Reversible space-time simulation of cellular automata. J r me O. Durand-Lose 1

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1 Lortoire Bordelis de Reherhe en Informtique, ur nrs 304, Universit Bordeux I, 35, ours de l Li rtion, Tlene Cedex, rne. Rpport de Reherhe Num ro Reversile spe-time simultion of ellulr utomt J r me O. Durnd-Lose LBRI, ur nrs 304, Universit Bordeux I, 35, ours de l Li rtion, Tlene Cedex, rne. We riey rell the denitions of Cellulr utomt (), simultion, reversiility nd Prtitioned ellulr utomt (p) s dened y Morit. We ll the sequene of the iterted ongurtions of ongurtion spe-time digrm. We dene n emedding reltion etween spe-time digrms nd spe-time simultion reltion etween. We uilt spe-time simultion of ny y reversile p (r-p). inlly, we stte our min result: there re reversile le to spe-time simulte ny of the sme dimension. Key words: Cellulr utomt, spe-time simultion, intrinsi universlity nd reversiility. Introdution Reversiility orresponds to the onservtion of informtion nd energy. It llows unmiguous ktrking. In omputer siene, reversile is studied in order to design omputers whih would wste less energy. In 973, Bennett [] proved tht ny Turing mhine n e simulted y nother one whih is reversile. Reently, Morit [5] proved tht ny twoounter utomt n y simulted y reversile one. Cellulr utomt () model mssively prllel omputtion nd physil phenomen. They work over mtries of innite size ut nite dimension; the underlying lttie is Z d. The elements of the mtries re lled ells. Eh ell tkes vlue from nite set of sttes S. A jdurnd@lri.u-ordeux.fr, Preprint sumitted to Elsevier Preprint Otoer 997

2 ongurtion is vlution of whole mtrix. A updtes ongurtion y synhronously hnging the stte of eh ell ording to the sttes of the ells round it nd lol funtion f. All ells use the sme lol funtion. This is prllel, synhronous, lol nd uniform proess. A is reversile when its glol funtion G is ijetion nd its inverse G? is it-self the glol funtion of some. Reserh on reversile (r-) egun in the 60's: Moore [] nd Myhill [6] proved tht if G is one-to-one then it is ijetion. Hedlund [6] nd Rihrdson [7] proved tht ny funtion over S Zd whih ommutes with ny shift nd whih is ontinuous for the produt topology is the glol funtion of some. As onsequene, for ny it is enough to e one-to-one to e reversile. In 97, Amoroso nd Ptt [] proved tht reversiility is deidle in dimension. In 990, Kri [9] proved tht it is not deidle ny more in dimension nd ove. In 977, Tooli [8] proved tht ny n e simulted y reversile (r-) one dimension higher nd tht there re r- of dimension nd ove whih re omputtion universl. It ws only in 99 tht the existene of omputtion universl r- ws proved in dimension y Morit [3]. To do it he introdued Prtitioned ellulr utomt (p). or p, only some prt of the stte is send to neighoring ells. Immeditely, p re nd reversile p (r-p) re r-. On the other wy round, p (r-p) re le to simulte ny (r-) [4]. In 995 the uthor [3] proved tht there re r- le to simulte ny r- of the sme dimension (greter thn ), over ny ongurtion (nite or not), in liner time. This result hs een extended to dimension in 997 [4] with the use of p. Yet, it is unknown whether it is possile to simulte ny with r- of the sme dimension. In 995, Morit [4] proved tht it is possile over nite ongurtions, i.e. ongurtions suh tht there exists some stte q suh tht there is nite numer of ell whih re not in stte q. inite ongurtions form strit suset of reursive ongurtions whih is it-self fr from eing the whole set of ongurtions. initeness is lso too restritive for physiins nd mthemtiins. Generlly, to simulte mens tht, up to some enoding, the result orresponds to ny possile initil ongurtion of the simulted mhine. On itertive systems, for indution purposes, one wnts ny itertion of the simulted mhine to e totlly enoded in one itertion of the simulting mhine. Generlly, it is not formlly onsidered tht simulted itertion might e enoded over vrious, mye n innite numer of, simulting itertions. In the theory of omputtion, one speks out the simultion of mhine/utomton/rewriting system/... y nother nd denes equivlene mong progrmming systems. Intrinsi universlity inside lss is dened s the ility to simulte ny mhines of the lss. The tion of mhine is generlly dened y indution. A mhine simultes nother if it goes through the sme steps of omputtion, whih is more thn just yielding the sme result.

