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
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Comments 0
Log in to post a comment