Aspects of Symmetry in Petri Nets
ANTHONY SPITERI STAINES
Department of Computer Information Systems,
University of Malta,
Msida MSD 2080,
MALTA
Abstract: - Symmetry is a fundamental mathematical property applicable to the description of various shapes
both geometrical and representational. Symmetry is central to understanding the nature of various objects. It
can be used as a simplifying principle when structures are created. Petri nets are widely covered formalisms,
useful for modeling different types of computer systems or computer configurations. Different forms of Petri
nets exist along with several forms of representation. Petri nets are useful for i) deterministic and ii) non-
deterministic modeling. The aspect of symmetry in Petri nets requires in-depth treatment that is often
overlooked. Symmetry is a fundamental property found in Petri nets. This work tries to briefly touch on these
properties and explain them with simple examples. Hopefully, readers will be inspired to carry out more work
in this direction.
Key-Words: - Computer Science, Graph Theory, Mathematics, Matrices, Petri Nets, Software Models,
Structural Representation, Symmetry.
Received: October 29, 2023. Revised: April 14, 2024. Accepted: May 29, 2024. Published: July 1, 2024.
1 Introduction
In everyday language symmetry expresses the
principles of balanced proportions and a sense of
harmony, [1], [2], [3], [4], [5], [6]. Symmetry deals
with things like coherence, orchestration,
consonance and respect for proportions. The very
opposite of symmetry is asymmetry which is
synonymous with disharmony in nature. Asymmetry
causes dissonance, disproportions, disunity, and
imbalances, [1], [2], [3], [4]. In the case of two-
dimensional shapes geometric symmetry at the most
basic level implies that a dividing line can be drawn
through the object generating two new shapes that
are mirror images or reflections of one another. This
is known as mirror symmetry, similarly to reflecting
an object in a mirror. Symmetry should be central to
describing graphical objects. This is even more so
for structures like Petri nets, which are types of
bipartite digraphs, [5], [6], [7], [8], [9], [10], [11],
[12], [13], [14], [15], [16], [17], [18], [19].
For symmetry in Petri nets, I give the following
top-level classification. i) Geometrical Symmetry,
ii) Matrix Symmetry, and iii) Operational or
executional symmetry. Geometrical symmetry can
be further subdivided into Euclidean, Reflectional,
Point Reflectional, Rotational, Translational, Glide
Reflectional, Helical, Double rotational, Shape etc.
However, in this work, the reference will be kept to
basic mirror and rotational symmetry. Matrix
symmetry refers to symmetry in the basic
representational matrices for Petri nets which are the
input, output, and incidence matrices. These can be
used for different classification of Petri net
properties, including symmetry. Matrix analysis of
Petri nets is very useful for limited-sized Petri nets.
Operational or executional symmetry can be derived
from the marking graph of the Petri net or some
other form of representation.
Petri nets have well over three decades of
coverage, apart from extensive uses to model
different system types, [7], [8], [9], [10], [11], [12],
[13], [14], [15], [16], [17], [18]. It is sometimes
argued that their composition and representation are
complex and incomplete. There are several ways to
represent Petri nets, however, the main two ways are
i) graphically and ii) mathematically. Graphical
representation implies having a graph structure to
which many aspects of geometric symmetry are
fully applicable. The mathematical representation
gives other interesting forms of understanding.
Much of the available work on symmetry in Petri
nets is not exclusive, [7], [8], [9], [12], [14], [15] but
focuses on other unrelated issues, barely touching
on the full possibilities of symmetry issues involved
in Petri nets. Symmetry has been used for
understanding sequential and parallel composition.
However these topics are not exclusively of Petri
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
164
Volume 23, 2024
nets and can be found in other graph based
structures.
The topic of symmetry in Petri nets is so vast
that it cannot be pinned down to a few topics only
like in [7], [8,] [9], [10], [12], [13], [14], [15], [16],
[17], [18], [19], [20], [21], [22]. This work will try
to summarize and classify these, however each of
the topics and sub-topics that will be referred to can
be used for in-depth research exclusively in their
own right. Volumes can be written on just a single
aspect. Some parts of these topics can prove to be
quite intuitive whilst other parts would be rather
abstract. For Petri nets just at a glance there can be
other explorable symmetry classifications, like i)
total or ii) partial symmetry. According to Plato,
symmetry should lie at the core or center of nature.
