Point Mutation Colonies with Restricted Rules
Adam Kožaný
Institute of Computer Science, Faculty of Philosophy and Sciences, Silesian University in Opava,
Bezručovo náměstí 13, Opava, Czech Republic
Keywords: PM Colony, Artificial Life, Multi-agent Systems.
Abstract: Point mutation colonies (hereinafter referred to as PM colonies) are multi-agent systems. Development of
the environment in these systems is determined by rewriting rules which allow the agent to influence other
agents and environmental symbols in its strict neighbourhood. The rules enable the agent to erase, substitute
or insert neighbouring agents/symbols, to change its position with neighbouring agents/symbols or to
disappear. In this paper we will focus on the impact of forbidding some of the rule-type or their combination
in the development of the entire family of PM colonies with such restriction and we will also look into the
impact of restrictions on the generative power of PM colonies.
1 INTRODUCTION
PM colonies were introduced in two papers (Martín-
Vide and Paun, 1998). This type of colonies
(Csuhaj-Varjú and Kelemenová, 1992) is motivated
by biology, or more precisely by communities of
organisms living in a common environment. In the
field of informatics this kind of co-existence
represents a multi-agent system.
In (Martín-Vide and Paun, 1998) there was an
open question of decidability results flowing into the
necessity of studying restricted classes of PM
colonies. Similar topics were studied in
(Kelemenová, 2002), (Kožaný, 2009) and (Kožaný,
2010).
In this paper we are restricting the set of possible
rule-types in PM colonies. Then we discuss
decidability problem if two agents can reach a
conflict in a given PM colony from a given starting
string. We will also discuss the generative capacity
of such PM colonies.
2 PM COLONIES
Colonies are grammar models of multi-agent
systems motivated by subsumptial architecture and
they are characterised as special forms of
cooperating grammars. A colony consists of a finite
number of simple components (agents) each
generating a finite language. More about grammar
systems and especially about colonies is presented in
(Harrison, 1978) and in (Csuhaj-Varjú et al., 1994).
Environment in colonies is represented by a
string of symbols, and it is influenced via
components which make changes in it. The set of all
possible states of the environment, which can be
generated from a given starting string, forms the
language of the colony.
In a PM colony environment the locations of
agents are fixed. The area, where the PM colony
works, is represented by a string of agents and
environment symbols (which can be changed) and
boundary markers of the environment. Boundary
markers label the beginning and the end of a word, it
is not allowed to erase them, to overstep or to
produce them.
The actions take place only in strict vicinity of
the symbol representing the agent. Each action can
add one environment symbol or one agent symbol (a
new agent can be also created), can move agent one
step to the left or right, can erase neighbouring agent
or environment symbol, or can substitute an
environment symbol to another one.
All agents work in parallel. The activity of an
agent depends totally on one symbol in front of it
and one symbol behind it. To solve a conflict, when
agents have a common neighbour, we arrange the set
of agents by a priority relation. An agent can‘t
change its own name or name of any other agent.
Agents with the same name may be present on more
than one position in the string. Formally:
245
Kožaný A..
Point Mutation Colonies with Restricted Rules.
DOI: 10.5220/0004180902450249
In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 245-249
ISBN: 978-989-8565-33-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Definition 1. PM colony is a construct C = (E, #, N,
>, R
1
, ..., R
n
), where
- E is the alphabet of the environment,
- # is the boundary marker,
- N is the alphabet of agents names,
- > is the partial order relation over N (the
priority relation for agents),
- R
1
, ..., R
n
are finite sets of action rules for agents
from N. The action rules can be of the following
forms:
- Deletion:
(a, A
i
, b)  (  , A
i
, b), where a  E  N, b  E 
N
 {#},
(a, A
i
, b)  (a, A
i
,  ), where a  E  N  {#}, b
 E  N,
- Insertion:
(a, A
i
, b)  (a, c, A
i
, b), where a, b  E  N 
{#}, c
 E  N,
(a, A
i
, b)  (a, A
i
, c, b), where a, b  E  N 
{#}, c
 E  N,
- Substitution:
(a, A
i
, b)  ( c, A
i
, b), where b  E  N  {#},
a, c
 E,
(a, A
i
, b)  (a, A
i
, c), where a  E  N  {#},b,
c
 E,
- Move:
(a, A
i
, b)  (A
i
, a, b), where a  E  N, b  E 
N
 {#},
(a, A
i
, b)  (a, b, A
i
), where a  E  N  {#}, b
 E  N,
- Death:
(a, A
i
, b)  (a,  , b), where a, b  E  N  {#}.
