Stochastic Properties of disturbed Elementary Cellular Automata - PDF document

Stochastic Properties of disturbed Elementary Cellular Automata Michał Posiewnik Institute of Theoretical Physics and Astrophysics University of Gdansk Gdańsk, 2005 1

Introduction • Physical background. Mechanics vs. statistical physics.Poins of view. eg. oscillators • Source of irreversibility and noises in physical sys- tems • Computability and algorithmic irreducibility • Poincare cycles ans phase space • Ergodicity

Elementary Cellular Automata Basic properties: • Elementary Cellular Automata consisting of an ar- ray of cells: A t i = { x t 0 , x t 1 , ...., x t N } where N is the size of the CA, t ∈ N stand for time and the local rule f defined on some neighborhood S such that: f ( O t ) = x t +1 n ≤ N ∈ N t ∈ n n O t are few cells from A i . • We can enumerate every state of an array A i by some number and build set of states S of CA such that S ⊂ N . • We can construct global function F : S → S . Reasons for using ECA

• simplicity of implementation • complexity of behavior

disturbed Elementary Cellular Automata Some explanations: • How to disturb cellular automata ? • Physical meaning of disturbance • Some intuitions Construction of our Model • I added to ECA two additional cells: A t i = { x t L , x t 0 , x t 1 , ...., x t N , x t R } x t L , x t R are updated at random at every time step, and takes values 0 or 1 with probability 0,5 at every time step t. So our automat became nondetermin- istic. • I used very small automaton consisting only of 8 cells, so it have 256 states.

Main Idea - Definitions Definition: 0.1 Let S = { 0 , ..., N } N = 255 be the state space of ECA. Remark: 0.1 Set S is decimal numeration for global state of ECA. Definition: 0.2 The matrix A = [ p i,j ] i,j ∈ N is called stochastic if it is non - negative and: � p ( i, j ) ≥ 0 , i, j ∈ N, p ( i, j ) = 1 , i ∈ N j ∈ S Definition: 0.3 Let M = [ p i,j ] i,j ∈ S be the matrix, con- sisting of probabilities of transition between states of cel- lular automata in one time step:   p 00 p 01 ..... p 0 N p 10 p 11 ..... p 1 N     . . ..... .     M = . . ..... .     . . ..... .       . . ..... .   p N 0 p N 1 ..... p NN it’s obvious that M is stochastic and that stochastic process generated by it is Markov chain

Remark: 0.2 Now we can represent every rule by the matrix M k , k=0,...,255 where k is decimal description of the rule. Discussion of advantages and disadvantages of idea

Tools definitions Stochastic parameters Definition: 0.4 We define three parameters: µ ( M ) = max 1 ≤ j ≤ N ( min 1 ≤ i ≤ N p ( i, j )) N � δ ( M ) = ( min 1 ≤ i ≤ N p ( i, j )) j =1 N � α ( M ) = min min( p ( i, k ) , p ( j, k )) 1 ≤ i,j ≤ N k =1 where M is some stochastic matrix. Remark: 0.3 µ, α, δ ∈ [0 , 1] . Short explanation of parameters.   p 00 p 01 p 02 p 03 p 10 p 11 p 12 p 13   M =   p 20 p 21 p 22 p 23     p 30 p 31 p 32 p 33

Dissipation Rate Definition: 0.5 Let S be the state set of CA. Let t ∈ N be some parameter (time, step). Let S t be the set of all possible, in sense of evolution, states on t - th step. Definition: 0.6 We call Dt = S t S dissipation rate of CA at time t. Remark: 0.4 Garden of eden of CA is set S − S 1 .