It is the author’s own opinion that the aspect of
symmetry in Petri nets, requires an in depth
treatment, that has often been overlooked. It is this
fundamental property that will be dealt with here.
2 Related Works
In [7], symmetries of Petri nets are described, these
are given for the reachability graph, structural
analysis, whilst symmetries are compared with
place/transition invariant computation. Algorithms
to compute these symmetries are described, the
main idea of using symmetries here is to reduce the
complexities of Petri nets and to solve issues like
deadlock, etc. In [7], high level abstractional
representational notations have been introduced.
The possible types and classifications of symmetries
that are given from the Petri net theory point of
view. This differs from the normal classical
approach because in [7] the start off is from the
mathematical aspects of symmetry and not Petri net
representation. No concrete practical examples of
symmetry are given in [7]. Compared with the
classic Petri net theory, there is the issue that several
other forms of symmetry are easily observable at a
glance. These are not considered. It is possible to
find many problems with the approach presented in
[7], e.g. even though symmetries are mentioned,
there is no specific explanation of what these are in
relation to Petri nets, symmetries are only seen as
useful for devising and creating algorithms for Petri
net analysis. This is just one aspect of Petri nets.
The symmetries of a system [6] are used for
understanding the state space of the Petri nets, [8].
The idea of using symmetries is to reduce the state
space of large Petri net structures. This would help
with the state explosion problem. An abstraction
technique called the state class graph (SCG) is used
for symmetry reduction for timed Petri nets. It is
evident that real symmetry is not the starting point
for this work. The limitation of this work is to timed
Petri nets, which is just one of the many classes of
Petri nets and the symmetry is used as a technique
only for simplification purposes.
In [9], it is shown that there are categories of
Petri nets with symmetry. The concept of symmetry
in Petri net unfolding is presented. The Petri net is
unfolded to create an occurrence net. This is a very
exclusive use of symmetry. The work in [9],
concludes by explaining that an implicit symmetry
on the unfolding of a generic Petri net does exist.
The topics addressed in [9], do not directly deal with
symmetry in Petri nets but explains the concepts of
symmetry, from the view of the unfolding process in
the net. There are several ways to include symmetry
in Petri and these are explained. However, the main
aspect of the work in [9], is not just about
symmetry.
The examined literature previously presented,
uses concepts of symmetry at the definition level of
the Petri nets. It is implied that the concepts of
symmetry presented in these works use high level
abstraction along with algebraic notations.
Petri nets are useful for describing supervisory
control systems such as those used in
manufacturing, [10]. The concept of symmetry is
presented as being useful to solve key issues in the
field of control synthesis. However the work in [10],
is rather vague how to actually and practically use
symmetry, even though it is referred to. In [10],
there is no real explanation of what symmetry is and
then again what type of symmetry is to be used. On
the other hand the work in [10], shows the practical
importance of symmetry when setting up Petri net
models for workstations to be controlled by a main
actor which is the supervisor actor, represented by
the supervisor Petri net structure. The authors in
[10], discuss the reduction of forbidden states in the
Petri net and how to improve the layout and design
of the models. The concepts of symmetry are briefly
mentioned and however they can be greatly
expanded upon.
Petri nets can also be abstracted to monoids,
[11]. Here sets of operations can be used for
transition composition. The work in [10], does not
deal directly with symmetry, however the fact that
new morphisms can be defined to represent Petri
nets and their respective structures is indicative that
Petri nets do have many symmetry properties. The
construction of new algebraic representations for
Petri nets presented in [10], is useful for creating
new theories. In [11], Petri nets are treated as
monoids. Even though the title of this paper does
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
165
Volume 23, 2024
not explicitly mention symmetry the models given
in this work exhibit some symmetrical properties.
In [12], a Petri net model with a repeated sub-
graph structure is given. The paper in part shows
how a repeated symmetrical sub-graph can be
removed, simplifying the net. This idea is very
good. The only issue is that this paper is restricted to
one type of symmetry used for reduction.
