=Paper=
{{Paper
|id=None
|storemode=property
|title=P systems: State of the Art with Respect to Representation of Geographical Space
|pdfUrl=https://ceur-ws.org/Vol-837/paper11.pdf
|volume=Vol-837
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==P systems: State of the Art with Respect to Representation of Geographical Space==
https://ceur-ws.org/Vol-837/paper11.pdf
P systems: State of the Art with respect to
P systems: State of the Art with Respect to
Representation of Geographical Space
Representation of Geographical Space
Zbyněk Janoška and Jiřı́ Dvorský
Zbyněk Janoška and Jiřı́ Dvorský
Department of Geoinformatics, Palacký University,
Department
Třı́da Svobodyof26,
Geoinformatics, Palacký
771 46, Olomouc, CzechUniversity,
Republic
Tř ı́da Svobody 26, 771 46, Olomouc,
zbynek.janoska@cdv.cz, Czech Republic
jiri.dvorsky@upol.cz
zbynek.janoska@cdv.cz, jiri.dvorsky@upol.cz
Abstract. Membrane computing is an emergent branch of natural com-
puting, taking inspiration from the structure and functioning of a living
cell. P systems, computing devices of this paradigm, are parallel, dis-
tributed and non-deterministic computing models which aim to capture
processes taking place in a living cell and represent them as a computa-
tion. In last decade, a great variety of extensions of model, introduced by
Paun in 1998, were presented. In this paper we present a comprehensive
review of current progress in the field of membrane computing, focusing
on representation of geographical space in P systems. Two approaches
are commonly used in Geographic Information Science (GIS) for repre-
sentation of entities: field-based and object-based. Both approaches are
discussed from the point of P systems and possibilities of using inherent
hierarchical structure of P systems in spatial modeling are mentioned.
Keywords: membrane computing, P systems, Geographical Information Systems, rep-
resentation of space
1 Introduction
Membrane computing represents new and rapidly growing branch of natural
computing, which starts from observation that the processes taking place in a
living cell can be understood as a computation. Membrane computing and its
computational device – P system – were introduced by Păun [32] and gained a lot
of interest in last decade. P systems start from observation, that membrane plays
a fundamental role in the functioning of a living cell. Membranes act as three-
dimensional compartments which delimit various regions of a living cell. They are
essentially involved in a number of reactions taking place inside cell and moreover
act as selective channels of communication between different compartments of a
cell [5].
P systems take inspiration from cell on two levels – the structure and the
functioning. Structure of cell is represented by its membranes and functioning is
governed by biochemical reactions. Every P system therefore has three main e-
lements: a membrane structure, where object evolve according to given evolution
rules [35]. Some authors add fourth basic element of membrane systems – com-
munication [5, 34]. Communication is always encoded in rules (they are called
J. Pokorný, V. Snášel, K. Richta (Eds.): Dateso 2012, pp. 13–24, ISBN 978-80-7378-171-2.
14 Zbyněk Janoška, Jiřı́ Dvorský
communication rules instead of evolution rules) and will be dealt with later
in the text. From the point of view of Geographical Information Systems (GIS),
communication (e.g. topology) is essential feature of most real world phenomena.
Simple example of P system is depicted
in Fig. 1. Membrane structure is hierarchi-
cally arranged set of membranes, contained
in a distinguished outer membrane, called
skin membrane. System is surrounded by
the environment, which may collect objects
leaving the system, or in some variants of P
systems, the environments can actively sup-
port system with objects [4, 11]. Membrane
structure can be represented in many ways
– as Venn diagram, as rooted tree, or by lin-
ear notation. Membrane structure of P sys-
tem depicted in Fig. 1 in linear notation is
written as [1 [2 [3 ]3 ]2 ]1 . Membranes de-
limit regions, with which they are in one-to-
one relation. Therefore the terms membrane
Fig. 1. Graphical representation and region are mostly interchangeable. Each
of P system, [46] membrane is identified by its label, which can
be with membranes in one-to-many relation.
The position of inner membranes does not
matter; we assume, that in membrane there is no ordering, everything is close
to everything else [35]. Please note the difference with the first law of geography:
everything is related to everything else, but near things are more related than
distant things [39]. This applies not only to membranes, but also to objects in
them. From the biological point of view, inner membranes are considered floa-
ting free in their parental membranes and therefore definition of topological or
metric relations between them makes no sense. This is of course not valid for
geographical space and we will discuss it later.
