On Routine Evolution of New Replicating Structures in Cellular

Automata

Michal Bidlo

Brno University of Technology, Faculty of Information Technology, IT4Innovations Centre of Excellence,

zet

echova 2, 61266, Brno, Czech Republic

Keywords:

Genetic Algorithm, Cellular Automaton, Transition Function, Conditional Rule, Replicating Loop.

Abstract:

This paper presents evolutionary design of two-dimensional, uniform cellular automata. The problem of repli-

cating loops is considered as a case study. Conditionally matching rules are used as a technique that is suitable

to the design of cellular automata state transition rules. A genetic algorithm is applied to the design of cellular

automata that satisfy the requirements of replicating loops. It is shown that such evolution is able to ﬁnd

various state transition rules that support replication of a given loop. Results presented herein demonstrate the

ability of derived cellular automata to perform replication not only from an initial instance of the loop but also,

that from a seed the loop can autonomously grow.

1 INTRODUCTION

Since the introduction of cellular automata (CA) in

(von Neumann, 1966), researchers have dealt, among

others, how to effectively design a cellular automa-

ton (and its transition function in particular) to solve

various problems. For example, cellular automata

have been studied for their ability to perform com-

putations, e.g. using principles from the famous Con-

way’s Game of Life (Berlekamp et al., 2004) or by

simulating elementary logic functions in non-uniform

cellular matrix (Sipper, 1995).

One of the topics widely studied in the area of ar-

tiﬁcial life is the problem of (self-)replicating loops.

Since the introduction of probably the most known

loop by Langton (Langton, 1984), which is able to

replicate in 151 steps in a CA working with 8 states,

some other researchers have dealt with this topic try-

ing to simplify the replication process or enhance the

abilities of the loop during replication. For example,

Byl introduced a smaller loop that is able to replicate

in 25 steps using a CA that works with 6 cell states

(Byl, 1989). Later, several unsheathed loops were

proposed by Reggia et al. from which the simplest

loop consists of 6 cells only and is able to replicate

using 8-state CA in 14 steps (Reggia et al., 1993). On

the other hand, Tempesti studied a possibility to intro-

duce construction capabilities into the loops and pro-

posed a 10-state CA that allows to generate patters in-

side the replicating structures (Tempesti, 1995). Per-

rier et al. created a “self-reproducing universal com-

puter” using 64-state CA by “attaching” executable

programs (Turing Machines) on the loops (Perrier

et al., 1996). Although the aforementioned solutions

were achieved using analytic methods, the process of

determining suitable transition rules for a given prob-

lem represents a difﬁcult task and requires an expe-

rienced designer (the process of “programming” the

CA is not intuitive). As the number of cell states in-

creases, the process of the CA design becomes chal-

lenging due to a signiﬁcant increase of the solution

space. Moreover, for some problems no analytic ap-

proach has yet been known to the design of the tran-

sition rules. In such cases various unconventional

techniques have been applied including Genetic Al-

gorithm (GA) (Holland, 1975)), possibly in combina-

tion with other heuristics.

For example, Mitchell et al. investigated a prob-

lem of performing computations in cellular automata

using GA (Mitchell et al., 1993). Their work con-

tains a comparison with the original results obtained

by Packard in (Packard, 1988) which can be consid-

ered as a milestone in applying evolutionary algo-

rithms (EA) to the design and optimisation of cellular

automata. In particular, the authors in (Mitchell et al.,

1993) claim: “Our experiment produced quite differ-

ent results, and we suggest that the interpretation of

the original results is not correct.” It may indicate

that the research of cellular automata (and their typ-

ical features like emergent behaviour or cooperative

Bidlo, M..

On Routine Evolution of New Replicating Structures in Cellular Automata.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 1: ECTA, pages 28-38

ISBN: 978-989-758-157-1

cell signalling by means of local rules) using various

computing techniques can provide valuable informa-

tion for advanced studies and applications in this area.

