7/27/2019 gosper-84

1/6

Physica 10D (1984) 75--80N o r t h - H o l l a n d , A m s t e r d a m

E X P L O I T I N G R E G U L A R I T I E S IN L A R G E C E L L U L A R S P A C E SR . W m . G O S P E RSymbolics Inc, 845 Page Mill Road, Polo Alto, California 94304, and Lawrence Livermore National Laboratory, Livermore,California 94550, USA

We i n f o r ma l l y de s c r i be a c ompu t e r a l go r i t hm f o r s i mu l a t i ng de t e r mi n i s t i c c e l l u l a r a u t oma t a . I t g r ows " s ma r t e r " bys el ec ti ve ly r e c o r d i ng i n t e r me d i a t e c om pu t a t i ons wh i l e ope r a t i ng on a c om pr e s s e d r e p r e s e n t a t i on o f t he c e l l u l a r spa c e - ti me . N oinformation is lost because configurations do not actually evolve. Instead, the sim ulator accomo dates random exploration ofplace-times in the fu ture o f init ial configurations emb edded in an effectivelyunbounded void.We use Conw ay's Life automaton as a m odel , but the idea extends easi ly to other geom etries, dimensions, and numbero f states.

1 . M o t i v a t i o n

I n 1 9 7 0, J . H . C o n w a y a n n o u n c e d a p a rt ic u l a r lyi n te r e st in g c e l lu l ar a u t o m a t o n w h i c h h e d u b b e d" L i f e " [ 1 - 3 ] . L i f e i s a t w o s t a t e , n i n e - n e i g h b o r h o o dr u l e a p p l i e d o n a n o r d i n a r y , t w o - d i m e n s i o n a l g r i d[4 ]. E a r l y m a n u a l a n d c o m p u t e r s i m u l a t i o n s s u g -g e s t e d t h a t i n i t i a l l y f i n i t e c o n f i g u r a t i o n s o f o n ce l l sr e m a i n e d f i n i t e . T h i s l e d C o n w a y t o c o n j e c t u r e t h en o n - e x i s t e n c e o f s u c h t h i n g s a s " p u f f e r t r a in s " , h i sn a m e f o r c o n f ig u r a t io n s w h i c h w o u l d t r a v eli n d e f i n i t e l y t h r o u g h e m p t y s p a c e l e a v i n g p e r -m a n e n t d e b r i s .

F i g . I . s h o w s f o u r s t a g e s o f w h a t s e e m s t o b e ap u f f e r t r a i n . E v e r y t e n t i m e s t e p s , t h e t o p m o s tt h r e e c o m p o n e n t s a d v a n c e f iv e c e ll s v e r t ic a l ly , a n dl e f t - f o r - r i g h t r e f l e c t t h e m s e l v e s , w h i l e t h e c e n t e rc o m p o n e n t e m i ts a p u f f o f d e b r i s ( in d i c a te d b y a na r r o w ) . T h u s , i n t e n m o r e t i m e s t e p s , t h ec o n f i g u r a t i o n w i l l t r y t o e m i t a m i r r o r i m a g e p u f ff iv e c e ll s f o r w a r d o f t h e p r e v i o u s p u f f . T h i s w o u l do b v i o u s l y b e a p u f f e r tr a i n , w e r e t h e p u f f s n o tt h e m s e l v e s v i g o r o u s l y in t e r a c ti n g , t h r e a t e n i n g t os p a w n a d i s t u r b a n c e w h i c h o v e r w h e l m s t h e p u f f e rf r o m b e h i n d .

I s fi g. 1 a p u f f e r t r a i n ?T h e r e i s p r o b a b l y n o w a y t o s e t t l e s u c h q u e s -

t i o n s e x c e p t b y e x p e r i m e n t . R u n t h e s i m u l a t i o nu n t i l e i t h e r t h e p u f f i n g s t o p s , o r t h e d e b r i s e x h i b i t s

e n o u g h r e g u l a r i t y f o r d e t e r m i n i s m t o f u r n i s h ap r o o f o f u n b o u n d e d g r o w th . ( T h e p ro v e n T u r i ng -u n i v e r s a l i t y o f L i f e o n i n f i n i te g r i d s i n t r o d u c e sa t h i r d , f o r t u n a t e l y r e m o t e p o s s i b i l i t y : t h ec o n f i g u r a t i o n m a y n e i t h e r r e g u l a ri z e n o r s e l f-d i s t r u c t , i n d e f i n i t e l y c o n c e a l i n g i t s u l t i m a t e b e h a v -i o r . )

