NONCONSTRUCTIBLE BLOCKS IN 1D CELLULAR AUTOMATA: MINIMAL GENERATORS AND NATURAL SYSTEMS. Andrew Adamatzky* and Andrew Wuensche**

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1 NONCONSTRUCTIBLE BLOCKS IN 1D CELLULAR AUTOMATA: MINIMAL GENERATORS AND NATURAL SYSTEMS Andrew Adamatzky* and Andrew Wuensche** * IAS Lab, University of the West of England, Coldharbour Lane, Bristol BS16 1QY, UK ai-adama@uwe.ac.uk ** Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA wuensch@santafe.edu ABSTRACT The paper concerns minimal nonconstructible blocks of cellular automata, i.e. minimal combinations of cell states that never appear in the evolution of an automaton. Using our previous work on the construction of predecessors of given configurations, we analyze with numerical experiments the sets of nonconstructible blocks for 1D cellular automata with binary and ternary neighbourhoods and binary cell states, and compare our results with the classification made by Voorhees; we also analyse the relationship of the Voorhees classes to the static parameter λ. On applying our algorithm to cellular-automata models of natural systems, we provide common sense interpretations of unreachable states of some biological distributed processes. Keywords: cellular automata, Garden-of-Eden configurations, irreversibility 1. INTRODUCTION A cellular automaton (CA) is an array of cells that have a finite set of states and evolve in discrete time. A cell updates its state as a function of the states of its closest neighbours. Various aspects of cellular automata theory and applications are discussed in the monographs [1-8]; however, only the books by Aladyev [4] and Voorhees [8] have separate chapters on nonconstructible configurations. The so-called nonconstructible, or Gardens-of-Eden, configurations were defined by Moore in 1962: "they are nonconstructible in the sense that there is no configuration at time T-1 which will give rise to the given configuration at time T by means of the function f which defines the rules for the transition from one state to another" [9]. Moore proved that cellular automata which have erasable configurations may have nonconstructible configurations and raised the question of how large the smallest array which has nonconstructible configuration can be [9, 10]. Myhill showed that Moore's sufficient condition - the existence of two mutually erasable configurations - for the existence of nonconstructible configurations is a necessary and sufficient condition [11]. In the seventies the constructibility/nonconstructibility problem was investigated in a series of papers in connection with the surjectivity/injectivity of CA mappings [12-15] and invertibility of CA [16, 17]. To date, two important results concerning the nonconstructibility problem have been obtained: the undecidability of the surjectivity (absence of nonconstructible configurations) problem for multi-dimensional CA proved in [18], and the classification of 3-site rules from the structure of the set of minimal nonconstructible blocks by Voorhees [8]. Kari's theorem covers a gap in CA fundamental theory, whereas Voorhees's classification uncovers new horizons for theory and practice. In the second section of the paper we provide the minimal background for cellular automata and nonconstructibility. The analysis of minimal nonconstructible blocks of 1D CA with binary cell states, for binary and ternary neighbourhoods, is given in section 3. The last section, 4, interprets the notion of nonconstructibility in terms of some natural systems. 2. BACKGROUND In this paper we consider a cellular automaton (CA) as a one-dimensional array of cells, each of which takes states from a finite non-empty set Q and changes its states in discrete time, depending on the states of its k neighbours, according to the local transition function f: Q k Q. We consider 2-site and 3-site rules that are applied to the neighbourhoods ux ( i) = ( xi 1, xi+ 1 ) and ux ( i) = ( xi 1, xi, xi + 1 ), respectively, where i N. The set Q is binary or ternary. For every function f we define the set of minimal nonconstructible blocks as Q * GE( f ) = { w Q * w Q * : f w = w w Q * : w w}, where Q is an infinite concatenation of the symbols from Q. This means that a string w of the symbols from Q represents the minimal block which is nonconstructible by function f if w does not have predecessors and is not a substring of any other nonconstructible block.

