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1

• On discovering gliders and glider guns
• with evolutionary algorithms to find
  computationally universal CAs
• Emmanuel Sapin, Pierre Collet, Olivier Bailleux,
  Larry Bull, Jean-Jacques Chabrier and Andy
  Adamatzky
• University of the West of England, Artificial
  Intelligence Group and Unconventional Computing
  Group

2
Introduction and Definitions
Our goal Find other universal 2 states 2D
cellular automata than the Game of Life
Our rationale Use stochastic search algorithms
(evolutionary algorithms) to find gliders and
glider guns, that can be used to implement logic
gates.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
3
Introduction and Definitions
Life is computationally universal
The Game of Life -gliders, glider guns and
eaters -An AND gate simulated by Life on the
next slide
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
4
Introduction and Definitions
Simulation of an AND Gate
Output stream (0101)
Glider gun
C1
A
Input stream A (1101)
C2
C2
after 176 generations
Eater
Collisions
Input stream B (0111)
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
5
Finding universal CAs
Evolutionary Algorithm No 1
Evolutionary Algorithm No 2
Set of automata
Subset of automata accepting gliders
Generalization to 3 state automata
Subset of automata simulating an AND Gate
Evolutionary Algorithm No 3
Subset of automata accepting guns
Evolutionary Algorithm No 2
Universality of an automaton
Universality of a automaton
..
Generalization to Highlife
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
6
Evolutionary Algorithm n1
-The algorithm tries to maximize the number of
gliders and the number of periodic patterns that
appear after the evolution of a random
configuration of cells.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
7
Results
This algorithm provided several automata
accepting gliders
Gliders with cardinal direction
Gliders with diagonal direction
Gliders with special angle
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
8
Finding logic-universal CAs
Evolutionary Algorithm No 1
Evolutionary Algorithm n2
Set of automata
Subset of automata accepting gliders
generalisation to 3 state automata
Subset of automata simulating an AND Gate
Algorithm
Subset of automata accepting guns
Evolutionary Algorithm n2
Universality of an automaton
Universality of a automaton
..
Generalization to Highlife
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
9
Evolutionary Algorithm n2
-This evolutionary algorithm searches for guns
emitting a specific glider. -The fitness
function tries to maximize the number of emitted
gliders while minimizing the total number of
cells (so as to maximize the chance that the
gliders are emitted by a gun).
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
10
Results

• - With this algorithm, for 50 different gliders
  we found 26 guns and over 25000 various guns were
  found for the following glider
• - Among the found guns, one looks for guns with a
  long period (so that the streams are usable for
  logic gates emulation).

Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
11
Periods of the found guns
10000
Three other guns of period 62, 74 and 243 are not
shown on the graphic.
Number of guns
2703
2557
1679
942
911
838
812
1000
133
79
100
50
45
42
43
38
28
28
25
17
11
10
3
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
1
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
Period of guns
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
12
Distribution of the number of guns emitting n
gliders during a period
Number of guns
Number of gliders
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
13
Discovered Guns
Emitting one glider
Emitting four gliders
Emitting eight gliders
Emitting two gliders
Emitting twelve gliders
Emitting six gliders
Emitting three gliders
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
14
Synthesis
- It is possible to discover new automata
supporting gliders, periodic patterns, and glider
guns with evolutionary algorithms .
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
15
Finding logic-universal CAs
Evolutionary Algorithm No 1
Evolutionary Algorithm n2
Set of automata
Subset of automata accepting gliders
generalisation to 3 state automata
Subset of automata simulating an AND Gate
Algorithm
Subset of automata accepting guns
Evolutionary Algorithm n2
Universality of an automaton
Universality of a automaton
..
Generalization to Highlife
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
16
Universality of an automaton accepting gliders
and glider guns
Among all the found automata, we chose the one
showed below (that we called R)
Glider
Glider gun
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
17
In order to show the universality of R, we chose
to simulate the Game of Life with R.
The first step is to simulate a cell of
Life Sn1 N2 . N1 . (Sn N0). - AND Gate and
Not Gate
Once a cell is created, one needs to tile a
surface with interconnected Life cells.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
18
R automaton
Glider
Eater found by evolutionary algortihm
-Glider gun
Glider gun
Glider gun
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England, Artificial Intelligence Group and
Unconventional Computing Group
19
Duplication and redirection
Glider guns can be used to duplicate or redirect
a glider stream (note that streams are
complemented by the duplication and redirection
operators)
Stream
Duplication
Redirection
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
20
Creation of a NOT gate using duplication and
redirection
C
C
Used scheme
C
C
C
C
C
C
New used pattern big gun
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
21
Stream a 10
Glider gun
NOT Gate
Stream way
Not(a)
Not(a)
a
a
Not(a)
Not(a)
Not(a)
Eater
Glider gun
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
22
Glider gun
Stream
Not(a)
Not(a)
a
a
Not(a)
Not(a)
Eater
Not(a)
Glider gun
Stream not(a) 01
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England, Artificial Intelligence Group and
Unconventional Computing Group
23
Simulation of a Life cell
The first eight input streams are the streams of
the eight adjacent cells numbered from C1 to C8.