The work in [13], is an important dissertation. It
does explain symmetry in Petri nets from a
particular perspective. The idea is restrictive to Petri
net main properties. There is much more to
symmetry that has not been explored. Even different
views to symmetry can be used. The work in [14],
focuses on algorithms used to analyze symmetry in
the Petri net. There is no focus on drawing
symmetry. In [15], again the work focuses on timed
continuous Petri nets. These are just a single class of
Petri nets.
There are many examples of Petri net uses. This
paper just contains some, [16], [17], [18], [19], [20],
[21], [22]. Petri nets are used in information
technology, hardware, software modeling etc. These
papers describe some substantial uses and
possibilities of Petri nets. E.g. in [20], Petri nets are
used for modeling the reliability and availability of
an electrical power system.
From the literature there is firsthand evidence
that the starting point of symmetry in Petri nets
should be the net structure itself. This has not been
done. The types of symmetry presented in these
publications is not real symmetry in the
mathematical sense.
This differs greatly from what is being
presented in this paper, which is symmetry from a
mathematical prerogative. The idea of this paper is
to examine directly the structure of the net and not
deviate using other forms of symmetry that do not
have a direct relationship to the Petri nets.
3 Problem Formulation
The issues of symmetry and its possible uses in Petri
nets is non-trivial. There is a problem where to start
using symmetry for Petri nets. This is directly
related to several types of symmetry found in
mathematics and group theory and how to classify
these forms. The research papers, [7], [8], [9], [10],
[11], [12], [13], [14], [15], clearly create additional
problems by considering singular or particular
aspects. To simplify the problem this work will be
restricted to symmetry from purely a mathematical
perspective. So in this part classification of
symmetry will refer to symmetry classification in
mathematics.
Shall the focus be on rotational symmetry,
incidence or input, output matrix symmetry? At the
simplest level of definition a geometrical object can
be called symmetrical if operations that are carried
out on it will leave the original structure unchanged.
The big problem is to devise correct ways of
classification and structure.
Classification issues relative to symmetry and
applicable to Petri nets are the first part of the
problem. Petri nets are widely covered formalisms
that are useful for modeling different types of
complete systems, [7], [8], [9], [10], [11], [12], [13],
[14], [15], [16], [17], [18], [19]. Their composition,
abstraction, and rules have various different types of
interpretation and representation. Petri nets are
representable in two main ways i) graphically and ii)
mathematically using things like equations,
matrices, etc. It is evident that matrices used to
represent Petri nets exhibit many features including
several aspects of symmetry. A simple classification
is to divide Petri nets into i) Petri net matrix
symmetry and ii) Diagrammatic or drawing
symmetry. The word diagrammatic symmetry is
used to represent all possible forms of geometrical
symmetry that would be possible for the Petri net
drawing. Figure 1 indicates this classification.
Fig. 1: Simplified Classification of Petri Net
Symmetry
4 Solutions
In classifying symmetry of the matrices in ordinary
Petri nets, the following basic types are possible. i)
total symmetry, ii) partial symmetry. These can be
established for the i) incidence matrix, ii) input
matrix and iii) output matrix. It is also possible to
find symmetry for the marking graph which can be
represented either i) as a graph or ii) using matrices.
In this case the symmetry would be in the firing
matrix.
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
166
Volume 23, 2024
Here several examples will be presented, but
readers of this work should note that indeed many
other examples are possible.
4.1 Input Matrix Symmetry
This is one of the simplest forms of symmetry in the
Petri net. It is quite easy to understand. A Petri net
has three types of matrices that describe its
structure. These are input, output and incidence
matrices. Many operations can be performed on
these matrices. For the example in this section,
matrix symmetry is considered from the point of
view of the input matrix generated for Figure 2.
For this to be possible the input matrix of the Petri
net must be square. I.e. the number of places =
number of transitions. In this case transposing the
matrix should leave it unchanged.
E.g. consider the following input matrix
I= [1 1 0
1 1 1
0 0 0]
for the Petri net depicted in Figure 2.
Fig. 2: Petri net no input matrix symmetry
Clearly the transposition of this matrix yields a
different matrix.
IT= [1 1 0
1 1 0
0 1 0]
Only one row and one column remain the same
thus there is no proper symmetry in this part.
