Journal of Software Engineering and Applications, 2013, 6, 653-661
Published Online December 2013 (
Open Access JSEA
Model Analysis of Equivalence Classes in UML Events
Nazir Ahmad Zafar
Department of Computer Science, King Faisal University, Hofuf, KSA.
Received November 4th, 2013; revised November 24th, 2013; accepted December 1st, 2013
Copyright © 2013 Nazir Ahmad Zafar. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor-
dance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual
property Nazir Ahmad Zafar. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
Unified Modeling Language (UML) has become a de facto standard for design, specification and modeling of object
oriented software systems. UML structures being graphical in nature lack defining semantics of the systems and are
prone to causing errors. Formal methods are proved to be a powerful tool for requirement analysis, design and specifi-
cation of software systems. Hence, linking UML with formal approaches will enhance modeling power of software sys-
tems. In this paper, an approach is developed by integrating UML and Z notation focusing on equivalence relation of
the state diagrams. The Z is used because it is based on the first order predicate logic having rigorous computer tool
support. The reflexivity, symmetry and transitivity properties, being important at design level, are identified and de-
scribed. It is believed that this approach will be effective and useful at both academics and industrial level. The need,
reasoning and benefits of the integrated approach are discussed. The resultant formal models are analyzed and validated
using Z/Eves tool.
Keywords: UML; State Diagrams; Equivalence Relations; Formal Methods; Z notation; Validation and Verification
1. Introduction
Requirements analysis and design specification play an
important role in software engineering. One of the ways
to overcome the above issues is to describe a formal
specification of the system which plays a vital role at
initial phases of software engineering. Formal specifica-
tion is the mathematical description that may be used to
construct a consistent system in a systematic and un-
ambiguous way. If formal specification of a system is
described, the correctness of the required system using
computer tools can easily be proved. By the use of for-
mal specification, an incorrect and inconsistent design
can be modified before its implementation reducing the
construction cost of software systems. Formal methods
which are based on discrete mathematics such as logic,
set theory, graphs, can be used to describe formal de-
scription ensuring quality of software. But these ap-
proaches are not as welcomed at industrial level as their
benefits which are observed. As software industry people
do not have much mathematical background as required
in real software engineering, this is one of the reasons for
ignoring use of formal methods at industrial level. How-
ever, the use of formal methods is recommended for
safety and security systems even by the opponent of for-
mal methods.
Unified Modeling Language (UML) which is based on
graphs and diagrams is much useful for requirements
analysis and presenting detailed design of a system.
UML, a multi-lingual graph based notation, has become
a de facto standard for design and deve lopment of object
oriented (OO) systems despite the fact that its semantics
is still semi-formal and allows ambiguities in design of a
system as in [1] and [2]. Some of the major issues in
modeling using UML diagrams are: 1) UML structures
are based on graphical notations and are prone to causing
errors, 2) The hidden semantics of UML allows am-
biguities at design level, 3) The same system needed can
be described by multiple notations or diagrams which
may cause inconsistencies or ambiguities and 4) UML
model may have multiple interpretations and someone
may not find what is put in the diagrams.
Modeling power of UML diagrams can be enhanced
Model Analysis of Equivalence Classes in UML Events Relations
by defining semantic rules in a formal way [3]. This is
because UML structures are based on graphical notations
and have informal or semi-formal definitions which are
prone to causing errors [4] as mentioned above. As a re-
sult, there is a need for formalizing UML diagrams, par-
ticularly, at design level capturing functionality of the
system to be developed. This integration of formal nota-
tions and UML diagrams will result in a complete, con-
sistent and correct modeling approach. Z notation is a
formal language used to describe and analyze the sys-
tems increa sing confidence at an abstract level of specifi-
cation. The Z is based on first order predicate logic hav-
ing rigorous computer tool support. There is a good rela-
tionship between state diagram and Z notation. This is
because states in state diagram can be described in terms
of relationship between schemas of Z. In this paper, the
relationships of the UML state diagrams are identified
and transformed to Z specification. The reflexivity, sym-
metry and transitivity properties of equivalence relation
being important at design level are identified and de-
scribed. This w ork is part of ou r ongoing project o n inte-
gration UML and formal methods. In this work, it de-
velops a conceptual model by capturing semantics hidden
under the diagrams instead of defining only syntactical
mapping among the approaches. Rest of the paper is or-
ganized as follows.
