Masaryk university
Faculty of Informatics
Conceptual modelling using the HIT method
Practical tutorial
Marek Winkler, Michal Oˇskera, Zdenko Stan´ıˇcek
Version 1.0
Contents
1 Introduction 3
1.1 Why to deal with conceptual modelling? . . . . . . . . . . . . 3
1.2 What is conceptual modelling? . . . . . . . . . . . . . . . . . . 5
1.3 What is a conceptual model? . . . . . . . . . . . . . . . . . . . 6
1.4 Relationship to data modelling . . . . . . . . . . . . . . . . . 6
1.5 What is the diﬀerence between conceptual modelling and HIT
method? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 What is the relationship between HIT method, HIT conceptual
model and E-R model? . . . . . . . . . . . . . . . . . . . 7
1.7 What is the diﬀerence between E-R model and E-R diagram? 7
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 HIT method – concepts and procedures 8
2.1 The steps of the modelling process . . . . . . . . . . . . . . . 8
2.2 Functional approach . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Modelling concepts using sorts . . . . . . . . . . . . . . . . . . 10
2.3.1 Entity sorts . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Descriptive sorts . . . . . . . . . . . . . . . . . . . . . 11
2.4 Modelling relationships among concepts using HIT attributes . 11
2.4.1 Graphical notation . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Linear notation . . . . . . . . . . . . . . . . . . . . . . 13
2.4.3 HIT attribute rotation . . . . . . . . . . . . . . . . . . 15
2.4.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Normalization of HIT attributes . . . . . . . . . . . . . . . . . 16
2.5.1 Deﬁnability . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.2 Decomposability . . . . . . . . . . . . . . . . . . . . . 18
2.5.3 The kernel of a set of HIT attributes . . . . . . . . . . 19
2.6 HIT conceptual model . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Transformation into ER model . . . . . . . . . . . . . . . . . . 20
2.7.1 Binarization principle . . . . . . . . . . . . . . . . . . . 21
2.7.2 The transformation procedure . . . . . . . . . . . . . . 23
Service Systems Laboratory, FI MU Brno 2
1 Introduction
This introductory section provides a brief explanation of the mutual relationships
between data modelling, conceptual modelling and the HIT method.
1.1 Why to deal with conceptual modelling?
For any kind of eﬀective teamwork the mutual understanding of all people
involved is essential. This applies to any form the team can take, e.g. SW
development team, organizational unit, company staﬀ, or even family. In
other words, mutual understanding is the prerequisite for any group of people,
that is expected to work together in synergistic way, to succeed. The
conceptual modelling discipline is about explicit and systematic facilitation
of such understanding process. One may ask: how conceptual modelling and
understanding process is related?
The explicit mutual understanding within a group of people is about
agreement or conformity in meaning of words or expressions of language being
used in mutual communication. One of the philosophical and logical
approaches1
at intuitive level explains that a concept may become meaning
of given language expression [?]. In other words, if we know the meaning of
the word ’car’, we know the concept of ’CAR’. Straightforwardly, the word
’car’ in spoken or written form represents the concept ’CAR’ in external
world, i.e. we people as lingual creatures use the words ’car’, ’auto’, ’das
Auto’ to represent the concept ’CAR’ outside of our minds. We do this in
order to communicate with other people. But the communication would be
useless or even harmful if the words received by our communication partner
represented diﬀerent concepts in their mind. Therefore, the conceptual modelling
discipline emerged to help people to achieve agreement on key concepts
(meanings) represented by words in their mutual communication.
There are endless situations in our lives in which we communicate with
other people and in many of them we think that we understand each other,
just because we know the meanings of words. But we know only one or few
of many possible meanings a single expression may have in minds of other
people. Very often, we do not notice or even care of shifts in the meaning
and in many ordinary situations we can handle this somehow. Actually, the
essence of whole category of jokes lies in the ambiguity of language – these
1
by coincidence underlying one for the HIT method
Service Systems Laboratory, FI MU Brno 3
are called puns2
.
However, there are many situations where one cannot successfully continue
in their work just sharing an approximate or even vague meaning. Consider
an emergency communication between medicine doctors. They need to
be as accurate as possible to avoid misunderstanding that can threaten one’s
life. Therefore, huge number of concepts (represented by unique language expressions)
was revealed, each of them identifying very single piece of human
body to enable clear communication.
Another example where accuracy in communication matters are all technical
and engineering disciplines. Both complex machines and SW programs
cannot be constructed just approximately. They need to be projected in an
exact way and they are usually results of team eﬀort. Therefore, people need
to clearly communicate about what they are doing. What’s more they also
need to clearly communicate with people whom they do it for; and this people
may be far from homo technicus. These people, let us call them users or
customers, have their own diﬀerent expertise and have no clue and also do
not care what for instance the term ’event handler’ denotes. They want to
be impressed by results not by technical means.
