ENTERPRISE SYSTEM DEVELOPMENT WITH INVARIANT
PRESERVING
A Mathematical Approach by the Homotopy Lifting and Extension Properties
Kenji Ohmori
Faculty of Computer and Information Sciences, Hosei University, Koganei-shi,Tokyo184-8584, Japan
Tosiyasu L. Kunii
Morpho, Inc. The University of Tokyo, Entrepreneur Plaza 5F, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Keywords:
Homotopy, HLP, HEP, π-calculus, Abstraction hierarchy, Invariant preserving.
Abstract:
In this paper, a theoretical method for developing enterprise systems represented by the π-calculus is intro-
duced. The method is based on the modern mathematics of homotopy theory. The homotopy lifting and
extension properties are applied to developing systems in bottom-up and top-down ways with the incremen-
tally modular abstraction hierarchy, where system development is carried out by climbing down abstraction
hierarchy with adding invariants linearly. It leads to avoid combinatorial explosions causing an enormous
waste of time and cost on testing. The system requirements and a state transition diagram drive the actions
of an event by applying the HEP. Then, the state transition diagram and actions bring π-calculus processes by
applying the HLP. These processes do not need testing because of invariant preserving.
1 INTRODUCTION
Why are tests necessary in the development process
of enterprise systems? The Rational Unified Pro-
cess (RUP) widely used as an iterative software de-
velopment process framework includes tests as en-
gineering disciplines. The designers preferably us-
ing the RUP believe that tests are inevitable, while
knowing that much time and cost are spent on tests.
When developing systems by the RUP, system re-
quirements are modeled using the Unified Modeling
Language (UML) consisting of several kinds of dia-
grams including class, use-case and activity diagrams.
The individual-dependent and non-theoretical profes-
sional work from system requirements to UML dia-
grams leads to the necessity of tests. Activity dia-
grams and class diagrams are naturally different when
designers are different. How do you verify theoret-
ically that these different diagrams meet the system
requirements? Without showing the correctness of
modeling process, tests are required as the last tools
for verification.
Invariants and invariant preserving are the most
important concepts in science. By defining the
most abstract invariants first and adding less abstract
or more specific invariants step by step with pre-
serving the previously added invariants, the correct-
ness of modeling process is guaranteed. The incre-
mentally modular abstraction hierarchy (IMAH) has
been introduced in these papers(Kunii, 2005), (Ku-
nii and Ohmori, 2006), (Ohmori and Kunii, 2006),
(Ohmori and Kunii, 2007a), (Ohmori and Kunii,
2007b), (Ohmori and Kunii, 2008a), (Ohmori and Ku-
nii, 2008b), where the IMAH has seven abstraction
levels from the homotopy level to the set theoretical
level, topological space level, adjunction space level,
cellular space level, presentation level and view level.
Invariants are added linearly while climbing down the
abstraction hierarchy. The IMAH has been applied
for architecture and modeling of cyberworlds.
On the way of climbing down the IMAH, methods
of top-down, bottom-up or mixture of them are re-
quired for architecture and modeling of systems. The
homotopy lifting property (HLP) and homotopy ex-
tension property (HEP) give theoretical backgrounds
(Havey, 2005), (Sieradski, 1992), (Spanier, 1966) for
a top-down and bottom-up method, respectively. Dif-
ficult problems can be solved by a divide and conquer
116
Ohmori K. and L. Kunii T. (2009).
ENTERPRISE SYSTEM DEVELOPMENT WITH INVARIANT PRESERVING - A Mathematical Approach by the Homotopy Lifting and Extension
Properties.
In Proceedings of the 11th International Conference on Enterprise Information Systems - Information Systems Analysis and Specification, pages
116-123
DOI: 10.5220/0001981501160123
Copyright
c
SciTePress
method in computer science, where a difficult prob-
lem is divided into two simple problems. The HLP
consists of a total, base and lifting space, where the
total space is complicated and is obtained in a bottom-
up way from two simple spaces of the base and lifting
space. As the HEP is dual to the HLP, the HEP gives
a top-down way.