Let (a, A
i
, b)   be an action rule of an agent, then
symbols a, b represent the context of agent A
i
.
PM colonies are devices, where agents work
parallel. Similarly as in the other parallel working
systems, conflicts can occur between agents.
Definition 2. If in a word w
 (E  N )
*
context
overlay of two agents A
i
and A
j
happens or if agent
A
j
takes part in context of agent A
i
, we call it direct
conflict between agents. If in w the pairs of agents
(A
1
, A
2
), (A
2
, A
3
)... (A
n
, A
n+1
) are in direct conflict
then the whole set of agents A
1
, A
2
, A
3
,..., A
n
, A
n+1
are
in conflict.
The conflict of agents A
1
, A
2
, A
3
,..., A
n
, A
n+1
in
PM colony can be solved by the agent with the
greatest priority, which takes action. So, to solve the
conflict, conflicting agents have to be ordered in
such a way, that there is an agent with priority
higher then all other agents in the conflict. Moreover
the agent with the greatest priority occurs in the
conflict set only once.
Definition 3. A configuration in a PM colony C is a
string #w#, where w
 (E  N )
*
.
Let A be its agent and #w# = xaAby be a
configuration in C, where a,b
 (E  N )  {#}. This
occurrence of agent A is active with respect to
configuration #w#, if (1) in C an action rule exists,
whose left side is in the form (a, A, b), and (2) A is
not conflicting with any other agent occurrence, or A
has the highest priority from all agents from those in
conflict.
An agent occurrence is inactive, if it is not
active.
Definition 4. A derivation step in a PM colony
denoted as
 is a binary relation on a set of
configurations. We write #w#
 #z# if and only if
each active agent A in the string w replaces its
context in w by corresponding rule and the resultant
string is #z#. Derivation

*
is the reflexive and
transitive closure of relation
 .
Definition 5. Deterministic PM colony is such PM
colony where each agent A has for any context
(a, A, b) at most one action rule.
3 (UN)DECIDABILITY RESULTS
IN PM COLONIES
In (Martín-Vide and Paun, 1998) there are several
problems focused on decidability mentioned. To
explain those problems we have to mention some
structural properties of PM colonies, which
determine structures in environmental states
introduced in the work above.
Definition 6. Let C = (E, #, N, >, R
1
, .. , R
n
) be a
PM colony. A state y  #(E
 N)
*
# is reachable in
C if there is a state z  y such that z  y with respect
to C. A state which is not reachable is said to be
unreachable. A state y  #(E
 N)
*
# is said to be
alive if there is a state z  y such that y  z. A state
which is not alive is said to be dead.
By intersecting the classes in the two
classifications above, we get four classes of states.
We denote by Reachable(C), Unreachable(C),
Alive(C), Dead(C) the languages of all reachable,
unreachable, alive, and dead states, respectively,
with respect to C. We also denote:
Garden-of-Eden(C) = Unreachable(C)  Alive(C)
Life(C) = Reachable(C)  Alive(C)
Doomsday(C) = Reachable(C)  Dead(C)
Non-Life(C) = Unreachable(C)  Dead(C)
Proposition 1 (Martín-Vide and Paun, 1998). All
the languages Reachable(C), Unreachable(C),
Alive(C), Dead(C), Garden-of-Eden(C), Life(C),
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Doomsday(C), Non-Life(C) are regular for PM
colony C.
This proposition implies that most problems
about the mentioned languages are decidable. For
instance, one can decide whether or not they are
empty, finite or infinite, equal to any given regular
language, included or including any given regular
language.
The previous decidability results hold with
respect to all states which are alive, reachable, dead,
etc, with respect to any given PM colony. Let us
assume PM colony and its starting string ω. We can
relate the reachable strings to be reachable from ω
and use following modifications of the above
structures
ωReachable(C) = {u: ω

C
*
u} ∩ Reachable(C).
This leads to a corresponding ωGarden-of-
Eden(C), ωLife(C), ωDoomsday(C), ωNon-Life(C).
When we consider the same problems with
respect to these ωstructures, then the results are
quite opposite, most problems are no longer
algorithmically solvable. We consider several
problems with such a status and which are of a clear
interest for predicting the development of a colony:
will a given agent become sometimes active, does
the colony reach a state when a conflict appears,
does the colony enter a deadlock? Unfortunately, as
we have mentioned above, if the starting state is
prescribed, these problems (and others similar) are
undecidable.