Although it could be argued that there is some form
of partial symmetry. However this is beyond the
scope of this paper. Now a second figure, Figure 3
is given. An input matrix is defined below.
Fig. 3: Petri net with input matrix symmetry
Consider the following input matrix for Figure 3.
I= [101
010
101]
Transposing this matrix leaves its values unchanged.
IT= [101
010
101]
In this case there is perfect symmetry where,
I= IT
4.2 Output Matrix Symmetry
Output matrix symmetry is similar to the input
matrix symmetry, but in this case the output matrix
for the Petri net is considered.
The following
O= [101
010
101]
is the output matrix for the Petri net depicted in
Figure 4.
Fig. 4: Petri net with output matrix symmetry
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
167
Volume 23, 2024
Transposing this matrix does not change the
values as shown below. i.e. 0=0T.
0𝑇= [1 0 1
0 1 0
1 0 1]
4.3 Incidence Matrix Symmetry
When the square incidence matrix is transposed, i.e.
the places are transposed with the transitions, the
resultant net is left completely and structurally
unchanged. This is why the author of this paper is
calling this total matrix symmetry. This can be
addressed as perfect symmetry too. The resultant net
is completely reversible too. It could be possible to
design the incidence matrix and the Petri net in such
a way as to give perfect symmetry.
The net in Figure 5 has the incidence matrix A.
Fig. 5: Petri net with symmetry in incidence matrix
A= [1 1 −1
1 2 0
−1 0 2 ]
Transposing this matrix AT leaves it unchanged.
Due to the simple design of this net it is possible to
draw it in several other ways. This is indicated and
shown in Figure 6.
AT= [1 1 −1
1 2 0
−1 0 2 ]
It should be clearly noted that for these forms to
be obtained, the Petri net structures need to follow
certain rules. E.g. The incidence matrix for the Petri
net has to be a square matrix. This places a
restriction on the types of Petri net structures that
can be drawn. The general conditions to allow this
would mean that the nP=nT (no of places = no of
transitions). However having an identical number of
places and transitions does not guarantee that these
types of matrix symmetries that are described do
exist.
Fig. 6: Redrawn Petri net identical to Figure 5
4.4 Other Non-Matrix Forms of Symmetry
There are other forms of symmetry that can be
applied to Petri net structures. These forms are
visual in nature and apply to the drawing of the net.
They are simpler to apply and interesting. The
practical examples given below show this. Due to
the size of this paper only some toy examples have
been presented. There are several forms of mirror
symmetry and drawing symmetry. The
mathematical formulae for these types of symmetry
are not explained in this paper. The concern is more
with the idea and possibility of uses.
4.5 Executional Graph Symmetry
Figure 7 is an execution graph or marking graph of a
typical Petri net. Figure 7 is used to explain mirror
symmetry.
For this type of symmetry it does not matter if
the net has an equal number of transitions at all. The
execution graph is relatively left unchanged from
the structural point of view if some of the entries in
the graph are put on the opposite side. This graph is
symmetric with respect to a line if reflecting the
graph over that line leaves the graph structurally
unchanged. This is indicative of execution
symmetry. I.e. it implies that there are alternate
execution paths, but the final outcome is unchanged.
In the example in Figure 7, the marking graph can
be rotated 180° leaving it essentially unchanged. I.e.
both sides can be flipped. There is much more to
this than meets the eye.
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
168
Volume 23, 2024
4.6 Mirror Symmetry
So much can be written and explained about mirror
symmetry. This type of symmetry is found in
different types of shapes. Mirror symmetry is
described in 4.5. A definition is not presented in
section 4.6. Rotational or line symmetry can be
considered although these are not necessarily
connected to mirror symmetry. The graph in Figure
8 is replicated from Figure 7 and has no node and
edge labeling. This graph is another example of
mirror symmetry that can be found in the Petri net
execution graph.
4.7 Drawing Symmetry
Drawing Symmetry in this work refers to symmetry
that is possible in the drawings of the Petri net.
Drawing symmetry is in short a reference to mirror
images or images that are rotated in a certain form
to create symmetry. They are being called drawing
symmetry here to simplify and generalize the idea.