Typically in IT, the business-IT consultants are the people who are responsible
for translation of business talk (spoken by people with non-IT expertise
– domain experts) to IT talk and vice versa. This process is called
analysis, it is iterative and requires at least good communication skills.
It is clear, that the translation may be successful only after clear understanding
of terms being translated. More precisely, consultants need to
understand terms such as ’budget’, ’rate’, schedule item in exactly the same
way as the domain experts do – they need to have the same concepts in
their minds. Otherwise, software conﬁgured or created according to diﬀerent
meanings of the same words will disappoint the customer very soon.
The more complex the domain is (by its scope, speciﬁcity, people involved,
...) the more diﬃcult is to understand it and come to an agreement among
all parties involved. In these situations the visual, schematic, intuitive but
not oversimplifying methods are needed.
The HIT method is one of the kind which may help to accomplish the
process of mutual understanding. And in accordance with aforementioned
facts, this method must be well understood before it can be applied correctly.
This tutorial is about explanation of concepts behind the the HIT method.
2
Did you hear about the guy whose whole left side was cut oﬀ? He’s all right now.
Service Systems Laboratory, FI MU Brno 4
Its goal is to reveal and explain key concepts to prospective conceptual modellers
and to support continuous understanding within already practising
community – everything within the meaning of the preceding paragraphs.
Facts to remember are as follows:
• The primary motivation for conceptual modelling is to understand and
not to design or to specify a database for instance3
.
• Conceptual modelling is very helpful and essential in interdisciplinary
cooperation as well as in team communication.
• You can do a lot of conceptual models by yourself and for yourself, but
the meaningfulness of these models will be checked only when they had
been accepted and followed by other people helping them achieve their
goals.
1.2 What is conceptual modelling?
More technically, the conceptual modelling can be described as the process of
conceptual understanding and rendering modelled part of reality. The modeller
tries to ﬁnd out which objects of interest the system should keep and
provide. Conceptual model should be totally independent of intended implementation
and intelligible for users. [?]
In other words, the conceptual model describes the concepts and their
relationships identiﬁed in the modelled domain. It is important to realize
that a conceptual model’s elements are the objects from the modelled domain,
not any database tables, programming language classes or any other artifacts
of the system being built for the modelled domain.
A good conceptual modelling technique should make the analyst reduce
the amount of the resulting model ambiguity as much as possible. For this
reason, the conceptual modelling method HIT is focused on well-formed
(semi-formal) deﬁnitions of all the concepts and the relationships in the
model.
3
However, the result of conceptual modelling can be very helpful for subsequent
database design.
Service Systems Laboratory, FI MU Brno 5
1.3 What is a conceptual model?
A conceptual model is a special case of a data model. As mentioned in the
previous section, a conceptual model describes the concepts and their mutual
relationships in the modelled domain.
1.4 Relationship to data modelling
The conceptual modelling is a special case of data modelling. Data modelling
is a discipline focusing on the description of data using diﬀerent kinds of
models. The models, along with the relevant data modelling sub-disciplines,
are usually grouped to the following three levels:
• conceptual models capture the elements of the modelled domain, they
are totally implementation independent; the entities have not been assigned
any attributes yet (unless an attribute is really necessary for the
description of the modelled domain), the many-to-many relationships
are not decomposed into associative entities;
• logical models describe the elements of the information system (IS)
which is being built for the domain, but they are independent of any
particular database or other means used to store IS data; a logical
model contains entities which can be speciﬁc for the IS implementation,
the entities have all their attributes deﬁned, etc.;
• physical models deﬁne the data structures with all details necessary
to create these structures in a particular database engine; in case of a
SQL database, this involves assigning an SQL type to each attribute,
decomposition of many-to-many relationships into associative entities,
deﬁning database indices, etc.
1.5 What is the diﬀerence between conceptual modelling
and HIT method?
Conceptual modelling is a discipline focused on understanding and description
of the concepts and their relationships in the modelled domain. HIT
method is one particular method or technique used to achieve the goals of
conceptual modelling.
Service Systems Laboratory, FI MU Brno 6
1.6 What is the relationship between HIT method,
HIT conceptual model and E-R model?
The HIT method provides a technique how to construct conceptual models.
The result of an application of HIT method is a HIT conceptual model which
consists of the base of sorts, the so-called kernel set of HIT attributes, and
the set of integrity constraints4
.
On the contrary, an E-R model (Entity-Relationship model) is not any
particular data modelling method, it is merely a standard notation for the
conceptual modelling product, i.e. the conceptual model. Therefore, it is
comparable to the HIT conceptual model rather than the HIT method itself.
In fact, every HIT conceptual model can be transformed into the appropriate
E-R model5
.
