sitabhra sinha- phase transitions in the computational complexity of "elementary'' cellular automata

8/3/2019 Sitabhra Sinha- Phase Transitions in the Computational Complexity of "Elementary'' Cellular Automata

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Chapter 33P h ase T r an s i t i o n s i n t h eC o m p u t a t i o n a l C o m p l e x i t yof "E lem e n ta ry ' ' C e l lu l a rA u t o m a t a

Sitabhra Sinha^Center for Condensed Matter Theory, Department of Physics,

Indian Institute of Science, Bangalore 560012, IndiaandCondensed Matter Theory Unit, Jawaharial Nehru Center forAdvanced Scientific Research, Bangalore 560064, [email protected]

A stud y of the comp utat io nal complexity of determ ining th e "Garden-of-Eden"sta t es ( i .e. , s ta te s witho ut any preimages) of one-dimensional ce l lular au to m at a (CA)is repor ted. The work a ims to re la te phase transi t ions in the computat ional complexity of decision problems, with the type of dynamical behavior exhibited by the CAtim e evolution . Th is is m otiv ated by the observa tion of critical behavior in severalcomputationally hard problems, e.g. , the Satisfiability problem (SAT) [1]. The focusis on "legal" CA rules, i.e ., those which obey the quiescence and sym m etry con dition s[2]. A rela t ion exists between the problem of "Garden-of-Eden" s ta tes determinationan d th e SAT proble m . Several CA rules (e.g. , CA rules 4, 22 and 54) are stud ied indeta i l to establish the occurrence of phase transi t io n. Fini te-size scaling expo nentscorresponding to the cr i t ica l behavior are obta ined. Based on these exponents , a newqu an tita tiv e classification of "elem entary " cellular au to m at a into 5 classes is propo sed.

^Present address: The Institute of Mathematical Sciences, C I T Campus, Taramani, Chen-nai 600113, India. {E-mail: [email protected])


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3381 I n t r o d u c t i o nThe spontaneous emergence of complexity in various physical systems is one ofth e mo st rem arkab le features of th e na tur al world. Th ere have been severalattempts to define and quantify "complexity" (in the dynamical sense). Measures such as information entropy, mutual information, etc., have been usedw ith varying degrees of success. As yet, there do es not seem to be an u nam biguous indicator of the degree of complexity in a system. However, in the fieldof computer science, a measure of complexity, called computational complexitywhich measures the increase in system resources required to solve a problem asthe problem increases in size, has been well-established since the 1960s [3]. Alimitation of using this measure on actual physical systems is that it is definedonly for processes taking discrete state values, as well as evolving in discretespace-t ime [4j . Further, in order to s tudy the computational complexity of aprocess, we need to pose a non-trivial decision problem for the system in question . T he simp lest class of mo dels th at shows interesting dy nam ical behav iorand for which "hard " d ecision problem s can also be posed is th at of cellularautomata (CA). They exhibit these desired properties with the least number oftheo retical assu m ptio ns and the least num erical effort. It is interestin g to notethat some CA are also the simplest systems capable of universal computation[2].

In this paper, we have studied the computational complexity of one-dimensional cellular auto m ata. In part icular , we have observed ph ase transit ionsin the computational hardness of a decision problem in CA. Previous work hasindicated a relat ion between phase transit ion and the occurrence of computational complexity in other decision problems, such as the Satisfiability problem(SAT) [1] and the n um ber-pa rt i t ioning problem [5]. W heth er th e phase transit ion in relat ion to computational complexity has any connection to quali tat ivechanges in th e dynam ical behav ior of CA is of par ticu lar in terest. For at leastone of the CA we have studied, a relat ion exists between the computationalphase transition and the existence of a dynamical phase transition [6].In Section 2, we have defined cellular automata and described the particulartype of one-dimensional CA that we have considered for our study. This sectionalso discusses the computationally hard decision problem for CA which is thefocus of our numerical investig ations. Th e back -tracking algo rithm used forob tainin g m any of th e num erical results is discussed in Section 3. Section 4contains the results of the computational study, with details for four particularCA ru les. A new classification schem e of CA , involving 5 classes, ha s beenpropo sed in this section. Finally, in Section 5, th e relevance of this stu dy toother systems is discussed.2 C e ll ul ar A u t o m a t aCellular automata (CA) are simple lattice models of dynamics evolving in discrete state space at discrete time intervals. One-dimensional CA is characterized