In this paper, the energy purchase problem is de-
signed from its specification to a model represented
by the π-calculus (Milner, 1999), (Sangiorgi and
Walker, 1999), (Hennessy, 2001) using the HLP and
HEP while climbing down the IMAH from the ab-
stract level of the system requirements to the spe-
cific level of a π-calculus model.@It is shown that the
most important concepts in this paper, that is, invari-
ants and invariant preserving are formally and theo-
retically described by modeling the energy purchase
problem (Havey, 2005) so that enterprise system de-
velopment is changed from a hand-made professional
job with testing to a theoretical and intelligent work
without testing.
2 MATHEMATICAL
BACKGROUNDS FOR
DEVELOPING ENTERPRISE
SYSTEMS
2.1 The Incrementally Modular
Abstraction Hierarchy
The IMAH has the following levels starting from
the most abstract to the most specific: the homo-
topy level; the set theoretical level; the topological
space level; the adjunction space level; the cellular
space level; the presentation level and the view level.
On the homotopy level, the most abstract invariants
with homotopyequivalenceare defined using the HLP
or HEP. On the set theoretical level, sets of spaces,
which are mostly discrete, are defined with logical
operations. On the topological space level, impor-
tant invariants with isomorphism is introduced. On
the adjunction level, invariants presenting the prop-
erties of dynamic changes, which are very important
characteristics for organizing the structure of infor-
mation, are defined using adjunction mapping. On
the cellular space level, abstract physical structures
such as frameworks of state transition diagrams are
represented. On the presentation level, which is the
starting point in the traditional architecture and mod-
eling, designing entities such as UML diagrams and
concrete state transition diagrams are defined. On the
view level, program codes including π-calculus pro-
cesses are obtained.
2.2 The Homotopy Lifting Property and
Homotopy Extension Property
The mathematical backgrounds for the HLP and HEP
are summarized as follows.
Def. 1: Continuous maps p, q are homotopic if there
exists a continuous map H : X × I Y such that
H(x,0) = p(x) and H(x,1) = q(x), where I is the
unit interval [0,1]. H is called homotopy of p and
q, denoted by p q.
Def. 2: A continuous map λ : I X yields a path.
λ(0) = x and λ(1) = y are called the initial and ter-
minal points. The path is denoted by w = (W,λ)
where W = λ(I).
Def. 3: A fiber bundle is a quadruple ξ = (E,B, F, p)
consisting of a total space E, a base space B, a
fiber F, and a bundle projection that is a contin-
uous surjection called Fbundle p : E B such
that there exists an open covering U = {U} of B
and, for each U U , a homeomorphism called
a coordinate chart ϕ
U
: U × F p
1
(U) exists
such that the composite U × F p
1
(U) U
is the projection to the first factor U. Thus the
bundle projection p : E B and the projection
p
B
: B × F B are locally equivalent. The fiber
over b B is defined to be equal to p
1
(b), and we
note that F is homeomorphic to p
1
(b) for every
b B, namely b B,F
=
p
1
(b).
Def. 4: Given any commutative diagram of continu-
ous maps as shown in Fig. 1, the map p : E B
has the homotopy lifting property if there is a con-
tinuous map
ˆ
H : Y × I E such that
ˆ
H × i
0
= h
and p
ˆ
H = H. The homotopy
ˆ
H thus lifts H
through p and extends h over i
0
where i
0
(a) =
(a,0).
Def. 5: A fibration is a continuous map p : E B
that has the homotopy lifting property.The homo-
topy extension property is dual to the homotopy
lifting property. The homotopy extension prop-
erty is defined as follows.
Def. 6: Given any commutative diagram of continu-
ous maps as shown in Fig. 2, there is a contin-
uous map
ˆ
K : X Y
I
such that p
0
×
ˆ
K = k and
ˆ
K × i = K. The homotopy
ˆ
K thus extends K over
i and lifts k through p
0
where p
0
(λ) = λ(0).
Def. 7: An inclusion of a closed subspace i : A ֒ X
is a cofibration if i has the homotopy extension
property. Y
I
is the path space on Y. The path
space is defined as follows.
ENTERPRISE SYSTEM DEVELOPMENT WITH INVARIANT PRESERVING - A Mathematical Approach by the
Homotopy Lifting and Extension Properties
117
Def. 8: The path space on X, denoted X
I
, is the
space {λ : I X|continuous} endowed with the
compact-open topology.
