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Searching for Glider Guns in Cellular Automata: Exploring Evolutionary and Other Techniques

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We aim to construct an automatic system for the discovery of collision-based universal cellular automata that simulate Turing machines in their space-time dynamics using gliders and glider guns. In this paper, an evolutionary search for glider guns with different parameters is described and other search techniques are also presented as benchmark. We demonstrate the spontaneous emergence of an important number of novel glider guns discovered by genetic algorithms.
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Searching for Glider Guns in Cellular Automata:
Exploring Evolutionary and other Techniques
E. Sapin and L. Bull
Faculty of Computing, Engineering and Mathematical Sciences,
University of the West of England,
Bristol BS16 1QY, UK
emmanuelsapin@hotmail.com, Larry.Bull@uwe.ac.uk
Abstract. We aim to construct an automatic system for the discov-
ery of collision-based universal cellular automata that simulate Turing
machines in their space-time dynamics using gliders and glider guns.
In this paper, an evolutionary search for glider guns with diﬀerent pa-
rameters is described and other search techniques are also presented as
benchmark. We demonstrate the sponstaneous emergence of an impor-
tant number of novel glider guns discovered by genetic algorithms.
1 Introduction
The emergence of computation in complex systems with simple components is
a hot topic in the science of complexity [1]. A uniform framework to study
emergent computation in complex systems are cellular automata [2]. They are
discrete systems in which an array of cells evolves from generation to generation
on the basis of local transition rules [3].
The well-established problems of emergent computation and universality in
cellular automata has been tackled by a number of people in the last thirty
years [4], [5], [6], [7], [8] and remains an area where amazing phenomena at the
edge of theoretical computer science and non-linear science can be discovered.
The most known universal automaton is the Game of Life [9]. It was shown
to be universal by Conway [10] who employed gliders and glider guns. Gliders
are mobile self-localized patterns of non-resting states, and glider guns are pat-
terns which, when evolving alone, periodically recover their original shape after
emitting some gliders.
The search for gliders was notably explored by Adamatzky et al. with a phe-
nomenological search [11], Wuensche who used his Z-parameter and entropy [12]
and Eppstein [13]. Sapin et al. have considered the emergence of gliders-based
universality by use of a genetic algorithm [14].
We aim to construct an automatic system for the discovery of computation-
ally universal cellular automata, spatially. Inspired by the link between universal-
ity and the presence of gliders and glider guns in cellular automata, we are here
interested in the emergence of glider guns. In this paper, three search methods
for glider guns are compared.
2 E. Sapin, L. Bull
The paper is arranged as follows: Section 2 describes previous related work.
Section 3 sets out the characteristics of the search methods. Then the result of
the best search method are described in Section 4. The last section summarizes
the presented results and discusses directions for future research.
2 Previous work
In this section, some previous work about cellular automata are presented. Brief
descriptions of some search methods are given. Then some previous work about
using an evolutionary approach to search for automata are presented.
2.1 Cellular automata
In [15], Wolfram studies the space Iof 2D isotropic CA, with rectangular 8-cell
neighbourhoods: if two cells have the same neighbourhood states by rotations
and symmetries, then these two cells take the same state at the next generation.
There are 512 diﬀerent rectangular 8-cell neighbourhood states. An automaton
of Ican be described as shown ﬁgure 1 by telling what will become of a cell in
the next generation, depending on its subset of isotropic neighbourhood states.
Fig. 1. The squares are the 102 neighbourhood states describing an automaton of I.
A black cell on the right of the neighbourhood state indicates a future central cell.
There are 102 subsets of isotropic neighbourhood states, meaning that there
are 2102 diﬀerent automata in I.
Searching for Glider Guns by Evolutionary Algorithms 3
2.2 Search methods
In order to search for universal automata, we have examined the search methods
such as monte carlo, taboo search [16] and an evolutionary algorithm [17], as
brieﬂy described here.
The monte carlo method consists solely of generating random solutions and
testing them.
Tabu search traverses the solution space by testing mutations of an individual
solution. Tabu search generates many mutated solutions and moves to the best
solution of those generated. In order to prevent cycling and encourage greater
movement through the solution space, a tabu list is maintained of partial or
complete solutions. It is forbidden to move to a solution that contains elements
of the tabu list, which is updated as the solution traverses the solution space.
Evolutionary algorithms have been used with cellular automata in a number
of ways, after [18].
