 Evolutionary Systems Agents’ Mathematical Models
Valentina N. Korzhova
1
, Viktor V. Ivanov
2
and Natalya V. Ivanova
3
1
Computer Science Department, University of South Florida, Fowler Ave. Street, Tampa, U.S.A.
2
Member Emeritus of AMS, Saint Petersburg, U.S.A.
3
Dental Care Alliance, Saint Petersburg, U.S.A.
Keywords:
Evolutionary System, Intelligent Agent, Mathematical Model, Problem if ... then ..., Problem Optimization.
Abstract:
General mathematical theory of evolutionary system developed earlier is implemented to various problems of
artiﬁcial intelligence and intelligent agent mathematical modeling. Examples of application of this general
theory to the evolutionary systems such as economics, education, and health care are also considered.
1 INTRODUCTION
The general mathematical theory and mathematical
models (MM) of the evolutionary system (ES) and
its various applications were launched by the pa-
pers (Glushkov, 1977), (Glushkov and Ivanov, 1977)
and can be seen in the monographs (Hitronenko and
Yatsenko, 1999),(Hitronenko and Yatsenko, 2003),
(Ivanov and Ivanova, 2006),(Korzhova et al., 2011).
The notation of intelligent agents (IA) can be seen
in (Wikipedia, 2011).
The present paper is devoted to another interpre-
tation of the notions of work place and active center
adapted to the notion of IA from the point of view of
artiﬁcial intelligence.
This interpretation allows researchers and con-
structors to apply the results in (Hitronenko
and Yatsenko, 1999),(Hitronenko and Yatsenko,
2003),(Ivanov and Ivanova, 2006) for the creation
of artiﬁcial intelligent systems and the respective
robotics.
2 THE BASE MM OF ES
The basic minimal or simplest MM has the form
m(t) =
R
t
a(t)
α(t, s)y(s)m(s)ds,
0 y 1, 0 a(t) t, α 0,
c(t) =
R
t
a(t)
β(t, s)[1 y(s)]m(s)ds, β 0,
R(t) =
Z
t
a(t)
m(s)ds, M(t) =
Z
t
0
m(s)ds,
G(t) = M(t) R(t),
f(t) = m(t) + c(t),t t
0. (1)
where m(t) is the rate of creation of the ﬁrst kind
new generalized product (resource) quantity at the
time instant t, which provides the fulﬁllment of the
internal functions of ES, that is, restoration of itself
and creation of the second kind product; y(t)m(t) is
a share of m(t) for fulﬁllment of internal functions
in the subsystem A of restoration and perfection of
the system as a whole; α(t, s) is the efﬁciency index
for functioning of the subsystem A along the chan-
nel α(t, s)y(s)m(s) m(t), i.e., the number of units
of m(t) created in the unit of time starting from the
instant t per one unit of y(s)m(s); a(t) is a special
temporal bound: the new product created before a(t)
is never used at the instant t, but created after a(t) is
used entirely; c(t) is the rate of creation of the second
kind new generalized product quantity at the instant
t, which provides the realization of the external func-
tions of ES;[1 y(s)]m(s) and β(t, s) are similar to ym
and α respectively but for the subsystem B of creation
of the second kind product; R(t) is the total quantity
of the ﬁrst kind product functioning at the instant t;
M(t) is the total quantity of the ﬁrst kind product to
be created during the time [0,t]; G(t) is the total quan-
tity of the obsolete product at the instant t; f(t) is the
rate of the resource inﬂow from the outside (m(t) and
c(t) are measured in the units of f(t)); t
is the start-
ing point for modeling; [0, t
] is the prehistory of ES,
for which all the functions are given (their values will
be noted by the same symbols but with the sign ”,
441
N. Korzhova V., V. Ivanov V. and V. Ivanova N..
Evolutionary Systems Agents’ Mathematical Models.