Def. 9: Let us start with a topological space X and
attach another topological space Y to it. Then,
Y
f
= Y
f
X = Y X/ is an adjunction space ob-
tained by adjunctingY to X by an adjunction map,
f (or by identifying each point y Y
0
| Y
0
Y with
its image f(y) X by a continuous map f).
denotes a disjoint union. The adjuncting map f
is a continuous map such that f : Y
0
X, where
Y
0
Y. Thus, the adjuncting space Y
f
= Y X/
is a case of quotient spaces
Y X/ = Y
f
X = Y X/(x f(y) | y Y
0
).
E
B
Y × I
p
H
ˆ
H
Y
i
0
h
Figure 1: Homotopy Lifting Property.
Y
I
Y
p
0
k
ˆ
K
A
i
K
X
Figure 2: Homotopy Extension Property.
3 APPLYING THE HOMOTOPY
THEORY TO AN ENERGY
PURCHASE PROBLEM
3.1 An Energy Purchase Problem
The energy purchase@problem is shown in the book
(Havey, 2005). The outline of this problem is as fol-
lows. A customer is registerd to an energy supplier.
The customer ordinarily buys energy from the energy
supplier. Because of deregulation, the customer is al-
lowed to buy energy from a retailer. As there are sev-
eral retailers, the custome is also allowed to switch
retailers. When the customer wants to switch retail-
ers, he has to notify the energy supplier of a retailer
switch and the energy supplier reports this switch to
the retailers. The customer also switches back to the
energy supplier
3.2 The π-calculus Process
The π-calculus is a parallel processing model where
communication links are dynamically changed. The
action prefixes π are a generalization of actions. An
action prefix expresses either sending or receiving a
message or making a silent transition. The π-prefixes
are defined as follows.
π ::= x(y) receive y along x
::= x(y) send y along x
::= τ unobservable action
The set P
π
of π-calculus process expressions is de-
fined as follows:
P ::= 0 |
λΛ
π
λ
.P
λ
| P
1
|P
2
| new a P | !P
0 is an inaction process that can do nothing. The pro-
cesses
λΛ
π
λ
.P
λ
are called sums. Each item is a pro-
cess and only one item is executed. Λ is any finite
indexing set. In a sum
λΛ
π
λ
.P
λ
, it is said that P
λ
is
guarded by π
λ
since the action by π
λ
has to proceed
before P
λ
becomes active. P
1
|P
2
can proceed indepen-
dently and interact by shared names. new a P restricts
the scope of the name a to P. !P repeats P infinitely.
Two examples are given for explaining the π-
calculus. The first example x(y).P
1
|x(z).P
2
shows
how a message is sent via a communication link.
x(y) of an action prefix π can receive any name and
y is bounded to the name or substituted by the re-
ceived name. x(y).P
1
|x(z).P
2
executes two processes.
x(y).P
1
, where x is used as a communication link,
sends y via x. x(z).P
2
receives y via x where z is sub-
stituted by y.
The second example x(y).P
1
|x(z).z(u).P
2
|y(w).P
3
shows how a communication link is dynamically
changed. The first process notifies the second process
of a communication link name, which will be used by
the second process for communicating with the third
process. That is, x(y).P
1
and x(z).z(u).P
2
changes a
message like the above explanation. x(z).z(u).P
2
re-
ceives y via x, it sends u via y since z is bounded to y.
Finally, y(w).P
3
receives u via y.
ICEIS 2009 - International Conference on Enterprise Information Systems
118
3.3 A Hep Application
The energy purchasing problem is solved by the fol-
lowing steps.
1) A state transition diagram is obtained by climb-
ing down from the homotopy level to the presentation
level.
2) Using the HEP, the actions for an event caus-
ing a situation change are obtained from the system
requirements and the state transition diagram. The
actions are also generated by climbing down the ab-
straction hierarchy.
3) Using the HLP, the processes for the energy
purchase problem are obtained.
At first, the system requirements are formed as a
topological space. Assuming that there are one cus-
tomer, one energy supplier and two suppliers without
losing generality, the system requirements consist of
the following elements.
x
1
: The customer ordinarily buys energy from the
energy supplier.
x
2
: The customer can buy energy from a retailer.
x
3
: When the customer changes his purchasing
place from the supplier to a retailer, he has to notify
the supplier of his enrollment.
x
4
: When the customer changes his purchasing
place from a retailer to another retailer, he has to no-
tify the supplier of his switch.
x
5
: When a customer changes his purchasing
place from a retailer to the supplier, he has to notify
the supplier of his drop.