Another conceptual model notation is a concept map. An example can
be found at http://www.w3.org/TR/ws-arch/#gsom.
1.7 What is the diﬀerence between E-R model and ER
diagram?
The E-R model consists of the E-R diagram and the text description of each
E-R diagram element.
1.8 Exercises
1. Find a word representing at least two diﬀerent concepts. How words
that possess such property are called?
2. Find two words representing at least the same concept. How words
that possess such property are called?
3. Form groups of three. Presuming that all members know the meaning
(concept) of word ’component’, let each team member independently
write his or her own representation of the concept ’COMPONENT’ to
the paper. Then compare and discuss the externalized concepts. What
do you observe?
4
More detailed HIT conceptual model deﬁnition can be found in the section 2.6 of this
document.
5
See section 2.7 of this document.
Service Systems Laboratory, FI MU Brno 7
2 HIT method – concepts and procedures
This section introduces the HIT conceptual modelling method – the process
of creating the HIT conceptual model, HIT method basic concepts, and ﬁnally
the transformation of the HIT conceptual model into the E-R model
notation.
2.1 The steps of the modelling process
Let us start with forming an initial idea what steps the process of making a
conceptual model according to the HIT method involves:
• the identiﬁcation and deﬁnition of sorts,
• the identiﬁcation and deﬁnition of HIT attributes,
• normalizing the set of HIT attributes,
• transforming the HIT conceptual model into ER model notation.
Individual steps are illustrated on a concrete domain in section ??. This
section continues with brief introduction of basic ideas behind HIT. You are
encouraged to read this section ﬁrst, but it is possible to skip directly to the
example application in section ?? and refer to the rest of this section when
needed.
2.2 Functional approach
Everything we encounter in HIT can be regarded as a function. In fact, HIT
deﬁnes all of its concepts as functions. Right now you have probably asked
’Why? The most common approach is the relational calculus which has been
widely used in databases nowadays.’. Let us try to shed some light on this
matter.
The functional approach (as compared to the relational approach) allows
for more natural mapping of the meaning (usually expressed in natural language)
of modelled domain elements onto the conceptual model elements
deﬁnitions. One reason is that a function is oriented while a relation is not.
Imagine a relationship between a student and a seminar. There is no
doubt that you can make up various kinds of such relationship – students
Service Systems Laboratory, FI MU Brno 8
who have enrolled in the seminar, or students who have successfully passed
the seminar. If we want to make clear what kind of relationship between a
student and a seminar was meant during the making of the model, we have to
provide a description telling the model reader what kind of the relationship
it is. We say that such a description deﬁnes the semantics or meaning of the
relationship.
Consider the semantics of the relationship from the example – students
who have enrolled in the seminar. This expression can be understood as a
description of a function which, being given a seminar, returns the collection
of all students who have enrolled in the given seminar. This function can be
perceived as a query which a user can resolve with using the knowledge from
the modelled domain.
The notion of a query is much more natural to domain experts, expressing
themselves in natural language, than relations. Therefore, HIT maps the
natural language expressions onto semi-formally deﬁned functions.
HIT distinguishes two kinds of functions: intensions, which depend on the
current state-of-aﬀairs, and extensions, which are the current state-of-aﬀairs
independent. But what is the current state-of-aﬀairs?
The current state-of-aﬀairs can be understood as a (potentially inﬁnite)
list of facts which are currently valid. Any intension is a function which is
able to access such a list of facts (you can imagine it as a method which can
query a database table, for instance). They are closely connected with the
so-called Entity sorts (see later). On the other hand, an extension can never
access such a list of facts, it can compute only information independent on
the current state-of-aﬀairs. Extensions are also called analytical functions
because they cannot access the empirical facts. They are connected with
Descriptive sorts (see later).
Because this tutorial is meant to serve as a practical guide to HIT method,
we are not going to formally deﬁne all HIT concepts6
. Instead, we provide an
explanation of the background concepts suﬃcient for the modelling purposes.
6
More detailed and rigorous deﬁnitions of the HIT method concepts can be found in [?]
or [?].
Service Systems Laboratory, FI MU Brno 9
2.3 Modelling concepts using sorts
In HIT, a concept is modelled as a sort. You can imagine a sort as a set of
all objects having a common property distinguishing the sort7
. An example
can be a sort of all individuals who are active students.
HIT distinguishes between entity and descriptive sorts. An entity sort
contains entities and is denoted with the hash character, e.g. (#Car),
(#Student). A descriptive sort contains descriptive attributes of entities
and is denoted without the hash, e.g. (Phone number).
2.3.1 Entity sorts
An entity sort E is deﬁned using a function which, being applied to an object
(or an individual) and the current state-of-aﬀairs, returns true, if the given
object is an instance of the sort being deﬁned, false otherwise. This function
can be viewed as a characteristic function of the sort E. Because this function
depends on the current state-of-aﬀairs, it is an intension.
