A compact topology for sand automata
In this paper, we exhibit a strong relation between the sand automata configuration space and the cellular automata configuration space. This relation induces a compact topology for sand automata, and a new context in which sand automata are homeomor…
Authors: Alberto Dennunzio (DISCo), Pierre Guillon (IGM), Beno^it Masson (LIF)
A ompat top ology for Sand Automata ∗ † Alb erto Denn unzio ‡ Pierre Guillon § Benoît Masson ¶ Abstrat In this pap er, w e exhibit a strong relation b et w een the sand automata onguration spae and the ellular automata onguration spae. This relation indues a ompat top ology for sand automata, and a new on text in whi h sand automata are homeomorphi to ellular automata ating on a sp ei subshift. W e sho w that the existing top ologial results for sand automata, inluding the Hedlund-lik e represen tation theorem, still hold. In this on text, w e giv e a haraterization of the ellular automata whi h are sand automata, and study some dynamial b eha viors su h as equion tin uit y . F urthermore, w e deal with the nilp oteny . W e sho w that the lassial denition is not meaningful for sand automata. Then, w e in tro due a suitable new notion of nilp oteny for sand automata. Finally , w e pro v e that this simple dynamial b eha vior is undeidable. Keyw ords: sand automata, ellular automata, dynamial systems, subshifts, nilp oteny , undeidabilit y 1 In tro dution Self-organized ritialit y (SOC) is a ommon phenomenon observ ed in a h uge v ariet y of pro esses in ph ysis, biology and omputer siene. A SOC system ev olv es to a ritial state after some nite transien t. An y p erturbation, no matter ho w small, of the ritial state generates a deep reorganization of the ∗ This w ork has b een supp orted b y the In terlink/MIUR pro jet Cellular Automata: T op o- logial Prop erties, Chaos and Asso iated F ormal Languages, b y the ANR Blan Pro jet Syo- more and b y the PRIN/MIUR pro jet F ormal Languages and Automata: Mathematial and Appliativ e Asp ets. † Some of the results of this pap er ha v e b een submitted at JA C 2008 and IFIP TCS 2008 onferenes. ‡ Univ ersità degli studi di Milano-Bio a, Dipartimen to di Informatia Sistemistia e Co- m uniazione, via Bio a degli Arim b oldi 8, 20126 Milano (Italy). Email: dennunziodiso.unimib.i t § Univ ersité P aris-Est, Lab oratoire d'Informatique de l'Institut Gaspard Monge, UMR CNRS 8049, 5 b d Desartes, 77 454 Marne la V allée Cedex 2 (F rane). Email: pierre.guillonuniv-mlv. fr ¶ Lab oratoire d'Informatique F ondamen tale de Marseille (LIF)-CNRS, Aix-Marseille Uni- v ersité, 39 rue Joliot-Curie, 13 453 Marseille Cedex 13 (F rane). Email: benoit.massonlif.univ-m rs.f r 1 whole system. Then, after some other nite transien t, the system rea hes a new ritial state and so on. Examples of SOC systems are: sandpiles, sno w a v alan hes, star lusters in the outer spae, earthquak es, forest res, load bal- ane in op erating systems [2 , 5, 4, 3 , 20 ℄. Among them, sandpiles mo dels are a paradigmati formal mo del for SOC systems [ 11 , 12 ℄. In [6℄, the authors in tro dued sand automata as a generalization of sandpiles mo dels and transp osed them in the setting of disrete dynamial systems. A k ey-p oin t of [ 6℄ w as to in tro due a (lo ally ompat) metri top ology to study the dynamial b eha vior of sand automata. A rst and imp ortan t result w as a fundamen tal represen tation theorem similar to the w ell-kno wn Hedlund's theo- rem for ellular automata [13 , 6 ℄. In [7, 8 ℄, the authors in v estigate sand automata b y dealing with some basi set prop erties and deidabilit y issues. In this pap er w e on tin ue the study of sand automata. First of all, w e in- tro due a dieren t metri on ongurations (i.e. spatial distributions of sand grains). This metri is dened b y means of the relation b et w een sand automata and ellular automata [8 ℄. With the indued top ology , the onguration set turns out to b e a ompat (and not only lo ally ompat), p erfet and to- tally disonneted spae. The strit ompatness giv es a b etter top ologial ba kground to study the b eha vior of sand automata (and in general of dis- rete dynamial systems). In fat, ompatness pro vides a lots of v ery useful results whi h help in the in v estigation of sev eral dynamial prop erties [1 , 16 ℄. W e sho w that all the top ologial results from [6℄ still hold, in partiular the Hedlund-lik e represen tation theorem remains v alid with the ompat top ology . Moreo v er, with this top ology , an y sand automaton is homeomorphi to a el- lular automaton dened on a subset of its usual domain. W e pro v e that it is p ossible to deide whether a giv en ellular automaton is in fat a sand automa- ton. Besides, this relation helps to pro v e some prop erties ab out the dynamial b eha vior of sand automata, su h as the equiv alene b et w een equion tin uit y and ultimate p erio diit y . Then, w e study nilp oteny of sand automata. The lassial denition of nilp oteny for ellular automata [10 , 14 ℄ is not meaningful, sine it prev en ts an y sand automaton from b eing nilp oten t. Therefore, w e in tro due a new denition whi h aptures the in tuitiv e idea that a nilp oten t automaton destro ys all the ongurations: a sand automaton is nilp oten t if all ongurations get loser and loser to a uniform onguration, not neessarily rea hing it. Finally , w e pro v e that this b eha vior is undeidable. The pap er is strutured as follo ws. First, in Setion 2 , w e reall basi de- nitions and results ab out ellular automata and sand automata. Then, in Se- tion 3 , w e dene a ompat top ology and w e pro v e some top ologial results, in partiular the represen tation theorem. Finally , in Setion 4, nilp oteny for sand automata is dened and pro v ed undeidable. 