Part
1
5
Theoretical
a
pproaches in
i
nformation
s
cience
Information science has for a long time been drawing on the knowledge
produced
in
psychology and related fields. This is reasonable, for the ce
n
tral issue in information science concerns individual use
rs navigating i
n
formation spaces such as libraries, databases, and the Internet. Because i
n
formation science is about the definition and location of information,
information seeking is the fundamental problem in information science,
while other problems, s
uch as document representation, are subordinate. In
this part we present some theoretical approaches in the information sc
i
ence. The papers cover the complexity of the auxetic systems, new verif
i
cation of reactive requirement for Lyee method, a tree deriva
tion proc
e
dure for multivalent and paraconsistent inference, some results about
feedback stabiliz
a
tion of matrix, Maxwell geometric dynamics, fractional
right ideals, concurrent password recovery of archived files, solutions for
an integral

differential eq
uation
,
development of genetic a
l
gorithms and C

method for optimizing a scattering by rough surface
.
Chapter 1
On t
he complexity of the auxetic systems
Ligia
M
unteanu, D
an
D
umitriu
,
Ş
tefania D
onescu
,
V
eturia
C
hiroiu
C
ontinuum Mechanics Departmen
t, Institute of Solid Mechanics,
Ctin Mille 15, Bucharest 010141
Mathematics Department, Technical University of Civil Engineering,
Bd. Lacul Tei nr.122

124, B
u
charest 020396
Abstract.
Two major levels of complexity are discussed in a way of
und
erstanding the structure and processes that define an auxetic sy
s
tem. The auxeticity and structural complexity is interpreted in the
light of Cosserat elasticity which admits degrees of freedom not pr
e
sent in classical elasticity, i.e. the rotation of poin
ts in the material,
and a couple per unit area or the couple stress. The Young’modulus
evaluation for a laminated periodic system made up of alternating
aluminum and an auxetic material is an example of computing co
m
plexity.
Keywords.
Auxetic material, Cos
serat elasticity, Young’s modulus,
Neg
a
tive Poisson’s ratio, Homogenisation technique
1.1
Introduction
Materials with a negative Poisson ratio are auxetic materials. The term
auxetic com
es
from the Greek word
auxetos
, meaning that
which may be
increase
. In
stead of getting thinner like an elongated elastic band, the au
x
etic material grows fatter, expanding laterally when stretched. All the m
a
jor classes of materials (polymers, composites, metals, ceramics hone
y
comb structures, reticulated metal foams, re

ent
rant structures, the skin
covering a cow’s teats, certain rocks and minerals, living bone tissue) can
2
1
2
1
1
1
1544
Ligia Munteanu, Dan Dumitriu, Ştefania Donescu, Veturia Chiroiu
exist in the auxetic form
.
Love [1] presents an example of cubic single
crystal pyrite as having a Poisson's ratio of

0.14, and he suggests the e
f
fect ma
y result from a twinned crystal
.
An
auxetic system
is composed
from different materials with different properties with a new mechanical
architecture based on tailored
properties
[2

5]
.
T
he idea is to transform a
non

auxetic material into auxetic forms as f
oams or cellular materials, or
to employ new architecture
techniques for
auxetic materials.
The class
i
cal
mechanics fail in describing the behavior of the auxetic material, b
e
cause
the chiral effects imply a fifth rank modulus tensor, which changes under
a
n inversion. The chirality effects imply the first major level of comple
x
i
ty, i.e.
the structural complexity
. This structural complexity is described by
Cosserat elasticity [
6
]
.
Phenomena associated with micropolar elasticity
are likely to be of larger mag
nitude, and therefore of greater interest in m
a
terials such as auxetic systems with larger scale structural features [
7

9]
.
The second level of complexity refers to
the computing complexity
. The
e
s
timation of the macroscopic Young modulus for a laminated p
eriodic
structure made up of alternating aluminum and an auxetic material layers,
by using the homogenization technique, is an example of the computing
complexity
.
1.2
Structural complexity
Consider a chiral Cosserat medium, in a Cartesian coordinates sy
s
tem
(,,)
x y z
. The equations of motion for the case without body forces and
body couples are [1
0

12]
,
0,
kl k l
u
,
0.
rk r klr lr k
m j
(1.1)
Here
kl
is the stress tensor,
kl
m
is
the couple stress tensor,
u
is the
displacement vector,
k
is the microrotation vector which in Cosserat
ela
s
ticity is
k
inematically distinct from the macrorot
a
tion vector
,
1/2
k klm m l
r u
, and
klm
is the permutation symbol.
Note
that
k
r
e
fers to
the rotation of points themselves, while
k
r
refers to the rotation assoc
i
ated
with
the
movement of nearby points. In (
1.
1)
is the mass density and
j
the microinertia. The constitutive equations are
1,2,3,
(2 ) ( )
,
kl rr kl kl klm m m
r r kl k l l k
e e r
C C C
(1.2)
On the complexity of the auxetic systems
1545
,,,1
2 3 3 2
( ) ( ) ( ),
kl r r kl k l l k rr kl
kl klm m m
m C e
C C e C C r
(1.3)
where
,,
1/2( )
kl k l l k
e u u
is the macrostrain vector
,
and
are Lamé
elastic constants,
is the Cosserat
rotation modulus,
,,
, the
Co
s
serat
rotation gradient moduli, and
,1,2,3
i
C i
,
are the chiral ela
s
tic constants associated with noncentrosy
m
metry. For
0
i
C
the equations
of isotropic micro
polar elasticity are reco
v
ered. For
0
,
(
1.2
)
is
reduce
d
to the constitutive equations of classical isotropic linear
elasticity theory
[
1
3

1
5
]. Consider the case of the laminated plates made
up of a p
e
riodic layering of sheets norm
al to the direction
x
of wave pro
p
agation, each of elastic material with constant properties. For simplicity,
the pa
r
ticular 2D case in which all quantities depend only on
x
and
z
is
consi
d
ered.
Let
F
{,,,,,1,2,3}
kl kl k k
m u k l
be a set composed of
the asymme
t
ric tensors
kl
,
kl
m
,
,1,2,3
k l
, and the vectors
k
u
,
k
. We
call
F
an
elastodynamic state
on the bounded medium, if it satisfies (
1.
1)

(
1.3
)
and appropriate initial conditions
.
The
re is
one

by

one transfo
r
mation
, which
transforms the ela
s
todynamic state
F
into another elast
o
dynamic state
ˆ
F=
ˆ
ˆ ˆ ˆ
{,,,}
kl kl k k
m u
,
,1,2,3
k l
,
composed by
the sy
m
me
t
ric tensors
ˆ
kl
,
ˆ
kl
m
,
,1,2,3
k l
, and
by the vectors
ˆ
k
u
,
ˆ
k
, satisf
ying
(
1.
1)