Properties - Definitions Definition: 0.7 Stochastic matrix is called stable if p i,j = p k,j for every i � = k and j ∈ S . Definition: 0.8 If for some stochastic matrix M, M n for n ∈ N n → ∞ is a stable stochastic matrix we called that it posses ergodic property. Definition: 0.9 Biggest subset of the state space S in which every state communicate ( p n ( i, j ) � = 0 , p m ( j, i ) � = 0 ) is called class. Definition: 0.10 A subset C ⊆ S is called closed if and only if � j ∈ C p ( i, j ) = 1 for every i ∈ C . Definition: 0.11 If C=S we say that the state space is irreducible and we called markov chain generated by it irreducible (ergodic) Markov chain. If C ⊂ S than C is absorbing. Definition: 0.12 Let A = [ a ( i, j )] be a nonnegative, quadratic matrix. We say that A is regular if there exists k ≥ N such that A k is positive. Definition: 0.13 A Markov chain is called regular if it’s transition matrix is regular.

Definition: 0.14 A stochastic matrix M that for some n 0 ∈ N , α ( M n 0 ) > 0 is called mixing. Definition: 0.15 A stochastic matrix is mixing if markov chain generated by it is irreducible or regular.

Calculations Now we are able to calculate defined parameters and check every property for all dECA represented by stochastic matrices. Procedure: • Construct a stochastic matrix for dECA • Count n-th power of it, and calculate the parame- ters α, µ, δ and check rank of the matrix • Get dissipation rate and parameters at every time step

First results • Every CA settle down on some stable set S t 0 . The set S t 0 form one or more classes. • We can distinguish three classes of dECA: ergodic, posses ergodic property and reducible. • There is no straight connection between symmetries and our classes

Classes Ergodic dECA in this class are ergodic in the sense of previous definition. Properties: • Dissipation rate at every time step is 0%. So full state space S form one class, and every state is avail- able during evolution • Automats in this class can be either symmetric (90, 105) or not (30, 45) • We find that for this class so called Langton para- meter is always equal 4. But in fact there is lot of dECA which are not ergodic and posses this prop- erty (23,43,77). • The parameters µ, α, δ are 0 during first few steps and than "jumps"to values µ = 0 . 003906 and α = δ = 1 (6 steps for automaton 30, 2 steps for 90). • The stable matrix:   p 00 p 01 ..... p 0 N p 10 p 11 ..... p 1 N     . . ..... .     M stab = . . ..... .      . . ..... .      . . ..... .   p N 0 p N 1 ..... p NN where p i,j = 0 . 003906 for all i,j.

If we check dependence of final state (local rule) from the neighborhood of predecessor, we find out that there is a linear dependence in at least of cells. Remarks • Problems with global evolution • Irreversibility of the rule • Physical behavior of ergodic dECA, averages (time of get to state). • Complexity of the rules

Satisfy Ergodic Property • Like in ergodic class there is only one atractor set C ⊂ S , forms a one class. • Set C can be very small, even consists only one state (f.e. automats 8,40 (dissipation rate D101:99 %)) or almost full set S (f.e 26,41,62 (dissipation rate D101 < 20 %)). • We can distinguish three subclasses of this kind of dECA depends on the evolution of parameters µ, α, δ . 1. µ = α = δ grows together 2. α = δ grows together, and µ grows in its own way 3. Every parameter grows in it’s own special way in general all of parameters grows (because of nu- merical calculations) asymptotically to 1. We can say that automats from this class can be called dissipative. The main parameter is dissipation rate. Lo- cal interactions of this class shows a variety of different behaviors. Probabilities are smeared all around the ma- trix but are not equal (dissipation). It’s a nice question if this kind of automata are ergodic on the subset C.

Reducible • Disturbed automata from this class are reducible in the meaning that there are few different attractors C i ⊂ S, i ∈ N , that forms different classes. • All parameters α, δ, µ are equal to 0 • Dissipation rate is always greater than 50 %. • The final matrix (after remove the states that dis- sipate) looks like:   1 0 0 0 0 0 0 0 0 , 5 0 0 0 0 , 5 0     0 0 0 , 5 0 0 , 5 0 0     M stab = 0 0 0 1 0 0 0     0 0 0 , 5 0 0 , 5 0 0      0 0 , 5 0 0 0 0 , 5 0    0 0 0 0 0 0 1

Conclusions