Five Lectures on CA 0. Preliminaries Thomas Worsch Department of - PowerPoint PPT Presentation

Five Lectures on CA 0. Preliminaries Thomas Worsch Department of Informatics Karlsruhe Institute of Technology at Hiroshima University, January 2012

Contents Five lectures on cellular automata A little bit of purely mathematical notation Basic definitions for cellular automata

Five lectures on cellular automata A little bit of purely mathematical notation Basic definitions for cellular automata

Topics of the five lectures on CA You have already had the short introduction by Prof. Morita. We will ◮ look at some topics in more detail ◮ look at some new topics ◮ mostly from an algorithmic point of view Topics of the lectures: Monday: Signals and markings Tuesday: Simulations Wednesday: Synchronization Thursday: Sorting Friday: Asynchronous and probabilistic CA

Where to get the pdf files ◮ slides and homework problems are available from http://liinwww.ira.uka.de/~thw/vl-hiroshima/ ◮ use a search engine to find “homepage Thomas Worsch” and look for “Five Lectures on CA”

Five lectures on cellular automata A little bit of purely mathematical notation Basic definitions for cellular automata

Total functions ◮ A and B sets ◮ a total function f : A → B assigns to each a ∈ A exactly one function value f ( a ) ∈ B ◮ A is called the domain of f ◮ B is called the codomain of f ◮ If A is finite, it can be completely specified by a function table ◮ for example ◮ A = {− 1 , 0 , 1 } and B = { 0 , 1 } ◮ some function called ℓ 5 : A → B : − 1 0 1 x ℓ 5 ( x ) 1 0 1

Sets of functions ◮ A and B sets ◮ the set of all total functions from A to B is denoted B A ◮ (possible) reason for this notation: if A , B are finite, there are | B A | = | B | | A | such functions ◮ just look at an example ◮ A = {− 1 , 0 , 1 } and B = { 0 , 1 } ◮ all functions ℓ : A → B :

Sets of functions ◮ A and B sets ◮ the set of all total functions from A to B is denoted B A ◮ (possible) reason for this notation: if A , B are finite, there are | B A | = | B | | A | such functions ◮ just look at an example ◮ A = {− 1 , 0 , 1 } and B = { 0 , 1 } ◮ all functions ℓ : A → B : x − 1 0 1 ℓ 0 ( x ) 0 0 0 ℓ 1 ( x ) 0 0 1 ℓ 2 ( x ) 0 1 0 ℓ 3 ( x ) 0 1 1 ℓ 4 ( x ) 1 0 0 ℓ 5 ( x ) 1 0 1 ℓ 6 ( x ) 1 1 0 ℓ 7 ( x ) 1 1 1

Sets of functions ◮ A and B sets ◮ the set of all total functions from A to B is denoted B A ◮ (possible) reason for this notation: if A , B are finite, there are | B A | = | B | | A | such functions ◮ just look at an example ◮ A = {− 1 , 0 , 1 } and B = { 0 , 1 } ◮ all functions ℓ : A → B : ℓ ( − 1) ℓ (0) ℓ (1) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

A special notation for addition ◮ if N is a set of numbers and x is a number we write x + N = { x + n | n ∈ N } ◮ example: if N = {− 1 , 0 , 1 } then 42 + N = { 41 , 42 , 43 } ◮ generalized even more to two sets of numbers: M + N = { m + n | m ∈ M ∧ n ∈ N }

Five lectures on cellular automata A little bit of purely mathematical notation Basic definitions for cellular automata

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

An example CA you have already seen for all x , z , y ∈ { # , 1 , 2 } f ( x , # , y ) = # f ( # , 1 , y ) = 1 f ( x , 2 , # ) = 2 f ( 2 , 1 , y ) = 2 f ( x , 2 , 1 ) = 1 f ( x , z , y ) = z otherwise t = 0 # 2 1 1 2 1 # t = 1 # 1 2 1 1 2 # t = 2 # 1 1 2 1 2 # t = 3 # 1 1 1 2 2 # t = 4 # 1 1 1 2 2 #

Definition of CA you have already seen A k -dimensional cellular automaton is a quintuple A = ( Z k , Q , N , f , q 0 ) ◮ Z k : k -dimensional discrete space of cells ◮ Q : non-empty finite set of states ◮ N : finite subset of Z k called a neighborhood ◮ f : local function f : Q | N | → Q ◮ q 0 : a quiescent state satisfying f ( q 0 , . . . , q 0 ) = q 0 ◮ α : Z → Q are global configurations ◮ the set of all global configurations is Conf ( Q ) let’s reformulate that just a little bit

Notation for CA (1) ◮ R : discrete space of cells ◮ we will always look at R = Z or R = Z 2 ◮ Q : non-empty finite set of states ◮ N : finite subset of R called a neighborhood ◮ the elements of N are used as differences of coordinates ◮ if x ∈ R , cells with coordinates x + n are its neighbors ◮ example: if N = {− 1 , 0 , 1 } then the neighbors of cell 42 are the cells in 42 + N = { 41 , 42 , 43 }

Notation for CA (2) ◮ a (global) configuration is a function c : R → Q describing the current states of all cells ◮ the “whole state” of a CA ◮ c ∈ Q R

Notation for CA (3) ◮ a local configuration is a function ℓ : N → Q , i.e. ℓ ∈ Q N ◮ an example ◮ N = { (0 , 1) , ( − 1 , 0) , (0 , 0) , (1 , 0) , (0 , − 1) } and Q = { a , b , c } ◮ (0 , 1) ( − 1 , 0) (0 , 0) (1 , 0) (0 , − 1) n ℓ ( n ) a b a c b ◮ or simply: a b a c b ◮ if cell (0 , 0) is indicated one also knows the others ◮ for each CA there is always only a finite number of local configurations

Notation for CA (4) ◮ local function f of a CA specifies how one cell changes its state depending on the local configuration it observes: f : Q N → Q ◮ compare that with f : Q | N | → Q

Notation for CA (4) ◮ local function f of a CA specifies how one cell changes its state depending on the local configuration it observes: f : Q N → Q ◮ compare that with f : Q | N | → Q ◮ example ◮ N = {− 1 , 0 , 1 } and Q = { 0 , 1 }

Notation for CA (4) ◮ local function f of a CA specifies how one cell changes its state depending on the local configuration it observes: f : Q N → Q ◮ compare that with f : Q | N | → Q ◮ example ◮ N = {− 1 , 0 , 1 } and Q = { 0 , 1 } ℓ ( − 1) ℓ (0) ℓ (1) f ( ℓ ) ◮ 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1

Notation for CA (4) ◮ local function f of a CA specifies how one cell changes its state depending on the local configuration it observes: f : Q N → Q ◮ compare that with f : Q | N | → Q ◮ example ◮ N = {− 1 , 0 , 1 } and Q = { 0 , 1 } ℓ ( − 1) ℓ (0) ℓ (1) f ( ℓ ) ◮ 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 ◮ this is the so-called rule 110 = ( 0 1 1 0 1 1 1 0 ) 2

Notation for CA (5) ◮ The global function F describes one step of the whole CA. ◮ F : Q R → Q R ◮ in a global configuration c ∈ Q R each cell x ∈ R “observes” a local configuration L ( c , x ) in its neighborhood defined as L ( c , x ): N → Q : n �→ c ( x + n ) ◮ the transition from c ∈ Q R to F ( c ) is defined by ∀ x ∈ R : F ( c )( x ) = f ( L ( c , x ))