2 2 Denitions F or all a, b ∈ Z with a ≤ b , let [ a, b ] = { a, a + 1 , . . . , b } and g [ a, b ] = [ a, b ] ∪ { + ∞ , − ∞} . F or a ∈ Z , let [ a, + ∞ ) = { a, a + 1 , . . . } \ { + ∞} . Let N + b e the set of p ositiv e in tegers. F or a v etor i ∈ Z d , denote b y | i | the innite norm of i . Let A a (p ossibly innite) alphab et and d ∈ N ∗ . Denote b y M d the set of all the d -dimensional matries with v alues in A . W e assume that the en tries of an y matrix U ∈ M d are all the in teger v etors of a suitable d -dimensional h yp er-retangle [1 , h 1 ] × · · · × [1 , h d ] ⊂ N d + . F or an y h = ( h 1 , . . . , h d ) ∈ N d + , let M d h ⊂ M d b e the set of all the matries with en tries in [1 , h 1 ] × · · · × [1 , h d ] . In the sequel, the v etor h will b e alled the or der of the matries b elonging to M d h . F or a giv en elemen t x ∈ A Z d , the nite p ortion of x of referene p osition i ∈ Z d and order h ∈ N d + is the matrix M i h ( x ) ∈ M d h dened as ∀ k ∈ [1 , h 1 ] × · · · × [1 , h d ] , M i h ( x ) k = x i + k − 1 . F or an y r ∈ N , let r d (or simply r if the dimension is not am biguous) b e the v etor ( r , . . . , r ) . 2.1 Cellular automata and subshifts Let A b e a nite alphab et. A CA ongur ation of dimension d is a funtion from Z d to A . The set A Z d of all the CA ongurations is alled the CA ongur ation sp a e . This spae is usually equipp ed with the T y hono metri d T dened b y ∀ x, y ∈ A Z d , d T ( x, y ) = 2 − k where k = min | j | : j ∈ Z d , x j 6 = y j . The top ology indued b y d T oinides with the pro dut top ology indued b y the disrete top ology on A . With this top ology , the CA onguration spae is a Can tor spae: it is ompat, p erfet (i.e., it has no isolated p oin ts) and totally disonneted. F or an y k ∈ Z d the shift map σ k : A Z d → A Z d is dened b y ∀ x ∈ A Z d , ∀ i ∈ Z d , σ k ( x ) i = x i + k . A funtion F : A Z d → A Z d is said to b e shift- ommuting if ∀ k ∈ Z d , F ◦ σ k = σ k ◦ F . A d -dimensional subshift S is a losed subset of the CA onguration spae A Z d whi h is shift-in v arian t, i.e. for an y k ∈ Z d , σ k ( S ) ⊂ S . Let F ⊆ M d and let S F b e the set of ongurations x ∈ A Z d su h that all p ossible nite p ortions of x do not b elong to F , i.e. for an y i, h ∈ Z d , M i h ( x ) / ∈ F . The set S F is a subshift, and F is alled its set of forbidden patterns. Note that for an y subshift S , it is p ossible to nd a set of forbidden patterns F su h that S = S F . A subshift S is said to b e a subshift of nite typ e (SFT) if S = S F for some nite set F . The lan- guage of a subshift S is L ( S ) = U ∈ M d : ∃ i ∈ Z d , h ∈ N d + , x ∈ S, M i h ( x ) = U (for more on subshifts, see [ 17 ℄ for instane). A el lular automaton is a quadruple h A, d, r , g i , where A is the alphab et also alled the state set , d is the dimension, r ∈ N is the r adius and g : M d 2r + 1 → A is the lo al rule of the automaton. The lo al rule g indues a glob al rule G : A Z d → A Z d dened as follo ws, ∀ x ∈ A Z d , ∀ i ∈ Z d , G ( x ) i = g M i − r 2r + 1 ( x ) . 3 Note that CA are exatly the lass of all shift-omm uting funtions whi h are (uniformly) on tin uous with resp et to the T y hono metri (Hedlund's theorem from [13 ℄). F or the sak e of simpliit y , w e will mak e no distintion b et w een a CA and its global rule G . The lo al rule g an b e extended naturally to all nite matries in the fol- lo wing w a y . With a little abuse of notation, for an y h ∈ [2 r + 1 , + ∞ ) d and an y U ∈ M d h , dene g ( U ) as the matrix obtained b y the sim ultaneous appliation of g to all the M d 2r + 1 submatries of U . F ormally , g ( U ) = M r h − 2 r ( G ( x )) , where x is an y onguration su h that M 0 h ( x ) = U . F or a giv en CA, a state s ∈ A is quies ent (resp., spr e ading ) if for all matries U ∈ M d 2r + 1 su h that ∀ k ∈ [1 , 2 r + 1] d , (resp., ∃ k ∈ [1 , 2 r + 1] d ) U k = s , it holds that g ( U ) = s . Remark that a spreading state is also quiesen t. A CA is said to b e spreading if it has a spreading state. In the sequel, w e will assume that for ev ery spreading CA the spreading state is 0 ∈ A . 2.2 SA Congurations A SA ongur ation (or simply ongur ation ) is a set of sand grains organized in piles and distributed all o v er the d -dimensional lattie Z d . A pile is represen ted either b y an in teger from Z ( numb er of gr ains ), or b y the v alue + ∞ ( sour e of gr ains ), or b y the v alue −∞ ( sink of gr ains ), i.e. it is an elemen t of e Z = Z ∪ {−∞ , + ∞} . One pile is p ositioned in ea h p oin t of the lattie Z d . F ormally , a onguration x is a funtion from Z d to e Z whi h asso iates an y v etor i = ( i 1 , . . . , i d ) ∈ Z d with the n um b er x i ∈ e Z of grains in the pile of p osition i . When the dimension d is kno wn without am biguit y w e note 0 the n ull v etor of Z d . Denote b y C = e Z Z d the set of all ongurations. A onguration x ∈ C is said to b e onstant if there is an in teger c ∈ Z su h that for an y v etor i ∈ Z d , x i = c . In that ase w e write x = c . A onguration x ∈ C is said to b e b ounde d if there exist t w o in tegers m 1 , m 2 ∈ Z su h that for all v etors i ∈ Z d , m 1 ≤ x i ≤ m 2 . Denote b y B the set of all b ounded ongurations. A me asuring devi e β m r of preision r ∈ N and referene heigh t m ∈ Z is a funtion from e Z to ^ [ − r, r ] dened as follo ws ∀ n ∈ e Z , β m r ( n ) = + ∞ if n > m + r , −∞ if n < m − r , n − m otherwise. A measuring devie is used to ev aluate the relativ e heigh t of t w o piles, with a b ounded preision. This is the te hnial basis of the denition of ylinders, distanes and ranges whi h are used all along this artile. In [6℄, the authors equipp ed C with a metri in su h a w a y that t w o on- gurations are at small distane if they ha v e the same n um b er of grains in a nite neigh b orho o