In Section 2, related work is discussed. Approaches
used are presented in Section 3. Integration of state dia-
grams and Z is given in Section 4. Model analysis is pre-
sented in Section 5. Conclusion and future work are dis-
cussed in Section 6.
2. Related Work
Although there exits a lot of work [5-11] on integration
of approaches but there does not exists much work on
linking UML diagrams with formal approaches. This is
because the hidden semantics under the UML diagrams
cannot be transformed easily into formal notations. It is
mentioned that only closely related work is discussed in
this section. For example, [12] has developed Alloy Con-
straint Analyzer tool supporting the description of a sys-
tem whose state space involves relational structures
which are complex in nature. By the tool it is possible to
analyze and develop a model by investigating the cones-
quences of given constraints by an incremental approach.
An approach is demonstrated using XML which is in fact
a transformation tool to analyze visualize TCOZ models
into various UML diagrams animating specification with
a multi-paradigm programming language as discussed in
[13]. In [14], it is described a way of creating tables and
SQL code for Z specifications according to UML dia-
grams. In another work, a relationship is investigated
between Petri-nets and Z notation [15]. An integration of
B and UML is presented in [16]. Formalization of the
UML is proposed by focusing on basic constructs of
class structures by taking simple case studies in [17]. A
tool is developed in [18] which takes UML class diagram
in the form of Rational Rose petal files and evaluates it
automatically and produces a list of comments on the
diagrams. A comparison of UML, state-charts, Z notation,
petri nets, fuzzy logic and finite state machines is pre-
sented by taking a simple case study on commerce sys-
tem in [19]. An approach is developed by integrating
UML and Z focusing on protocols of state diagram in
[20]. Some other relevant work can be found in [21-28].
3. Why UML and Formal Methods?
Designing has an important role in development process
of complex systems. UML diagrams have various bene-
fits for designing and modeling of systems. This is be-
cause UML is a semi-formal language in which each
element is strongly defined. And you are confiden t that it
will not be misleading when you are modeling a parti-
cular facet of a system. Further, UML is a concise and
easy to understand language. Alth ough UML is not a for-
mal language but it has enough exp ressive power to han-
dle massive and complex systems when viewed at an
abstract level of engineering a software. It is the result of
existing practices in design and modeling using object-
oriented concepts, consequently, it has proved a success-
ful modeling tool. Unified Modeling Language has be-
come a de facto standard for designing of systems using
object orient ed techn ol o gy [29].
On the other side, UML lacks with some important
concepts and, for a moment, cannot be used for the com-
plete design, specification and modeling of a system [30].
For example, UML has a lack of capturing formal se-
mantics of the diagrams. Meanings are hidden under the
UML diagrams which create misinterpretations and am-
biguities at the implementations level. That is why link-
ing and integration of UML with other languages, having
expressive power of capturing semantics, is required.
The use of formal methods is motivated by a belief
that appropriate mathematical analysis can contribute to
the reliability and robustness of software d esign and spe-
cification. Despite the differences over applications of
formal methods the use in the development of high inte-
grity safety or security systems is recommended [31].
Formal methods can be used at different levels from re-
quirements engineering to maintenance of software sys-
tems [32].
At basic level of application of formal approaches,
formal specification may be described and then program
can be developed or generated in an informal or auto-
mated way. This is assumed as most cost-effective option
in applications of formal methods for systems develop-
ment. In other benefits of formal methods, development
and verification may be used to produce a program in a
Open Access JSEA
Model Analysis of Equivalence Classes in UML Events Relations 655
formal manner. At this level of applications of formal
methods, proofs of properties from the specification to a
program may be conducted. This way is considered as a
most effective and appropriate level of applications in
high integrity software systems. Furthermore, theorem
proving techniques can be used to conduct formal proofs
which are fully checked by machines in a systematic and
ordered manner. As this is an expensive validation tech-
nique, that is why, it is only applied if the cost of failure
is very high. Formal methods may be classified in terms
of property or model oriented [33]. Property oriented are
used to describe software in terms of properties and
constraints whereas model oriented are used to construct
a model of a system [34]. Although there are various
tools and techniques available for formal notations but at
the current stage of their development in formal methods,
it needs an integration of formal and traditional approa-
Z is a model oriented specification language based on
set theory and first order predicate logic used at an abs-
tract level [35]. Z is used in this research to link with
UML because of a natural relationship which exists be-
tween these approaches. The Z is based upon set theory
including standard set operators, comprehensions, Carte-
sian products and power sets. Logic of Z is formulated
using first order predicate calculus. The Z allows organi-
zing a system into its components which are known as
schemas and helpful at design level for managing the
complex systems. The schema defines a way in which
state of a system can be described and can be used for
modeling the statics and dynamics of a system. A
promising aspect of Z is its stepwise refinement which is
verifiable and can be used from specification into an
executable code.