The possibilities of creating symmetry using this
approach are so vast that it is possible to write
several publications just to deal with this aspect
only. On the other hand, this work just briefly
touches upon these properties. Drawing symmetry
can imply geometrical or rotational symmetry.
This type of symmetry is different from matrix
symmetry. Consider the diagram in Figure 9, a mid-
line can be drawn vertically exactly in the middle
dividing it into two perfectly symmetrical halves.
Fig. 7: Mirror Symmetry in Execution Graph
5 Results and Observations
i) In the majority of ordinary Petri net structures, it
is difficult to find symmetry in the Petri net matrices
unless the net is restricted. I.e. the number of places
= the number of transitions (Table 1). However
other forms of symmetry like mirror symmetry are
still possible and do not depend at all on the
structural matrices (Table 2).
Fig. 8: Replicated Graph for Figure 7
ii) It is possible for the net to have symmetry in all
or some of its matrices. However, this might not be
evident in the drawing or representation of the net.
iii) Diagrammatic symmetry or drawing symmetry
does not necessarily depend on the structural
matrices of the Petri net. I.e. it is independent of the
matrices. iv) The principles of symmetry apply both
to the drawing and the structure of the net using
matrices.
Fig. 9: Petri net model for Drawing Symmetry
v) It is possible to find many different classes of
diagrammatic or drawing symmetry. E.g. rotational
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
169
Volume 23, 2024
symmetry, plane symmetry, mirror symmetry, etc.
Even something like rotational symmetry can be
further classified into the order of 1,2,..n. This part
on its own would require detailed further
investigations and studies beyond the scope of this
paper. vi) Operational symmetry or execution
symmetry is observable from the marking graph or
the firing matrix. Even this part requires in-depth
treatment. This finding is very interesting because it
shows that certain fundamental structures of the net
are preserved even if the firing order can be slightly
varied. These principles are not exclusive only to
the Petri net marking graphs but are found in other
graph structures too.
Ordinary Petri net structures share the principle
of duality and preserve it concerning their input,
output, and incidence matrices.
Creating the Petri net diagrams for showing the
input, output, and incidence matrix symmetry has
been a selective process that is limited to certain
structures only. It follows that most naturally
occurring models will not exhibit these forms of
symmetry by default. From the experiments in this
paper, it is obvious that if a Petri net has the
symmetry property in the incidence matrix then the
input and output matrices will also exhibit similar
properties. But does this imply that there is a
repeatable pattern? Possibly the inverse will also be
possible. If both the input and output matrices of
the Petri net have symmetry then the resultant
incidence matrix will have symmetry too. This
property could be useful to reduce the size of the
Petri net model. Another feature is that a large Petri
net model can be decomposed into smaller parts or
sub-nets that can exhibit symmetry individually.
An important finding of this study is that the
Petri net shape-related symmetry does not depend at
all on the matrix symmetry. The shape-related
symmetry is shown in Table 2 and this list is by no
means exhaustive. So this is an important property
where further work can be carried out.
Table 1. Petri Net Matrix Symmetry
Geometrical or rotational symmetry is different
from matrix symmetry, however, there could be
some form of overlap where matrix symmetry co-
exists with geometrical symmetry.
The composition of the net identified through
symmetry will have profound implications on other
things like parallel, concurrent, and even sequential
processing. So what comes first? Is it the
composition of the net or symmetry?
The results and observations also show
promising ideas on how to use symmetry for
visualization of the nets.
Table 2. Petri Net Shape-Related Symmetry
There are forms of symmetry that can be
identified through perception and not
mathematically. In [2], [5], [6] there are several
examples of this. I.e. you can have two triangles that
look visually symmetrical to one another. Symmetry
would simplify design and repeatability. It could be
possible to create a recursive form of design in
principle. Symmetry in Petri nets could be used for
pattern and complexity identification. A software
system described via Petri nets could be
decomposed into manageable components. When
creating a software or hardware system
asymmetrical components would be identifiable.
The structures could be decomposed into
symmetrical components. This would facilitate their
understanding. For having a telecommunications
network modeled using Petri nets the components of
the network, i.e. the subnets can be checked for
symmetry. This could provide valuable insight into
the fundamental design properties of the system.