The set of these elements forms a topological
space by introducing discrete topology.
A state transition diagram is constructed by ex-
tracting states and transitions from the system require-
ments. This process is achieved by providing two ab-
stract states, e
0
1
for purchase on standard supply and
e
0
2
for purchase by taking advantage of deregulation
and three state transitions, e
1
1
from e
0
1
to e
0
2
caused by
an enrollment, e
1
2
from e
0
2
to e
0
1
by a drop and e
1
3
from
e
1
2
to e
0
1
by a switch. As states and transitions are or-
dinary represented as vertexes and edges, e
0
1
and e
0
2
are defined as 0-dimensional spaces and e
1
1
,e
2
1
and e
3
2
as 1-dimensional spaces. The set of {e
0
1
,e
0
2
,e
1
1
,e
1
2
,e
1
3
}
is formed as a topological space by introducing dis-
crete topology. States and transitions are connected to
form a state transition diagram by applying adjunct-
ing maps. For this problem, both boundaries of e
1
1
are attached to e
0
1
and e
0
2
. In the same way, the bound-
aries of other state transitions are attached to the cor-
responding states.
To form the state transition diagram as a CW-
complex, two 2-dimentional spaces: e
2
1
and e
2
2
, are
provided and these boundaries are attached to the cor-
responding edges. The CW complex is further trans-
formed to a more specific state transition diagram
on the presentation level as shown in Fig. 3. The
state transition diagram is a subset of the system re-
quirements where e
0
1
,e
0
2
,e
1
1
,e
1
2
and e
1
3
correspond to
x
1
,x
2
,x
3
,x
5
and x
4
, respectively.
Enroll
Drop
Switch
Supplier
Retailer
e
0
1
e
0
2
e
0
3
e
1
1
e
1
2
e
1
3
e
2
1
e
2
2
The cellular space level
The presentation level
Figure 3: State transition diagram.
The energy purchasing problem has three situations:
Z
1
for purchasing from the energy supplier,Z
2
for pur-
chasing from retailer A and Z
3
for purchasing from
retailer B. Sometimes, a situation is called a use-
case. As the customer is allowed to change a pur-
chasing place, the sequence of situations generated by
changing purchase places is represented in the form of
(Z
1
(Z
2
|Z
3
)
)
. One example sequence is shown from
Fig. 4 to Fig. 7. The change of a purchasing place
is caused by events such as an enrollment, switch and
drop. Each of them has a series of actions.
Now, let’s consider obtaining actions in an event
using the HEP. Later, the actions are transformed into
π-processes. In the HEP, X is the system requirements
and A is the state transition diagram. Y is a series of
actions for an event. There are three events: enroll-
ment E
1
, switch E
2
and drop E
3
. E
1
, E
2
and E
3
cause
transitions of Z
1
Z
2
|Z
1
Z
3
, Z
2
Z
3
|Z
3
Z
2
and Z
2
Z
1
|Z
3
Z
1
, respectively. Each event has a
series of actions such that Y
1
= Y × E
1
has {1) Mike
notifies Sup of the purchase enrollment to RetA, 2)
Sup receives from Mike the purchase enrollment to
RetA, 3) Sup reports to RetA the purchase enroll-
ment of Mike, 4) RetA receives from Sup the pur-
ENTERPRISE SYSTEM DEVELOPMENT WITH INVARIANT PRESERVING - A Mathematical Approach by the
Homotopy Lifting and Extension Properties
119
Retailer A
Retailer B
Supplier
Customer
Standard supply
Figure 4: Customer is not enrolled with a retailer, on stan-
dard supply.
Retailer A
Retailer B
Supplier
Customer
Enroll A
Figure 5: Customer enrolls with retailer A.
chase enrollment of Mike} where the customer, the
energy supplier and two retailers are represented by
Mike,Sup,RetA and RetB. Actions are obtained from
the system specification. In the above case, x
3
defines
its actions.
For each event, the following actions are obtained.
1. Event: Y
1
(Purchase change from the energy
supplier to a retailer)
1) Sub event: Y
11
(Change from Sup to RetA)
y
1
11
: Mike notifies Sup of a purchase enrollment
to RetA.
y
2
11
: Sup receives from Mike a purchase enroll-
ment to RetA.