The Z/Eves tool is used here because it is a powerful
one and for analysis of Z specification [36]. It includes
declaration of constants of the standard mathematical
toolkit and provides useful theorem proving facility. The
Eves is used to analyze schema expansion, precondition
calculation, domain checking, syntax checking, type
analysis, and general theorem proving mechanisms. Any
specification written in a formal notation does not mean
to be correct, complete and mean ingful. That is why it is
user responsibility to make an appropriate use of the
tools for analysis insuring correctness of the model to be
developed. The remarkable feature of formal specifica-
tion is that it can be checked, analyzed and verified for
the presence of typographical and syntactical errors by
the tools. The Z/Eves provides various exploration tech-
niques to prove the properties of the system.
4. Formal Model of Equivalence Classes
In this section, identification and formal analysis of ex-
isting relationships in UML state diagrams is presented.
At first, approach used is discussed. Then formal defini-
tions used in the model are described. Finally, statics and
dynamics of the UML state diagrams are given in terms
of formal models.
The integrated approach is given in this section. Al-
though formal methods have a well-defined syntax and
semantics but are at the early stage and, hence, it needs
their integration with the existing approaches for a com-
plete and consistent development of software systems.
UML has become a de facto standard for design of object
oriented systems. Therefore, a relationship between
UML diagrams and formal techniques is analyzed and
established. State diagrams are selected because of their
importance for link ing UML and Z notation syntactically
and semantically. A mapping defining relationship be-
tween these approaches is established. Initially, we have
UML state diagrams which are transformed to Z notation
considering both the syntax and semantics. Then rela-
tionships of the state diagrams are identified to be useful
in design of a system.
The state identifier and event are represented as X and
E respectively both as set types. For simple specification,
the basic set types are used. In the definition of a transi-
tion from one state to another the guard is defined as
Boolean type. A state can have two possible values that
are active or passive represented by Active and Passive
respectively. The type of state can be simple, non-con-
current, initial or final.
In modeling using sets, we do not impose any res-
triction upon the number of elements and a high level of
abstraction is supposed. Further, we do not insist upon
any effe ctive procedur e for deciding whether an arbitrary
element is a member of the given collection or not. As a
consequent, our sets X and E are sets over which we
cannot define any operation. For example, cardinality to
know the number of elements in a set cannot be defined.
Similarly, subset and complement operations over sets X
and E are not defined.
The state diagram is a collection of states related by
certain types of relations. In the definition of a state, state
identifier, its type and status is considered. The state is
represented by a schema which consists of three compo-
nents described above which are encapsulated and put in
the Schema State.
Open Access JSEA
Model Analysis of Equivalence Classes in UML Events Relations
The collection of states is represented by the schema
Diagram which consists of four variables that are initial,
states, substates and final. The mapping substates from
State to power set of State describes types of the states.
Invariants: 1) The initial state is not in the collection
of states. 2) The initial state cannot be the final state. 3)
The initial state does not belong to domain of substates
mapping that is it has no sub-state. 4) The set of states is
non-empty. 5) For any state such that it is in the domain
of sub-states mapping, it is in the collection of states. 6)
For any state, s, if it is in the collection of the states and
is not the initial or final state and not the simple state
then it belongs to domain of su b-states. 7) The final state
does not belong to domain of the mapping sub-states.