Note that Mitchell et al. considered binary (i.e. 2-

state) 1D cellular automata only which represent a

fundamental concept for advanced models. Sipper

proposed a technique called Cellular Programming

(a spatially distributed and locally interacting GA)

that allows for the automatic design of non-uniform

CA that are well suited to various problems (Sipper,

1997). Sapin et al. introduced a GA-based approach

to the design of gliders and glider guns in 2D cel-

lular automata (Sapin and Bull, 2008)(Sapin et al.,

2010). It was shown that a spontaneous emergence of

glider guns in CA can occur with a signiﬁcant number

of new gun-based and glider structures discovered by

EA. The aim of the glider research was to construct a

system for collision-based computationally universal

cellular automata that are able to simulate Turing ma-

chines (Sapin and Bull, 2008). In recent years, several

solutions emerged that aim to optimize the CA de-

sign by introducing various evolution-based and soft-

computing techniques in combination with suitable

representations of the transition functions. For ex-

ample, Elmenreich et al. proposed an original tech-

nique for the calculation of the transition function us-

ing neural networks (NN) (Elmenreich and Feh

erv

ari,

2011). The goal was to train the NN by means of Evo-

lutionary Programming (Fogel et al., 1966) in order to

develop self-organising structures in the CA. Condi-

tionally Matching Rules (CMR) are a representation

of the transition rules, and were introduced, together

with some early results related to binary CA, in (Bidlo

and Vasicek, 2013) and (Bidlo, 2014).

Whilst the most of the aforementioned studies

considered binary CA (i.e. those working with two

cell states only), that may be suitable for straightfor-

ward hardware implementations (e.g. Sipper’s Fire-

ﬂy machine (Sipper et al., 1997)), multi-state CA can

provide a more efﬁcient way for the representation

and processing of the information thanks to the abil-

ity of the cells to work with more than two states.

This feature is important for studying complex sys-

tems that are in most cases described by integer (or

real-valued) variables. In addition, the introduction

of more than two states per cell in the CA may allow

to reduce the resources needed to solve a given prob-

lem (e.g. the size of the cellular array or dimension

of the automaton). For example, Yun

es studied com-

putational universality in multi-state one-dimensional

cellular automata (Yun

es, 2010). A technique for the

construction of computing systems in 2D CA was

demonstrated by Stefano and Navarra in (Stefano and

Navarra, 2012) using rules of a simple game called

Scintillae working with 6 cell states. Their approach

allows to design components (building blocks) for the

construction of bigger systems, e.g. on the basis of

gate-level circuits.

The goal of this paper is to demonstrate an ability

of the CMR approach to automatically design tran-

sition rules for CA that support replication of given

structures in uniform, multi-state 2D array. A genetic

algorithm will be applied in order to discover suitable

transition rules that perform replication of the given

loop-like structure according to the designer’s speci-

ﬁcation. It will be shown that novel replication sce-

narios can be found in CA that can copy the given

loop not only from its initial instance but also, from a

seed the loop can autonomously grow.

2 FUNDAMENTALS OF

CELLULAR AUTOMATA

The original concept of cellular automaton introduced

in (von Neumann, 1966), that will be considered in

this paper, assumes a 2D matrix of cells, each of

which at a given moment acquires a state from a ﬁ-

nite set of states. The development of the CA is

performed synchronously in discrete iterations (time

steps) by updating the cell states according to local

transition functions of the cells. Uniform cellular au-

tomata will be investigated in which the local transi-

tion function is identical for all cells and hence it can

be considered as a transition function of the CA. The

next state of each cell depends on the combination of

states in its neighbourhood. In this paper von Neu-

mann neighbourhood will be assumed that includes a

given (Central) cell to be updated and its immediate

neighbours in the North, South, East and West direc-

tion (i.e. it is a case of a 5-cell neighbourhood).

Since the CA behaviour can practically be evalu-

ated in the cellular array of a ﬁnite size, boundary con-

ditions need to be speciﬁed in order to correctly de-

termine cell states at the edge of the array. In this pa-

per, cyclic boundary conditions will be implemented

which means that cells at an edge of the CA are “con-

nected” to the appropriate cells on the opposite edge

(i.e. these cells are considered as neighbours) in each

dimension. In case of the 2D CA the shape of such

cellular array can be viewed as a toroid.

The transition function is usually deﬁned as a

mapping that for all possible combinations of states

in the cellular neighbourhood determines a new state.

This mapping can be represented as a set of rules of

the form N

→ C

t+1

where N

, W

, C

, E

and

denote cell states in the deﬁned neighbourhood at

a time t and C

t+1

is the new state of the cell to be up-

On Routine Evolution of New Replicating Structures in Cellular Automata

dated. It means that for every possible combination of

states N

a new state C

t+1

needs to be spec-

iﬁed. However, if the number of cell states increases,

the number of possible transition rules grows signiﬁ-

cantly which is inconvenient for efﬁcient CA design.

Of course, not all transition rules need to be speciﬁed

explicitly but the problem is how to choose the rules

which modify the central cell in the neighbourhood.

Therefore, an advanced representation of the transi-

tion rules was proposed and denominated as Condi-

tionally Matching Rules (Bidlo and Vasicek, 2013).

