Long Memory Properties and Complex Systems
• urban and population growth as in Almeida and Gleriani ( 2005 ) and Mavroudi ( 2007 );
• social change ( Nowak & Lewenstein , 1996 );
• consumer |
behavior |
( Rouhaud , |
2000 ); |
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• city traffic ( Rosenblueth & Gershenson , 2011 ).
Thus , if in the previous model all agents behave heterogeneously and globally — since their respective state do not depend on any specific neighborhood range , but rely on all agents states ( global interaction )— in cellular automata occurs the opposite situation . All agents behave locally and homogenously , where each cell represents an agent , so the focus in the study of the longrange dependency rise is the underlying state transition rule in each automaton .
In this section of the work , only Elementary Cellular Automata are studied . Despite the word “ elementary ”, they are still capable of displaying behaviors far from “ elementary ”. In fact , they can exhibit even chaotic behavior . For example , the Rule 110 is one of the most intriguing and beautiful pieces of computer software ever written . With a set of simple eight bitwise rules , one is able to , in principle , to compute any calculation or emulate any computer program , as conjectured by Wolfram ( 1985 ) and proved by Cook ( 2004 ). ( Note that rule numbers are encoded according to the 8-bit sequence of combinations , as shown in Figure 11 .)
Figure 11 . Encoding of Rule 110 .
In Figure 11 , for example , the first transition rule implies that , if the three cells in the previous line are black , the next one will be white ; if the previous left-most and center cells are black and the right-most cell is white , the next will be black , and so on . These eight transition rules define a specific cellular automaton . In addition to that , it is worth noticing that the outputs , when interpreted as bits , encode an 8-bit number — in this given example , the number 110 in a binary base .
Consequently , Figure 11 illustrates a mathematical operation of the type :
where denotes the value of the state at time step t , at position i , with .
Hence , with this set of simple rules , it is possible to generate very complex patterns that seem to evolve chaotically , depending on the initial conditions of the system . According to Wolfram ( 2002 ), this kind of system
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