Retailer A
Retailer B
Supplier
Customer
Switch B
Figure 6: Customer switches to retailer B.
Retailer A
Retailer B
Supplier
Customer
Drop B, back to
Supplier
Figure 7: Customer drops retailer, back to standard supply.
y
3
11
: Sup reports to RetA a purchase enrollment of
Mike.
y
4
11
: RetA receives from Sup a purchase enroll-
ment of Mike.
2) Sub event: Y
12
(Change from Sup to RetB)
y
1
12
: Mike notifies Sup of a purchase enrollment
to RetB.
y
2
12
: Sup receives from Mike a purchase enroll-
ment to RetB.
y
3
12
: Sup reports to RetB a purchase enrollment of
Mike.
y
4
12
: RetB receives from Sup a purchase enroll-
ment of Mike.
ICEIS 2009 - International Conference on Enterprise Information Systems
120
2. Event: Y
2
(Purchase change among retailers)
1) Sub event: Y
21
(Change from RetA to RetB)
y
1
21
: Mike notifies Sup of a purchase switch from
RetA to RetB.
y
2
21
: Sup receives from Mike a purchase switch
from RetA to RetB.
y
3
21
: Sup reports to RetA a purchase drop of Mike.
y
4
21
: Sup reports to RetB a purchase enrollment of
Mike.
y
5
21
: RetA receives from Sup a purchase drop of
Mike.
y
6
21
: RetB receives from Sup a purchase enroll-
ment of Mike.
2) Sub event: Y
22
(Change from RetB to RetA)
y
1
22
: Mike notifies Sup of a purchase switch from
RetB to RetA.
y
2
22
: Sup receives from Mike a purchase switch
from RetB to RetA.
y
3
22
: Sup reports to RetB a purchase drop of Mike.
y
4
22
: Sup reports to RetA a purchase enrollment of
Mike.
y
5
22
: RetB receives from Sup a purchase drop of
Mike.
y
6
22
: RetA receives from Sup a purchase enroll-
ment of Mike.
3. Event: Y
3
(Purchase change from the retailer to
the energy supplier)
1) Sub event: Y
31
(Change from RetA to Sup)
y
1
31
: Mike notifies Sup of a purchase drop out of
RetA.
y
2
31
: Sup receives from Mike a purchase drop out
of RetA.
y
3
31
: Sup reports to RetA a purchase drop of Mike.
y
4
31
: RetA receives from Sup a purchase drop of
Mike.
2) Sub event: Y
32
(Change from RetB to Sup)
y
1
32
: Mike notifies Sup of a purchase drop out of
RetB.
y
2
32
: Sup receives from Mike a purchase drop out
of RetB.
y
3
32
: Sup reports toRetB a purchase drop of Mike.
y
4
32
: RetB receives from Sup a purchase drop of
Mike.
Let’s consider constructing Y
I
. The interval I is rep-
resented by (E
1
E
2
E
3
)
. Y
I
is a set of paths, each of
which is the sequence of the i
th
action along I such
that
Y
I
= { ((y
1
11
|y
1
12
)(y
1
21
|y
1
22
)
(y
1
31
|y
1
32
))
,
((y
2
11
|y
2
12
)(y
2
21
|y
2
22
)
((y
2
31
|y
2
32
))
,...}.
However, a path yielded by Y
I
is not allowed to in-
clude y
k
1m
y
k
2n
and y
k
2m
y
k
2n
where m 6= n, and y
k
1m
y
k
3n
and
y
k
2m
y
k
3n
where m = n as it part.
It is not hard to show that A is included in X, K
and
ˆ
K are homotopies and
ˆ
K extends K over i.
3.4 A HLP Application
Now, consider obtaining π-calculus processes of the
energy purchase problem using the HLP where B and
Y × I are described as a state transition diagram and
events, and π-calculus processes are accommodated
in E. E consists of layers for agents in the energy pur-
chasing problem, where the customer, supplier and
customers are agents. When solving this problem, the
agent for the customer is categorized into two types:
the customer purchasing on standard supply and the
customer purchasing from a retailer. The obtained
processes are shown from Fig. 8 to Fig. 11, where
there are four processes: P
1
for the supplier, P
2
for re-
tailers, P
3
for the customer on standard supply and P
4
for the customer purchasing from retailer. Each pro-
cess is represented in the same CW-complex on the
cellular space level as the state transition diagram.