Proposition 2 (Martín-Vide and Paun, 1998). Given
a PM colony C, an agent A
i
, and a state w, we cannot
decide whether or not a state z can be derived from
w with respect to C such that the agent A
i
is active
on z.
4 DECIDABILITY RESULTS IN
PM COLONIES WITH
RESTRICTED RULE-SETS
Problems mentioned in the paragraph above can be
represented as a group of problems on the same
base. (Un)Decidability status of any one of these
problems can be quiet easily transformed to that one
of any other problem from that group. The questions
we are dealing with are: “Will a given agent in a PM
colony with given initial state become active?” “Will
a PM colony reach a state from which it is not
possible to continue in its development?” “Will a
PM colony reach a state in which a conflict
happens?” In (Martín-Vide and Paun, 1998) authors
indicate, that problem can become solvable when we
consider modifications in the PM colony. By the
change it can be understood application of different
kind of restrictions on the PM colony: determinism,
restricted rule-set or reduced parallelism.
In this paper we decided to investigate
decidability results in PM colonies with restricted
rule-set. From the group of decidability problems we
selected the one considering if a PM colony will
reach a state in which a conflict happens: “Is in a
PM colony C with a given initial string w
0
the
problem if two agents A and B will reach conflict
decidable?” This problem we want to discuss on PM
colonies with specific restrictions in its sets of rules.
In PM colonies there exists five types of rules.
As a restricted rule-set we consider each set of rules,
where there is at least one rule-type missing. In the
subsections we will study PM colonies with no
deletion and move as well as PM colonies with no
insertion.
4.1 PM Colony No Deletion Rules and
No Moving
Assume PM colonies, where moving of agents and
deletion of the agents and environment is forbidden.
The derived environment in such a system cannot be
reduced even if the agent itself can die. New agents
can appear but the mutual positions of the already
existing agents do not change.
Theorem 1. In a PM colony C with no deletion rules
and with no rules for move and with a starting string
w
0
it is decidable that two agents A, B will reach a
conflict.
Proof. Consider an algorithm simulating the
development of given (deterministic) PM colony.
The inputs of the algorithm are: C = (E, #, N, >, R
1
,
.., R
n
), starting string w
0
and agents A and B. We
have to consider the longest substring of conflicting
agents in the starting string – we denote it by s.
Outputs of the algorithm are messages if conflict
happens or not, number of derivation steps and
reached string (state of the colony).
The agents A and B will enter the conflict if w
0

*
uAvBw 
*
u´Av´Bw´. In the string uAvBw there
was at least one of agents A, B inserted during the
last derivation step. In the next derivation step the
agent cannot be rewriten or erased, but it still has not
to be in conflict with the second agent even during
the next derivation step. In the string u´Av´Bw´
agents A and B are in conflict for the first time. For
these cases there must exist variables k, l such that
w
0

k
uAvBw a uAvBw 
l
u´Av´Bw´ (k, l depends
PointMutationColonieswithRestrictedRules
247

on properties of PM colony C and on its starting
string w
0
).
In PM colony without rules for erasing agent or
environmental symbol and without rule for moving
of agent, we consider an algorithm, that for k+l
derivating steps simulates the development of the
colony. We have to determine the value of k and l
(respectively k+l).
Consider deterministic PM colony first. We
assume that the conflict between agents A a B
appears in finite number of derivation steps (the
exact number depends on properties of C and w
0
). If
the conflict does not appear in number of derivation
steps counted below, then it does not appear at all
(all the possible parts of string allowing changes in
the development of the colony – originating from w
0
– will be exhausted by changes caused by
development of the colony and existing agents will
repeat the same actions in cycles). Awaited output of
the algorithm is a message telling if conflict happens
or not.
This would be attended with information about
number of done derivation steps and reached string.
In case of development not reaching conflict of
specified agents includes output message the last
algorithmically reached string, number of derivation
steps done and message that conflict would not
come.
When analysing development of PM colony from
its initial string we are interested into the whole
string w
0
. In the colony, there can be some parts of
the string causing complicated development (e. g.
collision of more agents brings complications with
determining which agent is active and which agent
will be active in the next derivation step), we have to
focus on these parts. In the development plays the
role parts consisting of environmental symbols only.
