, roughly corresponding to life and death. The rules of the gameregulate justhowdeaths andbirthsinagiven cell shall occur in terms of theoccupancy of theneighboringcells. The stateofthe system atany one time t is,therefore, determined by thesequence {

ni(t)

}, where i labels the cells, and

pi

can be either 1 or 0 (for occupied or empty). The rules are fully

deterministic;

a specification ofthe state at time t gives an

unambiguous, precise, determination

of

the state at time

(t+ 1).

MorerecentlyEigen (4)in apaper

on,

theoriginof

biological

information referred to the "life" game as an interesting exampleof adeterministic scheme, which, although basedon verysimple rules,yetgivesriseto anextraordinary varietyof

configurations

andtypes ofbehavior.In fact, the behavior in

time and thegeometric patterns

of

thegameare so

involved,

thatnogeneral way tosurvey them seems to

exist;

the bulk of the surprising and at times amazing results have been obtainedfrom computer studies. As stressedagain by Eigen

(4),

the further

utility

ofthe "life" game in

studying

actual (biological)

evolutionary

processes is limited because of the absence of stochastic elements in the scheme. In actual

biological

processes a

fluctuating

environment is

always

present. Given the incompletenessofthepresent

description,

it would appear of great interest to obtain a formal or an

analytical

description

of this and related games, to

supple-ment the computer information.

General

resultsare

usually

more easily obtained from formal mathematical consider-ations than from computer programs. There are

interesting

andintricategeometrical questions which needstudy, before

a

thorough

understandingcanbe obtained. Finally,

stability

questions are of great significance; these again require ana-lytical tools. By combining thesemethods one might obtain new insights in the detailed operation and development of lifeprocesses.

The main point of this paperisthe observation that the formalism and the methods developed in connection with problems in nonequilibrium statistical mechanics (5-7), are particularly well suited for an analytic description of the "life" games. Moreimportant than that, analyticformulation provides (at least inprinciple) the possibility of a statistical treatment, the possibility of the systematic introduction of stochastic elements, and the possibility of substantial ex-tensionsand alterations of these games. It is notreally sur-prising that results of thestudy ofnonequilibriumbehaviorof models would be of relevance in "life" games. The "rules" ofa game are just a particular version of a discrete dynamics. It must therefore be possible to transcribe the game rules asmechanical equations of motion, yielding a direct relation between the life games and statistical models.

In the statistical problems the time dependence of various physical quantities (such as the total number of particles in a given state) are of great interest. The same type quantity is also of great importance in life-type games. For example,the total number of individuals at time t of the "life" game

N(t) =

E

qv(t)

p [1]

governs the indefinite growth, extinction, or oscillatory pattern of thesystem. Fromcomputer studies all three types of behavior are known to occur for appropriate initial con-figuration; however, computer studies by themselves cannot yield a connection between the nature ofN(t) and thenature ofthe initial state.

Inthenextsection (section 2) the methods of refs.5-7 are used to obtain the equations of motion for a "one-dimen-sional" version of the lifegame. This is done to illustrate the procedure andmethod;"one-dimensional life"byitself isnot particularly exciting.Thesamemethods, however, areuseful in the construction of the equations of motion for the full "Conwaylife" game. Theseequationsare oneof theprincipal results ofthispaper;theyarecontained in section 3. Once the formal structure of these equations is recognized, several generalizations suggestthemselves, which seem toyieldmore appropriate descriptions ofevolutionary processes. The most interesting generalizations result from the introduction of stochasticelements, orthe introduction ofcompetingspecies. It is not toodifficult to constructthe

appropriate

equations,

LifeGames and Statistical Models 957 buttheirmathematical discussionisfar fromstraightforward.

Some comments together with some questions raised by the equations are contained in section 4. One of the interesting insights to emerge from this discussion is the distinction between the model (or mechanical) rules, which presumably express physical or chemical laws; the evolutionary pattern, which is a mathematical consequence of the dynamical equations; andthe rulesofreward,which define the notion of success or survival in competing systems. It is conjectured that the essence of the notion of life is to be found in the abilitytoalter the rules of reward.