2 2 3. CA WITH BINARY CELL STATE The procedure for the computation of nonconstructible blocks is based on the method for the computation of preimages invented in [6-8]. The number of nonconstructible blocks of lengths for the 3-site 2- state transition functions is presented in Fig. 1. The first naive conclusion we can draw from the figure is that nonconstructible blocks accrue disproportionaly, nonlinearly, and at different rates for different functions. The functions which have no nonconstructible blocks are indicated by solid vertical lines. Proposition 1 (Voorhees classification). Sets of minimal nonconstructible blocks for 3-site rules can be divided onto the four classes: EM={f {,} 01 { 013,} GE(f )={ } }, FI={f {,} 01 { 013,} 0 < GE(f ) < }, SI={f {,} 01 { 013,} GE(f ) On () }, LI={f {,} 01 { 013,} GE(f ) On ( c ) }, where n is the length of a minimal nonconstructible block and c is a constant. The classes EM, FI, SI, and LI were discovered, clearly defined and completely analysed for 3-site rules by Voorhees [8]. However, there is no analytical proof for this proposition yet, only a computational demonstration. We have checked the Voorhees hierarchy carefully for nonconstructible blocks of size not more than 18 cells, and have verified that under these conditions the set of 3- site 2-state rules is subdivided into the four classes. The first class, EM, includes the functions that have no nonconstructible blocks at all; the global transition graphs are just cycles. Functions of the second class, FI, are characterized by a finite number of minimal nonconstructible blocks. The third class, SI, is built from the functions which have minimal nonconstructible block sets; the power (or cardinality) of the sets grows linearly with the increase in block length n. And the last class, SI, is characterized by polynomial growth (with respect to n) of the number of nonconstructible blocks. The exact correspondence between the number of nonconstructible blocks and the functions from 1 to 128 is shown in Fig. 2. The 'proof' of proposition 1 depends on the generation of nonconstructible blocks for certain values of n, and an induction hypothesis stating that if the Voorhees classification is correct for a given, sufficiently large n, it will be true for any other length of block. In addition we can produce two results that may be very useful in proving proposition 1. Hypothesis. For a 1D CA with q cell states and a k -site neighbourhood, the local transition function f lies in the EM class if there are no minimal nonconstructible blocks of length more than ( q k +1). The hypothesis rests on two facts: (i) it is true for q =2, k =23,, and (ii) the minimal length of a block containing all possible states of the neighbourhoods is q k. Let, e.g., for some f GE(f )= for n q k +1. This means that all blocks of length n have their predecessors. If some block v of n symbols is nonconstructible, then it contains a nonconstructible substring of length n that is in conflict with (ii) above. Interestingly, we found that one of the bounds relating to nonconstructible blocks was announced by V. Aladyev more than a quarter of century ago [19]. He stated that the minimal nonconstructible configuration of a 1D CA q k 1 has a size less than ( 2 1) + 1. This is close to our bound for the 2-state 3-site rules, but does not work for the 2-site rules. Proposition 2. SI is generated in linear time. For every element (group) of SI we found a set of CA rules that generates only blocks of this group from some specified block (the initial state). In other words, for any set of the minimal nonconstructible blocks GE(f ) we can find a cellular automaton R as follows: with local transition function g, and neighbourhood u, specify the boundary conditions, and choose the initial configurations c such that all configurations appearing in the evolution of R starting from c belong to GE(f ). Each group of blocks is therefore represented by a set of CA rules, the initial block, and one block that does not participate in the generation (can not be generated) (Table 1). This enriches Voorhees's proposition stating that for the SI class it is possible to identify the conditions for a sequence to be in SI, and stating the dependence of the number of blocks on the length of the blocks. How does the ability of a function f in generating nonconstructible blocks affect the global behaviour of the CA defined by f? One of the superficial features is that the transition from the EM class to the LI class through FI and SI is characterised by increasing morphological diversity of the graphs of global transitions, as portrayed in Fig. 3 for some particular examples. Let us assume for the moment that the number of minimal nonconstructible blocks for the function f characterizes a degree of irreversibility of f Then Voorhees's hierarchy describes the evolution of irreversibility in CA with increasing lattice size. Obviously, the number of nonconstructible blocks for a given rule determines the density of nonconstructible configurations appearing in the evolution of the CA (G-density). It was shown by Ikaunieks [20] that for d-dimensional CA, d 1, any local transition function has, with a very high probability, a nonempty set of nonconstructible blocks. The correspondence between G-density and the static