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339by fc, th e nu m ber of stat es available to each au to m ato n (located a t each cell),and r , the radius of the neighborhood whose states determine the state of anautomaton at the next t ime step. The mapping between a part icular input, i .e . ,the state configuration of (2r + 1) cells at time step n, to a particular output (attime step n + 1), is determined by the CA rule governing the system evolution.Different rules give rise to a wide variety of dynamical behavior, from a spatiallyhomogeneous, t ime-invariant pattern, to spatial ly inhomogeneous, t ime-varyingpa tte rn s. Ex am ples of pa tte rn s gen erated by different C A rules are shown inFig. 1. In the following we shall consider k = 2,r = 1 CA (the so-called "elementary" CA). In particular, the "legal" rules will be considered, i .e., those rules forwhich the state configuration 000.. .000 is a null configuration (i.e., invariantunder application of the rule), and for which the rule gives identical output forsymmetrically related input.

Figure 1: CA time evolution for CA Rules 4, 22 and 54.Cellular automata models are studied as models for emergence of complexdynamical behavior. There have been some efforts at establishing the existenceof a phase transition from ordered to chaotic behavior with complexity emerging

at the transition [7]. One-dimensional CA have been classified morphologicallyinto 4 classes by Wolfram [2], w ith C lass IV correspo nding to th e m ost com plex,marked by the presence of coherent, propagating structures (e.g., "gliders") [8].Langton's work seemed to indicate that , by varying a parameter , one couldget rules in Class I and II (which have simple dynamics), and then rules inClass III (which are "chaotic") through a transition, where the Class IV rules


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are obtained. However, as dynamical "complexity" is not well-defined, this hasbeen subjected to criticism. As computational complexity is quite a well-definedcon cept, we have tried to use this concept to classify CA rules. Th e no tioninvolves how computation resources used for solving a computational problemscales with problem size [3].

Recently, it has been shown that several NP-complete problems show phasetransition behavior, such that, not all instances of these problems are uniformlyhard. Typically, they show a transition from easy to hard to easy instances as asystem parameter is varied [1]. This has motivated us to find similar behaviorin the case of CA.

Th e part icular com putatio nal question th at we shall be focusing on is wh ethera given random binary string is a "Garden-of-Eden" configuration, i .e., one whichhas no pre-image. This is the simplest query that is very hard to answer, butgiven th e answer, it is very easy to verify t he co rrectness of th e solution. T heprob lem of finding a pre-image for a length A^ sequence und er cellular au to m ato nevolution is in the class NP [9]. Therefore, the problem can be written in termsof a known NP-complete problem, e.g., the Satisfiability (SAT) problem.

The rules for Cellular Automata are normally expressed in terms of Disjunct ive Normal Form (DNF). However, to express them in terms of the iC-SATproblem, one must convert these into Conjunctive Normal Forms (CNF). Immediately, one notices that the pre-image formula for any of the elementary CAcan be expressed as a suitable combination of 1-SAT, 2-SAT and 3-SAT.Take for example Rule 22. The existence of 0 at the 2-th site implies thatthe pre-image x, if it exists, must satisfy the following condition: {^xi-i V x^ VXiî) A {xi-i V Xi V -^Xi+i), which is a 3-SAT with a = 3, while the presence of1 implies the logical expression (x^_i V xt V Xj+i) A {^Xi-i V -^x^-î) A (-"X^-iV-^Xi) A {^Xiî V ^Xi) , which is a mixture of 2-SAT and 3-SAT. For Rule 54,Zi = 0 ^ {^Xi-i V Xi) A {-^Xijî V -^Xi) A {xj-i V -^Xi V Xj+ i), an d, z = 1 =^{xi-i V x^ V Xiî) A {-^Xi-i V -^Xi) A (-"Xi-fi V -^Xi), which are a mix of 2-SA Tand 3-SAT. Note that, the existence of a pre-image for evolution under Rule 90can be shown to be equivalent to the existence of a satisfiable assignment for apu re 2-SAT ( a = 2) prob lem. Th is indicates th at th e prob lem is solvable using apolynomial time algorithm. Indeed, a simple linear-time algorithm can be usedto determine the existence of a pre-image under CA Rule 90 evolution whichtakes 4A^ steps in the worst case.

