Cellular Automata with

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Cellular Automata with
Strong Anticipation
Property of Elements



Alexander Makarenko, prof.,dr.




NTUU “KPI”, Institute for of Applied System Analysis,

Kyiv, Ukraine

makalex@i.com.ua

2





Part I. Introduction.
Strong and Weak
Anticipation

INTRODUCTION


The presentation is devoted to the description of
rather new mathematical objects


namely the
cellular automata with anticipation.



Mathematically such objects sometimes
frequently have the form of advanced equations.



Since the introduction of strong anticipation by
D.Dubois the numerous investigations of
concrete systems had been proposed.


4

Anticipation (0)


an•tic•i•pa•tion



1. the act of anticipating or the state of being anticipated.



2. realization in advance; foretaste.



3. expectation or hope.



4. intuition, foreknowledge, or prescience.



5. a premature withdrawal or assignment of money from a trust
estate.



6. a musical tone introduced in advance of its harmony so that it
sounds against the preceding chord.


[1540

50; (< Middle French) < Latin]


Random House Kernerman Webster's College Dictionary, © 2010 K
Dictionaries Ltd. Copyright 2005, 1997, 1991 by Random House,
Inc. All rights reserved.

5




Models and Mathematics of
Anticipation

6

Examples of Problems with
Anticipation


Optimal control problems


Nerve conduction equations


Economic dynamics


Travelling waves in spatial lattice


The slowing down of neutrons in nuclear
reactor


Large social systems (A. Makarenko)


Sustainable development (A. Makarenko)

7

Mathematical Objects


Advanced differential equations


Mixed type differential equations


Advanced difference equations


Mixed type difference equations


Equations with deviated arguments


Fixed points


Periodic solutions


Theorems of existence and uniqueness



So in proposed talk the new examples of
models with anticipation had been
considered


namely the cellular automat.

STRONG ANTICIPATION


Since the beginning of 90
-
th in the works
by D.Dubois


the idea of strong
anticipation had been introduced:
“Definition of an incursive discrete strong
anticipatory system …: an incursive
discrete system is a system which
computes its current state at time, as a
function of its states at past times, present
time,, and even its states at future times




10





(1)




where the variable x at future times is
computed in using the equation itself.

),
1
(
),
2
(
(...,
)
1
(




t
x
t
x
A
t
x
)
),...,
2
(
),
1
(
),
(
p
t
x
t
x
t
x


WEAK ANTICIPATION


Definition of an incursive discrete weak
anticipatory system: an incursive discrete
system is a system which computes its
current state at time, as a function of its
states at past times, present time, , and
even its predicted states at future times




12






(2)




where the variable at future times are
computed in using the predictive model of
the system” (Dubois D., 2001).

),
1
(
),
2
(
(...,
)
1
(




t
x
t
x
A
t
x
)
),...,
2
(
),
1
(
),
(
*
*
p
t
x
t
x
t
x


13



Part II. Cellular
Automata with
Anticipation

14


(
Martinez G.J.,

et all, 2012) ‘One
-
dimensional CA is


represented by an array of
cells

where (integer



set) and each cell takes a value from a finite alphabet
.


Thus, a sequence of cells of finite length represents


a string or
global configuration

on . This way, the set


of finite configurations will be represented as .


An evolution is represented by a sequence of



configurations given by the mapping ;
thus their global relation is following




(3)



where time step and every global state of are is
defined by a sequence of cell states.

1
)
(



t
t
c
с
i
x


i

c

n

}
{
t
c
n
n




:
c
15


Also the cell states in configuration are updated at the


next configuration simultaneously by a local function as
follows’



(4)


Also for further comparing and discussion we show the
description of CA with memory from
(
Martinez G.J.,

et all,
2012)
:


‘CA with
memory

extends standard framework of CA by
allowing every cell to remember some period of its
previous evolution. Thus to implement a memory we
design a memory function, as follows:




(5)



such that determines the degree of memory


backwards and each cell is a state function of


the series of the states of the cell with memory up
to time
-
step.