2. Formalismof the one-dimensional "life"game

Inthe "life"game as originally formulated,a

given

cellin a two-dimensional grid has 8 neighbors. The survival, death, andbirthrulesare

phrased

intermsofthenumber of

neighbors

of a cell. If acell isoccupiedattimet,andhas0,or 1neighbors it willbeunoccupied attimet + 1 (the cell dies from isola-tion). If a cell is occupied attimetand has 4, 5, 6, 7, and 8 neighbors, itagain will beempty attimet + 1 (the cell dies from overcrowding). If a cell is occupied and has 2 or 3 neighbors it will survive, from time t to t + 1. A cell un-occupied attime tand

having

exactly

three occupied

neigh-borswillbeoccupiedattimet+ 1. Todescribethis* introduce thevariables

...

.p(t)

= 1 if pisoccupiedattimet [2a]

.7.

rp(t)

= 0 if p is empty attime t [2b] Clearly, what is wanted is an

equation

of motion for

sp(t

+

1)

intermsofqp

(t)

which

incorporates

therules of the game. To illustrate the

method,

considerfirst a one-dimen-sionalversion ofthe "life"game.

t

Inthat

special

caseeachcell p hasjusttwoneighborsp-1,p+ 1.

Introduce the number ofneighbors ofp:

Vp(t) -p +

I(t)

+

np_(t)

[3] Theone-dimensional "life" rulesare now:

Ifp isoccupiedattime tand Vp = 0, or

vp

2, then p will be empty attime t+ 1.

Ifpisoccupiedattimetand

vp

= 1,p willremainoccupied attime t + 1.

Ifp isunoccupiedattimetandVP = 2,p willbeoccupiedat timet + 1.

It is notdifficulttoverify that these rulesareallcontained inthe equation:

77(t

+ 1) = 7P-1(t)

70(t)

+

-qp

-

i(t)

77p

+i(t)

+ 7p(t) 77p+i(t) -

377p

-

1(t)

7ip(t)

tip +

1(t)

[41

Eq. 4 represents a set of coupled nonlinear difference equa-tions. Before commenting on these equations, it isfor later applicationsuseful torewritethegame rules in terms of

vp:

7tP(t

+

1)

=

vp(2

-

Vp)f7p

+

'/2vp(vp

-

1)(1

-

tip)

[5]

Eq.5expressesthat independentof

-qp(t),

tip(t

+ 1)vanishes if VP = 0. (If there are no neighbors there can be neither

sur-vival nor birth.) If

vp

= 2,

qp(t

+ 1) = 1 -

qp(t),

again in harmony with the rules. vp = 1 gives

qp(t

+ 1) =

+p(t)

one neighbor perpetuates the status quo. It is generally easier to phrase the game rules in terms of the v variables,

thendirectly in terms of the n variables. It was possible to guess Eq. 4, but it is very hard to guess higher dimensional equations.

Eq. 4 is a well-defined deterministic set of equations; it is notdifficult to see that the state at p at t isdetermined bythe initial values-qp-t(O).

p(O). *p

+t(O)accordingto

P(t)=

co(t) +

EcI(t)al

+

E'cij(t)aiaj

+ ...

+

a~p

aP.

ap.

e +t [6] In

[61

7p

-

I(O)

=

ap

-t, the

cl(t)cij,

aretime-dependent func-tions, whichcanbe shownto

satisfy

linear

algebraic

relations. Thesummationsin [6]all run overij,fromp - ttop + t.

A state of the system is described by a sequence ofone's and zero's;forexample,

state = 1 0 0 1 1 1.. .1 [7] Asnoted insection1, thegeometricalpatternswhichdevelop inthisgame are notterribly interesting. Onecanshow either from the rules or from [4] that an initially finite configu-ration remains finiteor decays, butnevergrows indefinitely. Forexample, thepatterns

11 0 0 1 1 0 0 0 11 0 0 11 remainconstant intime while

1 1 1...1 decaysin twosteps.

Aconfigurationsuch as

[8a

]

[8b

I

1 1 0 1 0 0 1 1010[8c]

inwhichtwoemptyplaces(two zero's)occurconsecutively isa separated

configuration.

It is easy to see that an initially

separated

configuration

remains separated.

Thus,eventhisverytrivialcaseexhibitsavariety of possible

behaviors.

It is interesting to note that it is very easy to

accommodate two

competing

species within this formalism. Assumethateachlocationpcouldbe occupiedby two differ-ent types of entities. Denote their numbers by

tp(t)

and

tip'(t).