The CA system we have considered for our numerical studies is a chain ofN si tes , with periodic boundary condit ions. All s i tes in the lat t ice are updatedaccording to the same CA rule. A variable parameter p (0 < p < 1) is introduced,which gives the probability for each site to independently assume the value 1;else the site has value 0.3 B a c k t r a c k i n g A l g o r i t h mTo check if a randomly generated binary string has a pre-image under a givenrule is in gen eral a difficult tas k. For mo st (A: = 2 ,r = 1) CA ru les a simp le


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341algo rithm ex ists. However, in general, we have to use a mod ification of th eCA Reverse Algorithm [8]. This is a backtracking method where a possible pre-image is gradually constructed, while checking for consistency at each step of theconstruction. To find whether a given configuration C has a pre-image, a part ialpre-image P is constructed, where 2r bi ts , up to and including the start ing si tePi, is given. The next unknown bit to the right, Pj-i-i, is chosen consistent withthe CA rule-table. If this canno t be done, then Pi is rejected, and anotherpossible pa rtial pre-imag e is chosen. Th e process is continue d u ntil either, avalid pre-image sequence P has been found, or, all possible partial pre-imageshave been exhausted. Note that the choice of P^+i may not be unique, as manypre-images can lead to the same string C. If multiple alternatives are present,a branching is made at this point, and the process is continued along one of thebranches until a full, valid pre-image sequence is obtained. If at some point, thestr ing can no longer be continued, the algori thm returns to the branch point andresumes one of the remaining alternative choices. This procedure is continueduntil th e algorithm arrives at the bit P^_i p receding the star tin g bit. It is checkedw heth er th e periodic bou nd ary cond itions are satisfied. If no t, th e proce dureis begun anew with a new partial pre-image, unless all such alternatives havebeen exhausted; in which case, the string C is a Garden-of-Eden state. In theworst case, the algorithm will still have to go through 2^ possible pre-images,as is th e case with ex hau stive search. However, in most cases, the n um ber ofpossibilities to be checked is much sm aller in th e back tracking algo rithm . Th is isbecause a candidate string is quickly rejected if it is found that it can no longerbe continued.

4 R e s u l t sIn the numerical studies of phase transitions, the variable is p, the fraction ofIs in the given binary string whose pre-image (if it exists) is to be determined.The variable p can also be looked upon as the probability of occurrence of 1 atany given site within the string. The order parameter is the fraction of Gardenof Ed en (G OE ) configurations, / ( G ) . Th is is analogous to th e magnetizationparameter (M) observed in models of magnetic systems, while the variable insuch systems is the temperature (T). Note that , in such systems, when T ^ Tc{T c is the critical temperature, where the phase transit ion occurs) the orderparameter scales with the variable according to the relation

M^\T-Tc\^where /? is the scaling or critical exponent characterizing the phase transit ion.Another parameter that shows similar characteristic scaling relation is the correlation length (


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[10].Finite-size scaling is done by observing the order parameter for various sys

tem sizes, A^. In cases where a simple polynomial algorithm for finding pre-images could not be obtained, we used the backtracking algorithm described inthe previous section. Th e med ian num ber of com puta tional s teps required toobtain the solution using this algori thm gave a measure of the computationalcost involved in solving th e problem . Th is is analogo us to th e susceptibility(x) parameter observed in magnetic systems, which shows the following scalingrelation near the critical point:

In the following examples, we have obtained the value of 7 for CA Rule 22.Rule 0. This is possibly the simplest CA, where all input configurations giveth e ou tp ut 0. T he limit set is th e null configuration (. .. 000 .. .) which is reachedin one iteration starting from any initial string. It follows that all strings exceptthe null configuration are GOE states. Note that , under evolution with CA Rule0, any output string in which the block ' 1 * occurs anyw here is forbidden. So ' 1 ' isthe smallest irreducible forbidden word in the formal language generated by the

CA, or it is the 'kernel ' . As the probability of the non-occurrence of 1 anywherewith in a string of length A^ is (1 - p ) ^ , the fraction of G O E s tate s is given by/ ( G ) = 1-{1-pf ^Np

Fin ite-size scaling da ta for CA R ule 0 ob tain ed using sys tem sizes A^ = 100, 200and 4 00, showed th a t th e value of th e critical expo nen t, i/ = 1 (/5 = 0). Th isagrees with the simple argument described above which shows how /(G) scaleswith A^.Rule 4. Th is simp le rule only ou tp ut s 1 for th e inpu t triple t 010, all othe rin pu t sets resultin g in an ou tp u t of 0. W olfram classified it in Class 11. T h elimit set of this CA rule consists of configurations with only isolated Is, and

is achieved in one iteration from any initial configuration. In particu lar, anyconfiguration is either an attractor (i.e., invariant with respect to CA evolution)or a G O E configuration. T he decision problem of w heth er a pa rticu lar bina rystring h as a pre-imag e or not u nder CA R ule 4 evolution , can be decided bysimply iterating it once in the forward direction and checking whether the stringrem ains invariant un der CA evo lution. If it is indeed invarian t, then it is anattractor, and therefore its own pre-image; if not, it is a GOE configuration.Therefore, the algorithm for solving the decision problem is a simple linear timealgorithm, belonging to the class P.