To move from one state to another, a transition must
be fired. The transition con sists of three components that
are event, guard and action. Action is in fact a sequence
of events described below. The transition is defined by a
schema Transition in Z which consists of three variables
which are event, guard and action as given blow.
Action == seq E
The complete set of transitions of the state diagram is
represented by the schema Transitions which consist of
set of all possible events and the transition function de-
fined over set of states. The transition function takes a
state and transition and returns the same or a new state.
Invariants: 1) For every event in the set of possible
events, there must be two states and a transition over
these states such that the event is in the transition and it
is also included in the sequence of events called action
which must be executed after the guard condition of the
transition is true. 2) For any two states and a transition
defined over the states, there exists an event, guard and
an action such that the event is in the transition and it is
also included in the sequence of events, which must be
executed to move from one to the other state after the
guard cond ition is true.
The set of events of the state diagram is represented by
the schema Events which includes two schemas that are
Diagram and Transitions. The Diagram represents to set
of states and Transitions represent to all the possible
transitions among the states as defined above. The set of
events gives the relationship between the states of the
state diagram and transitions among the states.
Invariants: 1) For every non-final state, there is a
transition which can be fired over it. 2) For every state
with a transition, must be in the collection of states o f the
Open Access JSEA
Model Analysis of Equivalence Classes in UML Events Relations 657
state diagram. 3) For every non-final state there is a tran-
sition which acts on this state and results the same or a
new state.
In the state diagram, it is possible that when a transi-
tion is fired it results the same state. That means there
exists a set of events over which the reflexive relation is
satisfied. The reflexive relation over the set of events is
defined in terms of the schema ReflexiveEvents. It takes
the schema Events and verifies the above property of
A relation R over a set A is symmetric if for all x, y in
the set A, if (x, y) is in the relation R then (y, x) is also in
R. In case of state diagrams, it is possible that when a
sequence of events is executed and control moves from
one state S1 to another state S2 then symmetry forces and
there exists an inverse of the sequence of events which
results S2 to S1. The symmetric relation over the set of
events of the state diagram is defined below by the
schema SymmetricEvents which takes the schema Events
and verifies the above symmetric property.
Invariant: 1) The events relation is symmetric over a
set of states if for every sequence of events which move
from S1 to S2 there exists a new sequence of events
which return S2 to S1. The sequence of events, is defined
in order by event, guard and action as is the case of UML
state diagrams.
Invariant: 1) For any two states s1 and s2, and a se-
quence of events which move the state s1 to s2 after
execution in the state diagram, for another state s3, and a
sequence of events which move the state s2 to state s3,
there exists another sequence of events which is con-
catenation of the above sequences and it moves the state
s1 to s3. The property is defined by decomposing into
three parts. The first one is in which control moves from
initial to next possible state. The second part is in which
control moves in all possible states except initial and
final state.
A relation R over A is transitive if for any x, y, z in A,
and (x, y) in R and (y, z) in R the order pair (x, z) in the
relation R. To define transitivity in state diagrams, if a
transition is fired from one state S1 to another state S2
and then a new transition is fired from S2 to S3 then a
composite transition can be fired from S1 to S3 that
means the transitive relation exits over the state diag ram.
The formal description of transitivity over the set of
events of the state diagram is defined below by using the
schema TransitiveEvents which takes the above schema
Events and verifies the transitive property.
Based on the definition of reflexive relation over the
state diagram, null actions over the diagram are com-
puted in the schema GenerateNullactions as described
below. The schema consists of two components which
are reflexive events and null actions. The first one is
given as input and second one is generated as output of
the schema. The null action returns the same state after
its execution.
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Model Analysis of Equivalence Classes in UML Events Relations
To generate the set of possible collection of undoable
events, a schema GenerateUndoables is described below.
The schema consists of two components which are sym-
metric events and a collection of undoable actions. The
undoable action is one which reverses the previous action.
The collection of symmetric events is given as input and
set of undoable events is generated as output of the
The set of accessible events can be computed by the
schema GenerateAccessibles is described below. The
schema consists of four components which are transitive
events and a collection of accessible actions, start and
target state. The collection of transitive events, start and
target states are given as input and set of accessible
events is generated as output of the schema.