C8
C7
C6
C5
C4
Input
C3
C2
C1
Sn
Output
Sn1
This simulation uses 33412 cells and produces the
stream Sn1 after 21600 generations.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
24
Once a cell is created, one needs to tile a
surface with interconnected Life cells.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
25
Interconnexion of simulation of cells
Simulation of cell 3
Simulation of cell 1
Simulation of cell 2
Problem interconnect the nine cells
S C3
S C1
C2 S
Simulation of cell S
Simulation of cell 5
Simulation of cell 4
C5 S
S C4
C6 S
C8 S
S C7
Simulation of cell 7
Simulation of cell 8
Simulation of cell 6
It is straightforward for a cell to send its
state to its cardinal neighbors C2, C4, C5, C7.
Sending its state to neighbors C1, C3, C6, C8 is
however more tricky, since they are situated
diagonally.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
26
All cells being identical, the inputs of a cell
must physically correspond to the outputs of its
neighbours. Therefore, the way a cell
receives the state of its neighbours can be
induced from the way it sends its own state to
its neighbours.
1 2
3
Simulation of a cell of the Game of Life
C1
C2
C3
C4
S
C5
4
5
C6
C7
C8
6 7
8
Stream Duplication
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
27
1 2
3
Simulation of a cell of the Game of Life
C1
C2
C3
C4
S
C5
4
5
C6
C7
C8
6 7
8
Stream Duplication
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
28
C7
1 2
3
S
S
C2
C2
Simulation of a cell of the Game of Life
C1
S
C2
C3
C4
S
S
C5
5
4
C6
C7
C8
S
C7
S
S
C7
6 7
8
Stream Duplication
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
29
1 2
3
C2
C2
C2
C3
C2
C2
C1
Simulation of a cell of the Game of Life
C3
C4
C2
C1
C4
C5
S
C6
5
4
C7
C5
C8
S
C7
C8
C6
C7
C7
C7
C7
6 7
8
Stream Duplication
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
30
Simulation of a cell of the Game of Life
Simulation of a cell of the Game of Life
Simulation of a cell of the Game of Life
C
C5
C5
C
C
C4
C
C
C2
C3
C2
C2
C1
C3
C3
C2
C2
C2
C1
C1
Simulation of a cell of the Game of Life
Simulation of a cell of the Game of Life
Simulation of a cell of the Game of Life
C3
C4
C2
C1
C4
C5
C
S
S
C6
C7
C5
C8
S
C7
C8
C7
C8
C6
C7
C6
C7
C7
C7
C6
C8
C
C5
C
C
C4
C5
C5
C
C4
Simulation of a cell of the Game of Life
Simulation of a cell of the Game of Life
Simulation of a cell of the Game of Life
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
31
Conclusion
- Life was shown to be computational universal by
Conway et al. in 1982 and by Rendell in 2002. - R
can implement a tilling of interconnected Life
cells ( R can implement Life) - R is therefore
also universal - A set of two genetic algorithms
has allowed to find another computational
universal CA than Life. Question How many 2D two
states CAs are universal ?
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
32
Annexe 1 Definitions
Turing Machine A MODEL OF COMPUTATION that
uses an underlying FINITE-STATE AUTOMATON but
also has a infinite tape to use as memory. Turing
machines are capable of UNIVERSAL
COMPUTATION. Universal Computer A computer
that is capable of UNIVERSAL COMPUTATION, which
means that given a description of any other
computer or PROGRAM and some data, it can
perfectly emulate this second computer or
program. Strictly speaking, home PCs amd
Macintoshes are not universal computers because
they have only a finite amount of memory.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
33
Annexe 2 Evolutionary Algorithm
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
34
Annexe 3 Evolutionary Algorithm n1
Number of gliders Number of periodic patterns
that appear during the evolution of a random
configuration of cells
Fitness Function
Transition rule are initialised at random
Initialisation
Mutating the value of one neighborhood Single
point crossover with a locus situated on the
middle
Genetic operators
(mu lambda) Evolution Strategy
Evolution Engine
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England, Artificial Intelligence Group and
Unconventional Computing Group
35
Conservation des 10 meilleures règles
Initialisation des 50 règles
Création de 40 règles par croisement et mutation
Évaluation des règles par la fonction de fitness
Constitution dune nouvelle population de 50
règles avec les règles conservées et celles
créées par croisement et mutation
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
36
Fitness
On génère une soupe primordiale
200
Évolution pendant 150 générations
40
Glisseur
Forme périodique
Chaque forme est isolée
Puis testée
Fitness glisseurs ? formes périodiques
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England, Artificial Intelligence Group and
Unconventional Computing Group
37
Annexe 4 Generalization to 3 state automata
Von Neumann neighboorhood
Representation of an automaton
0 1 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0
This space contains 320 automata about 3.48 109
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England, Artificial Intelligence Group and
Unconventional Computing Group
38
This algorithm found the number of 5944 gliders.