3 There re vrious wys to ompute with. Input nd result of omputtion re usully enoded in portion of the initil nd nl ongurtions. Considering the suession of ongurtions, dt n e set nd result retrieved in sequentil or in prllel (see [] for disussion out this). One n dene lnguge to e the words suh tht given ell, or ny ell, enters given stte. Some uthors onsider the spe-time mtrix s tool for onstrution. Heen [7,8] developed sttistil positioning to elerte omputtions. Mzoyer [] uses dynmil genertion of ltties to led omputtion inside digrms. Other uthors onsider the whole spetime digrm (or orit of ongurtion) s the result nd not just the nl ongurtion. untions onstrutile or isher-onstrutile [5,0] orrespond to geometril properties of spe-time digrms. Construted ojets re not given in the output ut onstruted through the itertions. The whole digrm hold the result or the proof of orretness. Bsed on these oservtions, we dene n emedding reltion etween spe-time digrms. We sy tht A spe-time simultes nother B when ny spe-time digrm generted y B n e emedded in one generted y A. Within this denition, we prove tht there re r- whih re le to spe-time simulte ny of the sme dimension, reversile or not. We prove this theorem in dimension with onstrution of r- whih progressively genertes ny digrm of given on ny strting ongurtions. This onstrution is sed on the the movements of signl on limited portion of the ongurtion. When the signl goes towrd the enter of the portion, it moves over more nd more iterted ells tht it updtes. When it goes wy from the enter to order, it moves over less nd less iterted ells without modifying them. The rtile is rtiulted s follows. The denitions of Cellulr Automt (), Prtitioned (p) nd reversiility re gthered in set.. In set. 3, we rell the usul denition of simultion nd explin how to simulte ny y of neighorhood f0; g d, then spetime digrms nd spe-time simultion re dened. In set. 4, we onstrut spe-time simultion of ny -dimensionl (-) y some -r-p. Denitions Congurtions re innite mtries of nite dimension d. Points of ongurtions re lled ells. Eh ell tkes vlue from nite set of sttes S. The set of ll ongurtions is denoted C (C = S Zd ). Cellulr utomt () nd Prtitioned ellulr utomt (p) updte ll the ells of ongurtion in prllel nd synhronous wy. 3

4 . Cellulr utomt A d-dimensionl Cellulr utomton (d-) is dened y (S; N ; f ). The neighorhood N is nite suset of Z d. The lol funtion f : S N S yields the new stte of ell in funtion of the sttes of the ells in its neighorhood. The glol funtion G : C C updtes ongurtions s follows: 8 C; 8x Z d ; G() x = f ( x+ ) N : The new stte of ell only depends on the sttes round it.. Prtitioned ellulr utomt Aording to Morit's denition [3,4], d-dimensionl Prtitioned ellulr utomton (d-p) is dened y (S; N ; ). The set of sttes is produt of sets indexed y the neighorhood, i.e. S = Q N S (). The omponent of stte s is denoted s (). The funtion opertes over S. The lol funtion f is dened y: 8 C; 8x Z d ; f () x = Y N () x+ : Equivlently, eh stte is the produt of the informtion to e send round. Eh omponent is send to only one ell. The funtion uses wht is left nd wht is reeived. The ell only keeps prtil knowledge out its own stte nd only reeives prtil knowledge out the sttes of the ells in its neighorhood, s depited in the right prt of g.. R? R? R? R? R G? R? R? R? R? R ig.. Updting of nd p..3 Reversiility A (p) is reversile if its glol funtion G is ijetion nd its inverse G? is the glol funtion of some (p). We denote r- (r-p) reversile (p). or p the following lemm is true in ny dimension. Lemm (Morit) A p is reversile if nd only if its funtion is permuttion, whih is deidle. 4