(
1.3
).
The state
ˆ
F=
can be decomp
o
sed in
1 2
ˆ ˆ ˆ
F F F
, where
1 11 13 33 22 1 3 2
ˆ
ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
{,,,,,,},
m u u
F
2 22 11 13 33 2 1 3
ˆ
ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
{,,,,,,}.
m m m u
F
After a proper combination, the following equations in
1 2 3
ˆ ˆ ˆ ˆ
(,,)
u u u u
and
1 2 3
ˆ ˆ ˆ ˆ
(,,)
are found
2 2
0 0
ˆ
ˆ ˆ ˆ
( 2 ) ( ) (1 ),
u K u K u
(1.4)
2 2 2
0 0 0
ˆ ˆ ˆ ˆ
ˆ
( ) (1 ) 2 (1 ),
K K u K j
(1.5)
with
the
coupling coefficient
0
K
d
efined as
2
2
1 2 3
0
( )
1
( 2 )( )
C C C
K
.
(1.6)
1546
Ligia Munteanu, Dan Dumitriu, Ştefania Donescu, Veturia Chiroiu
One can
see that (
1.4) and (1.5
) are decoupled into two sets of equations
in
1
ˆ
F
and
respectively
2
ˆ
F
.
W
e will concentrate o
nly on
the set of equ
a
tions
correspon
d
ing to
1
ˆ
F
, the other set being solved in a similar way.
1.3
Computing complexity
To solve the equations (
1.4
)

(
1.6
)
with proper initial conditions
,
let us a
p
ply
the Laplace and Fourier
transforms
, as follows
*
2
4
1 1 3 2
2 2 2
i (1 )
s
b
s
a
v v v
a a a
,
* 2
3 3 4 2 1
i i (1 )
s
v c v s a v
,
2
2
1 2
s
p
b
s s
,
2
2 2
1 2
s
p
c a
s s
,
2 2
2
2
1 0
2 2
0 4
2 (1 )
s
c K
d
K s
.
The
eigenvalue problem is obtained by ta
k
ing the solutions of the form
(,,) (,)exp( ),
W z p X p qz
(1.7)
where
1 3 2
(,,) {,,}
W z p v v
. The characteristic equation b
e
comes
3 2
1 2 3
0
q q q
.
(1.8)
The r
oots of (
1.8
) are
i
q
,
1,2,3
i
, with real parts
positive. The eige
n
vector
(,)
X p
is
1
(,)
i i
i i
a q
X p b
,
2
2
(,)
i i
i i i
i
a q
X p bq
q
,
1,2,3
i
.
Consequently
, the solution
s
(
1.7
)
can be written as
3
1
(,,) (,)exp( (,) )
i i i
i
W z p B X p q p z
,
where
i
B
,
1,2,3
i
, are arbitrary constants.
On the complexity of the auxetic syst
ems
1547
1.4
C
ase study and conclusions
Let us
consider
a laminated 2D composite plate which occupies the r
e
gion
[0,]
x L
,
[,]
z c c
, and made up
by
alternating the
N
aluminum and
auxetic material layers, no
rmal to the direction
x
of wave propagation
(
F
ig
ure 1.1
). The layers are parallel, planar
and
periodic
, t
he displacements
being
continuous
across them
. The length of each layer is
l
. The interfaces
between l
ayers are located at
nl
,
1,2,...,
n N
, each joint having two faces
ident
i
fied by
and
.
The
coordinates
are chosen
so that the waves lie in
the
(,)
x z
plane. The plate is assumed to support
in plane strains and
waves
running in the
x

direction. The Bécus homogenization techniq
ue
i
s
a
p
plied via mu
l
tiple scale expansion [
16].
We are interested in knowing the influence
of
the
Cosserat
rotation
modulus
, the Cosserat
rotation gradient moduli
,,
, and the ch
i
r
al elastic constants
,1,2,3
i
C i
, on the effective Young’ modulus value
of the laminated plate. T
he material constants for this laminated composite
are periodic functions of
x
, i.e.
( ) ( )
C x P C x
, where
the period
P
equals to the length of the basic cell for the composite
2
P l
, with
2
l
the
period represented by the length of the basic cell for the compo
s
ite.
Fig.
1.1
.
The composite plate
The Bécus homogenization via multiple scale expa
n
sion
yields
( )
E F C E
,
2
1
2
E p
,
2
2
0
2
( 1)
aux
p
K
,
2
0
1
K
,
1548
Ligia Munteanu, Dan Dumitriu, Ştefania Donescu, Veturia Chiroiu
(2 )(3 2 )
(2 2 )
aux
aux
,
2
1 2 3
( )
( 2 )( )
aux aux aux
aux aux aux aux
C C C
.
Here
al
C
are the aluminum constants and
aux
C
the auxetic consta
nts.
The function
( )
F C
is numer
i
cally determined only
. In the simulation, the
Young’s moduli of aluminum and respe
c
tively of the auxetic material are
109 GPa, and 1.55 GPa. We observe that the Young’s modulus is increa
s
ing with respe
ct to
the volume fraction
from 2 GPa to about 140 GPa,
having a maximum value for
0.75. For
above this value, the
Young’s modulus is decreasing with respect to
from 1
40
GPa to about
90 GPa.
Acknowledgement
.
The authors acknowledge the financial support of the
National University Research Council (NURC

CNCSIS) Romania, Grant
nr. 55/2007 and Postdoctoral CEEX Grant nr. 1531/2006.
References
1
.
Love
A
EH (1926)
A treatise on the mathematical theory of elasticity
. 4
th
ed.,
Dover, New York
2
.
Lakes R
S (1991)
Experimental micro mechanics methods for conventional
and negative Poisson's ratio cellular solids as Cosserat continua
. J. Enginee
r
ing Mat
e
rials and
Technology 113:148
–
155
3
.
Lakes R
S (1987)
Foam structures with a negative Poisson's ratio
.
Science
235
:1038
–
1040
4
.
Lakes
RS (1986)
Experimental microelasticity of two porous solids
. Int. J.
Solids, Structures 22:55
–
63
5.
Chiroiu
V (2004)
Identification a
nd inverse problems related to material pro
p
erties and behaviour
. In:
Chiroiu V, Sireteanu T
(eds
)
Topics in Applied M
e
chanics. Ed. Academiei Buch
a
rest,
1:83
–
126
6.
Cosserat E and F (1909) Theorie des Corps Deformables
. Hermann et Fils,
Paris
7.
Hlavacek
M
(1975)
A continuum theory for fibre reinforced composites
. Int.
J. Solids and Structures 11:199
–
211
8.
Hlavacek
M (1975)
On the effective moduli of elastic composite materials
.
Int. J. Solids and Structures 12:655
–
670
9.
Berglund
K (1982)
Structural model
s of micropolar media
. Mechanics of M
i
cropolar Media, World Scientific, Singapore
On the complexity of the auxetic systems
1549
1
0.
Eringen
AC (1968)
Theory of micropolar elasticity
.
In: Liebowitz
R (ed)
Fra
c
ture. Academic Press, 2:621
–
729
1
1.
Mindlin
RD (1964)
Microstructure in linear elasticity
. Arc
h. Rat. Mech. Anal.
16:51
–
78
1
2.
Mindlin
RD (1965)
Stress functions for a Cosserat continuum
, Int J. Solids
Structures
1
:
265
–
271
1
3.
Gauthier
RD (1982)
Experimental investigations on micropolar media
. M
e
chanics of Micropolar Media, World scientific, pp 395
–
463
1
4.
Teodoresc
u
PP, Munteanu L, Chiroiu
V (2005)
On
the wave propagation in
chiral media
.
New Trends in Continuum Mechanics, Ed. Thetha Foundation,
Bucharest, pp 303
–
310
1
5.
Teodorescu PP, Badea T, Munteanu L, Oni
s
oru
J (2005)
On
the wave prop
a
g
a
tion i
n composite materials with a negative stiffness phase
.
New Trends in
Continuum Mechanics, Ed. Thetha Foundation, Bucharest, pp 295
–
302
16.
Bécus
GA (1979)
Homogenization and random evolutions: Applications to the
mechanics of composite materials
.
Quarterly
of Applied Mathematics
XXXVII(3):209
–
217
Chapter 2
Ne
w verification of reactive requirem
ent for Lyee
m
ethod
Osamu A
rai,
Hamido F
uj
ita
Faculty of Software and Information Science
, Iwate Prefectural
University
Morioka, Iwate
,
JAPAN
arai
@
fujita.
soft.iwate