F i g . 2 s h o w s t h a t f i g. 1 i s i n d e e d a p u f f e r t r a i n ,b y v i r tu e o f t h e d e b r i s p l u m e a c h i e v i n g b o t ht e m p o r a l a n d s p a t i al p e r io d i c i t y , b u t o n l y a f te rs o m e 2 0 0 0 s t e p s a n d a s e v e n f o l d m u l t ip l i c a t io n o ft h e o ri g in a l p u f f p e r i o d . S u b s e q u e n t e x p e r i m e n t so n s i m i l ar p u f f e r t r a in c a n d i d a t e s h a v e s h o w np e r i o d d o u b l i n g t o b e f a r l i k e l i e r t h a n s e p t u p l i n g .A f e w c o n f i g u r a t i o n s , h o w e v e r , h a v e p u f f e d f o rs e v e r a l h u n d r e d s t e p s, o n l y to d i s in t e g r a t e w i t h o u te s t a b l is h i n g a n o v e r a l l p e r i o d , t h e r e b y u n d e r -s c o r i n g t h e n e c e s s it y o f e x p e r i m e n t a l p r o o f o fa p p a r e n t u n b o u n d e d g r o w t h .

A n d s h o u l d o n e tu r n f r o m n a t u r a l i s t t o e n g i n e er ,b u i ld i n g l a r g e c o n f ig u r a t io n s o f m o d u l e s w h i c hc o n s t r u c t o r c o m p u t e t h i n g s [ 3] , o n e s h o u l d s t il le x p e r i m e n t t o v e ri fy t h a t t h e " h a r d w a r e " w o r k s .

I n e i t h e r c a s e , t h e e x p e r i m e n t s w i l l u s u a l l y r e -q u i r e a l a rg e a n d u n p r e d i c t ab l e a m o u n t o f " r e a le s t a t e " a n d t im e , y e t m o s t o f th i s re a l e s ta t e w i llr e p e a t i t s el f m o s t o f t h e t im e .

T h e f o l l o w i n g a l g o r i t h m w i ll le t u s f re e l y e x p l o r et h e f u t u r e s p a c e - t i m e s o f la r g e i n it ia l c o n -

0 1 6 7 - 2 7 8 9 / 8 4 / $ 0 3 . 0 0 E l s e v i e r S c i e n c e P u b l i s h e r s B . V .( N o r t h - H o l l a n d P h y s i c s P u b l i s h i n g D i v is i o n )

7/27/2019 gosper-84

2/6

7 6 R. Wm. Gosper /Exp lo i t i ng r egular i t ies i n l arge ce llu lar spaces

Center (B e) Time 8

]enter ( 0 ~ ~) Tl~e IB

]enter (B, O ) Time 2B

P

@

'4

Center (0, 55) Time238

O

F i g . 1 , F o u r s t a g e s in t h e e v o l u t i o n o f a p r o b a b l e p u f f e r tr a i n .

f i g u r a t i o n s , p r o v i d e d t h a t t h e y a r e s u f f i c ie n t l y r e-p e t i t i o u s , b o t h s t r u c t u r a l l y a n d b e h a v i o r a l l y . I t( a n d e n o u g h c o m p u t a t i o n a l h a r d w a r e t o s u p p o r ti t) c o u l d h a v e s p a r e d C o n w a y o n e o f hi s f e wu n l u c k y c o n j e ct u re s .