5 PROO. If is permuttion then the inverse p is Q?N S (?) ;?N ;? where?n = f?j N g. The tion of is undone nd the dierent piees re send k to where they me from. Otherwise sine works over nite set, it is not one-to-one. It is esy to onstrut two ongurtions whih re mpped in the sme ongurtion. Deidility omes from the niteness of S. In 990, Kri [9] proved tht the reversiility of of dimension greter or equl to nd ove is not deidle. As fr s reversiility is onerned, nd p fundmentlly dier. 3 Simultion 3. Itertive pproh Cellulr utomt itertively updte ongurtions. We ll ny ongurtion generted y nite numer of itertion over ongurtion n iterted imge. or ny initil ongurtion, one wnts to nd eh iterted imge of y the simulted A wholly enoded inside n iterted imge of ongurtion e y the simulting B. The denition of Tooli [8] is: Denition A A simulte (itertively) nother B if there exist three funtions : C B N N, : C B C A nd : C A C B suh tht: 8 C B ; 8t N; G t (;t) B () = G A () : The funtions, nd must e of lower omplexity thn the simulted B in order to insure tht they re not doing the omputtion. Generlly nd re projetions or injetions. When is xed, (; t) my e undened for mny t s long s it is dened for n innity of t. This is required to llow speed-up: to simulte 3n itertions with n itertions, there re n itertions whih nnot e dened. A simultion is in liner time if (; t) = t for ny ongurtion. If = the simultion is in rel time. Sine d-p re d-, they n e simulted in rel time y d-. Identilly, d-r-p n e simulted y d-r-. In [4], there is onstrution of simultion of d- (d-r-) y d-p (d-r-p) in liner time. The following lemm gives n exmple of simultion. Lemm 3 Any d- n e simulted y d- with neighorhood f?; 0; g d in rel time. 5

6 N = f?; 0; g R? R? R? R? R? R? ig.. Grouping ells y. PROO. The ells re gthered in loks of djent ells. Let B = (S B ; N ; f B ) e ny d-. Let r e the rdius of B, i.e. the mximum solute vlue of ll the omponents of ll the elements in N. Let A = (S rd ; f?; 0; B gd ; f A ). The funtion is dened y (() x ) i = rx+i, then =? nd f B is dened y indution from f A s illustrted in g. for N = f?; 0; g. A simultes B in rel time. 3. Spe-time pproh Denition 4 A spe-time digrm A is the sequene of the iterted imges of ongurtion y. Put it dierently, let G e the glol funtion of some d- A nd ongurtion ( S Zd ). The ssoited spe-time digrm A : Z N d S is dened y A x;t = (G t ()) x. It is denoted (G; ) or (A; ). A spe-time digrm B is inserted inside nother spe-time digrm B when it is possile to `reonstrut' B from A nd the wy tht B is emedded inside A. To reover n emedded B-ongurtion eh ell hs to e tken t given itertion. A A-ongurtion is thus onstruted. This A-ongurtion is deoded to get n iterted ongurtion for B. More preisely, we dene this s follows: Denition 5 A spe-time digrm B = (B; ) is inserted in nother spe-time digrm nd : C A C B A = (A; ) when there exist three funtions : Z d N N, : C B C A suh tht: - = (); - 8(x; t) Z d N, let t e the ongurtion of A suh tht A t = x x;(x;t) ; - 8t N, G t B () = (t ). The ongurtion is enoded into d ording to. To reover n iterted vlue of, the funtion indites whih itertion is to e onsidered for eh ell nd deodes the generted ongurtion. 6