pu.ac.jp
,
issam@soft.iwate

pu.ac.jp
http
:
//www.
fujita.
soft.iwate

pu.ac.jp
/
Abstract.
Software development in
general
lacks simplicity and
richness in e
x
pressing
appropriately
the requirement, as well related
supportive
tools. Despite the advances in this
field, the solutions
have still not overcome and the proposed way of thinking is far from
resolving the problems on software development and maint
e
nance.
Recently, a new promising methodology, called Lyee, has been pr
o
posed. It aims to generate programs au
tomatically from user r
e
quirement. Program
s
tructure
in terms
of Lyee was formalized.
However,
when requirements are reactive type, the validation
m
e
c
h
anism
ha
s
not proposed yet to show correctness or
appropriat
e
ness
of the running requirement. For this pu
rpose, t
he propert
ies
that
r
e
quirement and operation should have
are
described as a
propos
i
tion
in terms
of te
mporal
logic
and are
verified by
such
logic.
New ver
i
fication of reactive requirement for Lyee Method
puts together as a
design rule.
Keywords.
Ly
ee, temporal logic, verification, reactive program, u
s
er requirement
1552
Osamu Arai, Hamido Fujita
2.1
Introduction
If r
equirement is
well
defined and
appropriately
expressed, software
met
h
odologies
can realize them
correct
ly for system built up. I
n order that the
program runs correctl
y and ends,
we should be able to
clarify
that
requir
e
ment and operation should have
a certain
property
for
evaluation
purposes.
The Lyee program generation principle [1] is applicable to Lyee r
e
quirement specification. This means that the program is either
providing
the user with the correct computation results, or indication to the user that
the specification is incorrect. In Lyee methodology, requirement specific
a
tion is
described
using Word and Process Root Diagram which is aut
o
ma
t
ically expanded to Rout
ing Vector (a kind of Word). The correctness of
Lyee in the fixed

point setting [2], extension of this for parallel and di
s
tributed computing [3] is already
discussed
.
Lyee
r
equirement
and pr
o
gram structure are
formalize
d [4

6
]
.
However, we think that almo
st all of
Lyee style program behaves as reactive system program. In the case with
multi screen and database management system or external equipments, the
b
e
havior of system is depend on the external
employment
such as human
or e
x
ternal equipment reaction.
Therefore, it is
necessary
that we apply a
n
other style of verification for such reactive requirement. A
s
an
example
,
Lyee
r
equirement
s
representing
“
The Light Control Case Study
”
[
7
]
are
d
i
s
cussed.
In the case of Process Control System such as energy savin
g
sy
s
tem of buildings, the users
’
intension (requirement) is to save energy.
Such intension (requirement) cannot verify without using process model.
In this paper,
we
consider the
whole system
including
not only the stru
c
ture of
its program
but also its
ex
ternal
requirement and operation
of h
u
man to behave
as
reactive system
. We formalize whole system by process
model, which
consist of
processes
,
states
and
shared variables
as a bull
e
tin board type
communicat
ion tool.
For this
purpose
,
a
language
called L i
s
defined
by shared variables declaration and process definition
e
m
bedded
in
its syntax.
T
he
semantics
is
represented
by the
labeled
transition system
extracted
from
the
program. By
construct
ing the
T
e
mporal
F
rame
(TF)
that
is the semantics
in terms
of te
m
poral
logic
extracted
from the calcul
a
tion sequence of
that system,
the linkage of semantics
of te
mporal
logic
and that of L, is presented.
Based on this provision, the property,
(i.e., u
s
ers
’
intension)
which
the
requirement
specification should have, is
d
e
scribed as a proposition
in
te
mporal
logic
.
Such proposition
is verified by
the theorem of te
mporal
logic, and
its
derivation
rule.
The remainder of this paper is organized as follows. In Section
2.
2
, we
introduce Lyee Methodology. In Section
2.
3
, we def
ine the program of r
e
active system in terms of language L. W
e
discuss the semantics of rea
c
tive
New verification of reactive requirement for Lyee method
1553
system in terms of language L and the semantics of temporal logic for va
l
idation purposes. In Section
2.
4
, we verify those properties in terms of R
e
active progr
am. In Section
2.
5
, we discuss classification of condition and
fairness, and transition of state and transition of SF in PRD in Lyee met
h
od. As summary of discussion, the design rule is shown. Finally, Se
c
tion
2.
6
provides concluding remarks on this work,
and related future work.
2.2
Lyee
methodology and
r
equirem
ent
In the Lyee methodology,
requirement specification is described using
W
ord and
P
rocess
R
out
D
iagram which is routing every SFs.
SF
(scenario
function)
is a unit in which input
s
, process
computat
ions
and output
s
are
re
ached
in fixed

point
representation.
If
SF
is an object for screen,
then the
output
is displayed
waiting for input. With
pushing
the enter button after a
value of variable input, it is re
start
ed and processing is performed.
U
sing
tra
nsition of screen or d
a
tabase management system
(DBMS), two or more
SF
are
required
.
O
ne SF is assigned
to
every
single button of
screen
and
to
every
single
table of database management system.
Accordingly
, Process
Route Diagram
(PRD) is
constructed
. PRD e
xpresses transition of two or
more screens, the correlation of a screen and DBMS, etc. as transition b
e
tween SF
s
. A routing vector is defined for this purpose. Transition of two
or more SF
s
is defined
as
PRD
.
For more details on Lyee
refer
to [
1
].
In the c
ase of screen and keyboard
communication
, Lyee requirement is
almost have reactive property in itself or whole system
including
enviro
n
ment such as human
operation
have such property.
2.
3
Reactive
s
ystem
s
In recent years,
not only Lyee requirement,
the co
mputing system has d
e
veloped into
online
system
style computation
.
Such
systems are called
r
e
active
s
ystem. Functional modeling cannot capture reactive system fe
a
ture.
Therefore, i
t is necessary to investigate all
states
during
calculation
. Since
the
seque
nce
of th
ese
state
s
continue
s
indeterministically
, a program does
not have
deterministic
output value at the time of program end, so the fo
r
mulization by input/output relation is
not
simple
.
A program continues
r
e
acting
over input from outside.
To confirm
whether the reactive system runs co
r
rectly as intended, e
ven
if
such
system itself does not have concurren
t property
, the
whole
system,
which consists of system,
and its
exterior
s
have concurrent property.
1554
Osamu Arai, Hamido Fujita
2.
3.1
Composition
of reactive system
A reactive s
ystem consists of the following.
Process
The subject
that
calculat
es
by
running
a program is called a process. Rea
c
tive system consists of two or more calculation subjects called process, and
each process
runs
in parallel, communicating with other processe
s.
State
The global
running status
of a process
that is nam
ed
as
state
is
used
to
grasp internal state of
the
system.
The state includes not only allocation of
value to variable but also
working
situation
of
process.
Inter