2.3 Evolving Cellular Automata
Previously, several good results from the evolution of cellular automaton rules
to perform some useful tasks have been published. Mitchell et al. [19–22] have
investigated the use of evolutionary computing to learn the rules of uniform
one-dimensional, binary cellular automata. Here a Genetic Algorithm produces
the entries in the update table used by each cell, candidate solutions being
evaluated with regard to their degree of success for the given task — density
and synchronization.
Sipper [23] has presented a related approach, which produces non-uniform
solutio¡ ns. Each cell of a one or two-dimensional cellular automata is viewed as
a genetic algorithm population member, mating only with its lattice neighbours
and receiving an individual ﬁtness. He shows an increase in performance over
Mitchell et al.’s work, exploiting the potential for spatial heterogeneity in the
tasks. Koza et al. [24] have also repeated Mitchell et al.’s work, using Genetic
Programming [25] to evolve update rules. They report similar results.
3 Search for Glider Guns
This section describes the used search for glider guns. To compare the parameters
of the search methods, glider guns that emitting the glider in ﬁgure 2 are searched
for. The ﬁrst search method is an evolutionary algorithm. Monte carlo algorithm
and tabu search are also used as benchmark.
3.1 Evolutionary Algorithm
The parameters of the evolutionary algorithm are described here.
Fitness Function
Two ﬁtness functions have been tried.
4 E. Sapin, L. Bull
Fig. 2. Glider.
First ﬁtness function
The computation of the ﬁtness function is based on the one used in [26].
A random conﬁguration of cells is evolved by the tested automaton. After
this evolution, the presence of gliders Gis checked by scanning the result
of the conﬁguration of the cells. The value of the ﬁtness function is the
number of gliders that appeared divided by the total number of cells.
Second ﬁtness function
The computation of the second ﬁtness function is based on the ﬁrst one.
A random conﬁguration of cells is evolved by the tested automaton.
After this evolution, the presence of gliders Gis checked by scanning the
result of the conﬁguration of the cells. The size of the biggest square S
without cells in the evolved conﬁguration of cells is computed. The value
of the ﬁtness function is the multiplication of the number of gliders that
appeared by the size of the square S.
Initialization
The search space is the set Idescribed in Section 2. Cell-state transition
table can describe an automaton of this space. An individual is an automaton
coded as a bit string of 102 Booleans representing the values of a cell at the
next generation for each neighbourhood state. The 102 bits of an automaton
are divided into two subsets. The ﬁrst subset, called invarious subset, is the
neighbourhood states used by the glider Gand their values are determined by
the evolution of G. The process that determines these neighbourhood states
is detailed for the glider of ﬁgure 2 in the ﬁgure 3. The other neighbourhood
states are in the second subset, called unused subset, and are initialised at
random. The population size is 100 individuals.
Genetic Operators
The mutation function simply consists of mutating one bit among the second
subset of the 102 bits. The rates 1,5 and 10 percent are tried, together with
three crossover operators.
No crossover
The genetic algorithm is tried without crossover.
Central point
A single point crossover with a locus situated exactly on the middle of
the genotype is tried.
Random point
Searching for Glider Guns by Evolutionary Algorithms 5
Fig. 3. Detail of the construction of set of neighbourhood states that are used by a
glider.
6 E. Sapin, L. Bull
A last kind of recombination is tried with a single point crossover with
a locus randomly situated.
A linear ranking selection and a binary tournament selection of size 2 are
tried.
Evolution Engine
An elitist strategy in which the best half of population is kept and a non-
elitist strategy in which the new population is made of only children are
tried.
Stopping Criterion
The presence of a glider gun is continuously checked. The test is inspired
by Bays’ test [27] and also used in [28]. After the evolution of the random
conﬁguration of cells, the pattern is isolated and tested in an empty universe.
If a pattern Preappears at the same place with gliders around then the
pattern Pis a glider gun. When a glider gun is found the algorithm stops.
The value of the ﬁtness function and the generation of the best rule are
memorized. If after ten new generations the algorithm has not found a better
rule the algorithm stops.
Thanks to these stopping criteria, an execution of the algorithm stops after
an average of 38 generations.
3.2 Monte Carlo method
In one million randomly generated automata, the presence of glider guns after
the evolution of a random conﬁguration of cells was tested. The test is the one
used for the stopping criterion of the algorithm described in Section 3.1. There
are not any guns found by this method.
