Cellular Automata with Strong Anticipation Property of Elements Alexander Makarenko, prof.,dr. NTUU “KPI”, Institute for of Applied System Analysis, Kyiv,

Cellular Automata with Strong

Anticipation Property of Elements

Alexander Makarenko, prof.,dr.

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

[email protected]

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• 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.

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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.

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• Models and Mathematics of Anticipation

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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)

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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

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• (1)

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

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

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

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

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•

• (2)

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

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

)),...,2(),1(),( ** ptxtxtx

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Part II. Cellular Automata with

Anticipation

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• (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)( tt cс

ix i

c

n

}{ tc nn :

c

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• 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

itri

ti

tri xxxx

iti

ti

ti sxxx ),,...,( 1

tc

tis

ix

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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.

kti

kti

ti

ti

ti

ti sxxxxx ),...,,,,...,( 11

),...,,(,...),,(..., 2111

kti

ti

ti

kti

kti

kti xxxsss

k

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• 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.

11),( ti

ti

ti sxx

111

111 ,...),,(...,

ti

ti

ti

ti xsss

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‘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.

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Game “Life”: a brief descriptionRule #1: if a dead cell has 3 living neighbors, it turns to “living”.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, Rule #2: if a living cell has 2 or 3 living neighbors, it stays alive, otherwise it “dies”.otherwise it “dies”.

Formalization:Formalization:

x 0 1 2 3 4 5 6 7 8

f0(x) 0 0 0 1 0 0 0 0 0

f1(x) 0 0 1 1 0 0 0 0 0}1,0{,

1

0

),(

),()(

1

0

kk

k

k

kkk F

C

C

Sf

SfSFF

NkSFC tk

tk ..1),(1

Next step function:Next step function:

- state of the k-th cell }1,0{kC

Dynamics of a Dynamics of a NN-cell automaton:-cell automaton:

t – discrete time

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“LifeA” = “Life” with anticipation

Conway’s “Life”Conway’s “Life”

NkFC tk

tk ..1,1

““Life” with anticipationLife” with anticipation

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

tk

tk SSFF)( t

ktk SFF

IRSSFF tk

tk

tk ),( 1

weightedweighted

additiveadditive

Dynamics:Dynamics:

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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).

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2 possible configurations at the same time moment

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

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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).

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Developed multivaluedness (multi-state)

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

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Developed multivaluedness (multi-state)

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

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Regularity in states

Regularity

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

Example with different number of configurations at different time moments

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LifeA: simulations

“Life”: linear dynamics“LifeA”: multiple solutions

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LifeA: simulations• The number of solutions reaches its maximum after several steps

and then remains constant, while the solutions themselves may change.

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• III. Examples of Applications and Further

• Research Problems

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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.

P1P3

P2

P4

kP )1( ,occkk PP PPkk – probability of moving in direction – probability of moving in direction kkPPk,occk,occ – probability of – probability of kk-th cell of the -th cell of the neighborhood being occupied (predicted)neighborhood being occupied (predicted)

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Anticipating pedestrians• Two basic variants of anticipation accounting were simulated:

)1( ,occkk PP ))1(1( ,max

occkk PvvP anan

dd

All pedestrians have All pedestrians have equal rightsequal rights

Fast moving pedestrians have Fast moving pedestrians have a prioritya priority

And two variants of calculation PAnd two variants of calculation Pk,occk,occ::

P1P3

P2

P4

P1P3

P2

P4

Observation-Observation-basedbased

Model-basedModel-based

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Anticipating pedestrians: simulations

E/P – equal rights/with priority; E/P – equal rights/with priority; O/M – observation-/model-based predictionO/M – observation-/model-based prediction

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Conclusions and further researchproblems

• Anticipation property may be quite naturally introduced

into CA models.

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• 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.

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• 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).

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• 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.

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REFERENCES1. 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

[email protected]://ceeisd.org.ua

http://www.summerschool.ssa.org

.ua