In HIT, this function is written as a Natural Language expression following
speciﬁc pattern. The reason is that providing logically exact deﬁnitions
built upon some very basic concepts would be so demanding, that such deﬁnitions
would be practically unusable.
Have a look at one possible deﬁnition of (#Car):
An object of category (#Car) is every road vehicle, typically
with four wheels, powered by an internal combustion engine and
able to carry a small number of people.
This is not the only “right” deﬁnition of (#Car), you could invent other
ones which would be “right” as well. The goal we are trying to achieve
with deﬁnitions is, that a deﬁnition is comprehensible to all intended model
readers and appropriately delimits the concept boundary in the modelled
domain. By “delimiting the concept boudary” we mean not only identifying
all objects that satisfy the deﬁnition, but also denying the objects that we
do not consider to be part of the concept.
The generic pattern of the Natural Language expression describing the
function is the following:
7
This deﬁnition is very vague, but suﬃcient for practical usage of HIT. An interested
reader should refer to the formal sort deﬁnition given in the PB114/PA116 lectures, or
in [?], [?].
Service Systems Laboratory, FI MU Brno 10
An object of category8
(#Sort name) is every/any . . . .
An entity sort cannot be directly represented in computers. The representation
is usually done by using a corresponding descriptive sort (ID).
2.3.2 Descriptive sorts
A descriptive sort D is deﬁned using a function which, being applied to an
object (or an element) returns true, if the given object is an instance of
the sort being deﬁned, false otherwise. This function can be viewed as a
characteristic function of the sort D. Because this function does not depend
on the current state-of-aﬀairs, it is an extension.
Descriptive sorts can be viewed as the sets of elements which can be
directly represented in computers. Consider the following example:
An element of category (Phone number) is every string value
having the maximal length of 13, containing only digits and optionally
a plus sign as the ﬁrst character.
The generic pattern of the expression describing the function is the fol-
lowing:
An element of category9
(Sort name) is every/any . . . .
2.4 Modelling relationships among concepts using HIT
attributes
A relationship among sorts T1, ..., Tn, S, where at least one sort Ti, 1 ≤ i ≤
n is an entity sort, is deﬁned using a function, which being applied to the
current state-of-aﬀairs and the given n-tuple (t1, . . . , tn), where ti ∈ Ti, 1 ≤
i ≤ n, returns a single s ∈ S, a set S ⊆ S, or is undeﬁned. Such functions
are called HIT attributes in HIT. Because these functions depend on the
current state-of-aﬀairs, they are intensions. The number n + 1 is called the
complexity of the HIT attribute.
A HIT attribute can be deﬁned either in graphical, or linear notation.
The linear notation is used in models, while the graphical notation is usually
more illustrative for discussions about the decomposability of a HIT attribute
(see later).
8
You can sometimes encounter type instead of category. Category is newer.
9
You can sometimes encounter type instead of category. Category is newer.
Service Systems Laboratory, FI MU Brno 11
2.4.1 Graphical notation
Graphical notation simply depicts a HIT attribute as a function (remember
that every HIT attribute is a function) illustrating the mapping of the function’s
input arguments to the output. The graphical notation consists of the
following elements:
• circles represent sorts; a crossed circle denotes an entity sort, an empty
circle represents a descriptive sort,
• the rectangle10
which represents the HIT attribute itself.
Have a look at Figure 1 displaying an example HIT attribute called Grad-
ing.
Figure 1: The HIT attribute example.
• The upper part declares the input arguments of the HIT attribute. Every
input argument is a sort accompanied with a compulsory textual
description of the semantics (the meaning) of this argument. The semantics
of the input argument can be imagined as the deﬁnition of the
role that the input argument plays in the HIT attribute.
• The lower part declares the result of the HIT attribute, optionally
accompanied with a textual description of its semantics.
10
sometimes an oval shape is used as well
Service Systems Laboratory, FI MU Brno 12
• The middle part (the oval) represents the HIT attribute itself. You
can notice the four values on the right side of the HIT attribute which
deﬁne the cardinality of the HIT attribute.
To sum it up, Figure 2 displays the generic form of a HIT attribute in
graphical notation. The card_orig is the cardinality of the function described
by the HIT attribute. The card_rev is the cardinality of the reversed
function. See the last paragraph of this section for the description of
cardinalities.
Figure 2: The generic form of HIT attribute graphical notation.
The meaning of the HIT attribute can be determined by reading the
diagram starting with the result sort and then reading the description of
each input argument semantics followed by the name of the sort:
Students who have passed the given course with given grade.
In fact, this procedure gives us (almost) the linear notation mentioned before.