Conditionally matching rules allows us to reduce the

size of representation of the transition functions espe-

cially with respect to the evolutionary design of cellu-

lar automata.

3 CONDITIONALLY MATCHING

RULES

The concept of conditionally matching rules showed

as a very promising technique in comparison with the

conventional (table-based) approach considering var-

ious experiments with binary cellular automata (e.g.

pattern development task (Bidlo and Vasicek, 2013)

or binary multiplication in 2D CA (Bidlo, 2014)). In

this paper, evolutionary design of the CMR-based rep-

resentation will be investigated in order to design cel-

lular automata with up to 10 cell states that support

replication of a given structure.

A conditionally matching rule represents a gener-

alised rule of a transition function for determining a

new cell state. Whilst the common approach speciﬁes

a new state for every given combination of states in

the cellular neighbourhood, the CMR-based approach

allows to encode a wider range of combinations into

a single rule. A CMR is composed of two parts: a

condition part and a new state. The number of items

(size) of the condition part corresponds to the number

of cells in the cellular neighbourhood. Let us deﬁne a

condition item as an ordered pair consisting of a con-

dition function and a state value. The condition func-

tion is typically expressed as a function whose result

can be interpreted as either true or false. The condi-

tion function evaluates the state value in the condition

item with respect to the state of the appropriate cell in

the cellular neighbourhood. In particular, each item

of the condition part is associated with a cell in the

neighbourhood with respect to which the condition is

evaluated. If the result of such evaluation is true, then

the condition item is said to match with the state of

the appropriate cell in the neighbourhood. In order to

determine a new cell state according to a given CMR,

all its condition items must match (in such case the

CMR is said to match).

The following condition functions will be consid-

ered: == 0, 6= 0, ≤, ≥. Note that this condition set

represents a result of our long-term experimentation

and experience with the CMR approach and will be

used for all the experiments in this paper. The condi-

tion == 0, respective 6= 0, evaluates whether the cor-

responding cell state is equal to 0 (i.e. a “dead” state),

respective whether it is different from state 0. Note

that the state value of the condition item for == 0

and 6= 0 is considered implicitly within the condition

itself. The conditions ≤ and ≥ represent relational

operators “less or equal” and “greater or equal” re-

spectively for which the state value of the condition

item must be explicitly speciﬁed.

Figure 1 shows an example of conditionally

matching rules deﬁned for a 2D CA with the 5-cell

neighbourhood together with the illustration of cells

the condition items are related to. CMR (A) is a

matching CMR since all the conditions of its condi-

tion part are evaluated as true with respect to the sam-

ple neighbourhood shown in the left part of Fig. 1.

On the other hand, CMR (B) does not match because

the second condition item ! = 2 evaluates as false with

respect to the west cell that possesses state 2. Simi-

larly, the third condition == 0 of CMR (B) is not true

as the central cell is in state 2.

not-

is0

is0 /=

(A)

(B)

Figure 1: Example of a conditionally matching rule speci-

ﬁed for 5-cell neighbourhood. The value of the new state is

written in bold. (A) example of a matching CMR, (B) ex-

ample of a CMR that does not match – the second and third

condition is evaluated as false.

A CMR-based transition function can be speciﬁed

as a ﬁnite (ordered) sequence of conditionally match-

ing rules. The following algorithm will be applied

to determine a new state of a cell. The CMRs are

evaluated sequentially one by one. The ﬁrst matching

CMR in the sequence is used to determine the new

state. If no of the CMRs matches, then the cell keeps

its current state. These conventions for evaluating and

applying the CMRs ensure that the process of calcu-

lating the new state is deterministic (it is assumed that

the condition functions are deterministic too). There-

fore, it is possible to convert the CMR-based tran-

sition function to a corresponding table-based repre-

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

Figure 2: Structure of a chromosome for genetic algorithm encoding a CMR-based transition function. cx denote a condition

for the cell at position x in the neighbourhood, sx represents the state value to be investigated using the appropriate condition

with respect to the state of cell at position x, ns speciﬁes the next state for a given CMR. All the conditions and state values

are represented by integer numbers.

sentation which preserves the fundamental concept of

cellular automata. Moreover, every condition set that

includes relation == allows to formulate transition

rules for speciﬁc combinations of states if needed (by

specifying == for all condition items of the CMR).