P
1
= enroll(ret,cus)ret(Add,cus).P
1
ret2(Add,cus).P
1
P
1
= drop(ret,cus)ret(Drop,cus).P
1
ret1(Drop,cus)
P
1
= switch(ret1,ret2, cus)
Supplier
Retailer
Figure 8: Process for the supplier.
P
2
= ret(mesg,cus).P
2
P
2
= ret(mesg,cus).P
2
P
2
= ret(mesg,cus).P
2
Supplier
Retailer
Figure 9: Process for the retailers.
ENTERPRISE SYSTEM DEVELOPMENT WITH INVARIANT PRESERVING - A Mathematical Approach by the
Homotopy Lifting and Extension Properties
121
+ebroll(RetB, Mike).P
4
(RetB)
P
3
= enroll(RetA,Mike).P
4
(RetA)
Supplier
Retailer
Figure 10: Process for the customer on standard supply.
P
4
(RetB) =
P
4
(RetA) = drop(RetA,Mike).P
3
P
4
(RetB) = drop(RetB,Mike).P
3
P
4
(RetA) =
switch(RetA,RetB,Mike).P
4
(RetB)
switch(RetB,RetA,Mike).P
4
(RetA)
Supplier
Retailer
Figure 11: Process for the customer purchasing from a re-
tailer.
However, these are different on the presentation level.
Arcs of the presentation level are obtained from the
events as follows. Firstly, Y
1
is mapped into these
state diagrams. As Y
1
has two subspaces Y
11
and
Y
12
, both of them are mapped step by step in the
following way. y
1
11
that is an element of Y
11
is
mapped as a process enroll(RetA,Mike). P4(RetA)
for the customer purchasing on standard sup-
ply. y
2
11
and y
3
11
are mapped as a process
enroll(ret,cus).ret(Add,cus). P1 for the supplier. y
4
11
is mapped as a process ret(message, cus). P2(ret) for
the retailer. Other mappings are carried out in the
same way for all events. It is not also hard to prove
that H and
ˆ
H are homotopies and
ˆ
H lifts H through
p.
4 REFINEMENT STEPS
A state diagram is a connected CW-complex that is
homotopy equivalent to a point. If a state transition
diagram is reduced to a point, then the processes dis-
tributed on arcs are assembled into a single expres-
sion. The processes for the supplier, retailers, cus-
tomer on standard supply and customer purchasing
from a retailer are obtained by the following expres-
sions.
P
1
= enroll(ret,cus)ret(Add,cus)
+switch(ret1,ret2, cus))
ret1(Drop,cus)ret2(Add,cus)
+drop(ret,cus)ret(Drop, cus)).P
1
.
P
2
= ret(mesg,cus).P
2
.
P
3
=
ret∈{RetA,RetB}
enroll(ret,Mike).P
4
(ret).
P
4
(ret1) =
ret2∈{RetA,RetB}∩(ret16=ret2)
switch(ret1,ret2, Mike).P
4
(ret2)
+drop(ret1,Mike).P
3
where ret1 {RetA,RetB}.
The generalized expression is also obtained using the
HLP. When the problem is extended to have several
customers and retailers, the above processes are gen-
eralized from the HLP as follows, where Cus and Ret
are the sets of the customers and retailers and cus and
ret1 are elements of Cus and Ret, respectively.
P
3
(cus) =
retRet
enroll(ret, cus).P
4
(cus,ret).
P
4
(cus,ret1) =
ret2Ret(ret16=ret2)
switch(ret1,ret2, cus).P
4
(cus,ret2)
+drop(ret1,cus).P
3
(cus).