Fig. 2 (left) shows the result of finite-size scaling for system sizes N = 100,200 and 400. The value of the finite-size critical exponent obtained from thisscaling is i/ = 2 (/? = 0 ). Th is is no t surp rising if we no tice th at un de r CARule 4, the block "U" is excluded. In other words, it is the smallest irreducibleforbidden word und er the formal language defined by CA Rule 4. Therefore,f{G) will grow as a function of Np"^ with higher order terms. Alternatively, theorder parameter increases as a function of py/N, which is observed numerically.


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punctuated by slowly moving defects. It has been conjectured that these defectsconvey information from one region of the CA to another.Although the dynamics seems to indicate that Rule 54 is in some sense a

marginal or critical CA, the decision problem is computationally easy to solvein this case. This is because the set of irreducible forbidden words for evolutionun de r R ule 54 is a finite set: 10110, 10101, 01101 and 1 0111101. To decidewhether a string has a pre-image or not, one simply has to check whether anyof th e above blocks occur anyw here with in the string. Therefore, th e prob lemcan be solved using a polynomial time algorithm. In Fig. 2 (right) one sees theresult of finite-size scaling which gives the value of the critical exponent z/ = 3(/? = 0).Rule 22. This rule is a member of the class of additive CA rules, as the rule

ou tp ut s a 1 if th e sum of th e bits in th e inp ut triplet is 1 (i.e., for the inp uttrip lets 00 1, 010 and 100); else it o ut pu ts 0. It has a uniq ue stable, invariantmeasure called the natural measure which is approached rapidly by any initialmeasure under evolution with CA Rule 22. The dynamical behavior of the CAis placed under Wolfram Class III, as the irregular spatio-temporal patternsgenerated by it are typical of the other members of this class.

In spite of the simple appearance of patterns created by this CA, non-tr iviallong-range effects have been reported, similar to critical phenomena [11]. Theresult ing patterns were shown to be neither periodic nor random, but somewh ere in between , due to th e presence of sub tle correlation s. La ter, ex tensiveMonte Carlo simulations have shown that the system behaves l ike a normal one-dimensional statistical ensemble with a critical point Pc = 0 [6]. As p > Pc,a critical slowing down to the asymptotic value of the average number of siteswith 1 is observed, h aving a dyn am ical critical exp one nt of unity.

The results of finite-size scahng can be seen in Fig. 3 (left). Note that unlikethe other rules for which results are given here, this CA has two finite criticalexponents, z/ = 4 and P ~ 1/3. T he latter exp one nt was zero for th e othe rCA, implying that, those were first order phase transitions, as the existence of anon-zero value of /3 is indicative of a second order phase transition. This can berelated to the fact that the set of irreducible forbidden words for this CA has aninfinite number of elements. The smallest of such excluded blocks are 10101001and 10010101. The number of such blocks or 'kernels ' increase rapidly with theblock length being considered [12], so th at , simple algo rithm s based on checkingfor the existence of such blocks cannot be devised to solve the decision problem.

The non-zero value of p can possibly be also related to the fact that thesequences generated by one iteration of Rule 22 do not constitute a finite-com plem ent regular lang uage [13], i .e., a regular langu age with a finite num berof excluded blocks. Note that these excluded blocks are the building blocks ofG O E state s. Fu rthe r, the finiteness of the num ber of excluded blocks wouldhave implied the existence of a computationally easy algorithm to identify GOEstates, namely, by simply checking for the existence of each of the words of theregular language. The presence of any one would mean that the configurationnecessarily lacks a predecessor and is therefore a GOE state.


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0 02 04 06 0.8 1 1.2 14 16 18 2

oO N = 100D N = eo0 N = 60A N = 40o

o

D O

0

Figure 3: (left) Finite-size scaling for fraction of Garden-of-Eden configurations forCA Rule 22 {/3 = l/3,iy = 4) and (right) the median computation cost for identifyingGOE configurations in CA Rule 22 using backtracking algorithm.