In order to obtain the conventional (table-based)

representation of the transition rules from an evolved

CMR-based solution, the following algorithm is ap-

plied using the same CA that was considered dur-

ing evolution. Let C

and C

t+1

denote states of a

cell in two successive steps of the CA at time t

and t + 1 respectively. A transition rule of the form

→ C

t+1

is generated for the combination

of states in the cellular neighbourhood if C

6= C

t+1

This process is performed after each step and for each

cell until the CA reaches a stable or periodic state.

The set of rules obtained from this process repre-

sents the corresponding conventional prescription of

the transition function. Note that only the rules that

modify the cell state are generated, all the other rules

are implicitly considered to preserve the current state.

4 EVOLUTIONARY SYSTEM

SETUP

A genetic algorithm is utilized for the evolution of

CMR-based transition functions in order to achieve

the given behaviour in cellular automata. Each chro-

mosome of the GA represents a candidate transition

function encoded as a ﬁnite sequence of CMRs. The

chromosome is implemented as a vector of integers

in which the condition items and next states of the

CMRs are encoded. Note that the population con-

sists of chromosomes of a uniform length (given by

the number of CMRs) which is speciﬁed as a parame-

ter for a speciﬁc experiment. The structure of a chro-

mosome is depicted in Figure 2.

The population of the GA consists of 8 chromo-

somes that are initialised randomly at the beginning

of the evolutionary process. In each generation, four

individuals are selected randomly from the current

population, the best one of which is considered as a

parent. In order to generate an offspring, the par-

ent undergoes a process of mutation as follows. A

random integer M in range from 0 to 2 is generated.

Then M random positions in the parent chromosome

are selected. The offspring is created by replacing

the original integers at these positions by new valid

randomly generated values. If M equals 0, then no

mutation is performed and the offspring is identical

to the parent. The process of selection and mutation

is repeated until the entire new population is created.

Crossover is not applied because no beneﬁt of this

operator was observed during the initial experiments.

Note that the same GA has successfully been applied

since the introduction of CMRs in various case studies

(Bidlo and Vasicek, 2013)(Bidlo, 2014). Although no

optimal (evolutionary) approach has yet been known

for uniform CA, our experiments indicate that small-

population EA (i.e. less than 10 individuals) with

a simple mutation operator may represent a suitable

class of algorithms to obtain working solutions with

a reasonable success rate and computational effort.

However, the detailed analysis and wider comparison

of different techniques is not a subject of this paper.

For each experiment, the GA is executed for 4 mil-

lions of generations. If no correct solution is found

within this limit, the evolution is terminated. The

evaluation of the chromosomes (i.e. the ﬁtness func-

tion) and details regarding various experimental set-

tings are described in the next section.

5 EXPERIMENTAL RESULTS

This section summarises statistics of the performed

evolutionary experiments and presents some results

Figure 3: Structure of the replicating loop and cellular au-

tomaton of the size that was evaluated in the evolutionary

experiments: (a) the initial CA state containing the loop to

be replicated, (b) the target state specifying the replica ar-

rangement.

On Routine Evolution of New Replicating Structures in Cellular Automata

Figure 4: Development of a CA performing replication of the loop from Figure 3. The sequence of steps reads from left to

right and top to bottom. The upper part of each step of the CA illustrates the replication of the initial loop. The bottom part

demonstrates a seed represented by a cell in state 5. Note that after the loop is ﬁnished, its replication continues in the same

way as from the initial instance (shown by the last CA state).

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

together with a more detailed analysis. Two sets of

experiments are considered in which the CA works

with 8 and 10 cell states. Moreover, different num-

bers of CMRs (varying from 20 to 50) encoded in the

GA chromosomes are considered. For each setup 100

independent evolutionary runs are executed. The ex-

periments were executed using the Anselm cluster

the time of a single run (4 millions of generations) is

approximately 12 hours.

A replicating loop is considered whose structure

consists of 6 different non-zero states as shown in

Figure 3a. The genetic algorithm is applied to the

design of a transition function for the CA that per-

forms the replication of the loop in a maximum of

30 steps. The required CA state, that contains the

replica, is depicted in Figure 3b. The following al-

gorithm is considered in order to evaluate the candi-

date solutions during evolution and calculate the ﬁt-

ness function. A partial ﬁtness function is evaluated

after each CA step as the number of cells in correct

states with respect to Figure 3b. The ﬁnal ﬁtness value

of a given candidate solution is deﬁned as the maxi-

mum of the partial ﬁtness values. It this case the repli-

cation can be considered as a pattern transformation

problem from a single (initial) loop onto two loops in

a given arrangement. However, the loop is required

to be able to replicate again and again during the sub-

sequent CA development. Moreover, an assumption

is considered that each newly created loop is shifted

by two cells down with respect to its predecessor (as

shown in Fig. 3b). Therefore, the obtained solutions

are further investigated using a visual software simu-

lator developed by the author of this paper in order to

check that. The goal of this approach is to determine

whether the GA is able to discover various new gen-

eral replication scenarios. Note that for the purposes

of this paper the term “general” means an ability of

a solution to repeatedly produce more replicas of the

given loop, not an ability to replicate arbitrary loops.