p
rocess communication
C
ommunicati
on
is done by
share
d
variable
s or channels [
4
] that is repr
e
senting
message exchange
among processes.
We use a
bulletin
board type
communication
that
use
s
the share
d
variable as communication
means.
2.3.2
R
eactive
system
on Lyee
s
pecification
W
e define a
program L for
reactive system
and give its syntax and sema
n
tics for formal representation
.
Program has declaration of shared variables
and definition of processes. Process
definition
has two attributes such as
state and action.
In declaration of a share
d
v
ariable, this program declares
to
us
e
the variable shared between each process. The other
part
of
this
pr
o
gram defines the
action
of a process. A process changes its state,
by upda
t
ing
share
d
variable
s
. An
action
of a certain process is defined
as in the f
o
l
lowing:
when
a process
is
in
state
α
, if the value of a shared variable
fulfills Conditions B
,
performing substitution A of
an assigned
value to a
share
d
variable
then such process transfers itself
to
state
β
. T
he below
fo
r
mula
shows such representation:
t
A
B
where
t is
the name
of
action
.
Here after, program expressed using this
re
p
resentation
, we call process model. T
he conditional expression B about the
value of a share
d
variable is called the execution conditions of t
.
T
he su
b
stitution of the value to a share
d
variabl
e is called the command
A
of t
.
In
Lyee Method structure, Scenario Function SF which has pallets W04

W02

W03 [1] in Fig
ure
2.
1 can be expressed by process model in Fig
ure
2.
2
.
Pallet
W
02 has two processes
;
one is computer (C), the other is exte
r
New verification of
reactive requirement for Lyee method
1555
nal event s
uch as human operation (H).
If processes have more than two
outputs, to describe which output
should
be selected, Process Rout Di
a
gram is
needed
. Generally, output is selected by human operation or ex
e
cuted result of W03.
The left part (a) of Fig
ure 2.
2 ca
n be
summarized
to
right part (b)
. This is
because the variable domain for routing is
limited
i
n
side of SF
; W
04

W
02

W
03. The
r
equirement of reactive type system
could be
d
e
scribed
as a transition of SF which is shown in (b)
.
Fig.
2.1
.
Lyee Program
Fig
.
2.2
.
Inner of SF
A sample program of reactive system type is shown as Fig
ure
2.
3 in full
description, where two variables
start
and
finish
(includes initial value) are
defined by {
start
:=undefined,
finish
:=false}. Two processes named H
(Human) and C (Com
puter) react with each other using shared variables
1556
Osamu Arai, Hamido Fujita
start
and
finish
.
At state
SF
, when shared variable
start
is false, process H
transfer its state from
a
to
itself
changing
start
variable.
Fig.
2.3
.
Reactive System Program
Then Process C can
transfer
i
ts state from
SF
to
Child SF
without
changing any variables. At state
Child SF
,
Process C can transfer its state
from
Child SF
to
itself
changing variable
finish
from false to true when a
f
ter
memory
reading is complete. In the case to
transfer
another SF,
var
i
a
ble
start
have value a
n
other.
2.
4
Verification
Lyee
r
equirement
by te
mporal
logic
To treat this representation more formally, we introduce
a
formal language
L and
temporal
logic [8

10]
.
In the
definition
of program C in language L,
its syntax and sema
ntics is defined using
L
abe
l
ed
T
ransition
S
ystem
(LTS). We introduce
temporal
logic to verify requirement
described as a
proposition of te
mporal
logic
. Time m
odality connector
○
◇
□
,
,
means
)
(

)
(

)
(

)
(
)
(


)
(
)
(

)
(

)
(
:
time
t
at the nex
P is true
P
some time
P is true
P
allways
P is true
P
Q
or
P
Q
and
P
Q
P
Q
P
Q
then
P
if
P
not
Q
P
P
tion
a proposi
on showing
The notati
p
P
○
◇
□
New verification of reactive requirement for Lyee method
1557
At definition of its semant
ics, we introduce
temporal
frame as a stru
c
ture.
Generally
, the fairness of processes
execution
should be maintained to
some extent by scheduling.
In Process route diagram, at the connecting SF,
this fairness of execution is important. If the execution of
routing is not
fair, user
’
s intension such as termination of program and saving energy r
e
sult of process execution cannot
achieved
. For the verification of requir
e
ment,
fairness
is
introduced
[10]
.
Based on this provision, the
intension
that
specification
should have is
described as a proposition of te
mporal
logic, and is verified by the
axiom
of te
mporal
logic, and
related
re
duc
tion rule
s
. Lyee
r
equirement
specific
a
tion
which
represents
Light Control Case Study
[
7
] is defined
as
an
exa
m
ple;
its intension
is to save energy. The
p
roperty
of Lyee requirement
(i
n
tension)
specification
is described as
proposition of te
mporal
logic
.
2.
4.1
Verification by te
mporal
logic
Verification that
a
C
program
in terms
of L

program
satisfie
s a pro
p
erty P
that is described b
y expression
in
temp
o
ral logical
,
means
that P is unive
r
sally valid in C.
In this case, we argue about the property "
to be
universally
valid
in the temporal frame made from the calculation sequence of LTS e
x
tracted from C
"; although it is not
universally v
alid
in any other temporal
frame.
For this reason, the following theorems are provided.
Theorem 1
Suppose that
,...,
i n IM
P P P
is
true
in
a s
et
IM
of te
mporal
frame and
logical expression
s
1
,,
n
P P
and
P
.
Then,
1
,,
m IM
Q Q Q
is
true if
Q
can be
e
x
tracted
from
1
,,
m
Q Q
by adding
1
,,
n
P P P
as
re
duc
tion rule.
IM
is the set of a temporal frame
.
Moreover, the following
general
the
o
rem
s are
used for our use
.
Theorem 2
Let
P, Q
be
logical formula.
Q
P
○
is universally valid formula on
C
,
when
the
stat
e
ment
[
If P is realized before
the
execution of
and
then
is
performed,
and
Q
will be
true
after execution
of
.
]
i
s realized
to
arbitrary
transition
in terms of
LTS
that
is extracted
from Program C.
1558
Osamu Arai, Hamido Fujita
Generally, liveness property is to prove the statement [a property Q will
be satis
fied at a certain time]
;
therefore the rules below are used.
If P is satisfied then P or Q will be satisfied at a certain time. If P co
n
tinues to be satisfied then
τ
can be done at a certain time. If P is satisfied
before execution of
τ
then Q is satisfied
after execution. Therefore,
it will
be
warrant
ed
that
if P is satisfied then Q will be satisfied at a certain time.
What formulized above
as
reduction
rule of liveness
shown below.
This
used to verify the property of specification that is
written
as a pro
position
in te
m
poral logic.
Theorem 3
(
rule of liveness)
The following
are
satisfied
when the set
which consists of transition
of the
LTS
extracted
from Program C
are
a weak fair
ness
set or a strong
fair
ness
set on
C
.
Q
P
Q
executed
P
enabled
Q
P
Q
P
P
C
M
C
◇
○
○
F
J,
,