3 3 parameters λ and Z [6, 21-23] is shown in Fig. 4. In Aladyev's book [4] one can find the asymptotic relation k Q { f Q : GE( f) = } q lim q k k Q k 1 q 1 { f Q : GE( f) } = ( 2πq ) irrespective of the dimension of a CA that proves Moor's hypothesis stating that k f f lim { Q Q : GE ( ) } k = 1. q Q { f Q } Notion 1. The classes in the Wolfram classification and in the Voorhees classification do not correspond exactly. From the fine structure of the correspondence between the FI, SI and LI classes and their representative λ parameters desplayed in Fig. 5, we may see that the distribution of λ within classes corresponds to the usual distribution, with the exception that the deviation of the λ value from 05. decreases when we pass from FI via SI to LI. Proposition 3. All 2-site rules lie in class FI. This is a direct consequence of the Voorhees hierarchy, because 2-site rules are elements of the set of 3-site rules. Let v=( v1,.., v4 ) and w=( w,..., w 1 8 ) be the vectors representing 2-site and 3-site rules respectively; then a 3- site rule is equivalent to a 2-site rule if w=( vvvvvv 1, 2, 1, 2, 3, 4, vv 3, 4 ). So, if GE(R) is a set of minimal nonconstructible blocks for the set R of 2-site rules, enumerated conventionally, then we have GE(0,15)={1}, GE(1,7,8,14)={10001, 10011, 11001, 11011}, GE(3,5,6,9,10,12)={ }, GE(2, 4,11,13)={101, 111}, where representatives of the rule set are underlined. In [7] we defined the so-called algebra of interacting CA as A =< Q,, u >, where Q is a finite nonempty set, u is a cell neighbourhood of size k, and : Q Qk Q Qk. Elements of Q Qk are the configurations of a CA with Q cell states and cell neighborhood u together with the vectors representing the cell state transition rules. For c = c c, the configuration c is obtained from c by applying the cell state transition rules encoded in configuration c. Proposition 4. Minimal generators of A are not elements of the set GE of 2-site rules. This is a direct consequence of the results of the investigations of minimal generating sets of algebra A, and proposition 3 of the present paper. The next three propositions concern CA with 3-site symmetric neighbourhoods. Proposition 5. For totalistic CA there are no rules of class SI. From the conventional enumeration of totalistic rules we have EM={(5,10)}, FI={(0,15), (3,12), (1,7,8,14), (6,9)}, LI={(2,4,11,13)}. In a CA of the class NSET [7] the cell state transition function is a mapping 2 Q \{ } Q whereas for the class POIN [7] it is f: Q k {,..., 1 k}. Every cell of a CA from class POIN takes the state of one of its neighbours, the index of which is computed from the state of the cell s neighbourhood. A 3-site CA is simulated in a POIN CA when q k. The classes NSET and POIN were invented to simulate reactiondiffusion media and flow graphs, respectively. For a POIN CA the complete Voorhees classification is found, with an increase in the number of groups or rules inside the classes because some 3-site rules have more than one corresponding rule from the POIN class. Proposition 6. All 3-site rules of NSET CA lie in class FI The FI class for NSET has the following structure: FI={(0,7), (1,5), (2,3,4), (6)}, where rule indices encode the vector ({0},{1},{0,1}). It is important to analyze nonconstructibility when, in the same structural condition, 3-site CA are simulated by an NSET 3-site CA. This is the case when q k 2 q 1, e.g. for q 10 where k =2, and would of course require enormously high computational resources. n We say that the configuration c Q of a CA of n belongs to the set C of complete configurations when k w Q: w c. Proposition 6. Let f f LI GE( f) C. k Q Q then It is evident that c C "c has at least q k + ( k 1)/ 2 symbols". Examples of complete configurations for several groups of class LI are shown in Table UNREACHABLE STATES IN NATURAL EVOLUTION The notion of nonconstructible configurations seems not to have found any applications in CA models of living or life-like processes, with the exception of the search for nonconstructible configurations for the game of Life [25]. We now discuss a 1D CA with a ternary symmetric neighbourhood including the central cell and ternary cell state elements, where nonconstructible configurations are interpreted in various ways. The models considered are as follows.