Table 1 summarises results of the evolutionary ex-

periments and provides an overview of some basic pa-

rameters of the CA that can be observed during its de-

velopment using the evolved transition functions. As

evident, the maximum success rate achieved during

the experiments is only 12% which is not very high.

Note, however, that the replication of the proposed

loop represents a problem for which no working solu-

tion was found during our previous experiments using

the table-based transition functions. More research

is needed in order to optimise the evolutionary algo-

rithm for this class of problems.

In addition to the results obtained for the CA

https://docs.it4i.cz/anselm-cluster-documentation/

hardware-overview

working with 8 cell states, some successful solutions

have even been obtained for 10 cell states which in-

dicates that the CMRs are an efﬁcient encoding of

the transition rules that allows for the design of more

complex multi-state CA. The solutions obtained in

this paper demonstrate a wide range of various repli-

cation schemes that can be performed using CA. For

example, a solution was found that is able to repli-

cate the loop in 16 steps (the best solution of this pa-

per) whilst some CA require 30 steps (the maximal

allowed number of steps) in order to ﬁnish the repli-

cation. Similarly, the number of transition rules gen-

erated from the CMRs varies from 84 to more than

1500 rules. These results indicate that cellular au-

tomata can in some cases exhibit behaviour that has

not yet been discovered which may be beneﬁcial not

only for the area of CA but also, for the study of com-

plex systems in general.

Figure 4 shows a CA development performed by

one of the successful transition functions obtained for

the replication of the given loop. It is one of the best

solutions discovered in this paper with respect to the

number of steps needed to create a copy of the loop.

The transition function was found with 30 CMRs in

the GA chromosomes and the corresponding conven-

tional representation contains 238 transition rules. If

the development of the initial loop is considered (see

the upper parts of each step in Figure 4), the CA needs

21 steps to create a complete replica. As shown by

the last step, more replicas can be created in the same

way according to the original speciﬁcation if the CA

development continues. However, a more detailed in-

vestigation of this result showed that the complete ini-

tial loop is not strictly needed in order to successfully

perform the replication. For example, the loop is able

to emerge even from a single seed – the lower parts of

each step presented in Figure 4 shows a development

of the loop from a single initial cell (a seed) in state

5. As marked by the up-most black arrow a complete

loop is developed from the seed after 18 steps which

is by 3 steps faster compared to the development from

the initial loop. This behaviour is caused by a need of

the initial loop to generate a cell in state 5 (i.e. the

same state as the seed) from which the replica can

be developed (it takes 3 steps – see the top-right CA

state in Figure 4). The process of ﬁnishing the replica

is identical with the development from the seed. Note

that the ability of the transition function to develop

and replicate the loop from a seed was not required in

the ﬁtness function. Hence it can be considered as an

additional feature of the evolved solution. However,

it is not a common behaviour because only a few of

the obtained transition functions are capable to do it.

Another result is presented in the form of an

On Routine Evolution of New Replicating Structures in Cellular Automata

Table 1: Results of the evolutionary experiments considering the design of transition functions for the replication of the loop

from Figure 3a. Success rate – the number of successful experiments out of 100 independent experiments performed that

has met the ﬁtness speciﬁcation in a limit of 4 millions of generations, Replicates repeatedly – the number of results from

the successful experiments that are able to produce more replicas during the subsequent CA development, Min. steps – the

minimal number of steps of the CA needed to create the replica (i.e. the lowest value of this parameter from the group of

“Replicates repeatedly” solutions, Min. rules – the minimal number of table-based transition rules obtained (i.e. the lowest

value of this parameter from the group of “Replicates repeatedly” solutions.

CA with 8 cell states CA with 10 cell states

Num. of Success Replicates Min. Min. Success Replicates Min. Min.