,
,
F
J,
,

C
M
C
｜
means that to the temporal frame
,
C
M
,
Q
P
◇
is
un
i
versally
valid in terms of
F
J
C
,
under strong and weak
fairness
where
is
a calculation sequence belon
g
ing to
F
J,
]
]
[
[
C
.
Theorem 4
(
r
ule of chain)
The following is
satisfied
on arbitrary natural number
i
.
Q
P
P
P
Q
P
P
P
P
Q
P
P
Q
P
i
i
i
◇
◇
◇
◇
1
0
1
1
0
0
1
0

Th
ese
two T
heorem
s 3 and 4
are
prove
n
by
d
eductio
n
.
According to the
Theorem 1
,
we can
use
the fo
r
mula, which used

instead
of

.
New verification of reactive requirement for Lyee method
1559
2.
4.2
Example
Our Example is a
part of
p
rogram
specification
which
represents
the Light
Control Case Study [
7
]. This case study is saving energy and security c
o
n
trol of a building.
Here we use energy saving part of this case study. That
is
«
If nobody is in the room, light is turned off in fixed time
»
. The Pr
o
gram is shown in Fig
ure
2.
4. Related processes and its states are shown in
Table
2.
1
.
Sh
a
red variables s
hared by processes
which
control
state transition and
common variables that are used plural states commonly
are
defined.
Decl
a
ration of share
d
variable
s and common variables
are
shown in Table
2.
2
.
The state which means somebody deemed in the room made by
events of
different equipments is introduced.
Process Definition is shown in Table
2.
3
. Since the variable which controls transition is expressed as a share
d
variable,
then
this variable
is
specified
in
command A
.
The shared variable
is cleared by transiti
on before
turning into
its initial state. s, t, z, x = 0, 0, 0,
0. States (SFs) c have two outgoing transitions (routes). In c, transition t6
and t8 are selectable. In this case, fare selection is needed. In order to save
energy, program selects transition
from c to a in a
certain
time.
2.
4.3
Proof of
e
xample
T
he property which specification should have is described
,
as a proposition
in terms of
te
mporal
logic, and is verified by the
axiom
of te
mporal
logic
and
its derivation
rule.
The r
equirement
that
rep
resent
s
Light control case
study
is defined as
an
example
. It
is
to verify that, "
i
ts L

program
surely
save energy
"
.
In this verification, we
clarif
y
the conditions of fairness, which
r
equir
e
ment and
o
peration should have. Liveness property of "
save energ
y
" is i
n
fluenced
by
fairness
property
. Verification of Liveness is the proof of log
i
cal formula deploying fairness set
s (defined in
section
2.
2)
. If people
go
out of
the
room
, a timer will surely be fin
ish
ed after a while and the light
will be put out. Tha
t is,
f
airness is required between the frequency f inco
m
ings and outgoings of people, and
interval
setting
of a t
i
mer.
1560
Osamu Arai, Hamido Fujita
Fig.
2.4
.
Example of pr
o
gram Light Control Case
Table
2.1
.
Process and state of exa
m
ple
name
description
Pr
ocess Q1
h
uman
detector on
Process Q2
h
uman
detector off
Process Q3
deemed

timer time up
Process Q4
light sensor scanning timer time up (repeat)
Process P
computer
whose role
is
governor
State a
nobody in one room
State b
somebody in the room
State
c
deemed

timer running
State a,b,c
deemed somebody in the room
Table
2.2
.
Declaration of share
d
and common
variable
s
Declaration of share
d
and common
variable
s
Description
shared variable
s = {0,1}
0: nobody in
1: somebody in
shared variable
t = {0,1
}
0: deemed

timer stopped
1: running
shared variable
x = {0,1
}
0: light sensor scan off
1: scan on
Table
2.3
.
Process definition
Process
B
A
Q1
t
1
a
a
s, t
=
0, 0
s, t, z
=
1, 0, 0:
some one d
e
b
t
10
Q2
b
c
Process Q1

4:
t1, t3, t5,t7,t9
Process P:
t2, t4, t6, t8, t10
a,b,c
a
t
４
a
t
3
c
a,b,c
t
8
c
t
5
t
7
t
6
t
9
Q1
Q1
Q3
Q4
t
2
P
P
t
1
New verif
ication of reactive requirement for Lyee method
1561
tected
P
t
2
a
b
s, t
=
1, 0
s, t, z
=
1, 0, 1: deemed
Q4
t
9
a,b,c
a,b,c
x = 0
x = 1: light

sensor

scan on
P
t
1
0
a,b,c
a,b,c
x
=
1
x = 0: light

sensor

scan off
control room light on z
As the prop
o
sition of temporal logic,
[
save energy
] is ex
pressed as:
a
P.
at
c
P.
at
◇
. At this time, Processes P and Q are located c, and uses
shared variables
t
(initial value of which is
0
:
t
). For this reason, h
ere
after,
[
a
P.
at
c
P.
at
◇
]
is verified.
Using
the state
c
of
p
rocess
P
and the value of
v
ariable
t
,
c
P.
at
is
cla
s
sified
as
in the following
two
case
s:
(( 0)
at P.c at P.c t
(2.1)
( 1)
at P.c t
(2.2)
Since
t
= {
0
,
1
}, this classification is comprehensive
ly
inclusive
and e
x
clus
ive
ly not overlapped internally
in this case. Since the value of
t
is
0
in
the case of (2.1), Process P can approach into
a
immediately. On the other
hand, (2.
2
) is far from
reaching the
end
In order to return to
a
,
at
timer is
over
, input
0
for
t
is requi
red
in
Process Q.
(
App
ly
Chain rule
)
The
name
1
0
P
,
P
is given to
each term according to
the order near the
goal
(
at P.a
)
for
(
2.1) to (2.
2
).
0
P t 0
at P.c
1
P t 1
at P.c
The following
reduction
will be
obtained
when
the
chain rule of
the
T
heorem 4 is applied to such logical
formula.
a
P.
at
a
P.
at
a
P.
at
◇
◇
◇
1
0
0
1
0
P
P

,
P
P
,
P
Liveness property is verifiable when the premise of rule from (
2.
1) to
(
2.
2) is derived b
e
cause
1
0
P
P
is
c
P.
at
.
(
Appl
y
Live rule
)
L
ogical expression
s P
0
, P
1
named above
are
derived
using the temporal
logic shown below.
1562
Osamu Arai, Hamido Fujita
1)
Derivation of (2.1)
Since P
0
is
0
t
c
P.
at
, when this is
true
,
8
t
P,
is performed
at
a
ce
r
tain
time
and the state of P should be set
to
a
.
That is, fair execution is
r
e
quired for t
8
as mentioned above
.
Therefore,
in
appl
ying
the
live
rule
of
T
heorem 3
to
a
P.
at
◇
0
P
and
8
t
P,
, the following derivation
can be
obtained
:
0 0
P,
P at P.a
(2.3a)
8
0
,
P ( ),
P t
at P.a enabled
(2.3b)
8
,
( )
0
P t
P executed at P.a
,T,F
0