Assume that all the rules are as

before,

except if

either typehasanyneighbor of the othertype, it will die, at the nextstep. Inthat casethesystemisdescribed

by

tiP(t

+ 1) = 1/2(P'p -2) (v'P - 1)

[vp(2

- Pp)np

+ '/2V (P-1)((- p)] [9a]

r7p'T

+

1)

=

1/2(Vpp-2)(vp

-

1)

[P'(2

-

'p)

77'p

+ 1/2VIp(Vp-1) (1 -t'p)] [9b]

The definition of v'p is of course

v'p

=

(n'p

+ 1 +

i'p

- 1).

Eq. 9

describes

a competing coupled system, whichis,

how-ever, still

completely deterministic.

It is also possible to introduce probability notions by introducing ameasure on the space ofsequences, or

by

de-fining

.(...

7t,,t),

the

probability

that the system is de-scribed

by

a

configuration

1...

i,,,

and

deriving

an

equation

for Walong the linesofrefs. 6 or7. This will becarried out in a later publication, but even these brief remarks should

*These variables are of the same type as those introduced in

refs. 6 and 7.

tJust the-one dimensional case is treated in the remainder of thissection.

958 Biophysics: Dresden and Wong

show that thesegeneralizationsare notdifficult to formulate within this framework.

3. Formalism of the full

"lifH"

game

Toconstruct the equations of motion for the full "life" game, it isbestto use thesecond method used in the last section, i.e., the formulationof the game rules, interms of the number of nearestneighbors Pp.In the general case

v,,

isdefined by

8 ip= , p+ i

i-= [10]

of motionin other cases. In the general case when the game rules are death survival death birth if v =0Q

1, 2,.

..a ifv =a+1

...b

ifv =b+1...n if v =

cc+1...c+A

theequationhas the samestructureas

[11

]:

7p(t

+

1)

=

N1(v)qp(t)P(l)(vp)

+

N2(P)[1

-P(2(Vp)I(

Thesumintended in [10] is over the nearest neighbors of p. Using [10] it can be shown that the full game rules are expressed bythefollowingequationsof motion:

[Ila] vp '= (E7p+i)

-,i

7p(t

+

1)

=

XlPl(vp)

+

X2-qpP2(vp)

Here

Xi

and X2arenumbers:

XI=1

1

Xl= \2=

-720 1440

[lb]

[12]

In

[11]

Pi

andP2 arepolynomialsin

vp:

Pi(v)

=

(8

-

v)(7

-

v)(6

-

v)(5

-

v)(4

-

v)(2

-

v)(1 -V)v

8 3

(v-

V)

= _n=O 3-v

[13a]