The most complicated development can be observed
in parts where more agents collide. There can be
more than one such part in the starting string. In all
of the parts many events can happen, but to
determine the “worst possible case of development”
(in the sense of highest number of derivation steps
which has to be done to determine if the conflict
between agents A and B arrive) we have to consider
the longest string of conflicting agents. Considering
the rules in colony it is pointless to think about
situation where two short conflicting strings become
one longer (such rule-type is in this type of PM
colony forbidden).
In this type of PM colony are only these rule-
types: substitution of environmental symbol,
insertion of an agent of environment symbol, death
of an agent. Only two types of rules can produce the
conflict 1.) death of an agent and 2.) insertion of an
agent.
When considering possibility of conflict due to
rule for agent insertion, we have to consider the
priority relation <. This rule-type can cause conflict
during a*a
2
derivation steps, where a is the number
of agent names in colony (each agent can produce up
to a agents on both is sides, but then the actions are
repeating. Repeated is also the whole life of the
colony and nothing new can happen).
In the case when conflict appears due to the rule
for death of an agent we have to consider s – the
longest substring of conflicting agents in the starting
string. In this case agents can create new agents or to
exchange environmental symbols on both its sides. It
means (e+a)
2
possible combinations of
development. If the conflict should appear, then it
has to happen no later than in s*a*(e+a)
2
derivation
steps and this is the value matching to k+l. If the
conflict does not appear in this number of derivation
steps, then it does not appear in this colony with
given starting string at all.
Running of this simulation gives sense only
when in colony exists a rule generating new copy of
agent A or agent B or in case of existing rule for
erasing an agent neighboring with any of these
agents.
In case of non-deterministic PM colony it is
necessary to bifurcate the computation every time
when it is possible to use more than one rewriting
rule for any context. It is also necessary to follow all
branches of computation until the number of
derivation steps mentioned before. If a conflict is
reached in any of the branches, the problem has
a solution for given non-deterministic PM colony
and given starting string. 
Note. Generative power of this restricted class of
PM colonies is lower than the power of original PM
colonies. The absence of a rule for deletion causes
that strings cannot be shortened. Example of a
language, which cannot be derived by this type of
PM colonies is a set {a, aa, aaa}.
4.2 PM Colony with No Insertion Rules
Assume PM colonies, where insertion of agents and
environment symbol is forbidden. No growth is
possible in these colonies.
Theorem 2. In a PM colony with no insertion rule
and with an initial string w
0
it is decidable if two
agents A, B will reach a conflict.
Proof. In this type of PM colony the length of string
cannot be prolonged. Because of finite language
produced by this type of PM colony, the problem if
agents A and B will enter a conflict is solvable. 
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In the topic of PM colonies with restricted rule-
set, there are more problems, which we are
interested in. We want to explore if the problem: „is
it decidable if two agents A, B will reach a conflict?“
is solvable in PM colony with given initial string w
0
.
In our „to-do list“ there remain PM colonies with no
rules for
a) deletion
b) substitution
c) move and
d) death of an agent.
4.3 Note on Generative Power
With restrictions on a generative system it is always
interconnected question of the impact on the
generative power. The generative power with
limitations on possible rewriting rules falls.
E.g. when we consider a restricted PM colony,
without rule for insertion, it is impossible to generate
any infinite language. A PM colony without this
restriction needs only one agent and one
environmental symbol to generate the infinite
language a
+
.
Without proof we present the theorem.
Theorem 3. Generative power of deterministic PM
colonies with restricted rule-set is lower then
generative power of deterministic PM colonies
without any restrictions on those types of rewriting
rules.
5 CONCLUSIONS
In this paper we focused on influences of restrictions
in the form of reduced rule-set on decidability
problems in PM colonies. As a part of the
restrictions influence we explored changes in the
generative power of restricted forms of PM colonies.
These restrictions give a possible algorithmic
solution to the problem if two agents will enter a
conflict. To find out if all suggested restrictions give
algorithmic solutions it is necessary to deal with the
resting – so far not solved restricted forms.
At this opportunity it is also necessary to inform,
that these restrictions cause declination of generative
power. The generative power of original PM
colonies is higher than generative power of PM
colonies influenced by restrictions introduced in this
paper.
ACKNOWLEDGEMENTS
This work was supported by grant SGS/5/2010 “Síla
a forma spolupráce v generativních systémech”.
(The power and forms of cooperation in generative
systems)
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