Second basic element of P systems are objects. By objects in biological sense
are meant chemicals, ions, molecules etc. Those substances are present in a cell
in enormous amount, but the ordering again does not matter. What matters is
the concentration, the population, the number of copies of each molecule [35].
Abstracting from biological reality, we represent each substance by a symbol
from given alphabet and since the multiplicity matters, instead of objects we use
multisets of objects. Common notation of multisets in P systems is following: if,
for example, objects a, b, c are present in 7,2 and 5 copies, they will be represented
by multiset a7 b2 c5 .
In basic variant of P systems, multisets of objects are considered to be floating
in inner regions of membrane systems. They evolve by the means of evolution
rules, which are localised with the regions of the membrane structure. There are
three main types of rules [35]: (1) multiset-rewriting rules, (2) communication
P systems: State of the Art . . . 15
rules and (3) rules for handling membranes. In this section only first type of
rules will be described.
Multiset-rewriting rules take form u → v, where u and v are multisets of
objects. For example, rule ab → cd2 says, that one copy of a and one copy of b
are consumed and one copy of c and two copies of d are produced. A number of
possible extensions of rules will be discussed later.
Two crucial features of P systems have to be mentioned at this point. As
mentioned earlier, in membranes everything is close to everything else. Therefore,
if one instance of an object can be processed by two or more rules, the rule to
be applied is chosen non-deterministically. All rules have the same probability
to be chosen. The rules also have to be used in maximally parallel manner.
More specifically, the objects are assigned to rules, non-deterministically
choosing the objects and the rules, until no further assignment is possible. An
evolution step in a given region of membrane system consists of finding the ma-
ximal applicable multiset of rules, removing from region all objects specified in
the left hand of the chosen rules and producing the objects on the right hand
side of the rules.
After giving short introduction to basic notions of P systems, let us continue
to more detailed survey on spatial properties of P systems. In Sect. 2 we will give
a formal definition of transition P system and in Sect. 3 some possible extensions
of P systems are presented. In Sect. 4 we will discuss how geographical space
is represented in Geographical Information Systems, in Sects. 5 and 6 we will
describe object-based and field-based representation of geographical space in P
systems and in Sect. 7 we will discuss how hierarchical structure of P systems
can be used to represent geographical phenomena. We will conclude with some
final remarks in Sect. 8.
2 Transition P system
P systems based on application of multiset-rewriting rules are called transition
P system. Formally, transition P system is a construct of the form:
Π = (O, C, µ, w1 , w2 , . . . , wm , R1 , R2 , . . . , Rm , io ), (1)
where:
– O is the finite and non-empty alphabet of objects,
– C ⊂ O is the set of catalysts,
– µ is a membrane structure, consisting of m membranes, labeled 1, 2, . . . , m;
one says, that the membrane structure, and hence the system, is of degree
m,
– w1 , w2 , . . . , wm are strings over O representing multisets of objects present
in regions 1, 2, . . . , m of membrane structure,
– R1 , R2 , . . . , Rm is finite set of evolution rules associated with regions 1, 2, . . . , m
of membrane structure,
16 Zbyněk Janoška, Jiřı́ Dvorský
– io is either one of the labels 1, 2, . . . , m and then the respective region is the
output region of the system, or it os 0 and then the result of the computation
is collected in the environment of the system.
A sequence of transitions of P system constitutes a computation. A computation
is successful if it halts, it reaches a configuration where no rule can be applied
to the existing objects, and output region io still exists [35].
The rules are of form u → v, where u ∈ O and v ∈ (O × T ar), where
T ar = {here, in, out}. Target indications T ar extend transition P system in
following way: rule ab → chere din eout consumes one instance of each a and b and
produces one copy of c in current membrane, one copy of d in a child of current
membrane and one copy of e in the parent of current membrane. If current
membrane is skin membrane, object e is send to environment of the system. If
current membrane does not have a child, rule can not be applied.
Another extension comes from the existence of catalysts. Catalysts are ob-
jects, which participate in a chemical reaction, but are not consumed or produced
by it. They just enable the application of rule. Rule with catalysts takes following
form: ac → bc, with object c being the catalyst.
One more extension must be mentioned at this place, and that is dissolu-
tion of membranes. During dissolution, membrane disappears and its content
(both objects and inner membranes) are left free in the surrounding membrane.
Dissolution rule takes form u → vδ, where δ denotes the action of dissolution.
3 Possible Extensions of P systems
In this section we will mention some elementary extensions of P system, which
however constitute only a fracture of possibilities. We refer reader to The P
systems Webpage [45] for complete list of publications and further information.