P2(v) = (8 - v)(7 -v)(6-v)(5

~~~~~8

-v)(4 - v)(3-v)(1 -v _ P

)

[13b]

-2 - v

Thesesomewhat curiouslooking polynomialsare constructed toexpress the game rule

conveniently.

Forexample

Pi(v)

is alwayszero

(i.e.,

forall

v)

unlessv = 3. In that caseP1(3) = 720. Clearly P2(3) = 0. Thus

[ii]

asserts that

v=

3--n(t+ 1) = 1

[14]

This is

precisely

what thegameruledemands. For ifphas three

neighbors

and is

unoccupied,

the model rulesdemand thatatthenextstep, pwill be

occupied.

Ifphas

three

neigh-borsand is

occupied,

it willremain sofor thenextstep,

again

yielding

q (t+ 1) 1.

Thisreasoningverified

[11

]

forv = 3; it should be

similarly

verified

for all other v. This is quite a

straightforward

pro-cedure;

for

example,

Pl(v),

P2(p)

both vanish when v=

8,

7,

6,5,4,1,0; then

according

to

[11

]

7p(t

+

1)

=

0,

independent

of

lip(t).

This expresses thegame rulethatforthose

particular

numbers of

neighbors

an

occupied

cell becomesempty

either

from

overcrowding

or

isolation.

The verification of

[11]

for

v = 2 followsthat same

pattern,

andis omitted.

Remark 1: It should

be

clear from

this

result,

that one

can construct

analogous

equations

forany

number

ofnearest

neighbors.

This number n has no

special

geometrical

signifi-cance; it is

only

related to the

dimensionality

of the space through the

particular

lattice structure

considered;

it could be8 as inthe

plane

lattice

considered

by

Conway,

and4ina

spatial

(tetrahedral)

arrangement.

It is a

straightforward

extension of the arguments

just

given

toconstruct

equations

-

lp)

[15]

Now thepolynomials havethe form:

P(l)(v)

= (v -1) ...(v

-a)(v

- b). [v- (b +

1)]

*(v-v

[16a]

P(2) y=

'(Y

- 1) ( -a)[v-(a+ 1)]

*. * [-(c+1) ][v-(c+A+1 )

*

(v-b)

... v)]

[16b]

The

N(v)

arenormalizationsasbefore; determined by NI(v) =

[P(l)(0)>1

fora + 1 < v < b - 1 [17a]

N2(v)

=

[P(2)(v)]1

for c < v < c + A

[17b]

In [15] the independent variable

Pp

is thesum [Ila] extended from 1 ...n. The normalization constant depends on the

number

ofnearest

neighbors.

Remark 2: Even though the

polynomials

in

[13]

appear tobe of

degree

8 it should be remembered that 7 2 =

So

that the 8thpowerof

(EZ

Fp +

{)

containsasmost

complicated

terma

product

of the qvariables of the8

neighbors

ofp.

TheEq.1lbthencontainsa

(symmetric)

sumof

products

of

the

vq

variables

of the

neighbors

ofp,with certain

numerical

constants. Thus

[lib]

is

the precise

counterpart of

Eq.

4for the

one-dimensional

(better

two-neighborl)

"life"game.

Remark 3: It is

worth

stressing

that the

techniques

de-veloped

in

this

section allow still more

general

systemstobe described

inmuch

thesame manner.Asan

example,

considera system where cells can be

occupied

in two states

(male,

female,

spin

up,

spin

down, charge

+,

charge

-).

The

appropriate

variables arenow

[18]

[19]

-1

7p= 4 0

t+1

pp =

E7ip

+, P = S1P+ i

Clearly, vp

measuresthetotal

number

of

neighbors,

irrespec-tive ofsex

(or charge)

while

O1,

measures the excess sex

(or

charge).

Further,

for

survival

it is

required

that

a< vp < b

and

-a X

Op

< +a survival

[20a]

c

Pvp

. c + / and -\ .

OP

+,

I birth

[20b]

p<

a,orvp > b

I@PI

> a death

[20c]

The sexofa birthshall be

(-0,,),

in case

0p

= 0 it shall be female

(negative).

Finally

a

natural

lifetime of eachstateis

introduced,

called

T, whichmeans

that

astateqpcannot

persist

longer

thanT. Proc.Nat.

Acad.

Sci. USA 72

(1975)

LifeGames andStatistical Models 959 Thus, acondition for survival ist, < T;anadditionalcause

of death islongevity:

tp> T

Theequationsforthissystem areobtained in thesameway andthey read:

,2 +

'(tp

+1) =

7p N(

p)P(

p)M(II(op)Q(')(OP)

X

{[1

-

0(tp

-

m)][e(t.

-

(m

-

1)T]}

- lion 0P+

E-

[1 -

(vip(m))2]N(2)P(2)M(2)Q(2)

[21]

ois the usualstepfunction. Here the Pand Qarepolynomials

PM(l)(p)=

p(vp

- 1)...

(vp

-

a)(v.

-

b)(vp

- b- 1) ...(,p- n) [22a]

pM(,V(P)

jVP.. (,VP - C)(,P - c- (,VP - c- A - 1) ...(,P-n) [22b]

Q(1)(Op)

=

(0p2

-

n2)

(02-

(n-

1)2)

...

(@p2

-Ca)

[22c]

Q(2)(0P)

(0P2

n2)

(Op2-

(n 11)2)...

(0P2

- 2) [22d]

TheN and M are again normalization constants

NWl)PM = 1 [23a]

N(2,p(2)

= 1 [23b]

M(1)Q(l)=

1 [23c]

M(2)Q(2)

= 1

[23d]

This system, still deterministic, has therefore two new features, the finite lifetime, and a "charge" limitation on life anddeath.Apreliminarycomputercalculationindicates that the total number increases sharply in the average and ap-proaches infinity, but there are strong oscillations super-imposedonthissmooth behavior.

4. Comments, unsettled questions

It would be misleading toimply that with the construction of the variousequations in section3 a great deal of under-standing has been gained aboutthelife-type games. There are