d of the pile indexed b y the n ull v etor. The neigh b orho o d is individuated b y putting the measuring devie at the top of the pile, if this latter on tains a nite n um b er of grains. Otherwise the measuring devie is put at 4 heigh t 0 . In order to formalize this distane, the authors in tro dued the notion of ylinder , that w e rename top ylinder . F or an y onguration x ∈ C , for an y r ∈ N , and for an y i ∈ Z d , the top ylinder of x en tered in i and of radius r is the d -dimensional matrix C ′ i r ( x ) ∈ M d 2r + 1 dened on the innite alphab et A = e Z b y ∀ k ∈ [1 , 2 r + 1] d , C ′ i r ( x ) k = x i if k = r + 1 , β x i r ( x i + k − r − 1 ) if k 6 = r + 1 and x i 6 = ±∞ , β 0 r ( x i + k − r − 1 ) otherwise. In dimension 1 and for a onguration x ∈ C , w e ha v e C ′ i r ( x ) = ( β x i r ( x i − r ) , . . . , β x i r ( x i − 1 ) , x i , β x i r ( x i +1 ) , . . . , β x i r ( x i + r )) if x i 6 = ±∞ , while C ′ i r ( x ) = β 0 r ( x i − r ) , . . . , β 0 r ( x i − 1 ) , x i , β 0 r ( x i +1 ) , . . . , β 0 r ( x i + r ) if x i = ±∞ . By means of top ylinders, the distane d ′ : C × C → R + has b een in tro dued as follo ws: ∀ x, y ∈ C , d ′ ( x, y ) = 2 − k where k = min n r ∈ N : C ′ 0 r ( x ) 6 = C ′ 0 r ( y ) o . Prop osition 2.1 ([ 6 , 8℄) With the top olo gy indu e d by d ′ , the ongur ation sp a e is lo al ly omp at, p erfe t and total ly dis onne te d. 2.3 Sand automata F or an y in teger r ∈ N , for an y onguration x ∈ C and an y index i ∈ Z d with x i 6 = ±∞ , the r ange of en ter i and radius r is the d -dimensional matrix R i r ( x ) ∈ M d 2r + 1 on the nite alphab et A = ^ [ − r, r ] ∪ ⊥ su h that ∀ k ∈ [1 , 2 r + 1] d , R i r ( x ) k = ⊥ if k = r + 1 , β x i r ( x i + k − r − 1 ) otherwise. The range is used to dene a sand automaton. It is a kind of top ylinder, where the observ er is alw a ys lo ated on the top of the pile x i (alled the r efer en e ). It represen ts what the automaton is able to see at p osition i . Sometimes the en tral ⊥ sym b ol ma y b e omitted for simpliit y sak e. The set of all p ossible ranges of radius r , in dimension d , is denoted b y R d r . A sand automaton (SA) is a deterministi nite automaton w orking on on- gurations. Ea h pile is up dated syn hronously , aording to a lo al rule whi h omputes the v ariation of the pile b y means of the range. F ormally , a SA is a triple h d, r , f i , where d is the dimension, r is the r adius and f : R d r → [ − r, r ] is 5 the lo al rule of the automaton. By means of the lo al rule, one an dene the glob al rule F : C → C as follo ws ∀ x ∈ C , ∀ i ∈ Z d , F ( x ) i = x i if x i = ±∞ , x i + f ( R i r ( x )) otherwise. Remark that the radius r of the automaton has three dieren t meanings: it rep- resen ts at the same time the n um b er of measuring devies in ev ery dimension of the range (n um b er of piles in the neigh b orho o d), the preision of the measuring devies in the range, and the highest return v alue of the lo al rule (v ariation of a pile). It guaran tees that there are only a nite n um b er of ranges and return v alues, so that the lo al rule has nite desription. The follo wing example illustrates a sand automaton whose b eha vior will b e studied in Setion 4. F or more examples, w e refer to [8 ℄. Example 1 [the automaton N ℄ This automaton destro ys a onguration b y ollapsing all piles to w ards the lo w est one. It dereases a pile when there is a lo w er pile in the neigh b orho o d (see Figure 1 ). Let N = h 1 , 1 , f N i of global rule F N where ∀ a, b ∈ ^ [ − 1 , 1] , f N ( a, b ) = − 1 if a < 0 or b < 0 , 0 otherwise. Figure 1: Illustration of the b eha vior of N . When no misunderstanding is p ossible, w e iden tify a SA with its global rule F . F or an y k ∈ Z d , w e extend the denition of the shift map to C , σ k : C → C is dened b y ∀ x ∈ C , ∀ i ∈ Z d , σ k ( x ) i = x i + k . The r aising map ρ : C → C is dened b y ∀ x ∈ C , ∀ i ∈ Z d , ρ ( x ) i = x i + 1 . A funtion F : C → C is said to b e verti al- ommuting if F ◦ ρ = ρ ◦ F . A funtion F : C → C is innity-pr eserving if for an y onguration x ∈ C and an y v etor i ∈ Z d , F ( x ) i = + ∞ if and only if x i = + ∞ and F ( x ) i = −∞ if and only if x i = −∞ . Remark that the raising map ρ is the sand automaton of radius 1 whose lo al rule alw a ys returns 1 . On the opp osite, the horizon tal shifts σ i are not sand automata: they destro y innite piles b y mo ving them, whi h is not p ermitted b y the denition of the global rule. Theorem 2.1 ([6 , 8 ℄) The lass of SA is exatly the lass of shift and verti al- ommuting, innity-pr eserving funtions F : C → C whih ar e ontinuous w.r.t. the metri d ′ . 6 3 T op ology and dynamis In this setion w e in tro due a ompat top ology on the SA onguration spae b y means of a relation b et w een SA and CA. With this top ology , a Hedlund- lik e theorem still holds and ea h SA turns out to b e homeomorphi to a CA ating on a sp ei subshift. W e also haraterize CA whose ation on this subshift represen ts a SA. Finally , w e pro v e that equion tin uit y is equiv alen t to ultimate p erio diit y , and that expansivit y is a v ery strong notion: there exist no p ositiv ely expansiv e SA. 3.1 A ompat top ology for SA ongurations F rom [8℄, w e kno w that an y SA of dimension d an b e sim ulated b y a suitable CA of dimension d + 1 (and also an y CA an b e sim ulated b y a SA). In partiular, a d -dimensional SA onguration an b e seen as a ( d + 1 )-dimensional CA onguration on the alphab et A = { 0 , 1 } . More preisely , onsider the funtion ζ : C → { 0 , 1 } Z d +1 dened as follo ws ∀ x ∈ C , ∀ i ∈ Z d , ∀ k ∈ Z , ζ ( x ) ( i,k ) = 1 if x i ≥ k , 0 otherwise. A SA onguration x ∈ C is o ded b y the CA onguration ζ ( x ) ∈ { 0 , 1 } Z d +1 . Remark that ζ is an injetiv e funtion. Consider the ( d + 1) -dimensional matrix K ∈ M d +1 ( 1 , . . . , 1 , 2) su h that K 1 ,..., 1 , 2 = 1 and K 1 ,..., 1 , 1 = 0 . With a little abuse of notation, denote S K = S { K } the subshift of ongurations that do not on tain the pattern K . Prop osition 3.1 The set ζ ( C ) is the subshift S K . Pr o of. Ea h d -dimensional SA onguration x ∈ C is o ded b y the ( d + 1) - dimensional CA onguration ζ ( x ) su h that for an y i, h ∈ Z d +1 , M i h ( ζ ( x )) 6 = K , then ζ ( C ) ⊆ S K . Con v ersely , w e an dene a preimage b y ζ for an y y ∈ S K , b y ∀ i ∈ Z d , x i = sup { k : y ( i,k ) = 1 } . Hene ζ ( C ) = S K . Figure 2 illustrates the mapping ζ and the matrix K = 1 0 for the dimension d = 1 . The set of SA ongurations C = e Z Z an b e seen as the subshift S K = ζ ( C ) of the CA ongurations set { 0 , 1 } Z 2 . Denition 3.1 The distan e d : C × C → R + is dene d as fol lows: ∀ x, y ∈ C , d ( x, y ) = d T ( ζ ( x ) , ζ ( y )) . In other w ords, the (w ell dened) distane d b et w een t w o ongurations x, y ∈ C is nothing but the T y hono distane b et w een the ongurations ζ ( x ) , ζ ( y ) in the subshift S K . The orresp onding metri top ology is the { 0 , 1 } Z d +1 pro dut top ology indued on S K . 7 (a) V alid onguration. (b) In v alid onguration. Figure 2: The onguration from Figure 2(a) is v alid, while the onguration from Figure 2(b) on tains the forbidden matrix K : there is a hole. Remark 1 Note that this top olo gy do es not oinide with the top olo gy obtaine d as ountable pr o dut of the disr ete top olo gy on e Z . Inde e d, for any i ∈ Z d , the i th pr oje tion π i : C → e Z dene d by π i ( x ) = x i is not ontinuous in any ongur ation x with x i = ±∞ . However, it is ontinuous in al l ongur ations x suh that x i ∈ Z , sin e ∀ k ∈ Z , ∀ x, y ∈ C , onditions π i ( x ) = k and d ( x, y ) ≤ 2 − max( | i | ,k ) imply that π i ( y ) = k . By denition of this top ology , if one onsiders ζ as a map from C on to S K , ζ turns out to b e an isometri homeomorphism b et w een the metri spaes C (endo w ed with d ) and S K (endo w ed with d T ). As an immediate onsequene, the follo wing results hold. Prop osition 3.2 The set C is a omp at and total ly dis onne te d sp a e wher e the op en b al ls ar e lop en (i.e. lose d and op en) sets. Prop osition 3.3 The sp a e C is p erfe t. Pr o of. Cho ose an arbitrary onguration x ∈ C . F or an y n ∈ N , let l ∈ Z d su h that | l | = n . W e build a onguration y ∈ C , equal to x exept at site l , dened as follo ws ∀ j ∈ Z d \ { l } , y j = x j and y l = 1 if x l = 0 , 0 otherwise. By Denition 3.1 , d ( y , x ) = 2 − n . Consider no w the follo wing notion. Denition 3.2 (ground ylinder) F or any ongur ation x ∈ C , for any r ∈ N , and for any i ∈ Z d , the ground ylinder of x enter e d on i and of r adius r is the d -dimensional matrix C i r ( x ) ∈ M d 2r + 1 dene d by ∀ k ∈ [1 , 2 r + 1] d , C i r ( x ) k = β 0 r ( x i + k − r − 1 ) . 8 F or example in dimension 1 , C i r ( x ) = β 0 r ( x i − r ) , . . . , β 0 r ( x i ) , . . . , β 0 r ( x i + r ) . Figure 3 illustrates top ylinders and ground ylinders in dimension 1 . Re- mark that the on ten t of the t w o kinds of ylinders is totally dieren t. (a) T op ylinder en tered on x i = 4 : C ′ i r ( x ) = (+1 , −∞ , − 3 , 4 , − 2 , − 2 , +1) . (b) Ground ylinder, at heigh t 0 : C i r ( x ) = (+ ∞ , − 2 , +1 , + ∞ , +2 , +2 , + ∞ ) . Figure 3: Illustration of the t w o notions of ylinders on the same onguration, with radius 3 , in dimension 1 . F rom Denition 3.1, w e obtain the follo wing expression of distane d b y means of ground ylinders. Remark 2 F or any p air of ongur ations x, y ∈ C , we have d ( x, y ) = 2 − k wher e k = min r ∈ N : C 0 r ( x ) 6 = C 0 r ( y ) . As a onsequene, t w o ongurations x, y are ompared b y putting b o xes (the ground ylinders) at heigh t 0 around the orresp onding piles indexed b y 0 . The in teger k is the size of the smallest ylinders in whi h a dierene app ears b et w een x and y . This w a y of alulating the distane d is similar to the one used for the distane d ′ , with the dierene that the measuring devies and the ylinders are no w lo ated at heigh t 0 . This is sligh tly less in tuitiv e than the distane d ′ , sine it do es not orresp ond to the denition of the lo al rule. Ho w ev er, this fat is not an issue all the more sine the onguration spae is ompat and the represen tation theorem still holds with the new top ology (Theorem 3.5 ). 3.2 SA as CA on a subshift Let ( X, m 1 ) and ( Y , m 2 ) b e t w o metri spaes. T w o funtions H 1 : X → X , H 2 : Y → Y are (top ologially) onjugate d if there exists a homeomorphism η : X → Y su h that H 2 ◦ η = η ◦ H 1 . 9 W e are going to sho w that an y SA is onjugated to some restrition of a CA. Let F a d -dimensional SA of radius r and lo al rule f . Let us dene the ( d + 1 )- dimensional CA G on the alphab et { 0 , 1 } , with radius 2 r and lo al rule g dened as follo ws (see [ 8℄ for more details). Let M ∈ M d +1 4r + 1 b e a matrix on the nite alphab et { 0 , 1 } whi h do es not on tain the pattern K . If there is a j ∈ [ r + 1 , 3 r ] su h that M (2 r +1 ,... , 2 r +1 ,j ) = 1 and M (2 r +1 ,... , 2 r +1 ,j +1) = 0 , then let R ∈ R d r b e the range tak en from M of radius r en tered on (2 r + 1 , . . . , 2 r + 1 , j ) . See gure 4 for an illustration of this onstrution in dimension d = 1 . Figure 4: Constrution of the lo al rule g of the CA from the lo al rule f of the SA, in dimension 1 . A range R of radius r is asso iated to the matrix M of order 4r + 1 . The new en tral v alue dep ends on the heigh t j of the en tral olumn plus its v ariation. Therefore, dene g ( M ) = 1 if j + f ( R ) ≥ 0 , g ( M ) = 0 if j + f ( R ) < 0 , or g ( M ) = M (2 r +1 ,... , 2 r +1) (en tral v alue un hanged) if there is no su h j . The follo