2.4.2 Linear notation
Linear notation of a HIT attribute is constructed using the following rule:
• if the HIT attribute returns at most one result, then the linear notation
follows this pattern:
Text0 (S) Text1 (T1) . . . Textn (Tn) Textn+1 / card_orig:card_rev
Service Systems Laboratory, FI MU Brno 13
• if the HIT attribute can return more than one result for given input,
then the linear notation follows this pattern:
Text0 (S)−s Text1 (T1) . . . Textn (Tn) Textn+1 / card_orig:card_rev
Using again the HIT attribute Grading as an example, the linear notation
is the following:
Students (#Student)-s who have passed the given course (#Course)
with given grade (Grade) / 0,M:0,M
There are a few rules that must be respected when constructing a linear
notation:
• every input argument is explicitely marked with the word given,
• the result of the HIT attribute must be stated in singular or plural
form according to the following rules:
– singular – when the function can return at most one instance of
the result sort,
– plural – when the function can return a set of instances of the
result sort.
The plural form must also be marked by adding the suﬃx “-s” to the
name of result sort (see the example).
Let us have a look at the same example HIT attribute linear form with
the parts aﬀected by the aforementioned rules underlined:
Students (#Student)-s who have passed the given course (#Course)
with given grade (Grade) / 0,M:0,M
To highlight the diﬀerences between singular and plural forms, imagine a
singular form of the same HIT attribute:
Student (#Student) who has passed the given course (#Course)
with given grade (Grade) / 0,1:0,M
Both graphical and linear notations are equivalent in terms that they
carry the same amount of information and one can be transformed into the
other without requiring any additional information.
Service Systems Laboratory, FI MU Brno 14
2.4.3 HIT attribute rotation
A HIT attribute rotation is every HIT attribute that is constructed from
the original HIT attribute by swapping the result sort with one of the input
argument sorts. HIT attributes that diﬀer only in the order of their
input arguments are considered equal, and therefore they are not diﬀerent
rotations.
The following example illustrates all admissible rotations of the HIT attribute
Grading:
• Students (#Student)-s who have passed the given course (#Course)
with given grade (Grade) / 0,M:0,M
• Grade (Grade) which has been achieved by the given (#Student) in
passing the given course (#Course) / 0,1:0,M
• Courses (#Course)-s which have been passed by the given student
(#Student) with given grade (Grade) / 0,M:0,M
Each rotation can be either singular or plural, thus the list of rotations
could be extended with rotations such as
Student (#Student) who has passed the given course (#Course)
with given grade (Grade) / 0,1:0,M.
However, such a rotation does not describe the modelled domain appropriately
– clearly, more than one student can pass a given course with a
given grade. Such rotations are called inadmissible. During the modelling
we concentrate on the admissible rotations only.
2.4.4 Cardinality
The cardinality of a relationship A provides information about the minimum
and maximum number of objects that can be involved in a single instance of
the relationship A. But how to interpret the cardinality in the context of a
HIT attribute, i.e. a function?
Consider our example HIT attribute Grading:
Students (#Student)-s who have passed the given course (#Course)
with given grade (Grade) / 0,M:0,M
Service Systems Laboratory, FI MU Brno 15
The cardinality information, located after the slash (“/”) character, consists
of two pairs of characters. The ﬁrst pair of characters is the cardinality
of the function directly described by the HIT attribute Grading:
• the ﬁrst element indicates if the function is partial or total; informally,
how many results at least the function returns for an arbitrarily chosen
input argument. The possible values are 0 (partial) and 1 (total).
In the example, (0,M:0,M) states that the function is partial, i.e. it can
be undeﬁned for some input arguments. Because it is possible that no
student passed the given course with the given grade, the function for
such input arguments returns nothing (it is undeﬁned).
• the second element deﬁnes if the function returns a single result or a
set; in other words, how many results at most the function returns for
an arbitrarily chosen input argument. The possible values are 1 (single
result) and M (many results)11
.
In our example, (0,M:0,M) indicates that the function returns a set
of students who have passed the given course with the given grade.
Indeed, several students might have achieved the same grade for the
same course.
The second pair of characters (in the example (0,M:0,M)) deﬁnes cardinality
of the so-called reversed function. The reversed function can be
obtained by swapping the input and output arguments of the function12
:
The pairs ((#Course), (Grade))-s which correspond to the
passing of this course with this grade by the given (#Student).
The cardinality is again 0,M: there can be zero or many such pairs returned
by the function for an arbitrarily chosen student.
2.5 Normalization of HIT attributes
Some of the identiﬁed HIT attributes can be in fact a composition of more
basic HIT attributes. Using a composed HIT attribute in the model is a
problem, because it results in the loss of information. Why?
Consider the following HIT attribute called Enrolls:
11
Sometimes, the value of N can be encountered as well – it has the same meaning as
M.
12
It is similar, but not completely identical, to the inverse function in mathematics.