Already in the text, three types of rules were mentioned. Evolution rules were
briefly covered in previous sections.
Communication rules were introduces in [34]. Basic idea of communicating
P systems is, that computation is achieved only by transporting object between
membranes. Direct inspiration from biology are symport and antiport. When two
chemical pass through membrane only together, in the same direction, the pro-
cess is called symport. When the two chemicals pass only with help of each other,
but in opposite directions, the process is called antiport [34]. Symport rules take
a form (ab, in) or (ab, out), and antiport rules take a form (a, out; b, in), where
a, b are object from alphabet of all possible objects. Meaning of rules is follo-
wing: for symport rule (ab, in) or (ab, out), if objects a, b are present in current
membrane, they are sent together into child (or parent, in second case) of the
membrane. For antiport rule (a, out; b, in), if a is present in current membrane
and b is present in the parent of a membrane, than a exits current membrane
and b enters it. Universality of P systems with symport and antiport have been
proven [34] and simplified version of communication, conditional uniport have
been studied [40]. Comprehensive review of communication strategies in P sys-
tems can be found in [41].
P systems: State of the Art . . . 17
Third type of rules are rules for handling membranes. Dissolution of mem-
branes has already been mentioned, but other ways to obtain dynamical mem-
brane structure, evolving during the course of computation, have been presented.
Most simple of those is assigning electrical polarization +, −, 0 to each mem-
brane. Polarization replaces target indicators in, out, here. Polarization of ob-
jects is introduced by the rules and polarized objects can enter only membranes
with opposite polarization. For example, ab → c+ d− means, that one instance of
c enters inner membrane with negative polarization and one instance of d enters
inner positive membrane. Rules can also be used to change the polarization of
membranes during the computation.
Another possibilities to alter membrane structure have been proposed. Di-
vision of membranes can be used to obtain exponential working space in linear
time [33] and have been used to solve NP-problems [31]. An optimalization
algorithms based on membrane computing were proposed [29, 24]. Also biologi-
cal processes of exocytosis, endocytosis and gemmation were translated into the
language of P systems and examined in detail [25].
Two issues seem essential, when P systems are used to simulate biological
phenomena rather than for computation. Non-determinism is first of the issues.
Some chemical reactions are more likely to occur than others. First attempt to
solve this is by assigning priorities to rules. Firstly, set of rules with the highest
priority is chosen and according to the principle of maximal parallelism, all rules
which can be applied, are applied. Then, the rules with second highest priority
are selected and the procedure repeats, etc.
Second approach is to assign probabilities to all rules. Probability can be
introduced to P system on different levels [30], but here we will mention only
probability on the level of rule selection. Different approaches have been proposed
[4, 13, 36]. Basic idea is to associate each rule with a constant k, so the rule takes
k
a form u − → v, where u, v are multisets of objects and k can be interpreted
either as a probability, or as a “stoichiometric coefficient”, using which the true
probability is calculated.
Last extension, which we will mention at this point, is representation of
time of P systems. In real world, every biochemical reaction takes some time.
Representation of time in P systems is similar to representation of probability.
t
A constant t is assigned to each rule, so the rule takes the form u → − v, where
u, v are multisets of objects and t is number of time units, which must pass to
complete the application of the rule [12]. In the first time step, multiset u is
consumed and removed from the current membrane. After t − 1 more time steps,
multiset v is introduced into the system. Time can also be introduced into P
systems as a lifetime of objects or even membranes [1].
For the sake of brevity, we will not discuss more extensions of P systems, al-
though many possibilities were explored within this framework. Every real-world
application of P systems requires careful and accurate definition of system to be
modeled and once the real-world system is defined, P system as a mathematical
model for simulation can be developed. It is very unlikely, that any of presented
variants of P systems would accurately describe the complex real-world pheno-
18 Zbyněk Janoška, Jiřı́ Dvorský
mena, however when considered as a modelling paradigm, P systems offer great
variety of extensions, and arbitrary P system for concrete application can be
developed.
4 Geographical Representation of Space
Object-based and field-based models of space are accepted as two alternative
approaches for conceptualization and geographical modelling [20]. Object-based
conceptualization understands geographical entities as sharply bounded and
therefore represented by mostly polygonal boundaries. Objects are located in
space, i.e. location is attribute. Fields are continuous phenomena and characte-
rize space by properties – functions and values – related to locations [42]. Raster
and vector represetations are dual to field-based and object-based conceptual-
ization of space [37].