Images or graphs represent abstractions of systems.
Some representations are universally better than
others. They can represent a pattern that possibly
repeats itself indefinitely even in other types of
systems or solutions. A case in point is the
producer/consumer pattern. Then is it possible to
find such types of patterns through symmetry?
Rotational symmetry might not obey algebraic
commutative rules like A.B = B.A. Then clockwise
rotations that are not shown in this work can be
considered. E.g. A clockwise rotation of 900 does
not necessarily give the same result as an
anticlockwise rotation of the same order. The results
that will be obtained depend on the net structure.
Another possibility is the fact that a net can be
reversed. I.e. the net could be inverted. This again
needs more exploration and experimentation.
System complexity would benefit from
understanding different forms of symmetry.
Symmetry has a profound effect on the thought
incidence Input Output
nP=nT Possible Possible Possible
nP≠nT No No No
Shape Related Symmetry Possible
Mirror Symmetry Yes
Drawing Symmetry Yes
Rotational Symmetry Yes
Reflectional Symmetry Yes
Point Symmetry Yes
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
170
Volume 23, 2024
processes of persons. So this is bound to have an
effect when describing computer systems.
6 Conclusion
This theoretical work about symmetry in Petri nets
just touches on this aspect. This work presents a
very short concise summary of the types of
symmetry. On each of the aspects that have been
presented volumes of research can be done. E.g. if
just mirror symmetry is considered it is possible to
delve into greater depths and come up with many
other works.
Symmetry is very difficult to define. It considers
how an object or entity may be moved or
transformed but the object or entity remains the
same after the transformation even though it might
look completely different from the observer’s point
of view.
Symmetry is one of the most important
principles that is used in physics. Symmetry governs
patterns in the real world.
As for the usefulness of this work in the real
world, the Petri net models that exhibit symmetry
properties are useful for representing systems. These
models could be used to present different
viewpoints and the fact that they are symmetrical
could mean that their properties will be used for
verification and analysis.
Key principles are often neglected when
designing software and software systems. The
neatness and diagram layout are visualization tools.
Good models exhibit good drawing characteristics
and layouts. Symmetry in the Petri nets focuses on
this aspect.
Some questions are important. Is it possible to
use symmetry instead of formal methods to describe
computer systems via Petri net structures? This
would require the creation of some notation for
representing this as the current notations given in
literature suffer from various drawbacks.
Symmetrical systems should exhibit well-balanced
properties. Symmetry could be used to create
reduced and restricted forms of Petri nets. Possibly
non-symmetrical Petri nets would have some more
difficulty to interpret. However, to understand and
check these conjectures more theoretical and
experimental work has to be undertaken.
Acknowledgement:
I dedicate this work to Haidakhan Babaji and I
thank Tracy Spiteri Staines for reviewing, correcting
and pointing out my writing mistakes in the paper.
References:
[1] H.R. Pagels, Perfect Symmetry In Search for
the Beginning of Time, Simon & Schuster,
2009.
[2] C.W. Tyler, Human Symmetry Perception and
Its Computational Analysis, Routledge Taylor
and Francis, 2002.
[3] I. Stuart, Why Beauty Is Truth: A History of
Symmetry, Basic Books; Illustrated edition,
2008.
[4] J. Rosen, Symmetry Rules How Science and
Nature Are Founded on Symmetry, Springer,
2008.
[5] C. Easttom, M. Adda, An Enhanced View of
Incidence Functions for Applying Graph
Theory to Modeling Network Intrusions,
WSEAS Transactions on Information Science
and Applications, Vol. 17, pp. 102-109, 2020,
https://doi.org/10.37394/23209.2020.17.12.
[6] R.J. Sánchez-García, Exploiting symmetry in
network analysis. Commun Phys., 3, 87
(2020). https://doi.org/10.1038/s42005-020-
0345-z.
[7] K. Schmidt, Symmetries of Petri Nets,
Computer Science, 1997.
[8] P. Bourdil, B. Berthomieu, S. Dal Zilio & F.
Vernadat. Symmetry reduced state classes for
Time Petri nets. 30th Annual ACM
Symposium on Applied Computing,
Salamanca, Spain.pp.1751-1758, 2015.
https://doi.org/10.1007/978-3-031-38821-7_6.