2 . T he algorithmT h e re a re t w o k ey c o m p o n e n t s - a h a s h m e ch -

a n i s m a n d m a c r o - c e l l s . T h e h a s h m e c h a n i s m p r e -v e n t s th e r e c o m p u t a t i o n o f i n d i st i n g u i sh a b l es c e n a r i o s . M a c r o - c e l l s m e c h a n i z e t h e i n f o r m a t i o n -c o m p r e s s i o n o f t h e s p a c e ti m e b e h a v i o r o fc o n f i g u r a t i o n s . A m a c r o - c e l l r e p r e se n t s a 2 " b y 2 "b l o c k o f a u t o m a t o n c e l l s , w h e r e n i s a n y n o n -

n e g a t i v e i n t e g e r . E a c h m a c r o - c e l l s e e k s t o d e t e r -m i n e i ts R E S U L T , n a m e l y t h e c o n c e n t ri c 2 "- 1 b y2" - Lm a c r o - c e l l w h i c h t h e p a r e n t m a c r o - c e l l e x c l u -s i ve l y d e t e r m i n e s a f t e r 2 " - 2 t i m e s t ep s . B u t i n s t e a do f o c c u p y i n g 2 2 " u n i t s o f s t o r a g e , a m a c r o - c e l l o fs i z e 2" ( n > 0 ) r e qu i r e s j us t f i ve un i t s . T he se ho ld( po in t e r s t o ) t he f o ur m a c r o- c e l l s o f si z e 2" - ~w h i c hc o m p r i s e t h e f o u r q u a d r a n t s , a n d , i f w e a r e l u c k y( a n d n > 1 ), t h e R E S U L T , a l s o o f q u a d r a n t s iz e.

T h e e n t i r e s t r u c t u r e a n d e v o l u t i o n o f a n i n i t i alc o n f i g u r a t i o n w i l l b e e n c o d e d i n t h e i n t e r l i n k i n g so f m a c ro - c e l ls , w h i c h a r e c o m p u t e d a s w e p r o b e i tsf u tu r e . T h i s w i l l usu a l l y r e qu i r e f e we r m a c r o- c e l l st h a n y o u m i g h t t h i n k , d u e t o t w o r e s t r i c t i o n s o nw h e n a m a c r o - c e l l c a n b e c r e a t e d . F i r s t , a m a c r o -c e l l i s n e v e r c r e a t e d i f o n e h a v i n g t h e s a m e q u a d -

7/27/2019 gosper-84

3/6

R. W m. Gosper/Exploiting regularit ies in large cellular spaces 7 7

. . ". ~ . ~, : , ~ . ~ " o

~ ' . . . ; .-:- .:." ~t,~ "~.. ":'.:. :.:,. ,

* 'o~" "e ". . ' . . . . ' , ( , ~ ' ~~ ' .~ . .b ' t - ~* :

.~-~... . ': , .*..-.~,; ; . . .

t . . . o ~ ' - o - - e- . - , - , . ' - . r ~

- . . . , . . . ' ~ . . ~ ,. .. ' i " " i ' - ' -'" .:.' -" 4, "'"~'~.; '.:." - ~'." i ' .-:- ~: .. . ...,. ~ ....

..., .-' -,:,. ~" ..",;, .' -. .. : ,.. "-:- -( ; . . . " ,; ,

: . ~ - : - ' - ? . . ' : , : , - . -, ; , . . . . : , ? . + . . '

- ,~ . . , : , . ' . o . . . . . ". .- :- ' ,? , . . : , .. . . . , . - - , . - , ,, ; ,~o o.~ o. ', : . . . : , . ' - . ? "' : ' . .. . . : .. ; " , . . .: - : . : . " . . : " .-

" ' : ' . " . . " . . : . ' ~ i . ": " ~ . - . " ~ .. . . - , . ' , . ' ~ : " . .

. i .. '! " ' . . ' - " ' "' " i SF i g . 2 . A b o v e t h e s t i l l - s m o l d e r i n g i n i t i a l e x p l o s i o n ( b o t t o m ) ,t h e r e f i n al l y a p p e a r g a s e g m e n t o f t e m p o r a l l y a n d s p a t i a l l yp e r i o d ic e x h a u s t , p r o v i n g t h a t a b o v e t h i s s e g m e n t o p e r a t e s at r u e p u f f e r t r a i n [ 5 ] .

