How does the neighborhood affect the global behavior of cellular - PowerPoint PPT Presentation

How does the neighborhood affect the global behavior of cellular automata? Hidenosuke Nishio (Kyoto) 11th Workshop on CA Gdansk University, September 3-5, 2005 Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 1/20

Summary Motivation: In order to clarify the significance of the neighborhood, we investigate the global behavior of CA when the neighborhood is differ- ently chosen for a fixed local function. To formalize: Given a local function f : Q s → Q of arity s , investigate the global behavior of CA= ( S, N, Q, f ) , where the neighborhood N = { n 1 , n 2 , ..., n s } of size s is differently chosen. Result: We report here a small result that a CA having the parity lo- cal function preserves the parity of global configurations in spite of any choice of neighborhoods within a simple algebraic constraint. We also consider subneighborhoods of Moore neighborhood in 2-dimensional Euclidean space—incomplete. This paper is along the line of the joint work with Maurice Margenstern and Fritz von Haeseler on the algebraic study of neighborhoods of CA. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 2/20

Neighborhood-sensitive and neighborhood-insensitive theories • The historical and most fundamental neighborhood is one defined by J. von Neumann which consists of 5 cells: Center, North, East, South and West. Using this neighborhood (called von Neumann ) J.von Neumann [5] inaugurated the study of cellular automata by finding the 29 states local rule which results in a self-reproducing machine with computation universality. —Golden rule. • The 9-cells neighborhood larger than von Neumann by 4 cells: North- East, South-East, South-West and North-West is called Moore after E. F. Moore who proved one of the earliest theorems for CA (Garden of Eden theorem) [2]. —Neighborhood-insensitive. • Game of Life [1] assumes Moore for the 2-dimensional binary state CA, where the local rule is cleverly chosen and several interest- ing phenomena like construction and computation universality have been proved to emerge. —Neighborhood-sensitive. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 3/20

Simulation study of natural phenomena When CA is used for the simulation in physics and biology, many au- thors assume the nearest neighbor like von Neumann and Moore . The lattice-gas theory is developed on the triangular grid with the nearest neighborhood of size 6. It is because not only of the intrinsic nature of the system to be simulated but also of the practical reason. It will be helpful to investigate each theory and application of CA with the naive question if the choice of the neighborhood affects the result or not. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 4/20

Cellular Automaton CA= ( S, N, Q, f ) • Cellular space S is expressed by a Cayley graph of a finitely gener- ated group with generators G and relations R . S = � G | R � , where G = { g 1 , g 2 , ..., g r } is a set of finite number of generators (symbols) and R is a finite set of relations (equalities) of words over G and G − 1 , where G − 1 = { g − 1 | g · g − 1 = 1 for g ∈ G } : R = { w i = w ′ i | w i , w ′ i ∈ ( G ∪ G − 1 ) ∗ , i = 1 , ..., n } . Every element of S is presented by a word x ∈ ( G ∪ G − 1 ) ∗ . For x, y ∈ S , if y = xg , where g ∈ G ∪ G − 1 , then an edge labelled by g is drawn from point x to point y . • Neighborhood (index) N = { n 1 , n 2 , ..., n s } ⊂ S . For any cell x ∈ S , the information of cell xn i reaches x in a unit of time. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 5/20

• Set of cell states Q = GF ( q ) where q = p n with prime p and positive integer n . • Local map f : Q N → Q , where Q N is the set of local configura- tions. • Global map F : C → C , where C = Q S is the set of global configurations. F is uniquely induced by f and N ; F ( c )( x ) = f ( c ( xn 1 ) , c ( xn 2 ) , · · · , c ( xn s )) , where c ( x ) is the state of the cell at point x for any c ∈ C and x ∈ S . When starting with a configuration c , the behavior (trajectory) of CA is given by F t +1 ( c ) = F ( F t ( c )) for any c ∈ C and t ≥ 0 , where F 0 ( c ) = c. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 6/20

Neighborhood and neighbors Given a neighborhood (index) N = { n 1 , n 2 , ..., n s } ⊂ S for a cellular space S = � G | R � , we recursively define the neighbors of CA. • Let p ∈ S , then 1-neighbors of p , denoted as pN 1 , is defined to be pN 1 = { pn 1 , pn 2 , ..., pn s } . • m + 1 -neighbors of p , denoted as pN m +1 , is defined by pN m +1 = pN m · N, m ≥ 0 , where pN 0 = { p } . Note that when computing a word pn i of new neighbors of p , the same relation R as in S is applied. The information of cells in pN m reaches p in m -units of time. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 7/20