1
)
,...,
,...,
(




t
i
t
r
i
t
i
t
r
i
x
x
x
x

i
t
i
t
i
t
i
s
x
x
x



)
,
,...,
(
1


t
c

t




i
s
i
x
16

Strong anticipation in CA


The key idea is to introduce strong anticipation into CA
construction. We will describe one of the simplest ways.
For such goal we will suppose that states of the cells of
CA can depend on future (virtual) states of cells. Then
the modified rules for CA in one of possible modifications
have the form:



(6)











(7)


where (integer) is horizon of anticipation.

k
t
i
k
t
i
t
i
t
i
t
i
t
i
s
x
x
x
x
x






)
,...,
,
,
,...,
(
1
1


)
,...,
,
(
,...)
,
,
(...,
2
1
1
1
k
t
i
t
i
t
i
k
t
i
k
t
i
k
t
i
x
x
x
s
s
s










k
17


Further we for simplicity describe the system of such CA without
memory and only with one
-
step anticipation. The general forms of
such equations in this case are:



(8)






(9)





The main peculiarity of solutions of (8), (9) is presumable multi
-
validness of solutions and existing of many branches of solutions.
This implies also the existence of many configurations in CA at the
same moment of time.


Remark that this follows to existing of new possibilities in solutions
and interpretations of already existing and new originating research
problems.

1
1
)
,
(



t
i
t
i
t
i
s
x
x

1
1
1
1
1
1
,...)
,
,
(...,







t
i
t
i
t
i
t
i
x
s
s
s

18

‘Anticipative’ modification may be
introduced to the game ‘Life’.


The suggested generalizations open the way for
investigations
---


of the
anticipatory cellular automata (ACA).


But the investigation of ACA is the matter of
future.


So, here we propose the description and first
steps of simplest example investigations


the
‘Life’’ Game with anticipation in elements (rules
for operating).


We name it as ‘LifeA’ Game.

19

Game “Life”: a brief description

Rule #1: if a dead cell has 3 living neighbors, it turns to “living”.

Rule #2: if a living cell has 2 or 3 living neighbors, it stays alive,
otherwise it “dies”.

Formalization:

x

0

1

2

3

4

5

6

7

8

f
0
(x)

0

0

0

1

0

0

0

0

0

f
1
(x)

0

0

1

1

0

0

0

0

0

}
1
,
0
{
,
1
0
),
(
),
(
)
(
1
0








k
k
k
k
k
k
k
F
C
C
S
f
S
f
S
F
F
N
k
S
F
C
t
k
t
k
..
1
),
(
1



Next step function:

-

state of the
k
-
th cell


}
1
,
0
{

k
C
Dynamics of a
N
-
cell automaton:

t



discrete time


20

“LifeA” = “Life” with anticipation

Conway’s “Life”

N
k
F
C
t
k
t
k
..
1
,
1



“Life” with anticipation

]
1
;
0
[
),
)
1
((
1










t
k
t
k
t
k
S
S
F
F
)
(
t
k
t
k
S
F
F

IR
S
S
F
F
t
k
t
k
t
k







),
(
1
weighted

additive

Dynamics:

21

One possible state of system in
‘LifeA’


Graphics of game’s states


The number of discret time step is represent in abscissa
axes


Ordinates represent the number of occupied cells. (Each
configurations of CA elements is represented by single
index).

22

2 possible configurations at the
same time moment

2 states (only the number of occupied cells
is represented)

23

3 and more states
(multivaluedness)

3 states (The sloping lines represent the
origin the configuration at next step from
given configuration. Each configurations of
CA elements is represented by single
index).

24

Developed multivaluedness (multi
-
state)

Multi
-
states (A large number of
configurations existing at the same
moment in model).

25

Developed multivaluedness (multi
-
state)

Multivaluednes (The same as at previous
slide but with lines connected
configurations).