P
C
M C
at P.a
(2.3c)
The logical expression
s
2.3a, 2.3b, 2.3c
are the
premise
of this
deriv
a
tion
and
verified
as below.
(
Verification of (2.3a)
)
In P
0
, since states of
p
rocess
P and Q are
both
c
= 0 and t = 0,
possible
transition is restricted to
8
t
P,
;
(case 1)
and the idling trans
i
tion
I
;
(case
2).
C
ase
1
:
When
8
t
P,
is performed, since
variable
c
P.
at
is true
, the
fo
r
mula (a)
is verified
using
T
heorem 2.
C
ase
2
:
When
I
is performed, since P
0
is
true
, the formula (
2.3
a)
is true
by
T
heorem 2.
Therefore, formula (2.3a)
i
s verified.
(2.3b), (2.3c) are verified same
way.
Therefore, i
t
i
s
verified that formula
(2.1),
(
i.e.
,
a
P.
at
◇
0
P
)
turns i
n
to a universally
valid formula
of
L

program
.
(
2.
1) is verified same way.
Since it
i
s verified that (2.1) and (2.
2
) are
universally valid formulas
, the
liveness
property
a
P.
at
c
P.
at
◇
which is our objective
,
i
s verified.
□
Consequently, when satisfying the following requirements,
the property
that energy saving is
performed correctly
is verified under the fair exec
u
tion.
1.
U
sing the combination of shared variables
,
a
special
cla
ssif
ication
is
necessary to carry out
such execution.
In this clas
s
if
ication
, condition
must be comprehensive
ly
inclusive
and exclusive
ly defined
.
This
New verification of reactive requirement for Lyee method
1563
means that
the conditions of transition of SF
is defined only once all
over domain of a vector.
2.
It is de
manded that fair execution
(
i.e.,
Timer time up
should surely
be
occurred
some
time
) need to be done
.
2.
5
Discussions
From the
state transition diagram
(STR), extract SF pattern of STR accor
d
ing to event. Unify
asynchronously simultaneous
events (SF patter
n of
STR) to one SF. This is
the SF division rule
.
One
Parent
SF is a group of
events that may happen there. A screen (SF) has some buttons, i.e. events
,
i
n
human interface with screen and keyboard
,
machine (sensor, actuator)
interface event
.
If these even
ts happen asynchronously,
it
needs to be r
e
ceived from one
parent
SF. That is, a group of events in which one of these
events happens and the other event does not happen, compose one
parent
SF. Event transfers process from one state to another state (event
driven).
Therefore, more than one state exists in one
parent
SF.
2.
5.1
Shared variable and
c
ommon variable
In Process Control Application,
there are
many events
such that
commun
i
cation
that
include internet
,
timer
,
human interface with screen and ke
y
board
,
machine (sensor, actuator) interface event
.
If these events happen
asynchronously, it needs to be received from one
parent
SF. That is, a
group of events in which one of these events happens and the other event
does not happen,
c
ompose one
parent
SF. Eve
nt transfers process from one
state to another state (
i.e.,
event driven). Therefore, more than one state e
x
ists in one SF.
The variable in one SF that is used commonly by plural
states is called common variable.
2.
5.2
Classification of condition
and Fair
ness
In classification of transition condition, using the combination of shared
variables
n
eeds fair classify
.
All conditions must be comprehensively i
n
clusive and exclusively defined. This means that the conditions of trans
i
tion of SF is defined only onc
e all over domain of variable.
F
air
e
xecution
should be done.
In the example, t
imer should surely time

up some time.
1564
Osamu Arai, Hamido Fujita
2.
5.3
Expression as a design rule
The program
in terms
of Lyee structure is correctly
,
performed only after a
definition and operation of
r
equirement of Lyee are
executed
correctly.
C
alculation condition
s
of word
s
and the conditions of transition of SF are
the
reference
of
requirement correct
ness
.
PRD of Lyee
specification i
s expressed
in L

program
by considering SF
as a state using the proc
ess model. Since a state is
an
atomic action, SF
cannot be partitioned
into
more than
one
atomic action. If SF
s
are
unif
i
ed
and
enlarge
d
, an excessive calculation will be carried out and
its
execution
speed
may
become slow. It is important to
extract
the
b
oundary
of SF a
p
propriately. For this reason,
it is effective to apply the process model pr
o
posed here for
safety checking
.
In order to apply the above discussion result from now on, it puts t
o
gether as a design rule.
1.
Describe a
state transition diagram b
ased on L

language.
2.
Verify Property which
design intension (
requirement
)
wants.
3.
Repeat 1) and 2) until fix the state transition.
4.
Divide system to program (PRD).
5.
Divide program (PRD) to SF based on the SF division rule above.
6.
Draw
PRD (transition between S
F) based on state transition diagram.
7.
Realize state as a set of
common
variable in SF by Lyee method such
as memory table, action vector, and word.
Terms from 1 to 3 are about process model, term from 4 to 7 is about
Lyee Methodology.
According to this De
sign rule, we have
developed
full
of
the Light Control Case Study
correctly and successfully by Lyee M
e
t
h
od.
2.
6
Conclusion
To verify the reactive property of requirement that multi screen, database
management and process control system have, we should app
ly to reactive
style of verification.
A
s
an
example
,
the
r
equirement
that represent
s
O
thello
game
is defined
to
verify
that
this program calculat
e
s correctly and surely stops
anyway.
Consequently,
we clarify the conditions of fairness
in
which the executi
on
conditions of
r
equirement and
o
peration should have.
In clas
s
if
ication presented at the
proof
of the example,
condition
s
must
be comprehensive
ly
inclusive
and exclusive
ly defined.
Moreover, oper
a
New verification of reactive requirement for Lyee method
1565
tion should be executed
fair
ly. These requirements are co
mmon in PRD of
Lyee methodology.
Especially, in Process control example, the Light Control Case Study,
the
trace of logic is difficult
,
and so
maintainability
is
poor, t
he concept of
state
of process model
is effective as an approach o
n
visualizing and cat
c
h
ing the program structure.
W
hen using
a
stru
c
tured program
such as
Lyee
or
PROLOG
, etc.
for
reactive
application
with many state transitions
, t
his
model is effective
.
In Lyee Methodology, requirement specification is not verified. To use
this methodolog
y for actual system development, system design intension
is desired to be verified beforehand. Using the process model, we can ve
r
i
fy other reactive property of intension (requirement) such as safety, partial
correctness, avoidance of dead lock, mutual exc
lusion of critical resource,
total correctness, in the future.
References
1.
Negoro
F
(2000)
Principle of Lyee
s
oftware
.
In:
Proceedings of 2000 Intern
a
tional Conference on Information Society in the 21st Century (IS2000), Aizu,
Japan, pp
441

446
2.
Gorlat
ch
S
(2003)
The Lyee programming model: Analysis correctness in a
fixed point setting. In
:
Fujita
H
,
Johannesson
P
(
eds
)
New Trends in Sof
t
ware
Methodologies, Tools and Techniques
.
IOS Press
,
p
p
214