5. Model Analysis
Since there does not exist any computer tool which may
assure guarantee about the complete consistency and
correctness of a computer software model. That is why
we believe even th e formal specification is written in any
of the formal notations, it may contain potential an d haz-
ardous errors. Such errors may range from syntax to
conceptual inconsistencies. The Z/Eves is one of the
powerful tools which can be used for analyzing formal
specification of a software system written in Z notation.
The tool is integrated with various facilities providing
rigorous analysis of the system to be developed and has
automated deduction capability. Because of the abstract
expressive power and model checking facilities, Z is
popular among all of the formal notations and techniques,
and is most widely used by the scientific community.
The syntax checking, type checking and theorem proving
facilities of the tool are used in this research. It is noted
that syntax and type checking facilities do not require
any interaction with the theorem proving facilit y.
Domain checking facility of the tool allowed us to
write the meaningful statements. We used Z/Eves to
check the specifications for identifying the domain er-
rors. It was observed that domain checking was much
harder than the syntax and type checking. This is because
the syntax and type checking is performed automatically
whereas one has to interact with the theorem prover to
perform the domain checking. We also observed that
proof ‘by reduce’ in the proof window of the tool was
sufficient for our formal specifications for domain
checking. If the specification passes the domain checking,
we get Y otherwise N as shown in Figure 1 which is a
snapshot of the model analysis using Z/Eves tool.
The schema expansion facility was used to unravel the
complex schemas. This facility simplified the model re-
sults which were not otherwise easy to understand the
detailed meaning of the given schema. Prove by reduce is
one of the most important facility in the toolset and is
used for analyzing the specification. The results of the
model analysis are shown in Table 1. In the Table, the
first column shows name of the schema analyzed and
evaluated, the second column is for syntax and type
check, third for domain checking, fourth for reduction
facility and the last one for the proof by reduction. The
symbol “*” after Y shows that proof is made by reduce-
tion technique.
6. Conclusion and Future Work
In recent years, integration of approaches has become an
important area of research because of developing auto-
mated tools supporting activities in software engineering.
Unified Modeling Language (UML) is used at initial
phases of software engineering because of having gra-
Open Access JSEA
Model Analysis of Equivalence Classes in UML Events Relations 659
Figure 1. Snapshot of the Model Analysis using Z/Eves.
Table 1. Results of Model Analysis.
Schema Name Syntax
Check Reduction Proof
State Y Y Y Y
Diagram Y Y Y Y
Transition Y Y Y Y
Transitions Y Y Y* Y
Events Y Y Y* Y
ReflexiveEvents Y Y Y* Y
SymmetricEvents Y Y Y* Y
TransitiveEvents Y Y Y* Y
GenerateNullactions Y Y Y Y
GenerateUndoables Y Y Y Y
GenerateNullAccessibles Y Y Y Y
phical representation whereas formal methods are useful
having rigorous mathematical and computer tools
support for capturing the semantics hidden under the
diagrams. Therefore, an integration of UML and formal
methods was needed for systematic development of
computer software systems. The first objective of this
research was to develop an approach by linking UML to
Z notation by defining a relationship among the
fundamentals of UML and formal techniques. To address
the reusability issue by defining the components and
developing recursive approach to be useful for easing the
development process was another objective.
In this paper, UML state diagrams are used to link
with Z notation by identifying and formalizing the rela-
tions among the states by focusing on the events and ac-
tions responsible for analyzing the state diagrams. The
resultant approach can be useful in development and
construction of automated computer tools for generating
the specification. For linking UML with Z, most abstract
view of the diagrams was perceived to define the generic
formal models independent of a system which will be
equally useful for any kind of domain problem.
It is mentioned that the most relevant work [37-41]
was considered as starting point for this research. An
exhaustive survey was done, and some interesting work
was found but our work is different because of concep-
tual and abstract level integration. In the existing work
either example is taken to make integration or only syn-
tactical mappings are defined. But we have defined both
the syntax and semantic analysis of both approaches.
The Z is used because every object is assigned to a
unique type providing useful programming practice. Se-
veral types of checking tools exist to support the specifi-
cation. The Z/Eves is a powerful tool to prove and ana-
lyze the specification which was used in this research.
The rich mathematical notations made it possible to rea-
son about behavior of a sp ecified system more rigor ously
and effectively.
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