DOI: 10.5220/0004219904410444
In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 441-444
ISBN: 978-989-8565-39-6
c
2013 SCITEPRESS (Science and Technology Publications, Lda.) e.g., m(t) = m
(t), t [0, t
]).
It is obvious that all the relations (1) are faithful
representations by deﬁnition. In a general case, the
indices α and β depend on m, c, a, y, R, M, G, and f.
It can be seen that (1) consists of 7 equalities and
7 inequalities connecting 14 values, namely: m, c, α,
y, β, 1 y, a, R, M, G, t, t
, f, and 0, all of which are
nonnegative. Usually, α, β, y, f, and/or R are given,
and the others are to be found. Even in the simpliﬁed
formulation, MM (1) is the system of nonlinear func-
tional relations, in which along with the nonlinear in-
tegral equation of the unusual form (the lower bound
a(t) can be unknown function) we have the system of
functional inequalities.
The respective new IA at the time t in the subsys-
tem A, B of ES are y(t)m(t), [1 y(t)]m(t), function-
ing in accordance to MM (1). It should be noted that
this magnitudesmost properly include combination of
IA and usual work places (WP) labor functions, which
are fulﬁlled by human beings. The base simplest self-
organized ES has the following MM:
α
(t) =
Z
t
a(t)
α(t, s)x(s)m(s)ds,
m(t) =
Z
t
a(t)
α(t, s)y(s)m(s)ds,
c(t) =
Z
t
a(t)
α(t, s)z(s)m(s)ds,
α(s,t) = α(s)e
d(st)
, d 0,
0 x, y, z 1, x+ y+ z = 1,
f(t) = α
(t) + m(t) + c(t),
t t
> 0, (2)
where xm is a share of m for creation in the subsystem
C new technology of ES.
Thus, the respective new IA and WP at the
time t in the subsystem A, B, and C of ES are
y(t)m(t), z(t)m(t), and x(t)m(t), functioning in ac-
cordance to MM (2).
3 MORE COMPLICATED MM OF
ES
The n-product MM, n > 2, can be formally written in
the same form (1), where m, a, and c are the vector
functions, and α, y, β, and z are the respective matri-
ces (where the inequalities for the vectors and matri-
ces are the same inequality for their appropriate com-
ponents). The continuous MM can be described in the
same form considering t and s as many-dimensional
variables and examining the appropriate integrals as
multivariate ones. The stochastic MM can be ob-
tained by considering α, β , and f as functions of a
random factor ω. The discrete MM can be represented
in the same form if the integrals in (1) are understood
in the sense of Stieltjes. The MM of ES (2) can be
generalized by the similar way. Thus, according to
those MM we have IA of the previous types for more
complicated systems.
4 PROBLEMS IF ... THEN ...
In the case of MM (1) the problem ’if ... then ...
means that, for example, when α, β, y, f and/or R are
given (and all the functions on the prehistory of ES
functioning), the other functions are to be found using
MM (1). In the case of MM (2), the problem means
that, for example, when x, y, f, and/or R are given,
and the other functions are to be found using MM (2).
For intelligence systems, those kind of problems are
rather important because they allow us to make theo-
retical experiments before practical realization.
5 AN OPTIMIZATION PROBLEM
One of the important typical optimization problems
for ES is maximization of the functional
I(y) =
Z
t
t
c(t)dt =
Z
t
t
(
Z
t
a(t)
β(t, s)[1 y(s)]m(s)ds)dt, (3)
over y with regard to MM (1).
The ﬁrst essential result on the properties of
solutions of the problem (3) has been obtained
in (Glushkov and Ivanov, 1977)(the ﬁrst law): The
record of an external function for any ES can be
obtained only under the conditions of its sufﬁciently
comfortable guarantee, that is, under the signiﬁcant
fraction of resources sent to the internal needs of ES.
As to the same problem (3) and MM of ES of (2)-
type, it was proven under certain conditions that the
second law (Korzhova et al., 2011) is valid:
The record of an external function for any ES can
be obtained only under the following priority of re-
source distribution: the highest priority has subsys-
tem C, then subsystem A, and then subsystem B.
ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence
442 6 ON APPLICATIONS
6.1 Economics IA Models
In the case of economics, the agents besides the skill
of using MM (1) or (2), have to know how to use the
base of data on labor functions and product and ser-
vices in order to properly change, e.g., the function
c(t) with regard to the up-to-date state of affairs.
6.2 Education IA Models
In this case, the external product c(t) is a rate of grad-
uate specialists number at the instant t, so that the re-
spective agents have to know how to use the base of
data on various tests and up-to-date needs of new pro-
fessionals.
6.3 Health Care IA Models
In this case, the external product c(t) is a rate of pa-
tients number subjected to prophylaxis, diagnosis, or
therapy at the instant t, so that the respective agents
have to know how to use the base of data on various
tests of the norm and pathology for human beings as
well as on various up-to-day remedies of prophylaxes,
diagnosis and curing. Let us dwell on this application
with much more detail.
6.3.1 MM of Organism Subsystems
For many applications the following MM of organism
subsystem (OS) should be considered:
m(t) =
Z
t
a(t)
α(t, s)u(s)v(s)v
1
(t)y(s)m(s)ds,
c(t) =
Z
t
a(t)
β(t, s)[1 y(s)]m(s)ds, β 0,
R(t) =
Z
t
a(t)
m(s)ds,
G(t) =
Z
t
a(t)
β(t, s)m(s)ds,
M(t) = R(t)+ G(t),
f(t) = m(t) + c(t) = k( f
,t)c(t),t t
0,(4)
where α, u, v, y, and β are matrices, 1 is the unit ma-
trix (inequalities for matrices mean the respective in-
equalities for their components) α, a, m, c, R, G, M,
and k are vectors, and the relations for f (f
is the
qualitative structure of f) should be replaced by
f(t) =
r
i=1
m
i
(t) +
s
j=1
s
j
(t) +
s
j=1
k
j
(t)c
j
(t),
r+ s = n. (5)
In addition, the equations of homeostasis should be
included; that is,
R
i
(t) = m
i
(t) m
i
(t)(a
i
(t))a
i
(t),
i = 1, ..., r. (6)
It should be emphasized that considering the given
subsystem of organism with the same kind of MM in
detail, we should include in its MM an aggregate MM
of the rest of the whole organism.
6.3.2 Conception of the Norm and Pathology
All the values in MM under consideration, (see, for
example, (1-3) have rather profound sense from the
point of view of the structure and functions of the sub-
system under consideration. So, if the vector of these
values is denoted by R = (r
1
, ..., r
q
), then it is natu-
ral to suppose that one of the conceptions of the norm
consists in the validity of the relations
r
s
r
s
r
+
s
, s = 1, ..., q, (7)
where r
s
and r
+
s
ations of the introduced values from point of view of
the norm. On the other hand, if at least one of the
relations in (7) is violated, then it is natural to speak
about conception of the pathology or the pathological
state of the subsystem.
6.3.3 Norm Restoration Problems
The most simple and expanded restoration is that the
determined values of the group A”) are injected into
the organism directly or, on the contrary, are removed
from the organism. The respective medicines should
be injected into the organism to achieve validity of
the necessary bounds in (7) relative to this group.
However, under the condition that the indices of vi-
olations of the group A”) are accompanied by those
of the group A’) and class B), such a simple method
is usually the temporal measure and does not deduce
restoration.
For increasing (decreasing) values of m(t)-type,
provided that the values of the other magnitudes are
not to be broken, it is usually necessary to increase
(decrease) the value components of the distributive
matrices.
Thus, the crucial condition for restoration is the
increase of a share of all resources of the subsystems
to its internal requirements.
The most serious and profound violations are con-
nected to the structural shift of the subsystem, i.e.,
with the violation of (7) for the indices of the class
B. Here only the interference into the structure of f
and the genome apparatus can likely restore the norm.