224
3.
Gorlatch
S
(2004)
Declarative
p
rogramming with Lye
e for
distributed sy
s
tem
s
.
In
:
Fujita
H
,
Gruhn
V
(
eds
)
New Trends in Software Methodologies,
Tools and Techniques
.
IOS Press, p
p
129

137
4.
Fujita
H, Mejri
M
,
Ktari
B
(
2004
)
A process algebra to formalize the Lyee
methodology
.
Knowledge

Based Systems 17
(
7

8
):
263

282, ISSN 0950

7051
5.
Arai Osamu, Fujita
Hamid (2003) Mathematical structure model for Word

Based Program. Knowledge

Based Systems 16(7

8):
399

411
6
.
Arai O
,
Fujita
H
(2006)
Verification of
the
Lyee requirement
.
In
:
Fujita
H
a
mid
,
Mejri
Mohamed
(
ed
s
)
New Trends in Software Methodologies, Tools
and Techniques
.
IOS Press,
(
Proceedings of the
5th
SoMeT
_06,
Quebec
),
p
p
340

361
7
.
Queins
S
et al. (2000)
Requirement
e
ngineering: The
light control case stu
dy
.
Journal of Universal Computer Science
6
(
7
)
8
.
M
anna
Z
,
Pnueli
A
(
1991
)
The
temporal logic of reactive and concurrent s
y
s
tems
.
Sringer

Verlag
9
.
Manna
Z
,
Pnueli
A
(
1995
)
The
temporal verification of reactive s
ystem; Saf
e
ty
.
Sringer

Verlag
10
.
Manna
Z
,
Pnueli
A
(
1996
)
Th
e temporal verification of reactiv
e sys
tem; Pr
o
gress
.
Draft
1566
Osamu Arai, Hamido Fuji
ta
Chapter 3
A tree derivation procedure for multivalent and
paraconsistent inference
David Anderson
School of Computing, University of Portsmouth, Portsmouth PO1 2EG,
Ph. (+44) 2392 84 6668, david.anderson@port.ac.uk
Abstract.
T
he use of trees as a derivation procedure, which is a r
e
finement of Beth’s semantic tableaux method and Gentzen’s sequent
calculus, is an elegant and well established technique. Despite wid
e
spread acceptance and application, little effort has been made to
e
x
tend the use of tree derivation procedures to mult
i
valent alternatives
to classical logic and none at all, so far as I am aware, to paraco
n
si
s
tent systems. The purpose of this paper is to outline briefly a si
n
gle tree derivation procedure which was desig
ned to work with a
pa
r
ticular multivalued paraconsistent system: Epsilon
442
[1]
,
but
will, with slight modification, work gene
r
ally.
Keywords.
Automated reasoning, Logic, Semantic tableaux, Par
a
consistency, S
e
quent Calculus
3.1
Valid arguments
In classical
systems, a valid argument may be distinguished from an inv
a
lid argument by virtue of the fact that the former has no counter exa
m
ples.
Hence, if an exhaustive search fails to produce a counter example we may
be certain that an argument is valid and otherw
ise invalid. The first method
for carrying out such a search which most of us encounter is the ‘truth t
a
ble’ but a much more elegant approach is the construction of log
i
cal trees.
In classical propositional logic the following familiar rules will su
f
fice:
1568
David Anderson
Fig.
3.1
.
Tree derivation rules for classical logic
3.2
Tree construction
A basic process for constructing a tree derivation is as follows:
Step 1:
Arrange the premisses in a column one above the other.
Step 2:
To the column formed in Step 1
, add the NEGATION of the co
n
clusion.
Step 3:
Using the decomposition rules presented above simplify each of
the expressions (or nodes) as far as possible.
Step 4:
When a node has been simplified, it is no longer of interest and
may be crossed out, ticked
or otherwise identified as no longer
forming an active part of the structure.
A trivial example follows. Given
,


, the application of
steps 1

4 yield the following stru
c
ture:
Fig.
3.2
.
An example of inference using tree d
erivation
with
c
lassical
l
ogic
Each path from top to bottom through the tree we have created repr
e
sents a possible valuation demonstrating the satisfiability of the set of
statements under consideration.
A tree derivation procedure for multivalent and paraconsistent inference
1569
The test for argument validity is carried out in
:
Step 5:
We put a cross at th
e bottom of each branch which contains any
given statement AND its negation. A branch so constituted is
called
closed
and any tree all of branches are closed is itself said to
be closed.
A valid inference will always give rise to a closed tree and invalid
infe
r
ences produce trees with one or more open branches. This is because the
decomposition rules are simply ways of displaying the conditions under
which a given complex expression may be true and, since we have i
n
clu
d
ed the negation of the conclusion as p
art of the structure, each branch re
p
resents a possible counter

example to the original inference. A closed tree
is one where all the possible counter

examples have been eliminated and
hence demonstrates validity.
3.3
Multivalent tree construction
If this
general approach is to be extended to cover multivalued systems
there are a number of considerations which need to be addressed. First, we
need to exercise care over the use of the negation sign, ‘ ¬ ‘. In the sy
s
tem
just described ¬ stands as a marker for
falsity (an its absence indicates
truth) as well as a logical operator. In a multivalued system matters are apt
to be more complicated and it will prove more effective to avoid any po
s
sible ambiguity and use a new piece of notation to indicate the valuati
on
b
e
ing asserted at any given point on the tree. To this end, I will employ
su
b
scripted parentheses containing the valuation. Thus to indicate that A is
true we would employ the following:
(true)
, and for A is false, we would
use
(false)
. In what follow
s I will be outlining a system designed to ha
n
dle
a four

valued system (more details of which are given in Appendix A)
whose possible valuations are 1,2,3 and 4 hence
(1)
,
(2)
,
(3)
and
(4)
will be used to make clear which of the available valuations A ta
kes at any
given point.
Another consideration to bear in mind is that multivalued systems give
rise to greater complication concerning how to interpret valuations. In pa
r
ticular, decisions about which valuations are to be desi
g
nated, anti

designated or non

designated mean that what counts as a counter

example
is not as clearcut as in the bivalent system. One possible approach to this
would be to construct a new tree for each of the sorts of counter example
but this leads to a more cumbersome system and the
preferred approach is
to rely on a single tree at the expense of slightly more convoluted tree d
e
r
1570
David Anderson
ivation process. The rules given below were developed for Epsilon 442, in
which {1,3} are designated values, {2} is anti

designated and {4} is non

designated.
The full matrices for Epsilon 442 are given in Appendix.
A tree derivation procedure for multivalent and paraconsistent inference
1571
Fig.
3.3
.
Tree derivation rules
for
a 4