7 As efore, the funtions, nd must e of lower omplexity thn the ones of the digrms nd t my e undened for mny t s long s it is dened for innitely mny t. Denition 6 A A spe-time simultes B when ny spe-time digrm generted y B n e inserted inside spe-time digrm generted y A nd ll insertions use the sme funtions nd. The funtion my depend on the initil ongurtion of the inserted digrm. If it only depends on the time nd the initil ongurtion, it is n itertive simultion. This new simultion reltion is n extension of the former. We refer the reder to the simultion in the next setion nd the simultion in [7] for exmples. 4 Spe-time simultion y reversile In this setion we give n expliit onstrution to prove the following lemm: Lemm 7 Any d- with neighorhood f?; 0; g d n e spe-time simulted y d-r-p. PROO. The proof is only detiled in dimension. We generlize it to ny dimension t the end. We explin how the simultion works efore going deeper nd deeper into detils. 4. Mro dynmis Let B=(S B ; f?; 0; g; f ) e ny - with the given neighorhood. We uild -r-p P = (S P ; f?; 0; g; ) whih spe-time simulte B. Let e ny ongurtion in C B nd B the ssoited spe-time digrm. We explin how the simulted digrm B is inserted into the digrm P generted y P nd n initil ongurtion p. The ide is to updte only nitely mny ells eh loop. Thus only nite prt of the ongurtion is not t itertion 0. Inside it, the loser to the enter ell is the higher its itertion numer is. As itertion goes y, this prt is inresing on oth sides (spe) nd in itertion numers (time). The simulted digrm B is generted ording to digonl lines, one fter the other. The updting lines of B re depited in gure 3 where the numers, the rrows nd the geometril symols on the lst olumn orrespond respetively to the order in whih updtes re mde, to their diretions nd to the identitions of the B-itertions (s on P in g. 4). 7

8 Itertion # N R R R R R Z -? The symol in the lst olumn identies the itertion. It orresponds to the insertion in g. 4. ig. 3. Order of genertion inside the simulted digrm B. The stte of ell x t itertion t in the inserted digrm B is denoted x n t ( x n t = G t B () x) nd the informtion needed to ompute x n t is denoted [ x n t ] ([ x n t ] = ( x? n t? ; x n t? ; x+ n t? )). Eh time ell is updted, [ x n t ] is generted to keep the dt needed for undoing the updte. Reordings of [ x n t ] re umulting. They nnot e disposed o euse G B is not one-toone nd the previous ongurtions nnot e guessed from the tul one. These needed ut umersome dt re evuted on oth sides of the ongurtion. A nite prt of the P-ongurtion represents B-ells whih re not in the initil stte, i.e. the ongurtion. We ll this prt the updting zone. The onstrution is driven y signl moving forth nd k on the updting zone. When signl goes from the left to the right for the n th time s in g. 4, its dynmis re s follows: Strting from the fr left of the updting zone, the rst ell enountered y the signl holds [ x n ]. The signl sets this dt moving to the left to evute nd sve it while it genertes xn. The next ell holds [ x n ] whih is lso set on movement to the right while x n is generted. This goes on until the signl rehes the middle of the updting zone (vertil line), then