In field-based model, every location in a spatial framework is associated with
a set of attributes. Fields are spatially continuous by definition. Field can be
viewed as a mapping between spatial location and an attribute domain [43]. The
most common field-types are scalar, vector or tensor; with scalar fields being the
most commonly used in GIS modelling. Representation of field must be always
approximate and rectangular, triangular or hexagonal tessellations are used [16].
Fields are usually stored as a georeferenced raster.
In an object-based perspective, space is viewed as a container populated by
objects. Location is an attribute of each object. Object’s spatial projection is
mostly represented in GIS environment by points, lines (polylines, networks) or
polygons [16].
Possible merge of field-based and object-based approaches have been dis-
cussed in literature [16, 42], but GIS applications include only two basic repre-
sentations of space.
5 Object-based Representation of Geographical Space in
P systems
Two basic terms in representation of geographical space are distance and topolo-
gy. The mathematical theory of metric spaces is well-known to be inadequate as
a formal foundation for distance measures in geographic spaces [28]. Contextual
knowledge is a key feature of human apprehension of geographic space [44]. For
example distance between cities A and B is different from the point of view of
cyclist and pilot of an airplane. There is an important distinction between global
view (top-table space) and geographical view (geographical space).
Top-table space can be viewed from one single point, whether geographical
space is context-based [44]. Therefore classic definition of metric in mathematical
space does not apply (geographical space is asymmetric and triangle inequality
does not apply).
In geographical space, neighborhood relations are commonly treated as prior
to metrics. For example we know that Austria and Germany share the border,
P systems: State of the Art . . . 19
but we are unable to estimate the length. Regarding the fractal nature of geo-
graphical boundaries [26], measuring the length may not even make sense.
Definition of topology between geographical entities is therefore essential for
any geographical analysis.
As mentioned earlier in the text, object-based representation of geographical
space understands space as a container with entities, which are defined by their
locations. The relation of entities is described by their topology. Nearness of two
neighboring entities can be quantified as a distance between them. Distance can
be context-based (i.e. time necessary to overcome distance between two points
can depend on the mean of transportation) and also dependent on direction.
Those relations can be formalized using graph theory. Entities are represented
as nodes of the graph and links represent topological relation between them.
Cost of links represents distance between geographical entities.
Special variant of P systems with membranes arranged in the net have been
proposed as tissue-like P system [27]. In this variant, membranes are arranged
in an arbitrary graph instead of in a hierarchy. The computation is achieved
using symport/antiport rules, but generalization using evolution rules can be
considered. Links between membranes are represented using synapses. Formal
definitions are here omitted and can be found in [19, 27, 35]. This formalization
is suitable for representing topological relations between entities in geographical
space. Entities can be represented as membranes (nodes of the graph) and their
topology could be stored in links. Adding cost to synapses will achieve further
representation of distances between entities. This approach has been adapted
by [6, 7] to simulate interaction between spatially separate regions (so called
metapopulations). This research was however only theoretical.
Cardona et al. in [8] started research on modelling population of Beraded
Vulture in the Pyrenees using model with several spatially separated regions,
which could however interact with each other by sending objects to the environ-
ment and retrieving them. This model was later expanded to model population
of 12 animals [9, 10], modules for modelling biomass of plants were added [14]
and model was also used for management of the area [15]. Also, [10] used similar
model to simulate population growth of invasive species of zebra mussel in water
reservoir. The water body was represented by 17 regions with different regime
of water temperature fluctuations with their topology represented by oriented
graph.
Object-based representation of space can be coherently represented using P
systems. Different geographical entities can be represented by membranes, topol-
ogy can be stored in a graph and distances between entities could be represented
using costs of links of graph. Moreover, each node – membrane – can have inner
hierarchical structure. Take agglomeration of larger city as an example. This
agglomeration is connected with other cities by roads and rails, therefore repre-
sented as a membrane with synapses to other membranes – cities. In the same
time the agglomeration can have inner structure, represented either by smaller
set of interconnected membranes (public transportation network with nodes rep-
resenting stations) or a hierarchy of membranes, representing for example ad-
20 Zbyněk Janoška, Jiřı́ Dvorský
ministrative zoning with several levels. Moreover, both representations can be
stored within this membrane separately.
6 Field-based Representation of Geographical Space in P
systems
Field-based representation of space understands geographical space as a contin-
uous field, where every location has a set of attributes (temperature, air pressure
or elevation, for example). For analytical purposes, continuous fields must be ap-
proximated by grid, mostly rectangular. This representation of space is similar
to cellular automata.
Recently, [3] introduced spatial P systems, which embody concept of space
and position inside membranes in similar manner that cellular automata do.