[9] J. Hayman & G. Winskel, The unfolding of
general Petri nets. Leibniz International
Proceedings in Informatics, LIPIcs. 2.
10.4230/LIPIcs.FSTTCS.2008.1755.
[10] S. Rezig, N. Rezg, & Hajej, Z. Online
activation and deactivation of a Petri Net
supervisor. Symmetry, 13(11), 2218. (2021).
https://doi.org/10.3390/sym13112218.
[11] J. Meseguer & U. Montanari, Petri nets are
monoids. Information & Computation, 88(2),
105–155, 1990. https://doi.org/10.1016/0890-
5401(90)90013-8.
[12] A. Meyer & M. Silva, Symmetry Reductions
in Timed Continuous Petri Nets Under Infinite
Server Semantics, IFAC Proceedings
Volumes, Vol. 45, Issue 9, 2012, Pages 153-
159, ISSN: 1474-6670. ISBN:
9783902823007,
https://doi.org/10.3182/20120606-3-NL-
3011.00067.
[13] A. Meyer, Symmetries and Bifurcations in
Timed Continuous Petri Nets, Ph.D.
dissertation, Fakultat fur Elektrotechnik,
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
171
Volume 23, 2024
Informatik und Mathematik der Universitat
Paderborn, 2012.
[14] K. Schmidt, How to calculate symmetries of
Petri nets. Acta Informatica, 36(7), pp. 545-
590,
2000. https://doi.org/10.1007/s002360050002
[15] A. Meyer, M. Dellnitz & M. H. Molo,
Symmetries in timed continuous Petri
nets. Nonlinear Analysis: Hybrid
Systems, 5(2), 2011, 125-
135. https://doi.org/10.1016/j.nahs.2010.05.00
5
[16] L. Gomes & J. Paulo Barros, Structuring and
Composability Issues in Petri Nets Modeling,
IEEE transactions on Industrial Informatics,
vol. 1, no. 2,2005, pp. 112- 123.
[17] R. Campos-Rebelo, A. Costa & L. Gomes.
Finding Learning Paths Using Petri Nets
Modeling Applicable to E-Learning
Platforms. IFIP Advances in Information and
Communication Technology, 2012, pp. 151–
160. https://doi.org/10.1007/978-3-642-
28255-3_17
[18] Zurawski, R., & Zhou, M. (1994). Petri nets
and industrial applications: A tutorial. IEEE
Transactions on Industrial Electronics, 41(6),
1994,
pp.567583. https://doi.org/10.1109/41.334574
[19] B. Berthomieu & M. Diaz, Modeling and
verification of time dependent systems using
time Petri nets. IEEE Transactions on
Software Engineering, 17(3), 1991, pp. 259–
273. https://doi.org/10.1109/32.75415.
[20] C. A. Pinto, J. Torres Farinha, S. Singh,
Contributions of Petri Nets to the Reliability
and Availability of an Electrical Power
System in a Big European Hospital - A Case
Study, WSEAS Transactions on Systems and
Control, vol. 16, pp. 21-42, 2021,
https://doi.org/10.37394/23203.2021.16.2.
[21] A. Spiteri Staines, Graph Drawing
Approaches for Petri Net Models, WSEAS
Transactions on Information Science and
Applications, Vol. 17, pp. 110-116, 2020,
https://doi.org/10.37394/23209.2020.17.13.
[22] A. Spiteri Staines, Alternative Matrix
Representation of Ordinary Petri Nets,
WSEAS Transactions on Computers, Vol. 18,
pp. 11-18, 2019.
Contribution of individual authors to the
creation of a scientific article (ghostwriting
policy)
All the work in the paper was carried out by
Anthony Spiteri Staines. Sections 1-6 and the
abstract are the complete work of Anthony Spiteri
Staines. AKA (Tony Spiteri Staines).
Sources of funding for research presented in a
scientific article or scientific article itself
All the funding for this work comes from the
University of Malta, Malta.
Conflict of Interest
The author has no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2024.23.15
Anthony Spiteri Staines
E-ISSN: 2224-2872
172
Volume 23, 2024
