Lecture ISI_10 CELLULAR AUTOMATA INTRODUCTION

OUTLINE OF PRESENTATION Some facts from history Definition of Cellular Automata Parameters of Cellular Automata Cellular Automata approach - examples Conclusions

HISTORY OF CELLULAR AUTOMATA The first Cellular Automata was formally established by John Von Neumann and Stanisław Ulam in late 1940 In 1970, John Conway published his Game of Life which popularized Cellular Automata among researchers The 80’s was the time, when Stephen Wolfram exhaustively examined linear CA, and introduced first formal classification of cellular automata

CELLULAR AUTOMATA - DEFINITION Regular lattice of cell in n-dimensional space Set of states of single cell, usually the same for all cells Set of local transition rules which define next state of cell as a function of cell and neighborhood previous states. CELLS STATES RULES

DIMENSION OF CELLULAR AUTOMATA One dimensional CA Best known Easy to implement Limited use Two dimensional CA Most use Some problems with implementation Three dimensional CA Worst known Needs a lot of computational power The future of CA Approach

SIZE OF CELLULAR AUTOMATA The bigger the best result is obtained The bigger... the more system resources is need the execution of single iteration takes more time the more output data to analyze Size L in 1D CA is the number of cells Size L in 2D CA is the number of columns, if n columns = m rows To compromise the size is need and the size we can handle we have to examine “Size Effect”

Elementary 1D automat komórkowy Elementary Cellular Automaton Najprostszy możliwy jednowymiarowy automat (1D_CA). Każda komórka ma dwa możliwe stany (0 lub 1), zaś reguły rządzące stanem komórki zależą od wartości w trzech sąsiednich komórkach. Istnieje 256 (2^8) różnych elementarnych CA. Wolfram dokonał klasyfikacji tych reguł. Np. rules {#30} = {(0, 0, 0) -> 0, (0, 0, 1)-> 1, (0, 1, 0)-> 1, (0, 1, 1)-> 1, (1, 0, 0)-> 1, (1, 0, 1)-> 0, (1, 1, 0)-> 0, (1, 1, 1)-> 0};

LATTICE GEOMETRY (periodic ones) Cartesian Rectangular Triangular Hexagonal Linear

LATTICE GEOMETRY (aperiodic ) Voronoi Tessellation

MAPPING PROBLEM IN NONE RECTANGULAR LATTICE GEOMETRY We implement Cellular Automata model using common data structure which are linear (vector) or Cartesian rectangular like (vector of vectors). So the input and output data are always array like. When one deals with none rectangular lattice then it is need to define a mapping scheme to rewrite e.g. triangular lattice data into rectangular like data structures.

SOME MAPPING SCHEMA

NEIGHBOURHOODS Von Neumann neighborhood of R=1 Wolfram neighborhood of R=1 Moore neighborhood of R=1 R=2

EXOTIC NEIGHBOURHOODS Arbitrary neighborhood Pair of Cells neighborhood Directed neighborhood

NEIGHBOURHOODS OF NONE RECTANGULAR LATTICE GEOMETRY Triangular lattice neighborhoods Hexagonal lattice neighborhoods

NEIGHBOURHOOD CATASTROPHE PROBLEM When rule is applied, one occurs problem for some cells, as follow:

NEIGHBOURHOOD CATASTROPHE PROBLEM There are many method to avoid this:

NEIGHBOURHOOD CATASTROPHE PROBLEM At most, the the first and the last cells are connected each other: Then one obtains periodic boundary condition:

NEIGHBOURHOOD CATASTROPHE PROBLEM Absorbing boundary condition: In boundary cells, rules are modified to not include missing cells

PERIODIC BOUNDARY CONDITION FOR TWO DIMENSIONAL AUTOMATA The concept is to stick together the first and the last column and rows respectively:

TIME EVOLUTION IN CELLULAR AUTOMATA Synchronous cell updating: apply rule to considered cell and the new state of cell write to secondary array (*) repeat * for each cell rewrite whole secondary array into primary array

TIME EVOLUTION IN CELLULAR AUTOMATA Asynchronous cell updating: apply rule to randomly chosen cell (*) If time domain is necessary, repeat * n-times, where n is the number of all cells iteration is done, after n samplings No need for secondary array, so no danger of lack of memory No artifact, pattern formation in some models Not so easy to implement in parallel system

TRANSITION RULES Rules based on: Differential equation Finite difference equation A function A logic statement Main classification: Deterministic rules Probabilistic rules Fuzzy rules

TRANSITION RULES - EXAMPLES Cooling of solid solid bar 400 iterations

TRANSITION RULES - EXAMPLES Simple Model of Forest Fire Propagation

INITIAL CONDITIONS Random initial condition In predator-prey CA Model, random initial condition leads to cluster formation. What is more random condition makes oscillations become more periodic. Clusters make chaotic oscillations.

INITIAL CONDITIONS Non random initial condition In some predator-prey CA Models, non random local cells configurations at initial state leads to propagate waves or spirals formation. This effect is also observed in some chemical reaction.

CELLULAR AUTOMATA APPROACH - EXAMPLES Hydrodynamics Lattice Gas Chemical reaction (Gray Scott Model)

CELLULAR AUTOMATA APPROACH - EXAMPLES Social model of two opinion formation dynamics. It starts from random initial condition, after a number of iterations: Segregation Suspicious like

CELLULAR AUTOMATA APPROACH - EXAMPLES Evacuation simulation Concrete destruction

PREDATOR-PREY MODEL - CELLULAR AUTOMATA APPROACH The predator-prey model introduced by Lotka and Voltera: (1)

PREDATOR-PREY MODEL - CELLULAR AUTOMATA APPROACH

PREDATOR-PREY MODEL - RESULTS COMPARISON Equation (1) solved by Runge-Kutta 4th order method:

PREDATOR-PREY MODEL - RESULTS COMPARISON Real observations:

PREDATOR-PREY MODEL - RESULTS COMPARISON Cellular Automata simulations:

TRAFFIC FLOW SIMULATION One dimensional synchronous linear cellular automata Boundary condition - periodic asymmetric neighborhood deterministic rules and fuzzy rules

TRAFFIC FLOW SIMULATION - NEW CONCEPT OF RULE Fuzzy rule based on fuzzy controlling:

TRAFFIC FLOW SIMULATION - RESULTS OF SIMULATIONS Real observations Cellular Automata simulations

OIL SLICK SPREADING SIMULATION Previous research: Karafyllidis, I. (1997): A model for prediction of oil slick movement and spreading using cellular automata. Environment International, 23 (6),

OIL SLICK SPREADING SIMULATION Our last research :

CONCLUSIONS We can model a wide range of physical (not only) phenomena Sometimes its much easier to describe complex system using real life language instead of dealing with differential equations This description can be easily translated into set of states and rules We don’t know directly how long is one iteration, how much it is a cell, what is the real character of the state Obtaining quantitative results is a problem Fortunately, sometimes qualitative results satisfy researcher as well Assistance of Mr. M.Burzyński from UKW, Bydgoszcz, in preparation of the presentation is highly appreciated.