Rules, as usual, specify the objects which are consumed and which are produced,
moreover, the position of produced objects can be specified. Although P systems
were used before to model processes in geographical space (see previous chapter),
[3] was first to inherently include space into P systems. In classical view of P
systems, position does not matter. Also [7, 9] and others, who worked with space,
did consider position of membranes only as an attribute of membranes, and
position of object inside membrane was never considered before.
We will not give formal definition of spatial P system and will focus on
possible applications instead, because up to date, there are none. Barbuti et
al. [3] defines spatial P systems in two-dimensional space, but living cells are
three-dimensional compartments, therefore extension to 3D would be suitable to
enhance expressiveness of the model.
Cellular automata are used in GIS for various applications, including mod-
elling of forest fires, urban growth and dispersion of pollution. Among these
application, modelling of spread of pollution seems as most promising ground
for P systems. Pollutants are mostly chemicals and their consumption, creation
and alteration can be naturally described using P systems. Their dispersion can
be described using spatial P systems.
Another example would be simulating growth of collonies of bacteria (both in
microscale and macroscale), where P systems can accurately describe behavior
of such simple organisms. In many other application, like modelling of defores-
tation, urban growth and qualitative changes in landscape, spatial P systems
could achieve similar results like cellular automata.
7 Using Hierarchy to Represent Space
Geographical space is without doubts structured in a hierarchical manner. Ad-
ministrative division of Czech republic is an example. Hierarchical data struc-
tures such as quadtree or octtree [38] are widely used in GIS and spatial databases.
Even methodology for adapting such data structures to globe was developed [21].
P systems: State of the Art . . . 21
Also hierarchical spatial reasoning gained interest in the community of geogra-
phers in last two decades [23]. This hierarchical approach mimics human rea-
soning when performing spatial operations: an appropriate scale is selected and
results are computed. The quality of results obtained is assessed, and if it is
satisfactory, the computation stops. If not, more detailed level of spatial data is
consulted.
Our literature search showed, however, that spatial modelling of hierarchi-
cally structured phenomena is rare. Eckhardt and Thomas [17] used multilevel
regression models for inspection of patterns of road accident occurrences. Other
publications dealing with hierarchical modelling also exist, but mostly use hie-
rarchical Bayesian models, where the structure of model is hierarchical, but not
in geographical sense [2]. Some research has been dedicated to possibilities of
visualization of geographical hierarchies [22].
P systems offer different ap-
proach, where modelling can be tak-
ing place on multiple levels of the
model simultaneously. In Fig. 2, a
simple system of forest is depicted deer 75
(rules are omitted). This model has
two levels. On upper level, the popu-
lation of animals (deers) is modelled. hardwood 150 coniferous 350
On lower level, the competition be-
tween two parts of forests coniferous
and hardwood is modelled. Because
population of deers is not dependent
on the inner structure of the forest,
it can be modelled separately, on up-
per level, and inherent hierarchical Fig. 2. Hierarchical structure of geo-
structure of P systems can be ex- graphical space in P systems
ploited.
In natural systems, ecosystems are hierarchically structured and this struc-
ture plays prominent role [18]. P systems for ecological application therefore offer
expressiveness, which most other computational models do not posses.
To our knowledge, no application, where multiple levels of a single system
were simultaneously modelled, were presented and also no theoretical research
of this topic was conducted. Hence, further research on application of P systems
should focus on this aspect of modelling.
8 Concluding Remarks and Future Work
We have presented brief introduction to membrane computing with emphasis on
description of handling of geographical space in P systems. Both representations
of geographical space – object-based and field-based were already discussed in
the literature and some application are available for object-based representation.
22 Zbyněk Janoška, Jiřı́ Dvorský
Only one publication so far was dedicated explicitly to computing with space in
P systems.
Currently, many extensions of classical transitional P system exist, and ex-
pressiveness of this model can be significantly enhanced. For real-world appli-
cation, however, unique models must be defined. We proposed some possible
application of P systems in geography, from which modelling the spread of pol-
lution seems the most promising, given the nature of the prenomenon.
Also perspectives of multi-level modelling were mentioned. This approach
exploits inherent hierarchical structure of P systems to simulate the behavior of
systems on multiple levels. Possible application can be seen in ecological studies,
since ecosystems are deeply hierarchized structures.
Our current research is focused on modelling of transportation using P sys-
tems. Individual based modelling, paralellism and evolution of components of a
system are key features needed to model complex behaviour of transport sys-
tems. However this research is in its initial stage and experimental results are
not available at the moment.
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