4 4 Neural Nets. Every cell is a neuron-like element that takes three states, rest ( ), excited (+) and refractory (-). The resting neuron becomes excited if the number of excited neighbours is equal to a specified number or exceeds some threshold. It switches from the excited state to the resting state, and from the refractory state to the resting state, unconditionally. We analyze three models of neural networks: NN1 when a neuron becomes excited if at least one neighbour is excited, NN2 when exactly two neighbours are excited, and NN3 when exactly three neighbours are excited. Populations. Every cell may contain a resource ( ) and/or creatures belonging to either species A or species B. The first model C1 illustrates competition between the two species for the resource. A cell currently containing a resource will be occupied after the next time step by A if the sum of A in the neighbourhood is greater then the sum of B (and vice versa for occupation by B), and will remain in a resource state if the sums of A and B are equal. Creatures A and B will remain alive if at least one neighbouring site contains a resource, and will die otherwise. In the second type of competition, C2, creatures die if the sum of neighbouring sites with resource is less than two. Symbiosis means that both creatures need each other to function or survive. In contrast with the two previous models, in the symbiosis model S a creature at a given site will live if at least one neighbouring site contains resource, and if at least one other neighbouring site contains a creature of the other species; otherwise, it will die. In the predator-prey system, PP, creatures A are prey and creatures B are predators: a resource site can be taken over only by prey, but a site with prey can be occupied by a predator when predator B eats prey A. Agents. Every cell is an agent which either holds one of two or three beliefs, or holds no beliefs at all. The model A1 assumes that an agent holds one of three beliefs a, b or c, and that it is a conformist, i.e. it adopts the belief held by the majority of the neighboring agents and does not change its belief when the beliefs of its neighbours are different from one another and from its own belief. In the models A2, A3 and A4 we require that every agent holds one of two beliefs, a and b, or no belief at all,. The first case, A2, is symmetric: if an agent does not hold any belief it adopts the belief of the majority of its neighbours if there is a majority, or continues to hold no belief if there is no majority; if it does hold a belief, it changes that belief only if both neighbours share a different belief. The agents in A3 are ordered on the lattice by authority from left to right. If an agent does not have a belief it adopts the belief of its left neighbour if the left neighbour holds any belief, or its right one otherwise, if its right neighbour holds any belief, but remains without any belief if both its right and left neighbours do not have any beliefs. An agent only changes its belief if both neighbours share the same belief, different from the agent's. The last model, A4, is a modification of the third model: an agent changes its belief only to that of its left neighbour, if the neighbour holds a belief. Proposition 7. NN1, NN2, NN3, PP, A4 FI, PP, S SI and C1, C2, A1, A2, A3 LI. The generators of some minimal nonconstructible blocks for these models of not more than 6 elements each are presented in Table 3. As may be seen from the table, CA models of natural systems do not belong to the class EM. This reflects the irreversibility of natural biological development. Of course, all nonconstructible blocks can be interpreted in natural terms. Thus a neural network can not be in a state where one or more of the neurons is excited whereas its neighbours are rest, or at least one neuron and both of its neighbours are excited. Neither prey nor predators can occupy a site encircled by empty (resource) sites; it impossible to find a configuration where a predator is sandwiched between two prey, or where two predators are between two prey. In the chain of agents ordered by authority, A4, the situation where an agent holds a belief but its left 'adviser' holds none is impossible. Proposition 8. The models are ordered by their membership of GE in the following hierarchy NN3<NN1<NN2<PP<A4 <S<A3<C1<C2<A1. This points to the fact that the number of unreachable states in neural networks increases with decreasing threshold of excitation, and is maximal for the minimal level of excitation. The power of the nonconstructible blocks increases when we pass from predator-prey systems to competition of populations for resources through symbiosis. Finally, we obtain the result that increasing 'freedom' of individual agents leads to the growth of unreachable distributed knowledge in intelligent systems. Several pictorial examples of these CA systems are shown in Fig. 6 (for periodic boundary conditions). We found that the neural networks are characterized by travelling waves of excitation (NN1) or damping of excitation (NN2 and NN3). In the predatorprey system, we observe travelling waves of predators in the presence of prey, and unbounded growth of prey in the absence of predators. The space-time patterns of the competing populations exhibit stable (C1) or oscillating (C2) islands of species A and B. ACKNOWLEDGMENTS Many thanks to Santa Fe Institute for financial support when AA visited AW and to O. Holland for assistance with the editing.