CMRs rate [%] repeatedly steps rules rate repeatedly steps rules

20 0 - - - 1 0 - -

30 10 6 19 84 12 9 21 146

40 9 4 20 139 12 6 16 186

50 10 6 18 130 12 6 21 177

>=6;>=1;>=3;==0;>=1|7 ==0;>=4;<=0;==0;!=0|1 N W C E S C N W C E S C N W C E S C N W C E S C N W C E S C N W C E S C

>=0;<=3;>=0;<=0;>=3|1 <=2;==0;==0;<=2;>=1|3 0 0 0 0 3 1 1 0 7 2 1 1 1 2 7 7 7 1 1 7 7 0 0 4 4 2 3 0 4 1 7 1 3 0 4 7

==0;==0;==0;>=3;!=0|6 >=7;==0;>=6;>=6;<=2|2 0 0 0 0 7 1 1 0 7 2 7 1 1 3 1 0 0 6 1 7 7 0 3 2 4 2 7 0 3 1 7 2 0 0 3 1

<=1;!=0;<=1;>=5;<=1|0 <=5;!=0;>=7;!=0;<=0|7 0 0 1 1 0 0 1 0 7 7 1 1 1 3 6 0 0 7 1 7 7 3 0 6 4 7 0 0 0 3 7 2 6 0 0 0

==0;!=0;>=3;<=2;!=0|5 >=0;==0;==0;!=0;!=0|1 0 1 0 0 2 7 1 0 7 7 7 1 1 3 6 7 1 1 2 1 3 0 0 1 4 7 3 0 0 7 7 4 3 0 0 0

>=4;>=0;>=7;==0;>=2|6 !=0;!=0;==0;==0;==0|3 0 1 0 0 4 1 1 1 0 0 3 1 1 3 7 0 3 1 2 1 4 0 1 1 5 0 6 2 5 1 7 6 2 6 1 6

<=7;>=5;!=0;<=4;!=0|2 <=1;<=0;!=0;!=0;>=6|7 0 1 1 1 7 3 1 1 1 7 1 0 1 3 7 4 0 6 2 1 7 3 0 6 6 0 0 0 0 5 7 6 4 4 1 2

<=5;<=3;==0;<=7;<=1|0 ==0;>=4;<=2;<=7;>=6|1 0 1 1 5 0 0 1 1 3 0 0 1 1 4 0 0 0 3 2 5 1 0 0 6 6 0 0 1 0 5 7 6 6 0 1 7

>=3;>=3;!=0;>=0;!=0|6 >=5;>=1;>=1;==0;==0|0 0 1 3 0 0 1 1 1 4 7 1 1 1 4 1 0 0 6 2 6 0 3 3 4 6 0 5 1 0 7 7 7 3 0 0 7

!=0;>=4;<=4;>=5;==0|2 <=7;==0;==0;<=5;<=3|5 0 1 3 1 6 5 1 1 7 4 0 6 1 4 3 0 0 6 2 6 0 4 3 3 6 1 0 0 0 3 7 7 3 0 4 7

<=3;<=1;!=0;==0;<=2|1 !=0;>=5;==0;!=0;!=0|3 0 1 7 0 2 5 1 1 7 7 2 1 1 4 4 0 0 6 2 7 4 1 1 2 6 1 0 0 4 1 7 7 7 0 0 4

==0;>=5;<=3;==0;<=3|4 ==0;==0;==0;>=5;!=0|6 0 2 4 0 1 5 1 2 1 0 0 6 1 4 6 0 0 7 2 7 4 3 5 2 6 1 1 0 0 0

==0;!=0;>=2;==0;!=0|4 ==0;<=3;==0;==0;!=0|7 0 4 0 0 1 1 1 2 3 0 0 6 1 5 5 0 1 2 2 7 6 0 0 4 6 3 0 0 0 3

!=0;>=5;>=6;==0;<=6|4 !=0;==0;>=5;!=0;>=7|3 0 4 4 6 0 2 1 2 4 0 0 6 1 6 0 0 0 3 2 7 7 0 0 4 6 4 3 0 0 0

<=7;==0;>=0;<=2;!=0|1 !=0;<=6;>=5;<=2;<=0|7 0 5 0 0 0 4 1 2 4 0 3 1 1 6 4 0 0 6 2 7 7 0 6 2 6 6 0 0 0 3

>=1;>=7;!=0;==0;==0|7 ==0;<=0;>=0;<=2;==0|0 0 5 0 0 1 4 1 2 4 6 1 2 1 6 4 3 1 2 2 7 7 3 6 2 6 7 0 0 0 3

<=7;>=3;<=6;<=3;>=3|4 ==0;==0;==0;>=2;<=4|5 0 5 1 0 4 2 1 2 4 6 7 2 1 6 4 3 3 2 3 0 4 0 0 1 6 7 3 0 4 7

