Entropy ISSN by MDPI

Transcription

1 Etropy 2003, 5, Etropy ISSN by MDPI O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100, Şalıurfa, Turey. E-mal: ahasa@harra.edu.tr Receved: 5 February 2003 / Accepted: 12 Jue 2003 / Publshed: 30 Jue 2003 Abstract: We show that for a addtve oe-dmesoal cellular automata o space of all doubly ftve sequeces wth values a fte set S = {0, 1, 2,..., r-1}, determed x + by a addtve automato rule f(x-,..., x + = (mod r, ad a -varat uform Beroull measure µ, the measure-theoretc etropy of the addtve oe-dmesoal cellular automata f wth respect to µ s equal to hµ ( = 2log r, where 1, r-1 S. We also show that the uform Beroull measure s a measure of maxmal etropy for addtve oedmesoal cellular automata f. Keywords: Cellular Automata, Measure Etropy, Topologcal Etropy. Itroducto Although the addtve cellular automata theory ad the etropy of ths addtve cellular automata have grow up somewhat depedetly, there are strog coectos betwee etropy theory ad cellular automata theory. We gve a troducto to addtve cellular automata theory ad the dscuss the etropy of ths addtve cellular automata. I [1] t was gve a algorthm for computg the topologcal etropy of postvely expasve cellular automata. For a defto ad some propertes of addtve oedmesoal cellular automata we refer to [2] (see also [1-6]. The study of the edomorphsms ad the automorphsms (.e. cotuous shft commutg maps, vertble or o-vertble of the full shft ad ts subshfts was taled by Hedlud ad coworers [3].

2 Etropy 2003, Shereshevsy has show [4] that f the automato map s bpermutatve the the assocated CAacto s strogly mxg ad t determes every drecto a Beroull trasformato wth respect to the uform Beroull measure. I [6] Ward has used (, ε-separated sets to calculate the topologcal etropy of cellular automata. The defto of addtve cellular automata that we have gve here dffers from the defto gve [6]. Detaled formato about cellular automata may be foud Wolfram's paper [7]. I order to state our result, we frst recall a formulato of our problem. We ca also calculate the topologcal etropy of addtve cellular automata. We show that there s a maxmal measure for addtve cellular automata. The orgazato of the paper s as follows: I secto 2 we establsh the basc formulato of problem to state our ma Lemma. I secto 3 we state our ma theorem ad prove t. Secto 4 cotas some remars. I secto 5 there s a cocluso. Formulato of the Problem ad Deftos Let S ={0, 1, 2,..., r-1} be a fte alphabet ad Ω = S Z be the space of doubly fte sequece x= ( x, x S. The shft σ : Ω Ω defed by (σx = x +1 s a homeomorphsm of compact metrc = space Ω. Assume that a fucto f(x -,..., x wth values S s gve. Ths fucto geerates a cellular automata of Ω by the formula: f x = ( y =, y = f (x -,..., x +. Cellular automata s cotuous ad commutes wth left shft (see [3]. Defto 1. (see [2] f s addtve f ad oly f t ca be wrtte as f(x -,..., x + = λ x + (mod r, where λ S. From ow o, we wll say that a cellular automata s addtve f the local rule o whch t s based s addtve. Let us cosder partcular case whe f(x -,..., x + = bloc (s,..., t S (t-s+1. We defe the cylder set x + (mod r. Gve teger s t ad a s[ s,..., t ] t = {x Ω: x j = j, s j t }. Let ξ(s, t deote the partto of Ω to the cylder sets of the form s [ s,..., t ] t, where ( s,..., t rus over S t-s+1. x + Lemma. Suppose that f(x -,..., x + = (mod r ad ξ(-, s a partto of Ω, where 2, the the partto ξ(-, s a geerator for addtve cellular automata. Proof. Let ξ(-, ={ - [ -,..., ] : -,..., S} be a partto of Ω. Evdetly the partto ξ cosst of r 2+1 elemets wth same measure r -(2+1. If we tae the local rule f(x -,..., x + = x (mod r, the t s easy to descrbe ( f -1 (ξ. Let us cosder a cetered + cylder set C = - [ -,..., ].