r a n t s a l r e a d y e x is ts T h i s a p p l i e s r e c u r s i v e l y t o t h eq u a d r a n t s A t t h e b o t t o m o f . t h e r e c u r s i o n a r e t h e2 o b y 2 0 ( i. e . 1 b y 1 ) c e l ls , o f w h i c h t h e r e a r e a t m o s tt w o , s i n c e L i fe i s a t w o s t a t e a u t o m a t o n L i k e w i s e ,a t m o s t 2 2 .2 = 1 6 t w o b y t w o c e l ls c a n b e c r e a t e d ,e tc . W h e n t h e a l g o r i th m t ri e s t o g r o u p f o u r q u a d -r a n t s t o f o r m a p r e - e x i s t i n g m a c r o - c e l l , t h e h a s hm e c h a n i s m n o t ic e s t h e c o in c i d e n c e a n d r e t u r n s t h eo l d ce l l i n s t e a d o f a n e w o n e . M o s t i m p o r t a n t l y ,t hi s ol d c el l m a y a l r e a d y k n o w i t s R E S U L T , w h i let h e n e w o n e c a n n o t ( I r o n i c a l l y , t h e " o l d " c e ll m a yw e l l h a v e b e e n c r e a t e d l a t e r in simulated t im e t h a nt h e " n e w " o n e . )

T h e s e c o n d r e s t r i c t i o n o n m a c r o - c e l l c r e a t i o n i si m p l i c i t i n t h e a l g o r i t h m : a m a c r o - c e l l o f s iz e 2 n,( n / > 2 ) c a n o n l y b e c r e a t e d w h e n i t s x , y , a n d t i m ec o o r d i n a t e s ( r e l a t i v e t o a l l i t s p a r e n t m a c r o - c e l l s )a r e m u l t ip l e s o f 2 " - 2 . T h u s , p r o l i f e r a t i o n s o f ce l lsi s l i m i t e d b y i n d i s t i n g u i s h a b i l i t y w h e n t h e y a r es m a ll , a n d b y i n f r e q u e n c y o f c r e a t i o n w h e n t h e ya r e l a r g e

A s a n e x a m p l e , a n e m p t y u n i v e rs e o f d i a m e t e r23 2 i s r e p r e s e n t e d b y a m a c r o - c e l l w h o s e f o u rq u a d r a n t s a r e a ll t h e s a m e , a m a c r o - c e U o f d i-a m e t e r 231 e t c . T h u s , t h e w h o l e t h i n g r e q u i r e s o n l y3 3 m a c r o - c e l l s . S i n c e th e r e p r e s e n t a t i o n s o f t h e" b i g " o n e s a re n o l a r g e r t h a n t h e " s m a l l " o n e s , o n ec a n b e c a s u a l a b o u t c r e a t i n g n e w u n i v e r s e s , a s l o n ga s o n e r e u s e s a l o t o f lo w e r - l e v e l c o m p o n e n t s T h u si f o n e w i s h e s t o " e d i t " a n e x i s t i n g 2 32 b y 232u n i v e r s e b y a l t e r i n g a s i n g l e 1 b y 1 c e l l, o n e m u s tc r e a t e a w h o l e n e w u n i v e r s e . B u t t h i s c a n e n t a i l n om o r e t h a n 3 3 n e w c e l ls (a t m o s t o n e o f e a c h s iz e ),n o m a t t e r h o w e x t e n s i v e t h e o r i g i n a l c o n f i g u r a t i o n

S i m i l a r l y , s i m u l a t i o n r a t e s c a n i n c r e a s e e x -p o n e n t i a l l y i n r e g i o n s o f r e p e t it i v e b e h a v i o r , b e -c a u se o f th e r e us e o f m a n y R E S U L T s in t h ec o n s t r u c t i o n o f l a r g e r R E S U L T s ( w h i c h r e p r e s e n tl a r g e r t i me - s t e p s ) .