3.3 Tabu Search
A random automaton Ais generated, two ﬁtness functions are tried to measure
the performance of this automaton:
All the automata obtained by mutating one bit among the unused subset, as
described section 2.5, are tested by the ﬁtness function. The best one who is not
in a list Lof the last chosen automaton is chosen to become the new automaton
A. The sizes of 10, 100 and 1000 are tried for the list L. The presence of glider
guns is checked in all the tested automata.
The algorithm stops when the best automaton, among the automata obtained
by mutation, who is not in a list Lis not better than the current automaton.
With this stopping criterion, an execution of the algorithm stop after an average
of 49, 42 and 35 generations depending on the size of the list L.
3.4 Discussion
For each of the values of the parameters, the number of executions which ﬁnd a
gun are shown in table 1.
Searching for Glider Guns by Evolutionary Algorithms 7
Table 1. Number of executions from a total of 100 per experiment that ﬁnd a gun
under a given combination of parameters or operators. The three numbers correspond
to the 1,5, and 10 mutation rates. No guns were found with the monte carlo algorithm.
The best parameters for the evolutionary algorithm, among the tested ones,
are a mutation rate of 1, a non elitist strategy, a tournament selection and a
central crossover and second ﬁtness function. The evolutionary algorithm with
these parameters have been chosen to obtain the glider guns described in the
next Sections.
The good results of the central crossover can be explained by the fact that
the ﬁrst 51 neighbourhood states determine the birth of cells, while the other
51 determine how they survive or die. The elitist strategy that kept half of the
population is worse than the non-elitist strategy. An elitist strategy that just
kept one parent could be tried, however.
The results of a monte carlo algorithm and tabu search,presented as bench-
marks, are not as good as the evolutionary approach. The results of the evolu-
tionary algorithms without crossover are about the same of the tabu search.
4 Results of the algorithm
The results of the genetic algorithm with the best parameters for the glider in
ﬁgure 2 are described here.
4.1 Number of guns
In order to determine how many diﬀerent glider guns were found, an automatic
system that determines if a gun is new is required. So, in order to determine if a
8 E. Sapin, L. Bull
gun is new, the set of neighbourhood states used by the given gun are compared
to the ones of the other guns. For each gun and each neighbourhood state, three
cases exist:
The neighbourhood state is not used by this gun.
The neighbourhood state is used by this gun and the value of the central
cell at the next generation is 0.
The neighbourhood state is used by this gun and the value of the central
cell at the next generation is 1.
Two guns are diﬀerent iﬀ at least one neighbourhood state is not in the same
case for the two guns.
Thank to this qualiﬁcation of diﬀerent guns leads, through the experimenta-
tions, 10008 diﬀerent glider guns were discovered. All these guns have emerged
spontaneously from random conﬁgurations of cells. The 10008 guns can be found
in [29] in Life format.
The total number of diﬀerent guns ﬁndable by this algorithm is unknown but
the evolution of the percentage of new guns among the last 1000 diﬀerent found
guns is given by the ﬁgure 4.
Fig. 4. Evolution of the percentage of new guns among 1000 diﬀerent found guns.
Suppose each gun has the same probability to be found.
Let Nbe the total number of guns ﬁndable by this algorithm. The probability
of a gun found by the algorithm to be new would be 1 10008/N. The number
of new guns among the last 1000 diﬀerent found guns is 755. So the total number
of guns ﬁndable by the algorithm could be estimated by N= 10008 1000/245
5 Synthesis and perspectives
This paper deals with the emergence of computation in complex systems with
local interactions. A search for glider guns has been presented, building on pre-
vious work in [26, 28].
Searching for Glider Guns by Evolutionary Algorithms 9
In particular, monte carlo method, tabu search and evolutionnary algorithms
are explored with diﬀerent parameters. The best results are found for an evo-
lutionnary algorithm. The experimentation showed that cross over in the evo-
lutionary algorithm plays a key role in the search process. Future work will
consider other search techniques.
The algorithm succeeded in ﬁnding 10008 glider guns [29] for the glider of
ﬁgure 2. The discovery of the emergence and existence of so many diﬀerent
glider guns for the same glider represent a signiﬁcant contribution to the area of
complex systems that considers computational theory.
Further goals can be to ﬁnd all the glider guns possible and to calculate how
many automata exhibit these guns. All these automata may be potential can-
didates for being shown universal automata thanks to an automatic system for
the demonstration of universal automata that can be developed. Then, another
domain that seems worth exploring is how this approach could be extended to
automata with more than 2 states.
