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

Copyright

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(s−t)

, 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.

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

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

are admissible bounds for vari-

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.

EvolutionarySystemsAgents'MathematicalModels

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

Care Intelligent Agent’s Business

The simplest, base MM of a doctor’s and/or health

care IA’s (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 ym–m,

[1 − y]m–c 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-

ical loading on a certain part of an organism and con-

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-

yond the examples already described.

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.

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Hitronenko, N. and Yatsenko, Y. (1999). Mathematical

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KAP.

Hitronenko, N. and Yatsenko, Y. (2003). Applied Mathe-

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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-

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