Service Systems Laboratory, FI MU Brno 16
Enrolls: Students (#Student)-s who have enrolled in a given
seminar (#Seminar) taught by a given lecturer (#Lecturer)
/ 0,M:0,M
It is tempting to believe that you can obtain the information which seminars
are taught by a given lecturer from this HIT attribute as well. However,
this is not completely true – what if there is a seminar taught by the given
lecturer but no student has enrolled in this seminar yet? This information
cannot be captured by the HIT attribute above.
This step of the modelling process aims at removing such compositions
and redundancies.
2.5.1 Deﬁnability
Let us begin with the HIT concept which allows to describe what it means
when we consider a HIT attribute redundant.
A HIT attribute A is said to be deﬁnable over the set of HIT attributes
{B1, . . . , Bn} when there is an analytical function that is able to compute
all values of A by using the HIT attributes B1, . . . , Bn.13
It is denoted as
A ← {B1, . . . , Bn}.
What is an analytical function? A function is analytical, if it does not
depend on any empirical knowledge, i.e. on the current state-of-aﬀairs14
. In
programming terms, you can imagine an analytical function as a procedure
which does not depend on any list of facts (for instance a database table of
current lecturers) about the “outside” world for its computation.
The fact that the function has to be analytical ensures that the function
itself is not another “hidden” HIT attribute bringing some additional
“unspotted” information into the model.
Let us make it clear on the example of HIT attribute from the introduction
to this section:
Enrolls: Students (#Student)-s who have enrolled in a given
seminar (#Seminar) taught by a given lecturer (#Lecturer)
/ 0,M:0,M
Now, imagine two simpler HIT attributes Teaches and Enrolls’:
13
For precise formal deﬁnition please see the PB114/PA116 lectures or the literature.
14
see Section 2.2
Service Systems Laboratory, FI MU Brno 17
Teaches: Lecturer (#Lecturer) teaching a given seminar (#Seminar)
/ 0,1:0,M
Enrolls’: Students (#Student)-s who have enrolled in a given
seminar (#Seminar) / 0,M:0,M
We can simply compute all values of the original HIT attribute Enrolls
using only the HIT attributes Teaches and Enrolls’ and some analytical operations.
The computation procedure indeed can be constructed as analytical
because all necessary empirical information is provided by the HIT attributes
Teaches and Enrolls’ which you can imagine as other procedures having access
to the appropriate data store.
Therefore, we can say that Enrolls ← {Teaches, Enrolls }.
2.5.2 Decomposability
HIT attribute decomposition essentially means reducing the original HIT
attribute to the set of its subattributes without losing any information. What
does it mean?
First, what is a subattribute? A subattribute A of HIT attribute A is
every HIT attribute constructed from A by omitting one of the sorts involved
in A.15
We can see that the HIT attributes Teaches and Enrolls’ are subattributes
of the HIT attribute Enrolls.
Next, we need to make sure that we do not lose any information if we
replace the original HIT attribute Enrolls with its subattributes. This follows
from the deﬁnability of Enrolls over the set {Teaches, Enrolls }. Why is this
true? Because we can create an analytical function which will compute all
values of the original HIT attribute by using the subattributes.
You have probably realized by now that decomposability is closely related
to deﬁnability which is reﬂected in the deﬁnition of decomposability itself:
A HIT attribute A is said to be decomposable into the set of HIT attributes
{B1, . . . , Bn}, when the following conditions hold:
1. every Bi, 1 ≤ i ≤ n is a subattribute of A,
15
This deﬁnition of subattribute corresponds to the notion of proper subattribute deﬁned
in the lectures. According to the HIT method, the deﬁnition of subattribute allows A to
be its own subattribute. For simplicity, we do not consider this possibility.
Service Systems Laboratory, FI MU Brno 18
2. A ← {B1, . . . , Bn}.
It is denoted as A ♦ {B1, . . . , Bn}.
Therefore, we can say that Enrolls ♦ {Teaches, Enrolls }.
2.5.3 The kernel of a set of HIT attributes
The kernel of a set of HIT attributes can be imagined as the normalized variant
of the original set. Thus the whole normalization step of the modelling
process can be rephrased as ﬁnding the kernel set of the initially identiﬁed
HIT attributes. This kernel set will be the part of the ﬁnal conceptual model.
Let K and A be sets of HIT attributes. The set K is called the kernel of
the set A, iﬀ the following conditions hold:
1. K and A are informationally equivalent; this means that every HIT
attribute from K is deﬁnable over A and every HIT attribute from A
is deﬁnable over K,
2. all HIT attributes in K are elementary; this means no HIT attribute
from K can be decomposed,
3. K does not contain any redundant HIT attributes; this means that
there is no HIT attribute A ∈ K such that K ← K − {A }.