Future work could also calculate for each automata some rule-based para-
meters, e.g., Langton’s lamda [30]. All automata exhibing glider guns may have
similar values for these parameters that could lead to a better understanding of
the link between the rule transition and the emergence of computation in cellular
automata and therefore the emergence of computation in complex systems with
simple components.
6 ACKNOWLEDGEMENTS
The work was supported by the Engineering and Physical Sciences Research
Council (EPSRC) of the United Kingdom, Grant EP/E005241/1.
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... So, after discovering gliders, glider guns were searched for with evolutionary algorithms. All the details about this algorithm can be found in [22,23]. ...
... 23 Gliders of period 2 and velocity 1 2 found by our algorithm ...
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These local proceedings hold the papers of two catgeories: (a) Short, non-reviewed papers (b) Full papers
Chapter
A number-conserving reversible cellular automaton (NC-RCA) is a computing model that reflects both reversibility and mass or energy conservation law in physics. We show that, despite strict constraints of reversibility and number-conservation, there exist simple two-dimensional NC-RCAs that are capable of universal computation. These automata are classified into two types. Automaton of the first type implements a Fredkin gate in its space-time dynamic. That is a Fredkin gate, as a universal logical element, is embedded in its cellular space. Automaton of the second type, incorporate a novel logical element, called a “rotary element” in its lattice dynamic. NC-RCAs of the second type simulate any reversible two-counter machine in their space-time configuration. In both types of cellular automata the computation is realized via appropriate control of “moving particles” in the cellular space.
Article
Natural evolution has "created" many parallel cellular systems, in which emergent computation gives rise to impressive computational capabilities. In recent years we are witness to a rapidly growing interest in such complex adaptive systems, addressing, among others, the major problem of designing them to exhibit a spe-cific behavior or solve a given problem. One possible approach, which we explore in this paper, is to employ artificial evolution. The systems studied are based on the cellular automata (CA) model, where a regular grid of cells is updated syn-chronously in discrete time steps, according to a local, identical interaction rule. We first present the application of a standard genetic algorithm to the evolution of CAs to perform two non-trivial computational tasks, density and synchroniza-tion, showing that high-performance systems can be attained. The evolutionary process as well as the resulting emergent computation are then discussed. Next we study two generalizations of the CA model, the first consisting of non-uniform CAs, where cellular rules need not be identical for all cells. Introducing the cellu-lar programming evolutionary algorithm, we apply it to six computational tasks, demonstrating that high-performance systems can be evolved. The second gen-eralization involves non-standard, evolving connectivity architectures, where we demonstrate that yet better systems can be attained. Evolving, cellular systems hold potential both scientifically, as vehicles for studying phenomena of interest in areas such as complex adaptive systems and artificial life, as well as practi-cally, showing a range of potential future applications ensuing the construction of adaptive systems, and in particular 'evolving ware', evolware.
Article
We study two- and three-dimensional latticesnodes of which take three states: rest, eccited, andrefractory, and deterministically update their states indiscrete time depending on the number of excited closest neighbors. Every resting node isexcited if exactly 2 of its 8 (in two-dimensionallattice) or exactly 4 of its 26 (in three-dimensionallattice) closest neighbors are excited. A node changesits excited state into the refractory state and itsrefractory state into the rest state unconditionally. Weprove that such lattices are the minimal models oflattice excitation that exhibit bounded movable patterns of self-localized excitation(particle-like waves). The minimal, compact, stable,indivisible, and capable of nonstop movementparticle-like waves represent quanta of information.Exploring all possible binary collisions betweenparticle-like waves, we construct the catalogue of thelogical gates that are realized in the excitablelattices. The space and time complexity of the logicaloperations is evaluated and the possible realizations ofthe registers, counters, and reflectors are discussed.The place of the excitable lattices in the hierarchy ofcomputation universal models and their high affinity to real-life analogues affirm that excitablelattices may be the minimal models of real-likedynamical universal computation.
Article
The existence of computation-universal one-dimensional cellular automata with seven states per cell for a transition function depending on the cell itself and its nearest neighbors (r=1), and four states per cell for r=2 (when next-nearest neighbors also are included), is shown. It is also demonstrated that a Turing machine with m tape symbols and n internal states can be simulated by a cellular automaton of range r=1 with m+n+2 states per cell.