5 CONCLUSIONS
A new development method using homotopy theory
has been introduced. The HEP and HLP are the most
abstract and powerful tools for categorizing mathe-
matical objects. Homotopy equivalence are main-
tained within the HEP and HLP. In this paper, the
HEP and HLP have been introduced for establishing
fundamentals in software science. Architecture, mod-
eling and designing are performed by climbing down
the abstraction hierarchy adding invariants incremen-
tally using the HEP and HLP. The IMAH has 7 lev-
els from the homotopy space level to the set theoret-
ical level, the topological space level, the adjunction
level, the cellular space level, presentation level and
the view level. The system requirements for the en-
ergy purchase problem on the set theoretical level is
transformed to the π-calculus processes on the view
ICEIS 2009 - International Conference on Enterprise Information Systems
122
level using the HEP and HLP. While climbing down
the abstraction hierarchy, invariants are incrementally
added from abstract to specific ones while keeping
homotopy equivalence on all levels and topological
equivalence on less abstract levels than topological
space level. The invariant preserving by homotopy
equivalence and topological equivalence makes test
unnecessary and enables to avoid combinatorial ex-
plosions that the conventional method is facing now.
The HEP and HLP have been introduced for a
top-down and bottom-up design, respectively. This
is a new application field of the HEP and HLP. When
solving the energy purchase problem, the state transi-
tion diagram and events, which are specific modeling
and designing entities from the system requirements
on the set theoretical level, have been obtained using
the HEP in a top-downway, where the system require-
ments, state transition diagram and events are homo-
topy equivalent. Then, the π-calculus processes on
the view level have been obtained from the state tran-
sition diagram and events using the HLP in a bottom-
up way. The introduced method gives a mathemati-
cal foundation for a top-down and bottom-up design,
which are carried out by individual know-how in the
conventional system. The top-down and bottom-up
design also keeps invariants with making tests neces-
sary.
The development steps introduced here start from
the system requirements, move to obtain the state
transition diagram and actions for events and end with
the π-calculus processes. These operations are au-
tomatic. By developing a development framework
along these steps, the development of enterprise sys-
tems will be automated from the system requirements
to its BPEL codes.
This paper has constructed a bridge between com-
puter science and the modern mathematics of homo-
topy theory. The introduced method is general and
applicable to any fields of computer science includ-
ing computer graphics, computer architecture, net-
work architecture and cyberworlds. In this paper, the
π-calculus is used to express processes for enterprise
systems. It is easily applicable to communicating se-
quential processes, which is another process algebra
and powerful tools in the field of embedded systems.
REFERENCES
Havey, M. (2005). Essential Business Process Modeling.
O’Reilly Media, Inc, Cambridge.
Hennessy, M. (2001). A Distributed Pi-Calculus. Cam-
bridge University Press, Cambridge.
Kunii, T. L. (2005). Cyberworlds -theory, design and
potetial-. The Transactions of The Institute of Elec-
tronics, Information and Communication Engineers,
E88-D(5):790–800.
Kunii, T. L. and Ohmori, K. (2006). Cyberworlds: Ar-
chitecture and modeling by an incrementally mod-
ular abstraction hierarchy. The Visual Computer,
22(12):949–964.
Milner, R. (1999). Communicating And Mobile Systems:
Pi-Calculus. Cambridge University Press, Cam-
bridge.
Ohmori, K. and Kunii, T. L. (2006). An incrementally mod-
ular abstraction hierarchy for linear software develop-
ment methodology. Int. Conf. on Cyberworlds 2006,
pages 216–223.
Ohmori, K. and Kunii, T. L. (2007a). Development of an
accounting system. ICEIS2007, pages 437–444.
Ohmori, K. and Kunii, T. L. (2007b). The mathematical
structure of cyberworlds. Int. Conf. on Cyberworlds
2007, pages 100–107.
Ohmori, K. and Kunii, T. L. (2008a). Mathematical model-
ing of ubiquitous systems. Int. Conf. on Cyberworlds
2008, pages 69–74.
Ohmori, K. and Kunii, T. L. (2008b). A pi-calculus model-
ing method for cyberworlds systems using the duality
between a fibration and a cofibration. Int. Conf. on
Cyberworlds 2008, pages 363–370.
Sangiorgi, D. and Walker, D. (1999). The Pi-Calculus: A
Theory of Mobile Processes. Cambridge University
Press, Cambridge.
Sieradski, A. J. (1992). An introduction to topology and
homotopy. PWS-Kent Publishing Company, Boston.
Spanier, E. H. (1966). Algebraic topology. Springer-Verlag,
New York.
ENTERPRISE SYSTEM DEVELOPMENT WITH INVARIANT PRESERVING - A Mathematical Approach by the
Homotopy Lifting and Extension Properties
123