wing diagram omm utes: C F − − − − → C ζ y y ζ S K − − − − → G S K , (1) i.e. G ◦ ζ = ζ ◦ F . As an immediate onsequene, w e ha v e the follo wing result. Prop osition 3.4 A ny d -dimensional SA F is top olo gi al ly onjugate d to a suit- able ( d + 1) -dimensional CA G ating on S K . Being a dynamial submo del, SA share prop erties with CA, some of whi h are pro v ed b elo w. Ho w ev er, man y results whi h are true for CA are no longer true for SA; for instane, injetivit y and bijetivit y are not equiv alen t, as pro v ed in [7 ℄. Th us, SA deserv e to b e onsidered as a new mo del. Corollary 3.3 The glob al rule F : C → C of a SA is uniformly ontinuous w.r.t distan e d . Pr o of. Let G b e the global rule of the CA whi h sim ulates the giv en SA. Sine the diagram (1) omm utes and ζ is a homeomorphism, F = ζ − 1 ◦ G ◦ ζ . Sine G is a on tin uous map and, b y Prop osition 3.2 , C is ompat, then the thesis is obtained. 10 F or ev ery a ∈ Z , let P a = π − 1 0 ( { a } ) b e the lop en (and ompat) set of all ongurations x ∈ C su h that x 0 = a . Lemma 3.4 L et F : C → C b e a ontinuous and innity-pr eserving map. Ther e exists an inte ger l ∈ N suh that for any ongur ation x ∈ P 0 we have | F ( x ) 0 | ≤ l . Pr o of. Sine F is on tin uous and innit y-preserving, the set F ( P 0 ) is ompat and inluded in π − 1 0 ( Z ) . F rom Remark 1 , π 0 is on tin uous on the set π − 1 0 ( Z ) and in partiular it is on tin uous on the ompat F ( P 0 ) . Hene π 0 ( F ( P 0 )) is a ompat subset of e Z on taining no innit y , and therefore it is inluded in some in terv al [ − l , l ] , where l ∈ N . Theorem 3.5 A mapping F : C → C is the glob al tr ansition rule of a sand automaton if and only if al l the fol lowing statements hold ( i ) F is (uniformly) ontinuous w.r.t the distan e d ; ( ii ) F is shift- ommuting; ( iii ) F is verti al- ommuting; ( iv ) F is innity-pr eserving. Pr o of. Let F b e the global rule of a SA. By denition of SA, F is shift- omm uting, v ertial-omm uting and innit y-preserving. F rom Corollary 3.3 , F is also uniformly on tin uous. Con v ersely , let F b e a on tin uous map whi h is shift-omm uting, v ertial- omm uting, and innit y-preserving. By ompatness of the spae C , F is also uniformly on tin uous. Let l ∈ N b e the in teger giv en b y Lemma 3.4 . Sine F is uniformly on tin uous, there exists an in teger r ∈ N su h that ∀ x, y ∈ C C 0 r ( x ) = C 0 r ( y ) ⇒ C 0 l ( F ( x )) = C 0 l ( F ( y )) . W e no w onstrut the lo al rule f : R d r → [ − r, r ] of the automaton. F or an y input range R ∈ R d r , set f ( R ) = F ( x ) 0 , where x is an arbitrary onguration of P 0 su h that ∀ k ∈ [1 , 2 r + 1] , k 6 = r + 1 , β 0 r ( x k − r − 1 ) = R k . Note that the v alue of f ( R ) do es not dep end on the partiular hoie of the onguration x ∈ P 0 su h that ∀ k 6 = r + 1 , β 0 r ( x k − r − 1 ) = R k . Indeed, Lemma 3.4 and uniform on tin uit y together ensure that for an y other onguration y ∈ P 0 su h that ∀ k 6 = r + 1 , β 0 r ( y k − r − 1 ) = R k , w e ha v e F ( y ) 0 = F ( x ) 0 , sine β 0 l ( F ( x ) 0 ) = β 0 l ( F ( y ) 0 ) and | F ( y ) 0 | ≤ l . Th us the rule f is w ell dened. W e no w sho w that F is the global mapping of the sand automaton of radius r and lo al rule f . Thanks to ( iv ) , it is suien t to pro v e that for an y x ∈ C and for an y i ∈ Z d with | x i | 6 = ∞ , w e ha v e F ( x ) i = x i + f R i r ( x ) . By ( ii ) and ( iii ) , for an y i ∈ Z d su h that | x i | 6 = ∞ , it holds that F ( x ) i = ρ x i ◦ σ − i F ( σ i ◦ ρ − x i ( x )) i = x i + σ − i F ( σ i ◦ ρ − x i ( x )) i = x i + F ( σ i ◦ ρ − x i ( x )) 0 . 11 Sine σ i ◦ ρ − x i ( x ) ∈ P 0 , w e ha v e b y denition of f F ( x ) i = x i + f R 0 r ( σ i ◦ ρ − x i ( x )) . Moreo v er, b y denition of the range, for all k ∈ [1 , 2 r + 1] d , R 0 r ( σ i ◦ ρ − x i ( x )) k = β [ σ i ◦ ρ − x i ( x )] 0 r ( σ i ◦ ρ − x i ( x ) k ) = β 0 r ( x i + k − x i ) = β x i r ( x i + k ) , hene R 0 r ( σ i ◦ ρ − x i ( x )) = R i r ( x ) , whi h leads to F ( x ) i = x i + f R i r ( x ) . W e no w deal with the follo wing question: giv en a ( d + 1 )-dimensional CA, do es it represen t a d -dimensional SA, in the sense of the onjugay expressed b y diagram 1 ? In order to answ er to this question w e start to express the ondition under whi h the ation of a CA G an b e restrited to a subshift S F , i.e., G ( S F ) ⊆ S F (if this fat holds, the subshift S F is said to b e G -in v arian t). Lemma 3.6 L et G and S F b e a CA and a subshift of nite typ e, r esp e tively. The ondition G ( S F ) ⊆ S F is satise d i for any U ∈ L ( S F ) and any H ∈ F of the same or der than g ( U ) , it holds that g ( U ) 6 = H . Pr o of. Supp ose that G ( S F ) ⊆ S F . Cho ose arbitrarily H ∈ F and U ∈ L ( S F ) , with g ( U ) and H of the same order. Let x ∈ S F on taining the matrix U . Sine G ( x ) ∈ S F , then g ( U ) ∈ L ( S F ) , and so g ( U ) 6 = H . Con v ersely , if x ∈ S F and G ( x ) / ∈ S F , then there exist U ∈ L ( S F ) and H ∈ F with g ( U ) = H . The follo wing prop osition giv es a suien t and neessary ondition under whi h the ation of a CA G on ongurations of the G -in v arian t subshift S K = C preserv es an y olumn whose ells ha v e the same v alue. Lemma 3.7 L et G b e a ( d + 1) -dimensional CA with state set { 0 , 1 } and S K b e the subshift r epr esenting SA ongur ations. The fol lowing two statements ar e e quivalent: ( i ) for any x ∈ S K with x (0 ,..., 0 ,i ) = 1 (r esp., x (0 ,..., 0 ,i ) = 0 ) for al l i ∈ Z , it holds that G ( x ) (0 ,..., 0 ,i ) = 1 (r esp., G ( x ) (0 ,..., 0 ,i ) = 0 ) for al l i ∈ Z . ( ii ) for any matrix U ∈ M d 2r + 1 ∩ L ( S K ) with U ( r +1 ,... ,r +1 ,k ) = 1 (r esp., U ( r +1 ,... ,r +1 ,k ) = 0 ) and any k ∈ [1 , 2 r + 1] , it holds that g ( U ) = 1 (r esp., g ( U ) = 0 ). Pr o of. Supp