3 Etropy 2003, The t ca be wrtte ( -1 (C = {{y Ω: y -2 = j -2,..., y 2 = j 2 }: j -2,..., j 2 S}, where y y 0 = - (mod r y y = 0 (mod r y y 2 = (mod r. The partto ξ ( -1 ( ξ cossts of as followg elemets; {y Ω: y -2 = j -2,..., y 2 = j 2 }. Smlarly, t s easy to show that partto ξ ( -1 ( ξ... ( -(-1 ( ξ cossts of as followg elemets; {z Ω: z - = -,..., z = }, where -,..., S. The t s evdet that ξ s a geerator, that s, ( f ( ξ = ε. Example. Let S = { 0, 1} ad ( f(x-2,..., x2 = x 2 = -2 = 0 (mod 2. The we ca wrte -1 ( -2 [01101] 2 = -4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4-4 [ ] 4. It s evdetly ( -1 ( -2 [01101] 2 ξ cossts of oly oe cylder set. Now we calculate the measure of the frst premage of cetered cylder set C = - [ -,..., ] uder addtve cellular automata. µ(( -1 ( C = r 2 µ{x Ω: x -2 = -2,..., x 2 = 2, -2,..., 2 S} = r 2 r -(4+1 = r -(2+1. Smlarly, we ca calculate the measure of the secod premage of cetered cylder set C = - [ -,..., ] uder addtve cellular automata. It s easy to see that the secod premage of C 4 has amout r as cetered cylder sets {x Ω: x -3 = -3,..., x 3 = 3, -3,..., 3 S}. The we have µ(( -2 ( C = r 4 µ{x Ω: x -3 = -3,..., x 3 = 3, -3,..., 3 S} = r 4 r -(6+1 = r -(2+1. If we cotue ths operato, a smlar way, we ca determe the measure of the (-1st premages of the cetered cylder set C = - [ -,..., ]. It s also easy to see that the (-1st premage of the cetered bloc C has amout r 2(-1 as cetered cylder sets

4 Etropy 2003, {x Ω: x - = -,..., x =, -,..., S}. We ca calculate the measure µ(( -(-1 (C = r 2(-1 µ{x Ω: x - = -,..., x =, -,..., S} = r 2(-1 r -(2+1 = r -(2+1. It s obvous that the addtve oe-dmesoal cellular automata s a uform Beroull measurepreservg trasformato. The Measure Etropy of the Addtve Oe-Dmesoal Cellular Automata I ths secto we troduce the measure etropy of the addtve cellular automata defed above. Now we ca state the ma result. Theorem. Let µ be the uform Beroull measure o Ω wth p( = r 1, for each =0,1,...,r-1 ad f(x -,..., x + = x + (mod r. The measure-theoretc etropy of the addtve oe-dmesoal cellular automata wth respect to µ s equal to 2log r. Proof. Now we ca calculate the measure etropy of the addtve cellular automata the Kolmogorov-Sa Theorem [5], amely, h µ ( f = hµ (, ξ, ad ξ s a geerator from Lemma. It s easy to see that f ξ = { - [ -,..., ] : -,..., S}, the we have H(ξ = -r (2+1 µ ( - [ -,..., ] log µ( - [ -,..., ] = -r (2+1 r -(2+1 log r -(2+1 hece = (2+1log r, ξ ( -1 (ξ = {{x Ω: x -2 = -2,..., x 2 = 2 }, -2,..., 2 S}. So H(ξ ( -1 (ξ = -r (4+1 µ ( -2 [ -2,..., 2 ] 2 log µ( -2 [ -2,..., 2 ] 2 = -r (4+1 r -(4+1 log r -(4+1 = (4+1log r. If we cotue, the we have the followg results: ξ ( -1 (ξ... ( -(-1 (ξ = {{x Ω: x - = -,..., x = }, -,..., S}. by usg Fally, t s ot hard show that H(ξ ( -1 (ξ... ( -(-1 (ξ = -r 2+1 µ ( - [ -,..., ] log µ( - [ -,..., ] So we have h µ ( f = 1 1 lm H ( f = 0 = -r 2+1 r -(2+1 log r -(2+1 = (2+1log r ( ξ = lm log r = 2log r.