The above conditions can be rephrased as follows: the kernel is every
minimal set of elementary HIT attributes deﬁning exactly the same information
as the original HIT attributes set. The properties following from the three
conditions given in the deﬁnition are underlined.
Unfortunately, no algorithm which would be capable of ﬁnding the kernel
is known. Determining the kernel also requires domain knowledge to decide
deﬁnability and decomposition of HIT attributes.
2.6 HIT conceptual model
Having found the kernel set of HIT attributes, we have constructed the HIT
conceptual model. It is deﬁned as the triple (B, K, C), where:
• B is the base of sorts that we have identiﬁed in the domain,
Service Systems Laboratory, FI MU Brno 19
• K is the kernel of the set of relevant HIT attributes identiﬁed in the
domain,
• C is the set of integrity constraints formulated over HIT attributes from
K.
Note that the model does not include any diagram. The next section describes
the procedure how to create an ER model (including an ER diagram)
according to the HIT conceptual model.
2.7 Transformation into ER model
The HIT conceptual model is transformed into the ER model notation mainly
for the following reasons:
• ER model includes ER diagram which is very useful tool for getting
quick awareness of the model,
• ER model is a well-known modelling artifact,
• ER model is often used at later phases of software design to accomodate
logical and physical data models.
Provided with the kernel set of HIT attributes, we are almost ready to
transform the HIT conceptual model into the ER model. All HIT attributes
are now either binary (of complexity 2), or cannot be further decomposed.
All HIT attributes add some information to the model, none of them can be
omitted.
The transformation task is to map the sorts and HIT attributes onto
a graph structure of nodes (entities in ER model) and their connections
(relationships in ER model). The ﬁrst intuition is straightforward: the entity
sorts are mapped onto entities and the HIT attributes are mapped onto
relationships16
.
But plain relationships can connect only two entities. This way we can
map only binary HIT attributes. The non-binary HIT attributes cannot be
decomposed because the HIT conceptual model includes only HIT attributes
from the kernel. Fortunately, the binarization principle deals with them.
16
The transformation of descriptive sorts and descriptive HIT attributes is omitted in
this ﬁrst intuition – see section 2.7.2.
Service Systems Laboratory, FI MU Brno 20
2.7.1 Binarization principle
The binarization principle transforms a non-decomposable HIT attribute
with complexity greater than 2 into one new created sort (called concatenated
type) and a set of the so-called projection HIT attributes.
Let us illustrate the binarization principle on our example HIT attribute
Grading shown in Figure 3:
Figure 3: The non-decomposable HIT attribute Grading.
First, the binarization principle instructs us to create a new sort17
called
concatenated type. It contains n-tuples corresponding to the original HIT
attribute; using the Grading HIT attribute, the concatenated type might
contain the following n-tuples:
(student A, PB114, A),
(student B, PB114, C),
. . .
where:
• student A and student B ∈ (#Student),
• PB114 ∈ (#Course),
17
More precisely a type rather than a sort – see lectures or references for the type theory
used in HIT.
Service Systems Laboratory, FI MU Brno 21
• A and C ∈ (Grade).
The concatenated type represents the original HIT attribute, therefore
the deﬁnition of concatenated type contains the semantics of original HIT
attribute in linear notation along with its cardinality.
In the example, the deﬁnition of concatenated type Grading looks like:
An object of category (#Grading) is every representation of a
relationship between (#Student) and (#Course) with the following
meaning:
Students (#Student)-s who have passed the given course (#Course)
with given grade (Grade) / 0,M:0,M
Now we have to map the n-tuples from the concatenated type onto the
referenced instances of sorts (#Student), (#Course) and (Grade). This
mapping is done by the so-called projection HIT attributes depicted in Figure
4.
Figure 4: The projections of the concatenated type onto the components of
the original HIT attribute.
Notice that there is no textual description of the semantics of the HIT
attributes – it seems as an exception to the rule that every HIT attribute
has to have its semantics deﬁned. In fact, the semantics of projection HIT
attributes is always the same – the projection of the concatenated type to
Service Systems Laboratory, FI MU Brno 22
one of its elements. The textual description could be for example “of given”
but is usually omitted.
Another consequence of the same semantics of all projection HIT attributes
is always the same cardinality of all projection attributes.
This section concludes with the illustration of binarization principle for
general HIT attributes – see Figures 5 and 6. The consequence of the existence
of the binarization principle is the ability to transform any HIT conceptual
model into a graph structure of nodes and edges which is required
by the transformation to ER model.
Figure 5: The non-decomposable HIT attribute with complexity > 2.
2.7.2 The transformation procedure
The process of transformation of the HIT conceptual model18
M = (B, K, C)
into ER model consists of the following steps19
:
• Derive entities:
1. For every HIT attribute A ∈ K having the complexity > 2 apply
the binarization principle, i.e. introduce new concatenated type
R and replace A with the set of projection HIT attributes Bi.