Fig. 3 (right) shows how the median computational cost of using the backtracking algorithm for identifying GOE states varies with p. The peak computational cost increases with N rapidly, while, the value of p at which the peakoccurs tends to p = 0. This suggests that , at the thermodynamic l imit , the peakcomputational cost will occur at Pc == 0- Finite-size scaling of the computationalcost data shows that the ratio of the critical exponents J = 2. This implies thatthere is no anomalous dimension for this system.

The above examples gave a detailed illustration of the nature of the phasetransit ion observed in these systems. Th e same analysis can be extended tooth er CA rules. In fact, all th e 32 one-dim ensional "legal" CA rules have acharacteristic value for the critical exponent i/, which fall in one of the followingclasses:z/ = 1: R ules 0, 94, 122, 126, 200, 222, 250 and 254.iy = 2: Rules 4, 32, 36, 128, 160, 178, 182, 232, 236.1/ = 3: Ru les 50 , 54, 72, 76, 108 , 132, 218 .iy = i: Rules 22, 104, 146, 164.z/ -^ oo: Rule s 90, 150 , 204.This suggests an extremely simple quantitative classification of one-dimensionalCA into Gve classes.

5 Discuss ionThe Wolfram classification scheme for one-dimensional CA, although the mostwell-known one, has fundam ental prob lems. No t only is it entirely q ualita tivein nature, but it is also extremely dependent on the choice of the initial configuration. Ba sed on W olfram's w ork, Culik an d Yu [15] ha d sugge sted a m orespecific definition of the four classes of CA according to their behavior on finiteconfigurations. It was shown th at t he m em bersh ip of a CA to a class in thisscheme is formally undecidable in general, for each of the four classes.


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Other classification schemes have also been proposed, which include, theclassification according to me an field app rox ima tion [16], th e types of pre-imageformu la [17], th e prop ertie s of finite sequen ces with ou t pre-im ages [12], etc . T heclassification proposed in this paper is the simplest quantitative scheme amongall thos e present in th e literatu re so far. Fu rthe rm ore , the nu m ber of classescomes out naturally from the value of the exponents , rather than being imposedarbitrarily, as is the case for most such schemes. The connection with criticalityalso makes the scheme appealing.

The relation of the classification on the basis of computational hardness toth e dy nam ical behavio r of a CA is not obvious. In fact, th e conn ection ofdynamics to the existence of Garden-of-Eden states have been explored before[18], wh ere th e concept of a "topo logical skeleton" was used to relate th e two . CArules which exhibit mostly skeletal structure in their state-transition graph (e.g.,Rule 30) generally show chaotic behavior while having very few GOE states. Atthe other end, rules which have mostly surface structure (e.g.. Rule 4) show moreordered b ehavio r and also have mo st configurations as G O E states . However,it remains an open question as to whether this suggested relation between thestate-transit ion graph structure and the dynamical propert ies hold generally inthe space of cellular automata.

Note that , a very interest ing connection can be made between the dynamicaland computational aspects of CA, through mapping them to equivalent neural network models. This can also be linked to Kaneko's work on informationtheory for mult i-at tractor dynamical systems, with focus on one-dimensionalcellular automata [19]. When viewed from the dynamical systems point of view,the two aspects of information processing that are important are informationgeneration and information storage. A dyn amical system with chaos can beviewed as an info rm ation sou rce beca use it amplifies microscopic fluctuationsinto macroscop ic informa tion. Inform ation can be stored in the large num berof at tractors which are usually found in spatial ly extended dynamical systems.An example is the Hopfield neural network model of associative memory [20],whose mult iple at tractors are used to s tore a large number of patterns, whichcan be recalled when the network is given a partial or corrupted pattern as inpu t. From this viewpoint of creation an d storage of inform ation, a CA can beplaced in one of four classes, the classification being essentially the same as thatof W olfram. It will be interesting to see w heth er a similar connection can b edrawn between the classification scheme proposed above and the properties ofinformation generation and storage by CA, in view of the connection to neuralnetwork models .

The problem of characterizing the complexity of CA through computationalhardness might have relevance to the question of applying CA for secure communication , i .e., for public-key encryp tion . Th is kind of app lication requires th e keyto be gen erated in such a m ann er th at is very easy to code bu t extrem ely difficultto decode. I t is obvious that NP problems are good candidates for generatingsuch keys. CA have been proposed as a possible mechanism for generating suchkeys. However, for successful application the decoding problem must not only


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be very hard to solve in the worst case, but should also be hard in the generalcase.