T h e s m a l l e s t m a c r o - c e l l w h i c h c a n h a v e a R E -S U L T i s 4 b y 4 . T h e s e a n d a ll l a r g e r c e ll s p r o d u c eR E S U L T S i n re s p o n s e t o m e s sa g e s, w h i c h u s u a ll yi s su e f r o m l a r g e r m a c r o - c e l l s in p u r s u i t o f t h e i ro w n R E S U L T s I n o r d e r t o m i n i m i z e s t o r a g er e q u i r e m e n t s o f in d i v i d u a l c e l ls , m e s s a g e h a n d l i n gc o d e is a s s o c i a t e d w i t h s i z e c l a s s e s o f ma c r o - c e l l s ,

7/27/2019 gosper-84

4/6

7 8 R. Win. Gosper /Exp loit ing regularit ies in large cellular spaces

rather than with the cells themselves. If the queriedmacro-cell already knows its RE SULT (from hav-ing computed it previously), it just returns it. If a4 by 4 cell doesn't know its RESULT, it computesit by brute force, i.e. by applying the Life rule [4]to the nine-neighborhoods of each of its fourcentral cells (bits). Larger cells determine theirRESULTs by a somewhat inelegant-looking recur-sion which involves combining a total of thirteenseparate RESULTs of quadrants, and otherquadrant-sized macro-cells formed by groupingRESULTS and regrouping quarants of quadrants.

Here is a mental picture of the recursion. Amacro-cell at time 0 is the top stratum of a patchof earth. Successively deeper strata hold futuretime-slices. We are trying to excavate down to theRESULT stratum, which is the fiat bottom of ahole with sides of slope 1, a frustum of theforeseeable future cone with the initial macro-cellas its base. (Actually, the future cone is the exteriorof the light cone of the exterior of the initialmacro-cell.) We daren't excavate any steeper, lestwe unearth the uncertainty that seeps sidewaysfrom the edges at slope 1. The depth of the hole is1 of its width, and half the width of the bottom. Toscoop out such a hole, we recursively scoop outholes of the next smaller size, i.e. one half thedesired diameter and depth. By taking the RE-SULTs of the four quadrants, we reach half of thedesired depth, but there remain dikes covering 5/9of this halfway bottom (see fig. 3). To excavatethese dikes, five artificial, shifted "qu adr ant s" mustbe constructed from quadrants' quadrants, andthen RESULTed. This will involve the re-excavation of some thin air, but at little cost, sincethe reexcavated cells will remember their RE-SULTs. (If they didn't, the recursion would actu-ally recompute the same results a number o f timesproportional to the diameter of the outermostmacro-cell!) We are now on the halfway bottom,composed of nine subresults, which are thengrouped in fours to form four overlapping squares.The grouping of the RESULTs of these foursquares is the grand RESULT.

Thirteen scoopings is not too inefficient, given

that each scoop is only 1/8 the volume of thedesired excavation.

Note that the algorithm is indifferent to the x, y,and time coordinates and even the sizes of themacro-cells. Their spatial coordinates are implicitin the quadrant structure of their owners, and theirtime coordinates are implicit in the RESULTstructure. Nothing is even checking whether aRESULT is to be computed by counting (4 by 4),or by recursion (> 4 by 4), since the knowledge ofhow to RESULT is in the cell classes themselves.The 16 by 16s send the same RESULT message totheir 8 by 8s as the 8 by 8s send to their 4 by 4s,without any knowledge of their own size or the sizeof their quadrants. There are 4 by 4s in thequadrants of the 8 by 8s by construction, and aproperly constructed universe will stay that way.

"Digital physicis ts" i.e. seekers of cellular mod-els of the real Universe, may be interested in thisalgorithm's decoupling of simulated time fromsimulator time, thus freeing their models from theirritating requirement of "pan-synchronicity" , andgiving time a rather more topological inter-pretation. But this very virtue becomes a drawbackwhen the simulator desires to view the progress ofthe simulation as if it were evolving flatly andparallel to simulator time. One approach is toartificially mainta in x and y coordinates through-out the recursion, and display RESULTs in theirproper position whenever they are obtained. Thisis especially tempting, in that it might have theadditional virtue of skipping over scenarios thathave been played before. Alas, the result, at leastif viewed fiat, is incomprehensible. Time not onlyruns at different speeds in different places, it evenappears to run backwards in fixed places. Thiscomes from the reexcavation of "thin air" de-scribed earlier, wherein overlapping events areexplored several times. A possible fix might be to"ratchet" time by maintaining, for each I by I cellposition, the largest time value reached. But thiswould prevent the simulation of those very largebut repetitive configurations for which this algo-rithm is well suited. What's worse, the ratchetingleaves an unflat time surface which still results in