no more updting is done until the signl rehes the right end. On its wy k, the signl updtes the other hlf of the updting zone The signl mkes n updtes one wy nd n updtes on its wy k. Then it mkes n + nd n + updtes, then n + nd so on. The ells orresponding to the itertion (, 3 nd 4 respetively) in B re generted on n hyperol indited y irles (tringles, lozenges, squres respetively) on the simulting digrm P in g. 4. This orresponds to the lyers onstrution of B depited in g. 3. igure 4 depits the evution of the [ x n t ] wy from the updting zone for the rst 00 itertions. Evuted dt never intert. N 4. Miro dynmis Let us go into the detils of how this mehnism works. Cells re orgnized in three lyers: the upper lyer holds the stte of the simulted ell, the middle one holds signls driving the dynmis nd the lower one ts like onveyor elt to evute the [ x n t ]. 8

9 Itertions or time? ig. 4. Sheme of the evution of dt([ x n t ]) on the simulting digrm P. The rst 6 itertions re depited in g. 5. In the upper lyer, the ells lterntively holds 3 times the sme stte ( x n t ) or the sttes of the ell nd its two losest neighors t the sme itertion ( x? n t? ; x n t? ; x+ n t? ), otherwise some mix over or 3 itertions. A ell n only e updted when it hs the informtion [ x n t ]. By indution from g. 5, the possiility to updte ell only depends on the prity of the sum of simulting nd simulted itertion numers (the full demonstrtion is esy ut very long euse of the mny ses to onsider). Let us dene the signl whih leds the dynmis. We ll it the suit signl. It is the tres of the su-sttes,, nd in P. The suit signl only moves forth nd k in the updting zone nd thus ppers s zigzg on gures 4 nd 5. It is delyed y one on the left side to keep synhronism with the presene of [ x n t ]. The updting zone is delimited y pir of nd its middle is indited y. The progressively move wy from eh other while the osilltes in the middle. Strting on the left, the suit signl is. While pssing, it mkes the simulted updtes of the ells until it rehes. Afterwrds it is nd just moves to the other. Eh time simulted updte is done, three vlues, x? n t?, x n t? nd x+ n t?, re `used up' nd eome useless. They re gthered in [ x n t ] nd moved to the lower lyer to e evuted. Three opies of the new stte x n t re mde. They will e used for the next updte of the simulted ell nd of its two losest neighors. 9

10 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 n 0 3n 0 3n 0 3n 0 4n0 4n 0 4n 0 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 n 0 3n 0 4n 0 4n 4n 4n 5n 0 5n 0 5n 0 6n 0 6n 0 6n 0 7n 0 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 4n 0 5n 0 6n 0 5n 0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 Time? 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 n 0 3n 0 3n 0 4n 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 0n 0 0n 0 0n 0 n 0 n 0 n 0 0n 0 0n 0 n 0 [4n ] [4n ] [4n ] [4n ] 4n 0 4n 4n 0 4n 5n 0 5n 0 6n0 6n 0 6n 0 7n 0 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 [4n ] n 0 3n 0 4n 0 4n 4n 4n 4n 0 5n 0 6n 0 n 0 n 0 n 0 3n 0 3n 0 4n 4n 0 4n 4n 0 4n 5n 0 5n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 n 0 3n 0 4n 0 4n 4n 4n 5n 5n 5n [5n ] 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 n 0 3n 0 3n 0 4n 4n 0 4n 5n 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 n 0 3n 0 4n 0 4n 4n 4n 