ose that (1) is true. Let U ∈ M d 2r + 1 ∩ L ( S K ) b e a matrix with U ( r +1 ,... ,r +1 ,k ) = 1 and let x ∈ S K b e a onguration su h that x (0 ,..., 0 ,i ) = 1 for all i ∈ Z and M − r 2r + 1 ( x ) = U . Sine G ( x ) (0 ,..., 0 ,i ) = 1 for all i ∈ Z , and M 0 2r + 1 ( x ) = U , then g ( U ) = 1 . Con v ersely , let x ∈ S K with x (0 ,..., 0 ,i ) = 1 for all i ∈ Z . By shift-in v ariane, w e obtain G ( x ) (0 ,..., 0 ,i ) = 1 for all i ∈ Z . Lemmas 3.6 and 3.7 immediately lead to the follo wing onlusion. Prop osition 3.5 It is de idable to he k whether a given ( d + 1 )-dimensional CA orr esp onds to a d -dimensional SA. 12 3.3 Some dynamial b eha viors SA are v ery in teresting dynamial systems, whi h in some sense lie b et w een d -dimensional and d + 1 -dimensional CA. Indeed, w e ha v e seen in the previ- ous setion that the latter an sim ulate d -dimensional SA, whi h an, in turn, sim ulate d -dimensional CA. F or the dimension d = 1 , a lassiation of CA in terms of their dynamial b eha vior w as giv en in [15 ℄. Things are v ery dieren t as so on as w e get in to dimension d = 2 , as noted in [ 19 , 18 ℄. The question is no w whether the omplexit y of the SA mo del is loser to that of the lo w er or the higher-dimensional CA. Let ( X, m ) b e a metri spae and let H : X → X b e a on tin uous appli- ation. An elemen t x ∈ X is an e qui ontinuity p oin t for H if for an y ε > 0 , there exists δ > 0 su h that for all y ∈ X , m ( x, y ) < δ implies that ∀ n ∈ N , m ( H n ( x ) , H n ( y )) < ε . The map H is e qui ontinuous if for an y ε > 0 , there exists δ > 0 su h that for all x, y ∈ X , m ( x, y ) < δ implies that ∀ n ∈ N , m ( H n ( x ) , H n ( y )) < ε . If X is ompat, H is equion tin uous i all elemen ts of X are equion tin uit y p oin ts. An elemen t x ∈ X is ultimately p erio di for H if there exist t w o in tegers n ≥ 0 (the prep erio d) and p > 0 (the p erio d) su h that H n + p ( x ) = H n ( x ) . H is ultimately p erio di if there exist n ≥ 0 and p > 0 su h that H n + p = H n . H is sensitive (to the initial onditions) if there is a onstan t ε > 0 su h that for all p oin ts x ∈ X and all δ > 0 , there is a p oin t y ∈ X and an in teger n ∈ N su h that m ( x, y ) < δ but m ( F n ( x ) , F n ( y )) > ε . H is p ositively exp ansive if there is a onstan t ε > 0 su h that for all distint p oin ts x, y ∈ X , there exists n ∈ N su h that m ( H n ( x ) , H n ( y )) > ε . The top ologial onjugay b et w een a SA and some CA ating on the sp eial subshift S K helps to adapt some prop erties of CA. In partiular, the follo wing haraterization of equion tin uous CA an b e adapted from Theorem 4 of [15 ℄. Prop osition 3.6 If F is a SA, then the fol lowing statements ar e e quivalent: 1. F is e qui ontinuous. 2. F is ultimately p erio di. 3. A l l ongur ations of C ar e ultimately p erio di for F . Pr o of. 3 ⇒ 2 : F or an y n ≥ 0 and p > 0 , let D n,p = { x : F n + p ( x ) = F n ( x ) } . Remark that C = S n,p ∈ N D n,p is the union of these losed subsets. As C is om- plete of nonempt y in terior, b y the Baire Theorem, there are in tegers n, p ∈ N for whi h the set D n,p has nonempt y in terior. Hene the onjugate image ζ ( D n,p ) has nonempt y in terior to o, and it an easily b e seen that it is a subshift. It is kno wn that the only subshift with nonempt y in terior is the full spae; hene D n,p = C . 2 ⇒ 3: ob vious. 2 ⇒ 1: Let F b e ultimately p erio di with F n + p = F n for some n ≥ 0 , p > 0 . Sine F, F 2 , . . . , F n + p − 1 are uniformly on tin uous maps, for an y ε > 0 there exists δ > 0 su h that for all x, y ∈ C with d ( x, y ) < δ , it holds that ∀ q ∈ N , q < n + p , d ( F q ( x ) , F q ( y )) < ε . Sine for an y t ∈ N F t is equal to some F q with 13 q < n + p , the map F is equion tin uous. 1 ⇒ 2: F or the sak e of simpliit y , w e giv e the pro of for a giv en one-dimensional equion tin uous SA F . Let G b e the global rule of the t w o-dimensional CA whose ation on S K is onjugated to F . By Denition 3.1 , and sine the diagram 1 omm utes, the map G : S K → S K is equion tin uous w.r.t. d T . So, for ε = 1 , there exists l ∈ N su h that for all x, y ∈ S K , if M − l 2l + 1 ( x ) = M − l 2l + 1 ( y ) , then for all t ∈ N , G t ( x ) 0 = G t ( y ) 0 . Consider no w ongurations ζ ( c ) , where c ∈ {−∞ , + ∞} Z has either the form ( . . . , −∞ , − ∞ , + ∞ , + ∞ , . . . ) or ( . . . , + ∞ , + ∞ , −∞ , −∞ , . . . ) . Sine ev ery ζ ( c ) are ultimately p erio di (with prep erio d n = 0 and p erio d p = 1 ) and G is equion tin uous, for an y k ∈ Z 2 and an y y ∈ S K with M k − l 2l + 1 ( y ) = M k − l 2l + 1 ( ζ ( c )) , it holds that the sequene { G t ( y ) k } t ∈ N is ultimately p erio di. F or an y U ∈ L ( S K ) ∩ M 2 2l + 1 , let x U b e the onguration su h that M − l 2l + 1 ( x ) = U , x ( i,j ) = 0 if − l ≤ i ≤ l and j > l , and x ( i,j ) = 1 otherwise. Exept for the nite en tral region, x U is made b y the rep etition of a nite n um b er of matries app earing inside on- gurations ζ ( c ) . Hene, x U is an ultimately p erio di onguration with some prep erio d n U and p erio d p U . Then, for an y y ∈ S K with M − l 2l + 1 ( y ) = U , the sequene { G t ( y ) 0 } t ∈ N is ultimately p erio di with prep erio d n U and p erio d p U . Set n = max { n U : U ∈ L ( S K ) ∩ M 2 2l + 1 } and p = lm { p U : U ∈ L ( S K ) ∩ M 2 2l + 1 } where lm is the least ommon m ultiple. Th us, for an y onguration z ∈ S K , w e ha v e that G n ( z ) 0 = G n + p ( z ) 0 . By shift-in v ariane, w e obtain ∀ k ∈ Z 2 , G n ( z ) k = G n + p ( z ) k . Conluding, G is ultimately p erio di and then F is to o. In [15 ℄ is presen ted a lassiation of CA in to four lasses: equion tin uous CA, non equion tin uous CA admitting an equion tin uit y onguration, sensitiv e but not p ositiv ely expansiv e CA, p