7/27/2019 gosper-84

5/6

R . Wm . Gosper /Ex plo i t ing reg ulari ties in larg e cel lular spacesR r t e r t h r e e = c a o ~ s R C t e r b v e ~ c o o p s

79

R F t e r n i n e u n a e t h o d i c a l ~ O O D ~ T ~ e l v e d o u n , o n e t o 9 0

Fig. 3 . The recurs ion s tep: d igging a hole in spacet ime wi th ha l f -s ize scoops . G reate r depths represent grea ter t ime. The th in ,c i r c umf e r e n ti a l f la nge i nd i c a ti ng t i me s t r a t um 0 e x t e nds i n t o n e i ghbo r i ng m a c r o - c el l s .

c o n f u s i n g d i s p la y s . F o r e x a m p l e , a t r a v e l i n g o s ci l-l a t o r m a y s i m p l y d i s a p p e a r f o r a n e x t e n d e d t i m e ,o n l y t o r e a p p e a r s i m u l t a n e o u s l y i n s e v e ra l p l a c e s,d u e t o m u l t i p l e i n t e r s e c t i o n s o f t h e o s c i l l a t o r ' ss t r a i g h t t r a c k w i t h t h e t e r r a c e d w a l l s o f t h e p a r t i a le x c a v a t i o n .

T h e r e s o l u t i o n o f t h i s d r a w b a c k i s t o l e t t h eo b s e r v a t i o n d r i v e th e c o m p u t i n g , i n s t e a d o f v ic ev e r s a . I n g e n e r a l , o n e w i s h e s t o h a v e o n e o r m o r e" w i n d o w s " i n t o t h e s p a c e t im e , t h a t i s, ( p ro b a b l y )r e c t a n g u l a r s l ab s o f I b y 1 c e ll s, o n e t i m e u n i t t h i c k .T o S H O W t h e i n te r s e c ti o n o f s u c h a s la b w i t h t h e

spa c e t im e , t e a c h the m a c r o- c e l l c l a s se s t o c he c kw h e t h e r t h e i r f u t u r e c o n e i n t e r s e c ts t h e s l a b , a n d i fs o , p r o p a g a t e t h e S H O W m e s s a g e , a l o n g w i t ha p p r o p r i a t e x , y , a n d t i m e o ff s e ts , t o t h e q u a d r a n t sa n d R E S U L T s , w h i c h a r e c o m p u t e d a s n e c e s s a r y .T h e n t e a c h t h e 1 b y l s t o s i g n a l t h e q u e r y i n gw i n d o w i f t h e y g et a S H O W m e s s a g e w i t h t i m e a n ds p a c e c o o r d i n a t e s a l l 0 . A l t h o u g h s i g n i f i c a n t l ym o r e c o m p l i c a t e d t h a n t h e R E S U L T a l g o r i t h m ,t h is S H O W a l g o r i t h m i s m o r e e ff ic ie n t t h a n i t m a ys o u n d . I f n o R E S U L T s n e e d to b e c o m p u t e d ,S H O W i s o n l y l o g a r i th m i c i n t h e t im e c o o r d i n a t e .

7/27/2019 gosper-84

6/6

80 R. W m . G os per /Exp lo i t i ng r egu lar i t i es i n lar ge ce l lu lar s pacesF u r t h e r m o r e , m o s t f u t u r e c o n e s ( a n d , b y th e g e o -m e t r y , t h e c o n e s o f a ll t h e ir c o m p o n e n t s , r e c u r -s i v e l y , ) w i l l n o t i n t e r s e c t n o n - g i g a n t i c w i n d o w s .

H o w e v e r , t h e o u t e r m o s t S H O W m e t h o d e n su r e st h a t t h e c o n f i g u r a t i o n b e i n g p r o b e d i s s u r r o u n d e db y e n o u g h ( i te r a ti v e ly d o u b l e d ) v a c u u m s o t h a t i tsf u t u r e c o n e e n t ir e l y c o n t a i n s t h e p r o b e w i n d o w , n om a t t e r h o w l a rg e o r r e m o t e i n s p a ce o r t im e . T h u s ,t h e r e a r e n e v e r a n y e d g e e f f e c t s .