always

certain advantages associated with analytic

formula-tions,

butonlyifit would be possibletodeduceresultsfrom

the equations, which couldnot sodirectly be obtained from a computer

study,

wouldgenuine progress have been made. It may, therefore, be useful to raise a number of specific questions, which ought to bestudied in general for the equa-tions which have been set

upt.

The real utility of the equa-tions will depend on the effectiveness with which they can handlethese questions.

(a)

Oneseries ofquestions refers to the time behavior of the totalnumber ofindividuals N(t). This can grow to infinity in somecases; in onedimensionit eitherdecays, or is constant. One knows from computer studiesthatfor8neighbors various

typesof behaviorcanoccur.But it should be

possible

froman examination ofthe

equations

togeta

systematic

answer.

(b)Inthe computerstudies of theConway "life"game very interesting and

special

geometric

configurations

arise as finalstablestates. Itwould be

interesting

tofirstseeandthen derive, whether among these states certain

patterns

occur

preferentially. One could further

study

whethersuchevolved figures or

geometrical

arrangements possess a greater or

differentdegree of

symmetry

than the

initial

configurations.

(c) Itwould be

interesting

and

important

tohavea system-atic study of the effect of random elements. As

explained

before, it isnotdifficult to

incorporate

this in the

equations,

butcertain important

questions

cannot

(or

donotappear

to)

be answered so

easily.

Suppose that without random ele-ments, certain

geometric

patterns

develop

preferentially,

as forexample, the benzene

ring hexagon

inthe

Conway

model§.

The

question

then

is,

which one of these patterns is most

(or least) affected

by

theintroduction of random

elements,

ormistakesormutations. This

question

suggestsa

larger

and moreambitious setof

problems;

the

appropriate

definition of stability and

especially

of structural

stability

in discrete dynamicalsystems.

(d) Another importantissueishowasystemwould

respond

tothe imposition ofanoverall constraint. Avery

simple

ex-ample would be a fixed maximum number

No

of

cells,

or a fixedmaximumaccessible

volume,

ina

typical

"life"game.It isconceivablethatundersuchcircumstances,the

evolutionary

patternfor

Ninitial

<N

(N,

issomecritical

number,

N,

<

No)

would be quite different from that for

Winitial

< N

(however

N

<No).

(e) A very trivial

example

was

presented

of the

life-type

game

equations

with two

interacting

species.

This seems an

important extension to consider

seriously.

The interaction between the two species

(or players

or

sexes)

could be pre-scribed

stochastically

or in a deterministic manner. It now also becomes

possible

to

prescribe

"scoring rules,"

or

"suc-cess"

or rules of reward if these two

species

are viewed as

competitors. One could defineassuccesslongevity, numerical superiority, or survival.

Alternately,

the attainment of a

particular

geometrical

pattern could be defined as success.

It would appear that a truelife-like feature will be added if the rules of reward could be altered while maintaining the physical (game-like) rules of operation. It isalsoconceivable thatastage isreached intheoperationorintheevolutionary process wheretherules of reward themselvesbecome

ambigu-ous, sothatabifurcation processmightoccur.

It

would

be very worthwhile if examples, no matter how

contrived, exhibiting this behavior could actually be con-structed. Some efforts inthis directionwill be reported in a

subsequent publication.

Some of the

questions

raised hereneed to be answered (or shown tobeirrelevant)iffurtherunderstandingof the nature of the evolutionary processes is to be gained. The equations setup in this papershould be auseful starting point toward thatgoal.

More detailed calculations, especially aboutthe statistical features of the systems, will with appropriate luck be pre-sentedindue course.

§Instudies made on games with 5 nearest neighbors, the square appears to be overwhelmingly prevalent among stable final

states.

tItmightwellbe wise to first study these questions for the case of3or 4neighbors, rather than the full"life"game.

This work was supported in part by Grant no. P4-P2656-00 from theNational Science Foundation.

1. von Neumann, A. J. & Morgenstern, 0. (1962) Theory of Games and Economic Behavior (Princeton Univ. Press, Princeton,N.J.).

2. Gardner,M.(1970)Sci. Amer.223(Oct.),120-123. 3. Gardner,M.(1971) Sci.Amer.224(Feb.),112-117.

4.

Eigen,

M.

(1973)

in The

Physicist's

Conception of Nature,

ed. Mehra, K. (Reidel Publishing, Dordrecht,

Holland),

pp. 594-635.

5. Kac,M.(1956)Bull.RoyalSoc.Belgium 42,356-361. 6. Dresden,M. (1962)inStudies inStatisticalMechanics. I,eds.

deBoer,J. &Uhlenbeck, G. E. (NorthHolland Publishing Co., Amsterdam),pp.303-350.