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 n 0 3n0 3n 0 4n 4n 0 4n 5n 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 3n 3n 3n [3n ] 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 3n 0n 0 0n 0 n 0 0n 0 n 0 n 0 [3n ] [3n ] n 0 n 0 3n 0 3n 3n 4n 4n 4n 4n 3n 0 3n 4n 4n 4n 4n [4n ] [4n ] 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 3n 3n 0 3n 4n 4n 4n 4n 4n 5n 5n 0 [3n ] [4n ] 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 3n 3n 4n 4n 4n 4n 4n 5n 5n [4n ] 0n 0 0n 0 0n 0 [4n ] n 0 n 0 n 0 n 0 n 0 3n 3n 0 3n 4n 4n 4n 4n 4n 5n 5n 0 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 3n 3n 4n 4n 4n 4n 4n 5n 5n 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 3n 3n 0 3n 4n 4n 4n 4n 5n 5n 5n [5n ] 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 3n 3n 4n 4n 4n 5n 0n 0 0n 0 0n 0 n 0 n 0 n 0 n 0 n 0 3n 3n 0 3n 4n 0n 0 0n 0 n 0 0n 0 n 0 n 0 n 0 n 0 3n 0 5n 0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 6n 0 6n 0 6n 0 4n 5n 5n 0 5n 6n 0 6n 0 7n 0 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 5n 0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 [5n ] 5n 5n 5n 5n 0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 [5n ] 4n 5n 5n 0 5n 6n 0 6n 0 7n 0 7n 0 7n 0 8n 0 8n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 [5n ] 9n 0 9n 0 9n 0 5n 5n 5n 5n 0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 4n 5n 5n 0 5n 6n 0 6n 0 7n 0 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 [5n ] 4n 4n 4n 4n 5n 5n 5n0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 4n 4n 4n 3n 3n 4n 4n 4n 5n 0n 0 0n 0 0n 0 n 0 n 0 n 0 n0 n 0 3n 3n 0 3n 4n 4n 4n 4n 0n 0 0n 0 n 0 0n 0 n 0 n 0 n n n [n ] 0n 0 0n 0 0n 0 n 0 n 0 n 0n 0 0n 0 n 0 [n ] [n ] 0n 0 0n 0 0n 0 n 0 n 0 n 0n 0 0n 0 n 0 [3n ] 0n 0 n 0 n 0 n n 3n [3n ] n 0 n 3n 3n 3n 3n [3n ] [3n ] 0n 0 n 0 n 0 n n 3n n 0 n 3n 3n 3n 4n 3 [4n 3 ] 3n 3n 4n 4n 4n 5n 4n 4n 4n 3n 3n 4n 4n 3 4n 3 4n 3 [4n 3 ] [4n 3 ] 5n 6n 0 6n 0 7n0 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 5n 0 6n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 5n 6n 0 6n 0 7n 0 7n 0 7n 0 6n 6n 6n [6n ] 4n 5n 5n 5n 6n 6n 5n 6n 6n 0 6n 7n 0 7n 0 [5n ] 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 [6n ] 5n 5n 5n 5n 6n 6n 0 6n 7n 0 7n 0 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 [5n ] 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 [6n ] 4n 5n 5n 5n 6n 6n 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 [5n ] [6n ] 8n 0 9n 0 9n 0 5n 5n 5n 5n 6n 6n 0 6n 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 4n 5n 5n 5n 6n 6n 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 5n 5n 5n 5n 6n 6n 0 6n 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 4n 5n 5n 5n 6n 6n 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 [5n ] 4n 4n 3 4n 4n 3 5n 5n 5n 6n 6n 0 6n 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 3n 3n 4n 4n 3 4n 3 4n 3 4n 5n 5n 0n 0 0n 0 0n 0 n 0 n 0 n n 0 n 3n 3n 3n 4n 3 4n 4n 3 4n 4n 3 5n 5n [4n 3 ] 0n 0 0n 0 n 0 0n 0 n 0 n 0 n n 3n 3n 3n 4n 4n 3 4n 3 4n 3 4n 5n 5n [4n 3 ] 0n 0 0n 0 0n 0 n 0 n 0 n n 0 n 3n 3n 3n 4n 3 4n 4n 3 4n 4n 3 5n 5n 0n 0 0n 0 n 0 0n 0 n 0 n 0 n n 3n 3n 3n 4n 4n 3 4n 3 4n 3 4n 5n 5n 0n 0 0n 0 0n 0 n 0 n 0 n n 0 n 3n 3n 3n 4n 3 4n 4n 3 4n 4n 3 5n 5n 5n 6n 6n 6n0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 5n 6n 6n 0 6n 7n 0 7n 0 8n0 8n 0 8n 0 9n 0 9n 