• ∞ -neighbors of p , denoted as pN ∞ , is defined by ∞ pN ∞ = � pN m . m =0 • ∞ -neighbors of 1 is simply called neighbors (of CA) and denoted as N ∞ . Here we note the following lemma, where generation of a semigroup is denoted by �·|·� sg while that of a group by �·|·� g . Lemma 1 � g 1 , g 2 , ..., g r | R � g = � g 1 , g 2 , ..., g r , g − 1 1 , g − 1 2 , ..., g − 1 r | R � sg . Example: Z 2 = � a, b | ab = ba � g = � a, b, a − 1 , b − 1 | ab = ba � sg Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 8/20

Horse power problem Definition 1 A neighborhood N is said to fill the space S , if � N | R � sg = S . Typically the 3-horse is proved to fill the infinite chess board Z 2 [3]. Theorem 1 The 3-horse N 3 H = { a 2 b, a − 2 , ab − 2 } ⊂ Z 2 fills Z 2 . We have a theorem which provides a method to decide whether a neigh- borhood fills a space or not [4]. Theorem 2 Let x 1 , ..., x s ∈ Z d , where s ≥ d + 1 . Then the neighbor- hood { x 1 , ..., x s } fills the space or � x 1 , ..., x s � sg = Z d , if and only if the following two conditions hold. condition 1: gcd( { det ([ x i 1 , ..., x i d ]) | i 1 , ..., i d ∈ { 1 , ..., s }} ) = 1 . condition 2: 0 ∈ int ( conv ( { x 1 , ..., x s } ) . ( The zero of R d should be in the interior of the convex hull of { x 1 , ..., x s } .) Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 9/20

Parity function This section is devoted to an example for showing how the neighborhood does not affects the global property of CA; The parity of global configura- tions is preserved by computation of CA having the parity local function in spite of arbitrary choice of neighborhoods besides a certain condition. Consider a CA= ( S, N, Q, f ) , where Q is the binary set GF (2) = { 0 , 1 } and S and N = { n 1 , n 2 , ..., n s } ⊂ S are arbitrary. As for the local rule f we consider the (binary) parity function defined by s � f = f BP,N ( n 1 , n 2 , ..., n s ) = c ( n i ) m od 2 , (1) i =1 where c ( n i ) is the state of cell n i . The global map F BP,N : Q S → Q S is induced by f BP,N as usual and also called the (global) parity function. A configuration c ∈ Q S is called finite if # { i | c ( i ) � = 0 , i ∈ S } < ∞ . The finiteness of configurations is preserved by the parity function. In Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 10/20

the following we assume finite configurations. The parity P ( c ) of a configuration c is defined by � P ( c ) = c ( x ) m od 2 , (2) x ∈ S where c ( x ) is the state of cell x . Theorem 3 The parity function preserves the parity of configurations if the neighborhood size is odd. That is, P ( F BP,N ( c )) = P ( c ) , (3) if and only if the neighborhood size is odd. Note that the parity function is not number conserving. Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 11/20

Proof : This theorem is a special case of the following generalized theo- rem. � Example 1 1-dimensional binary parity CA in Z = � a |∅� with a neigh- borhood of size 3 such as N 3 = { a − 1 , 1 , a } , N ′ 3 = { a − 2 , 1 , a 2 } and N ′′ 3 = { 0 , a, a 2 } preserves the parity, but one with N 2 = { 1 , a } does not. The theorem holds for finite spaces like Z ( n ) = � a | a n = 1 � . Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 12/20

Generalized parity function (Modulo p sum) Let Q = { 0 , 1 , ..., p − 1 , ... } = GF ( q ) and define the generalized parity function f GP,N by s � f GP,N ( n 1 , n 2 , ..., n s ) = c ( n i ) mod p. (4) i =1 The global map is denoted by F GP,N . The generalized parity GP ( c ) of a configuration c is defined by � GP ( c ) = c ( x ) mod p. (5) x ∈ S Theorem 4 GP ( F GP,N ( c )) = GP ( c ) , (6) if and only if N ∈ N s , where s = kp + 1 , k ≥ 0 . Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 13/20

Proof : � � GP ( F GP,N ( c )) = F GP,N ( c )( x ) = f GP,N ( xn 1 , xn 2 , ..., xn s ) (7) x ∈ S x ∈ S s s � � � � = c ( xn i ) = c ( xn i ) . (8) x ∈ S i =1 i =1 x ∈ S We note here that � � c ( xn i ) = c ( x ) , 1 ≤ i ≤ s. (9) x ∈ S x ∈ S Then, if s = kp + 1 , we have s � � � c ( xn i ) = c ( x ) = GP ( c ) . (10) i =1 x ∈ S x ∈ S As for the necessity of the condition s = kp +1 , we note the case where p = 2 and s = 2 . Any CA with parity function maps all configurations having parity 1 (and 0) into those of parity 0 and does not preserve the parity. � Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 14/20