ositiv ely expansiv e CA. This lassiation is no more relev an t in the on text of SA sine the lass of p ositiv ely expansiv e SA is empt y . This result an b e related to the absene of p ositiv ely expansiv e t w o-dimensional CA (see [19 ℄), though the pro of is m u h dieren t. Prop osition 3.7 Ther e ar e no p ositively exp ansive SA. Pr o of. Let F a SA and δ = 2 − k > 0 . T ak e t w o distint ongurations x, y ∈ C su h that ∀ i ∈ [ − k , k ] , x i = y i = + ∞ . By innit y-preservingness, w e get ∀ n ∈ N , ∀ i ∈ [ − k, k ] , F n ( x ) i = F n ( y ) i = + ∞ , hene d ( F n ( x ) , F n ( y )) < δ . An imp ortan t op en question in the dynamial b eha vior of SA is the existene of non-sensitiv e SA without an y equion tin uit y onguration. An example for t w o-dimensional CA is giv en in [18 ℄, but their metho d an hardly b e adapted for SA. This ould lead to a lassiation of SA in to four lasses: equion- tin uous, admitting an equion tin uit y onguration (but not equion tin uous), non-sensitiv e without equion tin uit y ongurations, sensitiv e. Another issue is the deidabilit y of these lasses. In [7 ℄, the undeidabilit y of SA ultimate p erio diit y w as pro v ed on the partiular subsets of nite and p erio di ongurations. It follo ws diretly that equion tin uit y on these subsets is undeidable. The question is still op en for the whole onguration spae C . 14 4 The nilp oteny problem In this setion w e giv e a denition of nilp oteny for SA. Then, w e pro v e that nilp oteny b eha vior is undeidable (Theorem 4.5 ). 4.1 Nilp oteny of CA Here w e reall the basi denitions and prop erties of nilp oten t CA. Nilp oteny is among the simplest dynamial b eha vior that an automaton ma y exhibit. In- tuitiv ely , an automaton dened b y a lo al rule and w orking on ongurations (either C or A Z d ) is nilp oten t if it destro ys ev ery piee of information in an y initial onguration, rea hing a ommon onstan t onguration after a while. F or CA, this is formalized as follo ws. Denition 4.1 (CA nilp oteny [10 , 14 ℄) A CA G is nilp otent if ∃ c ∈ A, ∃ N ∈ N ∀ x ∈ A Z d , ∀ n ≥ N , G n ( x ) = c . Remark that in a similar w a y to the pro of of Prop osition 3.6 , Denition 4.1 an b e restated as follo ws: a CA is nilp oten t if and only if it is nilp oten t for all initial ongurations. Spreading CA ha v e the follo wing stronger haraterization. Prop osition 4.1 ([9 ℄) A CA G , with spr e ading state 0 , is nilp otent i for every x ∈ A Z d , ther e exists n ∈ N and i ∈ Z d suh that G n ( x ) i = 0 (i.e. 0 app e ars in the evolution of every ongur ation). The previous result immediately leads to the follo wing equiv alene. Corollary 4.2 A CA of glob al rule G , with spr e ading state 0 , is nilp otent if and only if for al l ongur ations x ∈ A Z d , lim n →∞ d T ( G n ( x ) , 0 ) = 0 . Reall that the CA nilp oteny is undeidable [14 ℄. Remark that the pro of of this result also w orks for the restrited lass of spreading CA. Theorem 4.3 ([14 ℄) F or a given state s , it is unde idable to know whether a el lular automaton with spr e ading state s is nilp otent. 4.2 Nilp oteny of SA A diret adaptation of Denition 4.1 to SA is v ain. Indeed, assume F is a SA of radius r . F or an y k ∈ Z d , onsider the onguration x k ∈ B dened b y x k 0 = k and x k i = 0 for an y i ∈ Z d \{ 0 } . Sine the pile of heigh t k ma y derease at most b y r during one step of ev olution of the SA, and the other piles ma y inrease at most b y r , x k requires at least ⌈ k / 2 r ⌉ steps to rea h a onstan t onguration. Th us, there exists no ommon in teger n su h that all ongurations x k rea h a onstan t onguration in time n . This is a ma jor dierene with CA, whi h 15 is essen tially due to the un b ounded set of states and to the innit y-preserving prop ert y . Th us, w e prop ose to lab el as nilp oten t the SA whi h mak e ev ery pile ap- proa h a onstan t v alue, but not neessarily rea hing it ultimately . This nilp o- teny notion, inspired b y Prop osition 4.2 , is formalized as follo ws for a SA F : ∃ c ∈ Z , ∀ x ∈ C , lim n →∞ d ( F n ( x ) , c ) = 0 . Remark that c shall not b e tak en in the full state set e Z , b eause allo wing in- nite v alues for c w ould not orresp ond to the in tuitiv e idea that a nilp oten t SA destro ys a onguration (otherwise, the raising map w ould b e nilp oten t). An yw a y , this denition is not satisfying b eause of the v ertial omm utativit y: t w o ongurations whi h dier b y a v ertial shift rea h t w o dieren t ongura- tions, and then no nilp oten t SA ma y exist. A p ossible w a y to w ork around this issue is to mak e the limit onguration dep end on the initial one: ∀ x ∈ C , ∃ c ∈ Z , lim n →∞ d ( F n ( x ) , c ) = 0 . Again, sine SA are innit y-preserving, an innite pile annot b e destro y ed (nor, for the same reason, an an innite pile b e built from a nite one). There- fore nilp oteny has to in v olv e the ongurations of Z Z d , i.e. the ones without innite piles. Moreo v er, ev ery onguration x ∈ Z Z d made of regular steps (i.e. in dimension 1 , for all i ∈ Z , x i − x i − 1 = x i +1 − x i ) is in v arian t b y the SA rule (p ossibly omp osing it with the v ertial shift). So it annot rea h nor approa h a onstan t onguration. Th us, the larger reasonable set on whi h nilp oteny migh t b e dened is the set of b ounded ongurations B . This leads to the follo wing formal denition of nilp oteny for SA. Denition 4.4 (SA nilp oteny) A SA F is nilp otent if and only if ∀ x ∈ B , ∃ c ∈ Z , lim n →∞ d ( F n ( x ) , c ) = 0 . The follo wing prop osition sho ws that the lass of nilp oten t SA is nonempt y . Prop osition 4.2 The SA N fr om Example 1 is nilp otent. Pr o of. Let x ∈ B , let i ∈ Z su h that for all j ∈ Z , x j ≥ x i . Clearly , after x i +1 − x i steps, F x i +1 − x i N ( x ) i +1 = F x i +1 − x i