!=0;!=0;>=6;!=0;==0|6 ==0;>=5;>=0;>=5;>=2|7 0 5 4 0 1 5 1 2 5 5 1 2 1 6 4 4 1 2 3 4 4 0 0 6 7 0 0 0 0 1

==0;==0;>=0;==0;<=3|5 ==0;<=4;==0;>=6;<=2|6 0 6 0 0 0 4 1 2 6 0 0 7 1 6 6 0 0 4 3 5 0 0 0 3 7 0 3 0 0 1

<=1;<=3;>=3;>=7;!=0|1 ==0;<=0;>=1;>=4;==0|2 0 7 0 0 0 4 1 2 6 0 3 1 1 6 6 0 1 2 3 6 0 0 0 3 7 0 5 0 0 7

>=4;<=3;!=0;==0;!=0|2 >=5;==0;==0;<=2;>=6|4 0 7 0 0 1 4 1 2 6 6 1 2 1 6 6 7 1 2 3 7 0 0 0 3 7 1 0 0 0 1

==0;<=0;==0;<=6;<=1|7 !=0;!=0;<=7;>=5;==0|3 0 7 3 0 0 4 1 2 6 7 1 1 1 6 7 0 0 4 3 7 4 0 0 7 7 1 0 0 3 1

>=0;>=5;<=7;>=2;==0|6 !=0;>=2;!=0;==0;==0|6 0 7 4 3 0 6 1 2 7 0 3 1 1 7 0 0 0 3 4 1 0 0 4 1 7 1 1 0 1 2

<=1;>=2;!=0;>=5;<=7|2 <=0;<=2;<=5;!=0;>=7|3 0 7 6 6 3 2 1 2 7 4 0 6 1 7 3 0 0 7 4 1 1 0 2 2 7 1 2 0 6 1

>=7;<=2;==0;<=6;<=1|1 >=6;==0;<=3;<=5;<=1|1 1 0 5 2 6 1 1 2 7 7 0 6 1 7 4 0 0 7 4 1 5 0 0 7 7 1 3 0 0 0

(a) (b)

t t t t t t+1 t t t t t t+1 t t t t t t+1 t t t t t t+1 t t t t t t+1 t t t t t t+1

Figure 5: Transition function for the CA in Fig. 6 and 7: (a) the evolved representation with 50 CMRs, (b) the corresponding

conventional representation consisting of 130 rules. This result represents one of the best solutions discovered in this paper.

evolved transition function (Fig. 5) and the appropri-

ate CA development (Figures 6 and 7). This cellular

automaton demonstrates a development process from

a seed that at ﬁrst creates rather a chaotic structure

even larger than the required loop itself. A “mature”

loop is developed from this structure during the subse-

quent CA development that is able to replicate itself.

Whilst the replication of the initial loop takes 25 steps

(marked by the black arrow in Figure 6), the develop-

ment of the chaotic structure needs 36 steps. Starting

by step 37 (Fig. 7) the loop is developed from that

structure in the same way as from the initial loop. It

was veriﬁed that the loops are able to replicate repeat-

edly if the CA development continues.

For both the presented solutions the transition

function was identiﬁed as redundant (i.e. not all

the conventional transition rules generated from the

CMR representation are needed for the replication of

the initial loop required by the ﬁtness function). A

more detailed analysis showed that this redundancy

is caused by the ﬁnite CA size with cyclic boundary

conditions and by generating the transition rules from

the CMRs until the CA reaches a stable or periodic

state. Although this approach leads to more complex

table-based transition functions, in this case it showed

as very beneﬁcial for achieving some additional fea-

tures that were not required during evolution (espe-

cially the ability to develop the loops from a seed).

Advanced experiments with the resulting CA showed

that if the transition functions are optimized (i.e. only

the rules for the development of a single replica from

the initial loop are considered), the CA in most cases

loose the ability of the development from the seed.

It was also determined that the seed-based develop-

ment does not work in case of the known replicating

loops (e.g. Langton’s or Byl’s loop). In the future,

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

Figure 6: Part 1 of the replication according to the transition function from Figure 5. The sequence of steps reads from left

to right and top to bottom. The development shows a replication of the initial loop (the upper part of each step) and a growth

of a non-speciﬁc structure from a seed allowing to create the loop autonomously (the lower part of each step). The seed is

represented by a cell in state 7.

this ability may be beneﬁcial for the advanced study

of complex systems in which a given (complex) con-

ﬁguration needs to be achieved — distributed — from

a single cell or a simple initial conﬁguration. In addi-

tion to the results presented herein, various other so-

lutions were found that are able to replicate a given

structure. It indicates that the replication in CA is not

limited to known schemes only but can be performed

On Routine Evolution of New Replicating Structures in Cellular Automata

Figure 7: Part 2 of the replication according to the transition function from Figure 5. The sequence of steps reads from left to

right and top to bottom. The development shows an autonomous growth of the loop from a non-speciﬁc structure that emerged

in the last step of Figure 6 (the bottom part of each step). It was veriﬁed that the loop is able to replicate in the same way as

the initial loop during the subsequent CA development.

in many different ways.