valued logic Epsilon 442
How do we arrive at exactly these rules and how would we develop
rules for a different system?
Rather than painstakingly g
oing through each structure in turn I will
take a single example in order to demonstrate the principle. Take
. To devise the appropriate decomposition structure we need to
consult the E
p
silon 442 matrix for conjunction:
1572
David Anderson
AND
1
2
3
4
1
1
2
4
4
2
2
2
2
2
3
4
2
3
2
4
4
2
2
4
Fig.
3.4
.
Devising an appropriate tree structure from an Epsilon 442 m
a
trix
We can readily see that (A & B) is assigned the value 4 in just five ca
s
es:
i.
the first operand has a value of
(1)
and the second has a value of
(3)
.
ii.
the firs
t operand has a value of
(1)
and the second has a value of
(4)
.
iii.
the first operand has a value of
(3)
and the second has a value of
(1)
.
iv.
the first operand has a value of
(4)
and the second has a value of
(1)
.
v.
the first operand has a value of
(4)
and the sec
ond has a value of
(4)
.
And these five cases are captured precisely by the decomposition rule
where each of the alternatives is represented by its own branch. Each of the
other structures is arrived at in a similar fashion. Of course, different m
a
trices wo
uld give rise to a different decomposition rule but the process r
e
mains straightforward.
3.3.1
Multiple tree
derivation scheme
Step 1:
Conjoin all the premisses and mark the resulting conjunction as
having the lowest numbered available designated valuation
, e.g.
(1)
.
Step 2:
Below the expression formed in Step 1, add an expression which is
the original conclusion and mark it as having the lowest nu
m
bered
avai
l
able anti

designated valuation, e.g.
(2)
.
Step 3:
Using the decomposition rules presented above sim
plify each of
the expressions (or nodes) as far as possible.
A tree derivation procedure for multivalent and
paraconsistent inference
1573
Step 4:
When a node has been simplified, it is no longer of interest and
may be crossed out, ticked or otherwise identified as no longer
forming an active part of the structure.
Step 5:
Put a cro
ss at the bottom of each branch which contains any given
statement having two different valuations e.g., A
(3)
and A
(1)
A
branch so constituted is called
closed
and any tree all of branches
are closed is i
t
self said to be closed
When steps 1

5 have been com
pleted, if the tree is closed, the process
should be repeated with the next unused designated valuation (in this case
(3)
) being employed in Step 1. This process should continue until there are
no untested designated valuations. Then the whole cycle should
be gone
through again using the next unused anti

designated valuation (in this sy
s
tem there is only one anti

designated valu
a
tion) being employed in Step 2,
and so on until all the anti

designated valuations have been tested.
A valid inference will always
give rise to a closed tree for every test and
invalid inferences produce one or more trees with one or more open
branches.
3.3.2
Single tree derivation scheme:
Step 1:
Conjoin all the premisses and mark the resulting conjunction as
having the lowest numbe
red available designated valuation, e.g.
(1)
.
Repeat the process with each remaining unused available desi
g
na
t
ed valuation, the resulting conjunctions should be disjoined with
each other to form one complex disjunction.
Step 2:
Mark the original conclusio
n and as having the lowest numbered
available anti

designated valuation, e.g.
(2)
. Repeat the process with
each remaining unused available anti

designated valuation, the r
e
sulting expressions should be disjoined with each other to form
one complex disjunct
ion. Add the resultant expression below the
expression formed in Step 1.
Step 3:
Using the decomposition rules presented above simplify each of
the expressions (or nodes) as far as possible.
Step 4:
When a node has been simplified, it is no longer of inte
rest and
may be crossed out, ticked or otherwise identified as no longer
forming an active part of the structure.
Step 5:
Put a cross at the bottom of each branch which contains any given
statement having two different valuations e.g., A
(3)
and A
(1)
A
bran
ch so constituted is called
closed
and any tree all of branches
are closed is i
t
self said to be closed
1574
David Anderson
A valid inference will always give rise to a closed tree and invalid infe
r
ences produce one or more open branches.
It is worth noting that the tree deriv
ation scheme suggested here handles
a multivalent system which is also paraconsistent and in this respect it is,
to the best of my knowledge, unique.
References
1.
Anderson CDP (2002) Developing a framework for investigating inconsi
s
te
n
cy handling in autom
ated reasoning. In:
6th World Multi

Conference on Sy
s
temics, Cybernetics and Informatics (
SCI 2002),
O
r
lando, Florida
Appendix
Matrices for Epsilon 442
:
NOT
1
2
2
1
3
3
4
4
AND
1
2
3
4
1
1
2
4
4
2
2
2
2
2
3
4
2
3
2
4
4
2
2
4
¬(¬P & ¬Q)
¬(P &
¬Q)
OR
1
2
3
4
1
1
1
1
1
2
1
2
4
4
3
1
4
3
1
4
1
2
1
4
COND
1
2
3
4
1
1
2
4
4
2
1
1
1
1
3
1
4
3
1
4
1
2
1
4
NB. In Epsilon 442, {1,3} are designated values, {2} is anti

designated
and {4} is non

designated
.
Chapter 4
On the feedback stabi
lization of matrix
JianDa
.Han, Z
he
.Jiang, Y
iWen
.Zhao
,
Y
iYong
.N
ie
State Key
Laboratory
of
Robotics, Shenyang Institute of Automation,
A
c
ad
e
mia Sinica. 1
1
00
16
Shenyang
, China
A
bstract.
In this paper, several strategies for feedback stabilization
of
matrix ar
e presented via the numeri
c
al stabilization of polyn
o
mial
and matrix.
Keywords
.
F
eedback stabilization, numerical
stabilization, dete
r
mining coefficients, Hessenberg form,
Fr
o
benius

like form
4.1
Introduction
It is important to guarantee stability of a dy
namic system. In order
to d
e
sign a stable control system, the numerical stabilization of
system has b
e
come quite an interesting problem.
Every state equation of linear time

invariant system can be
described as
fo
l
lows,
( ) ( ) ( )
x t Ax t Bu t
(4.
1
)
Fo
r a system with
p
inputs, and
n
state variables,
A
and
B
are, r
e
spectively,
n n
and
n p
constant matrices.
In
state feedback, the input
vector
u
is given by
u r Kx
, where
K
is a
p n
gain matrix with
real elements
ij
k
. This is
the constant ga
in negative state feedback. We
may suppose
0
r
since only stability of the state feedback system is i
n
vestigated.
Substituting
u Kx
into the state equations of (4.1) yields
1576
JianDa.Han, Zhe.Jiang, YiWen.Zhao, YiYong.Nie
11 11 12 12 1 1
21 21 22 22 2 2
1 1 2 2
  
  
( ),
  
n n
n n
n n n n nn nn
a v a v a v
a v a v a v
x A BK x x
a v a v a v
(4.2)
where
1
,,1,2,,.
p
ij is sj
s
v b k i j n
(4.
3
)
The matrix
A BK
is referred to as stable if its each
eigenvalue has
negative real

part. The system (4.1) is asymptotically
stable if and only if
the matrix
A BK
is stabl
e; see, e.g.,
[3,5,6]. The matrix
A BK
is r
e
ferred to as critical if the
largest real

part of eigenvalues is equal to zero. If
the system (4.1)
is critical, then the second approximation may be co
n
structed such
that (4.1) is stable o
r unstable, see, e.g., [5]. So that the crit
i
cal
matrix is rather considered unstable. A useful matrix for control
system
must have sufficient stability, namely, the largest real

part
of eigenva
l
ues
must be less than or equal to an appropriate negative
num
ber.
Regarding each
sj
k
in (4.3) as a parameter, we can appropriately
select
( 1,,,1,,)
sj
k s p j n
such that the
feedback system (4.2) is stable.
This is referred to as
feedback
stabilization
. Evidently, simultaneously s
e
lecti
ng
sj
k
is too
complicated for general
A
and
B
. Fortunately, in order
to
stabilize the matrix
A
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