26

Regularity in states

Regularity

27

1
-
1
-
1
-
3
-
1
-
3
-
4
-
4 transitions

Example with different number of
configurations at different time moments

28

LifeA: simulations

“Life”: linear dynamics

“LifeA”: multiple solutions

29

LifeA: simulations


The number of solutions reaches its maximum after several steps
and then remains constant, while the solutions themselves may
change.

30





III. Examples of
Applications and Further


Research Problems

31

How anticipation can be introduced
into pedestrian traffic models?


One of the possible ways:

Supposition
: the pedestrians avoid blocking each other. I.e.
a person tries not to move into a particular cell if, as he
predicts, it will be occupied by other person at the next
step.

P1

P3

P2

P4

k
P
)
1
(
,
occ
k
k
P
P




P
k



probability of moving in direction
k

P
k,occ



probability of
k
-
th cell of the
neighborhood being occupied (predicted)

32

Anticipating pedestrians


Two basic variants of anticipation accounting were simulated:


)
1
(
,
occ
k
k
P
P




)
)
1
(
1
(
,
max
occ
k
k
P
v
v
P






and

All pedestrians have
equal rights

Fast moving pedestrians have
a priority


And two variants of calculation P
k,occ
:

P1

P3

P2

P4

P1

P3

P2

P4

Observation
-
based

Model
-
based

33

Anticipating pedestrians:
simulations

E/P


equal rights/with priority;

O/M


observation
-
/model
-
based prediction

34

Conclusions

and further
research

problems



Anticipation property may
be quite naturally introduced
into CA models.

35


1.

At first we remember the new possibilities in
considering of non
-
deterministic CA (and moreover
usual automata). Non
-
deterministic automata allow
few transition ways from one state to others. Usually it
is supposed that such structure is only theoretical and
in reality only one of the ways is used in each
transition.


CA with anticipation opens the natural possibility for
considering of the systems with many different ways in
parallel.


Accepting possibilities of physical realization of strong
anticipatory systems it may be accepted existence of
CA with many branches.


Also such systems are interesting as multi
-
valued
dynamical systems.

36


2.

In proposed paper we have considered only
the case of finite alphabet for indexing the cell’s
states. But previous investigations of dynamical
systems with strong anticipation show the
possibilities of existing the solutions with infinite
numbers of solution branches.


This allows introducing CA with infinite number
of cell’ states (or at least infinite alphabet for
CA).

37


3.

The generalizations from point 1 and 2
and analysis of automata and CA theories
origin follows to presumable considering of
aspects of computation theory.



The short list of topics may be the next:


computability;


Turing and non
-
Turing machines;


automata and languages;


recursive functions theory;


models of computation;


new possibilities for computations with
accounting possible branching.

38

REFERENCES

1. Dubois
D. Generation of fractals from incursive automata, digital
diffusion and wave equation systems. BioSystems,
43

(1997) 97
-
114.


2
Makarenko A
.,
Goldengorin B
. ,
Krushinski D
.
Game


Life

with
Anticipation Property
.
Proceed. ACRI 2008, Lecture Notes
Computer Science
,
N
. 5191,
Springer
,
Berlin
-
Heidelberg
, 2008.
p
.
77
-
82


3. Springer

B. Goldengorin, D.Krushinski, A. Makarenko
Synchronization of Movement for Large


Scale Crowd. In: Recent
Advances in Nonlinear Dynamics and Synchronization: Theory and
applications. Eds. Kyamakya K., Halang W.A., Unger H., Chedjou
J.C., Rulkov N.F.. Li Z., Springer, Berlin/Heidelberg, 2009 277


303


4.
Makarenko A., Krushinski D., Musienko A., Goldengorin B. Towards
Cellular Automata Football Models with Mentality Accounting. LNCS
6350m Springer


Verlag, 2010. pp. 149


153.

Thanks

for


attention

makalex@i.com.ua

http://ceeisd.org.ua

http://www.summerschool.
ssa.org.ua