5 5 REFERENCES 1. Neumann von J. Theory of Self-Reproducing Automata. Univesity of Illinois Press, Codd E. F. Cellular Automata. NY: Academic Press, Aladyev V. Mathematical Theory of Homogeneous Structures and Their Applications. Tallinn: Valgue Press, Aladyev V. Homogeneous Structures: Theoretical and Applied Aspects. Kiev: Technika Press, 1990 (In Russian). 5. Toffoli T. and Margolus N. Cellular Automata Machines. MIT Press, Wuensche A. and Lesser M. The Global Dynamics of Cellular Automata. Addison-Wesley, Adamatzky A.I. Identification of Cellular Automata. London: Taylor and Francis, Voorhees B. Computational Analysis of One- Dimensional Cellular Automata. Singapore: World Scientific, Moore E.F. Machine models of self-reproduction. Proc. Symp. Appl. Math. 14 (1962) Moore E.F. Mathematics in biological sciences. Scientific American 211 (1964) 3, Myhill J. The converse of Moore's Garden-of-Eden theorems. Proc. Amer. Math. Soc. 14 (1963) 4, Amoroso S. and Patt Y. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structures. J.Comput. Syst. Sci. 6(1972), Yaku T. The constructibility of a configuration in a cellular automaton. J.Comput. Syst. Sci. 7 (1973) Maruoka A. and Kimura M. Condition for injectivity of global maps for tessellation automata. Inform.Control 32 (1976) Sato T. and Honda N. Certain relations between properties of maps of tessellation automata. J. Comput. Syst. Sci. 15 (1977), Culik II K. On invertible cellular automata. Complex Systems 1 (1987) Toffoli T. and Margolus N. Invertible cellular automata: A review. Physica D 45 (1990), Kari J. Decision Problems Concerning Cellular Automata. University of Turku, Aladyev V.Z. Toward The Theory of The Homogeneous Structures. Moscow: VINITI. N 4204, 1971 (In Russian). 20. Ikaunieks E. On the information properties of cellular structures. Problemy Peredachi Informacii 4 (1970) 6, (In Russian). 21. Langton C.G. Computation at the edge of chaos: phase transitions and emergent computation. Physica D 42(1990) Wuensche A. Complexity in One-D Cellular Automata, Cognitive Science Research Paper 321, Univ. of Sussex., Wuensche A. Attractors Basins of Discrete Networks. Cognitive Science Research Paper 461, Univ. of Sussex, Wuensche A. Discrete Dynamics Lab (DDLab). Software and Manual Gardner M. Wheels, Life and Other Mathematical Amusements. NY: W.H. Freeman and Co, Table 1. Finite description of blocks from SI. (Row "Rules" presents vectors describing cell state transition rules; initial state is the starting point for the evolution of the CA that generates the block; the additional block is necessary but does not appear in the evolution of the CA; '+' means the operation of concatenation). Group of rules of SI CA that generates blocks of the group Additional block List of elements Representative Neighbourhood ux ( i ) Rules Initial state (46, 116, 139, 209) 46 ( xi 2, xi 1, xi) 01?11? (38, 44, 52, 100, 155, 38 ( x, x, x) i 2 i 1 i 203, 211, 217) (27, 39, 53, 83, 172, 172 (i) (i) (i) , 216, 228) ( xi 2, xi 1, xi, xi + 1) 00010?1?11??1??? (ii) 010?11?100?11?1? (58, 78, 92, 114, 141, 163, 177, 197) (ii) ( xi 3, xi 2, xi 1, xi) 58 ( x, x) i 1 i (i) 001? (ii) 0011 (ii) (i) (ii)