Most common gliders
Diagonal glider
Slowest glider
Dynamic glider gun
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
39
Annexe 5 Evolutionary Algorithm n2
Number of gliders / the total number of cells
that appear during the evolution of a random
configuration of cells
Fitness Function
Transition rule are initialised at random
Initialisation
Mutating the value of one neighborhood not used
by the given glider Single point crossover with a
locus situated on the middle
Genetic operators
(mu lambda) Evolution Strategy
Evolution Engine
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
40
Détermination des contextes critiques dun motif
Neighborhoods used by a glider
Exemple le glisseur de R0 sur une demi période
Ensemble des contextes encadrés ensemble des
contextes utilisés
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England, Artificial Intelligence Group and
Unconventional Computing Group
41
Annexe 6 Recherche d un effaceur
Formes périodiques apparaissant le plus souvent
dans lévolution de R0
Chaque forme est testée dans plusieurs positions
?
Aucun effaceur détecté ? Modification de R0 en
conservant le lance-glisseurs ? Introduction des
notions despace vital et de contexte critique
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England, Artificial Intelligence Group and
Unconventional Computing Group
42
Détermination des contextes critiques dun motif
Exemple le glisseur de R0 sur une demi période
Ensemble des contextes encadrés ensemble des
contextes utilisés
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England, Artificial Intelligence Group and
Unconventional Computing Group
43
Contextes utilisé par le canon de R0
81 contextes utilisés ? 221 automates acceptent
le canon ? Recherche dun automate acceptant un
effaceur
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England, Artificial Intelligence Group and
Unconventional Computing Group
44
Opérateur de l algorithme évolutionnaire
Forme-position-règle
Individu
50 individus engendrés - forme et position
aléatoires, - règle initialisée à R0.
Initialisation
Croisement
Pas de croisement
Mutation de - la forme, -
labscisse, - lordonnée,
- la valeur dun contexte non
critique.
Mutation
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England, Artificial Intelligence Group and
Unconventional Computing Group
45
Le nombre de glisseurs stoppé par le motif
Exemple
Deuxième glisseur
Motif
Résultat de lévolution par R après 11
générations
Après 16 générations
Position
Premier glisseur
Le premier glisseur  percute  le motif ? Ils
sont tous deux détruits ? la valeur de la
fonction dévaluation est 1
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England, Artificial Intelligence Group and
Unconventional Computing Group
46
Algorithme évolutionnaire recherchant un effaceur
Initialisation des 50 individus
Les 25 meilleurs sont conservés
25 individus sont créés par mutation
Evaluation des individus par la fonction
dévaluation
Constitution d une nouvelle population de 50
individus avec ceux conservés et ceux créés par
mutation
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
47
Distribution de la fonction dévaluation des
individus dans la population initiale
Fonction dévaluation après lévolution de
lalgorithme évolutionnaire
2 1
5 4 3
8 7
Maximum local
Extremum
17 16 15 14
14 13
11 10
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
48
Annexe 7 Highlife
Transition rule
The Gosper gun does
not work with the transition rule of Highlife
Highlife can not be shown universel by Conway
demonstration of universality of the Game of Life
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
49
Generalisation of the demonstration to Highlife
We show that an automaton that simulates these
four patterns is universel
Duplication
Nand Gate
Temporization
Redirection
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
50
Generalisation of the demonstration to Highlife
We show that Highlife simulates these four
patterns and is universal thanks to this gun
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
51
Annexe 8 How many guns can be found by this
algorithm?
Suppose each gun has the same probability to be
found.Let N be the total number of guns findable
by this algorithm. The probability of a gun found
by the algorithm to be new would be
1- 15363/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
153631000/245 about 62706.
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
52
Annexe 9 Glider gun for various gliders
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
53
Annexe 10 an And Gate found by evolutionary
algorithm
Gun
Eater
Stream A 0 1 1
0 0 1 Stream AB
1 0 1 Stream B
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group
54
Annexe 1
Emmanuel Sapin, University of the West of
England, Artificial Intelligence Group and
Unconventional Computing Group