5 Etropy 2003, Measures wth Maxmal Etropy The metrc o Ω s defed by = ρ ( x, y = 2 d( x, y, where d s a metrc o S ad Ω s a compact metrc space. Defto 2. Let be ay uformly cotuous map of a metrc space (Ω, ρ. A set E Ω s sad to be (, ε-separated uder f for every par x y E there s a m {0, 1,..., -1} wth the property that ρ( ( m (x, m (y > ε. For each compact set K Ω, let s K (, ε = max { E : E K s (, ε-separated uder }, 1 h K ( f,ε = lmsup log sk (, ε, ad h K ( f = lm h (, ε fally, defe the topologcal etropy of ε 0 K to be htop( f = sup h ( f. See [1,6] for the topologcal etropy of a addtve cellular automata. Corollary ([6, Corollary]. A addtve cellular automata : Ω Ω wth local rule f (x -u,..., x 0,..., x = ( a -u x -u a 0 x a x (mod r, where a -u,..., a 0,..., a S ad a -u, a 0, has topologcal etropy log r f u 0, h top ( = ( u + log r f, u 0, u log r f - u 0, I ths paper we assume that a -u =... = a 0 =... = a =1. Defto 3. The measure µ s maxmal the sese that the measure-theoretc etropy of respect to µ cocdes wth the topologcal etropy h top(. Remar 1. Let µ be a uform Beroull measure o Ω wth p( = r 1, for each =0, 1,..., r-1, ad f(x -,..., x + = x (mod r. Because measure etropy of addtve cellular automata f wth respect to µ s equal to + 2log r = h µ ( f = htop(, the addtve cellular automata has a maxmal measure. Remar 2. There are ( r defes the cellular automata r dfferet fuctos f :(x-,..., x + S. The fucto x + K f(x -,..., x + = (mod r wth maxmal etropy amog such fuctos. K wth Cocluso Ths paper cotas the followg results:

6 Etropy 2003, We have foud a geeratg partto for the addtve oe-dmesoal cellular automata (Lemma. We have calculated the measure-theoretc etropy of the addtve oe-dmesoal cellular automata (Theorem. We have compared the measure-theoretc etropy ad the topologcal etropy of the addtve oe-dmesoal cellular automata. We have see that the uform Beroull measure s a measure of maxmal etropy for addtve oe-dmesoal cellular automata. I vew of Remar 1, a terestg ope questo s whether there exsts a measure of maxmal etropy for D-dmesoal (D 2 addtve cellular automata over the rg Z r (r 2. Acowledgemets The author acowledges the support provded by TUBITAK, thas Professor Nasr Gahodjaev for hs ecouragemet ad helpful dscusso durg the preparato of the mauscrpt. I addto, he s grateful to the referees for ther may helpful commets. Refereces 1. M. D'amco, G. Maz, L. Margara. O computg the etropy of cellular automata. Theor. Comput. Sc. 2003, 290, P. Favat, G. Lott, L. Margara. Addtve oe-dmesoal cellular automata are chaotc accordg to Devaey's defto of chaos. Theor. Comput. Sc. 1997, 174, G. A. Hedlud. Edomorphsms ad automorphsms of full shft dyamcal system. Math. Syst. Theor. 1969, 3, M. A. Shereshevsy. Ergodc propertes of certa surjectve cellular automata. Mh. Math 1992, 114, P. Walters. A troducto to ergodc theory; Sprger-Verlag New Yor, 1982, T. Ward. Addtve Cellular Automata ad Volume Growth. Etropy 2000, 2, S. Wolfram. Statstcal mechacs of cellular automata. Rev. Moder Physcs 1983, 55, by MDPI ( Reproducto for ocommercal purposes permtted.