18
See section 2.6.
19
The precise algorithm can be found in the PB114/PA116 lectures.
Service Systems Laboratory, FI MU Brno 23
Figure 6: The resulting set of projections of the concatenated type onto the
components of the original HIT attribute.
– every HIT attribute in K is binary (its complexity = 2) now
– binarization principle does not operate with constraints, so
they need to be updated
2. Replace every constraint c ∈ C valid for A with corresponding
constraint(s) for Bi.
– the constraints match the results of binarization principle ap-
plication
3. Represent every entity sort as a kernel entity in ER model.
4. Represent every concatenated type as an associative entity in ER
model.
5. For every descriptive HIT attribute D ∈ K with a plural rotation
having a descriptive sort S as the result20
:
– represent descriptive sort S as a characteristic entity in ER
model,
– represent the HIT attribute D as a relationship in ER model.
The cardinality of the relationship is derived from the cardinality
of the HIT attribute.
20
In other words, the cardinality of this rotation is ?,M:?,?.
Service Systems Laboratory, FI MU Brno 24
• Derive relationships:
1. represent every HIT attribute in K which does not operate with
any descriptive sort as a relationship in ER model. The cardinality
of the relationship is derived from the cardinality of the HIT
attribute.
• Derive attributes (only when transforming into ERAM 21
):
1. Represent every descriptive HIT attribute D ∈ K, with a singular
rotation having a descriptive sort S as the result22
, as an ERAM
attribute of the corresponding kernel or associative entity.
• Provide deﬁnitions:
1. Write deﬁnitions of all kernel, associative and characteristic entities
by using deﬁnitions of sorts and concatenated types from
B.
2. Write deﬁnitions of all relationships by using linear notation of
HIT attributes from K.
The complete transformation procedure described in PB114/PA116 lectures
deals with the supertype-subtype hierarchy which is not covered by
this tutorial. The reason is the non-trivial amount of required theoretical
background, while the eﬀect for the practical utilization of HIT would not
be as signiﬁcant.
Let us illustrate the transformation procedure on a simple example. Imagine
the following HIT conceptual model:
The sorts:
An object of category (#Lecturer) is every person contracted
by the university as a teacher of a course.
An object of category (#Course) is every series of lectures or
lessons in a particular subject oﬀered by the university.
21
The transformation procedure can be used for transformation into EntityRelationship-Attribute
Model. This model type is often used for logical or physical data
models.
22
In other words, the cardinality of this rotation is ?,1:?,?.
Service Systems Laboratory, FI MU Brno 25
An object of category (#Student) is every person currently enrolled
in a university study programme.
An element of category (Grade) is every character from the following
list: A, B, C, D, E, F.
The kernel set of HIT attributes:
Teaches: Lecturer (#Lecturer) teaching a given course (#Course)
/ 0,1:0,M
Enrolls: Students (#Student)-s who have enrolled in a given
course (#Course) / 0,M:0,M
Grading: Students (#Student)-s who have passed the given
course (#Course) with given grade (Grade) / 0,M:0,M
The set of integrity constraints is empty.
First, we apply the binarization principle to the Grading HIT attribute
because it is the only HIT attribute with complexity greater than 2. Next,
we transform the entity sorts into ER model entities, and the HIT attributes
into the corresponding relationships.
The ﬁnal model consists of the ER diagram (see Figure 7) and the following
textual description. Notice that the projection HIT attributes (obtained
from the binarization principle application) are not part of the textual de-
scription:
Entities:
Name: Lecturer
Type: kernel
An object of category (#Lecturer) is every person contracted
by some faculty department to give lectures to students.
Name: Course
Type: kernel
An object of category (#Course) is every series of lectures or
lessons on particular subject approved by the faculty scientiﬁc
committee for being taught at the faculty.
Service Systems Laboratory, FI MU Brno 26
Name: Student
Type: kernel
An object of category (#Student) is every person signed for
study in a bachelor’s, master’s or doctoral study programme. The
study must not be currently interrupted.
Name: Grading
Type: associative
An object of category (#Grading) is every representation of a
relationship between (#Student) and (#Course) with the following
meaning:
Students (#Student)-s who have passed the given course (#Course)
with given grade (Grade) / 0,M:0,M
Relationships:
1: Lecturer (#Lecturer) teaching a given course (#Course) /
0,1:0,M
2: Students (#Student)-s who have enrolled in a given course
(#Course) / 0,M:0,M
Figure 7: The ER diagram part of the resulting ER model.
The description of the transformation procedure concludes the ﬁrst part
of the tutorial. The next section illustrates the application of HIT method
to the construction of conceptual model of the domain of a restaurant.
Service Systems Laboratory, FI MU Brno 27