6 6 Table 2. The examples of the complete configurations of 1D CA with the 3-site rules. The groups of the class LI The configurations with complete blocks (26, 74, 82, 88,...) , , , (41, 97, 107, 121,...) , , , (73, 109,...) , , , Table 3. Generators of cellular-automata rules corresponding to some natural systems. Neural Networks Population systems Agents NN1 NN2 NN3 C1 C2 PP S A1 A2 A3 A ABA BAB..AB..BA A.AB AB.. AB.B B.BA BA.. BA.A..A....A.A..B....B.B A.A.. A.A.A B.B.. B.B.B.ABBA..BAAB. ABA BAB A.AB A.BA AB.A AB.B ABBA B.AB B.BA BA.A BA.B BAAB A.A.A A.A.B A.AAB A.B.B ABB.B B.A.A B.B.A B.B.B B.BBA BAA.A A..AAB A..BBA A.AA.A ABB..A ABB..B ABBBBA B..AAB B..BBA B.BB.B BAA..A BAA..B BAAAAB.A..B..BA AB. ABA.ABBA. ABA BAB A.AB AB.B B.BA BA.A A.A.A B.B.B AB..AB BA..BA abac abcb acab acbc babc baca bcac bcba caba cacb cbab cbca abaab abbab acaac accac baaba babba bcbbc bccbc caaca cacca cbbcb cbccb abaaca acaaba babbcb bcbbab caccbc cbccac..a...ab..b...ba.a...a.a.aba.b...b.b.bab a.a. a.ab ab.. ab.b aba. b.b. b.ba ba.. ba.a bab. abaab abbab baaba babba..aa....aa.a..aaba..bb....bb.b..bbab a.aa.. a.aa.a a.aaba abaa.. abaa.a b.bb.. b.bb.b b.bbab babb.. babb.b.a..ab.b..ba ab. ba..aa..bb..aaba bbab ababb. ababba babaa. babaab.a..ab.b..ba.aa..aab.bb..bba

7 7 Fig. 1. Integral characteristics of the numbers of nonconstructible blocks of length n=1,..., 10 for 3-site rules A.

8 8 B. Fig. 2. Relationship between nonconstructible blocks of length 2 to 8 (A) and 8 to 10 (B) and 3-site rules. Asterisk being on the intersection of i -th block and j -th function means that i -th block can not be generated by j -th function. A. B. C. D. Fig. 3. Global transition graphs of 1D CA (n=11) for FI, rule 3 (A) and rule 23 (B), SI, rule 18 (C), and LI, rule 109 (D) (computed and drawn by DDLab [24]).

9 9 Fig. 4. Relationship between G-density and λ and Z parameters for all 3-site rules of 1D CA with binary cell state set (computed and drawn by DDLab [24]). Fig. 5. Relation of Voorhees classes, where the rules are grouped as in (Voorhees, 1996), to λ parameter. Diameter of black disc on the intersection of λ value and index of group is proportional to the number of elements of the group with specified λ value.

10 10 NN1 C1 C2 PP S A1 A4 Fig. 6. Space-time patterns of CA models of 1D natural systems: space index increases from left to right, time index top down.