EvolutionarySystemsAgents'MathematicalModels
443 Although in principle, in this case, there can be devel-
oped a theory similar to the above theory. It means the
following principle of the norm of organism restora-
tion:
The crucial condition for restoration (under weak-
ened functions of the classes A and B) is the increase
of the share of all the resources to the needs of a com-
fortable state of OS structure ﬁrst, then its internal
sphere, and after that its external sphere.
6.4 On MM of a Doctor’s and/or Health
The simplest, base MM of a doctor’s and/or health
care IAs (D and/or IA) business has the form
m(t) =
Z
t
0
α(t, u)λ(t,u)y(u)m(u)du,
c(t) =
Z
t
0
β(t, u)µ(t, u)[1 y(u)]m(u)du,
R(t) =
Z
t
0
λ(t, u)y(u)m(u)du+
Z
t
0
µ(t,u)[1 y(u)]m(u)du,
G(t) =
Z
a(t)
0
β(t, s)m(s)ds,
f(t) = M(t) R(t),
M(t) =
Z
t
0
m(u)du, f(t) = m(t) + c(t) =
kc(t), t t
0, t > 0, (8)
where m is the new resource (per unit of time) of a
D and/or IA WP, providing its internal functions; c
is some new resource providing external functions,
including the function of curing; λ and µ are the
intensities of functioning along the channels ymm,
[1 y]mc respectively; R(t) is the total amount of
functioning internal resource; y, α, β, G, k, f, and t
have the same deﬁnitions as many times above.
The function of curing decides a D and/or IA busi-
ness. The number of patients’ c
i
who have been cured
for i
th
disease in the unit of the time is actually de-
cided by the same function. Therefore, this function
should be well estimated. It should be estimated by
the total number of the patients who have been suc-
cessfully cured in a certain period of time. The meth-
ods of curing or restoring to the norm depend on many
factors. We would like to dwell only on some ap-
proaches to curing, following from the models.
To increase (decrease) the values of m(t)-type, if
the values of the other magnitudes are not broken, it
is usually necessary to increase (decrease) the values
y(s), s < t. Thus, the crucial condition for restoration
(under weakened functions of the class A) is the in-
crease of a share of all the resources of a D and/or IA
to his/her/IA internal requirements: development of a
new, much more effective technology.
One may hope that any modes of action, among
them nontraditional ones, including the various phys-
tributing to increasing of resource distribution in ad-
vantage to the internal sphere of the sick subsystem,
will usually bring an essential positive effect.
In conclusion of this subsection, we would like
to note that (Ivanov and Ivanova, 2006), (Korzhova
et al., 2011) contain MM and their applications to
many concrete diseases such as AIDS, cancers, dia-
betes, etc.
7 CONCLUSIONS
In conclusion, we would like to emphasize that the
theory and application of MM of ES and all the tech-
niques under consideration may be expanded far be-
However, the creation of the respective intelligent
systems and robotics requires still much combined ef-
fort of various specialists, which is much more ahead
of us.
REFERENCES
Glushkov, V. (1977). On a class of dynamic macroeconomic
models. Control systems and machines, 2.
Glushkov, V. and Ivanov, V. (1977). Modeling of optimiza-
tion of work places distribution between the branches
of production A and B. Kibernetika, 6.
Hitronenko, N. and Yatsenko, Y. (1999). Mathematical
Modeling in Economics, Ecology, and Environment.
KAP.
Hitronenko, N. and Yatsenko, Y. (2003). Applied Mathe-
matical Modeling of Engineering Problems.
Ivanov, V. and Ivanova, N. (2006). Mathematical Models of
the Cell and Cell Associated Objects. Elsevier.
Korzhova, V., Saleh, M., and Ivanov, V. (2011). Mathemat-
ical models of information systems developing. The
Proceeding of 2nd International Multi-Conference on
Complexity, Informatics, and Cybernetics, Vol. 1, pp.
223-228.
Wikipedia (2011). Intelligent agent. 23 July.
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