0 9n 0 5n 6n 6n 6n 0 7n 0 8n 0 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 5n 6n 6n 0 6n 7n 0 7n 0 8n 0 8n 0 8n 0 9n 0 9n 0 9n 0 5n 6n 6n 7n 7n 7n [7n ] 7n 0 8n 0 9n 0 8n 0 9n 0 9n 0 6n 6n 6n 6n 7n 7n 0 7n 8n 0 8n 0 9n 0 9n 0 9n 0 [6n ] [7n ] ig. 5. The rst 6 itertions of the spe-time simultion. The endless movement of the suit signl nd updtes (t the right prities) n e dedued y indution. Sine the intertion is only lol nd hs rdius, glol properties re not otherwise modied. All the neessry steps for the indution n e found on the two nd hlf loops of the suit signl in g. 5. 0

11 4.3 Lol funtion The set of sttes of P is detiled in g. 6. If the B hs m sttes, P hs 00 m 3 (m 3 + ) sttes. This represents ig inresing in the size of the tle of the lol funtion nd in omplexity. 0 - S B S B S B S 3 B [ f g _ S3 B [ f g ig. 6. Set of sttes of P: S P. Cells re depited s 3 3 mtries s in the rst line of g. 5. The upper lyer holds the sttes of P; ongurtion of B is enoded there s depited in g. 4. The middle lyer holds the suit signl nd the lower lyer is used to store the dt wy from the updting zone. The suit signl is lterntively equl to nd. While shifting, updtes ells while does nothing. The signl eomes nd to move respetively nd. equls ` ', `' or. d d d d d d with d = f (; ; ) nd = (; ; ). ( dely) d d d d d d with d = f (; ; ) nd = (; ; ).,, nd d elong to S B, nd elong to S 3 B [ f g. ig. 7. Denition of.

12 The trnsition rules re given in g. 7. The rst rule orrespond to the lk of ny signl. On the lower lyer, the two vlues on the side re swpped, this ts like onveyor elt. As soon s something is put on the lower lyer, it is shifted y one ell t eh itertion. This is used to evute the umersome dt. The rules whih orresponds to the updting re on the lines nd 5. The seond nd third lines of g. 7 depited how moves to the right nd updtes ells. When it rehes the middle frontier, it moves it one step to the right s nd turns to. Let us detil how the signl turns on the right side, s depited on the fourth line of g. 7. On rriving on from the left, gr it nd turns to. On the next itertion, turns to nd does nothing else. This is the dely of itertion needed to keep up with prity. Next itertion, regenertes the nd the signl whih goes k to the left. The signl turns k one itertion fster on the left side s depited on the lst line of g. 7. The stte does not pper. The dened rules re one-to-one, thus they n e ompleted so tht is permuttion, B is then reversile (lem. ). The initil ongurtion is depited on the rst line of g. 5. The stte of eh ell is opied thrie in the upper lyer. Mrkers, nd re lid in the enter of 3 djent ells nd the is together with the left. 4.4 Generliztion This onstrution n e generlized to ny dimension greter thn. The ent of the spe-time digrm is lwys done on the rst diretion. On this diretion, the dynmis re extly the sme s explined ove. On the other diretions, the sme spe-time lotion nd signls re found. The updting is still onditioned y the prity of the sum of the numers of the itertions. There is n innity of, nd suit signls. They re rrnged on hyperplnes orthogonl to the rst diretion nd re extly synhronized. Theorem 8 Any d- n e spe-time simulted y d-r-. PROO. Any d- n e simulted in rel time y d- whih neighorhood is f?; 0; g d (Lemm 3). The theorem omes from from previous lemm 7 nd the ft tht d-r-p re d-r-.