A n o t h e r v i r tu e o f t he m a c r o - c e l l a p p r o a c h i st h a t a l a r g e S H O W o r R E S U L T c o m p u t a t i o nw h i c h i s a b o r t e d b e f o r e c o m p l e t i o n i s n o t w a s t e d ,f o r a ll o f th e i n te r m e d i a t e R E S U L T s h a v e b e e np e r m a n e n t l y r e c o r d e d i n t h e i r o w n e r s . T h u s , ar e s t a r t e d c o m p u t a t i o n w i ll v e r y q u i c k ly r e g a i n t h es t a t e w h e r e i t b r o k e o f f . L i k e w i s e , i n d e p e n d e n te x p e r i m e n t s o n t h e s e q u e n c e o f s i m i la rc o n f i g u r a t i o n s m a y a c c e l e r a t e a s t h e h a s h - t a b l ea c c u m u l a t e s " s m a r t " m a c r o - c e l l s .

D i r e c t g e ne r a li z at io n s o f th e R E S U L T m e c h -a n i s m e n a b l e t h e g e o m e t r i c t r a n s f o r m a t i o n , a g i n g ,a n d m e r g i n g o f l a r g e c o n f i g u r a t i o n s , a s w e l l a se x t e n s i v e r e p l i c a t i o n s i n a r b i t r a r y g r i d s .

3. ConclusionE v e n i n s i m u l a t i n g s u c h a n u n p r e d i c t a b l e a n di r re v e r si b le a u t o m a t o n a s L i f e , c o n s i d e r a b l e e c o n o -

m i e s a r e p o s s i b l e . B y a t t r i b u t i n g s i m i l a r t h r if t i n e s st o w h a t e v e r i m p l e m e n t s o u r o w n r e a l i ty , o u r ( s im -u l a t e d ) i m a g i n a t i o n s m a y b e s t i m u l a t e d .

AcknowledgementsI f o u n d t h e R E S U L T a l g o r i t h m w h i l e a t t h e

X E R O X P a l o A l t o R e s e a rc h C e n t er . I ts o n l yi m p l e m e n t a t i o n s o f a r h a s b e e n i n t h e e x c e p t io n a l l yp o w e r f u l F l a v o r L i s p m e c h a n i s m d e v i se d b y H o w -a r d C a n n o n ( o f S y m b o l i c s , I n c .) . T h i s is al s o t r u eo f t h e s u b s e q u e n t l y d i s c o v e r e d d is p l a y , g e o m e t r i ct r a n s f o r m a t i o n , a g i n g , m e r g i n g , a n d g r i d r e p l i -c a t i o n a lg o r i th m s . J o h n L a m p i n g o f S t a n f o r d U n i -v e r s it y s a v e d m e m u c h w o r k b y s u g g e s ti n g a b i n a r yq u a d r a n t s t r u c tu r e i n p l ac e o f t h e t e r n a r y s t r u c t u r ei n m y o r i g i na l p l a n. A n e a r l y m o d e l S y m b o l ic s3 6 0 0 r a n t h e p u f f e r t r a i n o f f ig s . 1 a n d 2 s e v e r a lm i l l i o n st e p s , a t a r a t e w h i c h d o u b l e d e v e r y t w o o rt h r e e m i n u t e s , o n c e t h e i n i t i a l e x p l o s i o n s e t t l e di n t o o s c i l l a t i o n .

F i g. 3 w a s p l o t t e d b y M I T M a c s y m a .

References[1] M . Gardn er, Sci. Amer. 223, No. 4 (1970) 120; 224, No. 2(1971) 112.[2] Berlekamp, Conw ay, & G uy, Winning Ways, AcademicPress, Vol. 2, 817-849. Contains design of a Turing m achinein Life.[3] M. G ardner, W heels, Life and Other M athematical Am use-men ts, (W .H. Freem an, San Francisco , 1983).[4] The Life rule: If two of you r eight nearest neighbors are on ,don't change . If three are o n , turn on . Otherwise, turn off.[5] The initial explosion, actually ust a puff reacting against thevacuum instead o f previous puffs, eventually decays to aharmless cloud of oscillators (of periods one, two and three)at step number 5533.