The scheme of using computational hardness as a measure for complexitycan also have implication for other spatially extended dynamical systems. Thenext higher step is to characterize the complexity of couple map lattices (CML).However, this will involve having a proper definition of complexity classes forcomputation over real numbers [21]. It will be interesting to see whether broaduniversality classes will emerge, encompassing a large variety of dynamical systems.

A c k n o w l e d g m e n t sI would l ike to thank Chandan Dasgupta, Vital ly Mykhaylovskyy, Peter Grass-berger, Andreas Engel and Bikas Chakrabarti for helpful discussions. Financialsupport from JNCASR is acknowledged.

B i b l i o g r a p h y1] MoNASSON, R ., R . Z E C C H I N A , S . K I R K P A T R I C K , B . S E L M A N a n d L . T R O Y -

ANSKY, "Determining computational complexity from characteristic 'phasetransit ions ' ", Nature 400 (1999), 133-137.

2] W O L F R A M , S . , "Statis t ical Mechanics of Cellular Automata", Rev. Mod.Phys. 55 (1983), 601-644.

3] G A R E Y , M . R . , and D. S. J O H N S O N , Computers and Intractability, W. H.Freeman (1979).

4] Recently, a complexity theory for computation over real numbers has beendeveloped. See, e.g., L. B L U M , F . C U C K E R , M . S H U B and S. S M A L E , Complexity and Real Computation, Springer Verlag (1998).

5] M E R T E N S , S . , "Phase transit ion in the number part i t ioning problem", Phys.Rev. Lett. 81 (1998), 4281-4284.

6] Z A B O L I T Z K Y , J. G., "Critical properties of rule 22 elementary cellular automata" , J . S ta t . Phys. 50 (1988), 1255-1262.

7] L A N G T O N , C . G . , "Computation at the Edge of Chaos: Phase Transit ionsand Emergent Computa t ion" , Physica D 42 (1990), 12-37.

8] WUENSCHE, A. E., "Classifying C ellular A ut om ata A utom atically: Fin dinggliders, filtering, and relating space-time patterns, attractor basins, and theZ pa ramete r" . Complexity 4{3) (1999), 47-66.

9] W O L F R A M , S . , "Computa t ional Theory of Cel lular Automata", Comm.Math. Phys. 96 (1984), 15-57.


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[10] P R I V M A N , V. (ed.), Finite Size Scaling and Numerical Simulation of Statistical Systems, World Scientific (1990).[11] G R A S S B E R G E R , P . , "Long-range effects in an elementary cellular automaton' ' , J . Stat . Phys 45 (1986), 27-39.[12] VoORHEES, B. and S. B R A D S H A W , "Predecessors of cellular au to m ata states

III. Garden of Eden classification of cellular automata", Physica D 7 3(1994), 152-167.[13] JEN , E., "Enumeration of preimages in cellular automata", Complex Sys

tems 3 (1989), 421-456.[14] BOCCARA, N., J. N A S S E R and M. R O G E R , "Part iclel ike structures and theirinteractions in spatiotemporal patterns generated by one-dimensional deter

minist ic cellular-automaton rules", Phys. Rev. A 44 (1991), 866-875.[15] CULIK, K., L. P. HURD and S. Y u , "C om pu tation theo retic asp ects of global

cellular automata behavior", Physica D 45 (1990), 357-378.[16] GUTOWITZ, H. A., "A hierarchical classification of cellular automata",

Physica D 45 (1990), 136-156.[17] JEN , E., "Preimages and forecasting for cellular automata". Pattern For

mation in the Physical and Biological Sciences (H. F. NiJHOUT, L. NADELand D. STEIN eds.), Addison-Wesley (1997), 157.

[18] GUTOWITZ, H. A. and C. D O M A I N , "The topological skeleton of cellularautomaton dynamics" , Physica D 103 (1997), 155-168.

[19] K A N E K O , K . , "Attractors , basin structures and information processing incel lular automata", Theory and Applications of Cellular Automata (S .W O L F R A M ed.). World Scientific (1986), 367-399.

[20] HOPFIELD, J. J., "Neural networks and physical systems with emergentcollective computational abilities", Proc. Natl. Acad. Sci. USA 79 (1982),2554-2558.

[21] SiEGELMANN, H. T., A. B E N - H U R and S. F I S H M A N , "Computa t iona l Complexity for Continuous Time Dynamics", Phys. Rev. Lett. 83 (1999), 1463-1466.

sitabhra sinha- phase transitions in the computational complexity of "elementary'' cellular automata

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