13 5 Conlusion We hve proved tht ny n e spe-time simulted y reversile. There re d-r- le to simulte (itertively) ll d-r- over ny ongurtion [4]. Theorem 9 There re d-r- le to spe-time simulte ny d-. Unfortuntely, with the onstrution we gve, the inserted spe-time digrm is ent in.. This mkes it very hrd to ess geometril properties like isher-onstrutiility. In our spe-time simultion, it is not possile to go kwrd efore the rst ongurtion if no previous ongurtion ws previously enoded in the initil ongurtion. This would yield n too omplited nd moreover, there is no insurne tht there exists ny previous ongurtion t ll. In our onstrution, n innite time is needed to fully generte the ongurtion fter one itertion. When the signint prt of ongurtion represents only nite prt of the spe, the result of the omputtion is given in nite time. But this is not the se for the verge ongurtion. Spe-time simultion must hve dierent properties tht the usul simultion euse it is lrger reltion. Our denition keeps the lolity of the informtion proessing ut is not shift invrint. It would e interesting to know up to wht extend the tehniques nd results of this rtile n e dpted to the se where must e ounded when t is xed. or exmple, if they is n integer suh tht 8x; y; 8t; j(x; t)? (y; t)j, we elieve tht there is some itertive simultion vi some kind of grouping. Our onstrution relies on the inniteness of the spe to store dt for reversiility. It does not work on limited spe like torus. We do not know how to extend our result to suh underlying ltties. Referenes [] S. Amoroso nd Y. Ptt. Deision proedure for surjetivity nd injetivity of prllel mps for tesselltion struture. Journl of Computer nd System Sienes, 6:448464, 97. [] C. H. Bennett. Logil reversiility of omputtion. im Journl of Reserh nd Development, 6:5553, 973. [3] J. O. Durnd-Lose. Reversile ellulr utomton le to simulte ny other reversile one using prtitioning utomt. In ltin'95, numer 9 in Leture Notes in Computer Siene, pges Springer-Verlg,

14 [4] J. O. Durnd-Lose. Intrinsi universlity of -dimensionl reversile ellulr utomton. In sts'97, numer 00 in Leture Notes in Computer Siene, pges Springer-Verlg, 997. [5] P. C. isher. Genertion of primes y one-dimension rel-time itertive rry. Journl of the m, (3):388394, 965. [6] G. A. Hedlund. Endomorphism nd utomorphism of the shift dynmil system. Mthemtil System Theory, 3:30375, 969. [7] O. Heen. A liner speed-up theorem for ellulr utomt synhronizers nd pplitions. To pper in Theoretil Computer Siene. [8] O. Heen. Eonomie de Ressoures sur Automtes Cellulires. PhD thesis, litp, ip, Universit Pris 7, 996. In renh. [9] J. Kri. Reversiility of D ellulr utomt is undeidle. Physi D, 45:379385, 990. [0] J. Mzoyer. Signls in one dimensionl ellulr utomt. Tehnil Report 94-50, lip, ens Lyon, 46 ll e d'itlie, Lyon 7, 994. [] J. Mzoyer. Computtions on one dimensionl rrys. Annls of Mthemtis nd Artiil Intelligene, 6:85309, 996. [] E. Moore. Mhine models of self-reprodution. In Proeeding of Symposium on Applied Mthemtis, volume 4, pges 733, 96. [3] K. Morit. Computtion-universlity of one-dimensionl one-wy reversile ellulr utomt. Informtion Proessing Letters, 4:3539, 99. [4] K. Morit. Reversile simultion of one-dimensionl irreversile ellulr utomt. Theoretil Computer Siene, 48:5763, 995. [5] K. Morit. Universlity of reversile two-ounter mhine. Theoretil Computer Siene, 68:30330, 996. [6] J. Myhill. The onverse of Moore's grden-of-eden theorem. In Proeedings of the Symposium of Applied Mthemtis, numer 4, pges , 963. [7] D. Rihrdson. Tesselltions with lol trnsformtions. Journl of Computer nd System Sienes, 6:373388, 97. [8] T. Tooli. Computtion nd onstrution universlity of reversile ellulr utomt. Journl of Computer nd System Sienes, 5:33,