In order to perform a general evaluation of the ob-

tained results within to the context of computational

features of cellular automata and with respect to the

existing replicating loops, the following issues need

to be clariﬁed:

1. The objective was not to design self-replication.

The loops with the ability to self-replicate contain

the information of how to create a copy encoded

in their “body” as a suitable arrangement of cell

states. The transition rules interpret this informa-

tion and calculate the appropriate state transitions

of the CA in order to perform the replication pro-

cess. In this paper, however, the initial loop is con-

sidered as an object of a given shape that ought to

be transformed onto a CA state that contains the

copy of the loop. The goal was to ﬁnd both the

transition rules and the sequence of the CA states

that lead to the emergence of the replica.

2. The resulting CA do not represent universal com-

puting models (it was not a goal of the experi-

ments). It means that a speciﬁc transition func-

tion, that was obtained as a result of a success-

ful evolution, is dedicated to replicate only the

given loop that was a subject of evaluation in

the ﬁtness function. Nevertheless, as the results

showed, some transition functions are able to cre-

ate the loops from a seed which was not explicitly

required during evolution.

Although the shape of the proposed loop was in-

spired by the existing (self-replicating) loops and the

GA provided some successful results to replicate this

loop, no working solution has yet been achieved by

the GA to replicate the existing loops (e.g. Byl’s loop)

with the exact shape and arrangement of the replicas.

This issue can be caused by the fact that some of the

self-replicating loops are dynamical structures even

after the replica is ﬁnished (e.g. Byl’s loop exhibits

such feature). However, only static replicas were con-

sidered in our experiments. Another aspect may be

the size of the loop that requires a higher number of

steps to ﬁnish the replica (e.g. Langton’s loop needs

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

151 steps) which makes the evaluation of such solu-

tion very time-consuming (it is a case of the prob-

lem of scale during the ﬁtness evaluation). Finally,

the information encoded in the loop body, that spec-

iﬁes the self-replication features, actually determines

the replication algorithm (i.e. the CA development)

which is speciﬁc for the given loop. If no more valid

algorithms exist in the solution space for such loop,

then the GA may not be able to ﬁnd the solution in a

reasonable time. Despite this issue, the obtained re-

sults bring some open questions whose investigation

could be beneﬁcial for the self-replication as well as

cellular automata in general. For example, can the

seed-based development create a conﬁguration in the

CA that supports self-replication (or other useful fea-

tures)? Are there other (simple) structures that sup-

port development of more complex (self-)replicating

objects? Can evolutionary techniques be applied to

the design of computationally universal CA-based

models? Not only these questions represent ideas for

our future research.

6 CONCLUSIONS

In this paper the evolutionary discovery of new repli-

cation processes was proposed. It was shown that

conditionally matching rules are suitable for a rou-

tine evolutionary design of multi-state CA that per-

form replication of a given loop-like structure. The

experiments provided many different solutions how

the replication of an initial loop can be performed.

In addition, some of the transition functions demon-

strated that the loop can even autonomously grow

from a single-cell seed and subsequently replicate ac-

cording to the original speciﬁcation. It indicates that

CA may exhibit some features that has not yet been

known and has not been discovered so far using con-

ventional techniques. A disadvantage of the proposed

results may be seen in a low replication speed in com-

parison with some known replicating loops (the so-

lutions presented herein replicate the loop in one di-

rection only). However, optimization of the replica-

tion speed was not a goal of this paper. In general, it

was demonstrated that new techniques of replication

can be discovered automatically for a given loop-like

structure.

ACKNOWLEDGEMENTS

This work was supported by the Czech Science Foun-

dation via the project no. GA14-04197S Advanced

Methods for Evolutionary Design of Complex Dig-

ital Circuits, and the IT4Innovations Centre of Ex-

cellence, no. CZ.1.05/1.1.00/02.0070, funded by the

European Regional Development Fund and the na-

tional budget of the Czech Republic via the Research

and Development for Innovations Operational Pro-

gramme.

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