N ( x ) i = x i . By immediate indution, w e obtain that for all j ∈ Z there exists n j ∈ N su h that F n j N ( x ) j = x i , hene lim n →∞ d ( F n N ( x ) , x i ) = 0 . Similar nilp oten t SA an b e onstruted with an y radius and in an y dimen- sion. 16 4.3 Undeidabilit y The main result of this setion is that SA nilp oteny is undeidable (Theo- rem 4.5 ), b y reduing the nilp oteny of spreading CA to it. This emphasizes the fat that the dynamial b eha vior of SA is v ery diult to predit. W e think that this result migh t b e used as the referene undeidable problem for further questions on SA. Problem Nil inst ane : a SA A = h d, r, λ i ; question : is A nilp oten t? Theorem 4.5 The pr oblem Nil is unde idable. Pr o of. This is pro v ed b y reduing Nil to the nilp oteny of spreading ellular automata. Remark that it is suien t to sho w the result in dimension 1 . Let S b e a spreading ellular automaton S = h A, 1 , s, g i of global rule G , with nite set of in teger states A ⊂ N on taining the spreading state 0 . W e sim ulate S with the sand automaton A = h 1 , r = max(2 s, max A ) , f i of global rule F using the follo wing te hnique, also dev elop ed in [ 8℄. Let ξ : A Z → B b e a funtion whi h inserts mark ers ev ery t w o ells in the CA onguration to obtain a b ounded SA onguration. These mark ers allo w the lo al rule of the SA to kno w the absolute state of ea h pile and b eha v e as the lo al rule of the CA. T o simplify the pro of, the mark ers are put at heigh t 0 (see Figure 5): ∀ y ∈ A Z , ∀ i ∈ Z , ξ ( y ) i = 0 (mark er) if i is o dd , y i/ 2 otherwise. This an lead to an am biguit y when all the states in the neigh b orho o d of size 4 s + 1 are at state 0 , as sho wn in the piture. But as in this sp eial ase the state 0 is quiesen t for g , this is not a problem: the state 0 is preserv ed, and mark ers are preserv ed. Figure 5: Illustration of the funtion ξ used in the sim ulation of the spreading CA S b y A . The thi k segmen ts are the mark ers used to distinguish the states of the CA, put at heigh t 0 . There is an am biguit y for the t w o piles indiated b y the arro ws: with a radius 2, the neigh b orho o ds are the same, although one of the piles is a mark er and the other the state 0 . 17 The lo al rule f is dened as follo ws, for all ranges R ∈ R 1 r , f ( R ) = 0 if R − 2 s +1 , R − 2 s +3 , . . . , R − 1 , R 1 , . . . , R 2 s − 1 ∈ A , g ( R − 2 s + a, R − 2 s +2 + a, . . . , R − 2 + a, a, R 2 + a, . . . , R 2 s + a ) − a if R − 2 s +1 = R − 2 s +3 = · · · = R 2 s − 1 = a < 0 and − a ∈ A . (2) The rst ase is for the mark ers (and state 0 ) whi h remain un hanged, the seond ase is the sim ulation of g in the ev en piles. As pro v ed in [ 8℄, for an y y ∈ A Z it holds that ξ ( G ( y )) = F ( ξ ( y )) . The images b y f of the remaining ranges will b e dened later on, rst a few new notions need to b e in tro dued. A sequene of onseutiv e piles ( x i , . . . , x j ) from a onguration x ∈ B is said to b e valid if it is part of an eno ding of a CA onguration, i.e. x i = x i +2 = · · · = x j (these piles are mark ers) and for all k ∈ N su h that 0 ≤ k < ( j − i ) / 2 , x i +2 k +1 − x i ∈ A (this is a v alid state). W e extend this denition to ongurations, when i = −∞ and j = + ∞ , i.e. x ∈ ρ c ◦ ξ ( A Z ) for a giv en c ∈ Z ( x ∈ B is v alid if it is the raised image of a CA onguration). A sequene (or a onguration) in invalid if it is not v alid. First w e sho w that starting from a v alid onguration, the SA A is nilp oten t if and only if S is nilp oten t. This is due to the fat that w e hose to put the mark ers at heigh t 0 , hene for an y v alid eno ding of the CA x = ρ c ◦ ξ ( y ) , with y ∈ A Z and c ∈ Z , lim n →∞ d T ( G n ( y ) , 0) = 0 if and only if lim n →∞ d ( F n ( x ) , c ) = 0 . It remains to pro v e that for an y in v alid onguration, A is also nilp oten t. In order to ha v e this b eha vior, w e add to the lo al rule f the rules of the nilp oten t automaton N for ev ery in v alid neigh b orho o d of width 4 s + 1 . F or all ranges R ∈ R 1 r not onsidered in Equation (2 ) , f ( R ) = − 1 if R − r < 0 or R − r +1 < 0 or · · · or R r < 0 , 0 otherwise. (3) Let x ∈ B b e an in v alid onguration. Let k ∈ Z b e an y index su h that ∀ l ∈ Z , x l ≥ x k . Let i, j ∈ Z b e resp etiv ely the lo w est and greatest indies su h that i ≤ k ≤ j and ( x i , . . . , x j ) is v alid ( i ma y equal j ). Remark that for all n ∈ N , ( F n ( x ) i , . . . , F n ( x ) j ) remains v alid. Indeed, the mark ers are b y onstrution the lo w est piles and Equations (2) and (3 ) do not mo dify them. The piles o ding for non-zero states an hange their state b y Equation ( 2), or derease it b y 1 b y Equation ( 3), whi h in b oth ases is a v alid eno ding. Moreo v er, the piles x i − 1 and x j +1 will rea h a v alid v alue after a nite n um b er of steps: as long as they are in v alid, they derease b y 1 un til they rea h a v alue whi h o des for a v alid state. Hene, b y indution, for an y indies a, b ∈ Z , there exists N a,b su h that for all n ≥ N a,b the sequene ( F n ( x ) a , . . . , F n ( x ) b ) is v alid. In partiular, after N − 2 N r − 1 , 2 N r +1 step, there is a v alid sequene of length 4 N r + 3 en tered on the origin (here, N is the n um b er of steps needed b y S to rea h the onguration 0 , giv en b y Denition 4.1 ). Hene, after N − 2 N r, 2 N r + N 18 steps, the lo al rule of the CA S applied on this v alid sequene leads to 3 onseutiv e zeros at p ositions − 1 , 0 , 1 . All these steps are illustrated on Figure 6. 0 0 time i N j −2Nr+1,2Nr+1 N −2Nr+1,2Nr+1 + N Figure 6: Destrution of the in v alid parts. The lo w est v alid sequene (in gra y) extends un til it is large enough. Then after N other steps the 3 en tral piles (hat hed) are destro y ed b eause the rule of the CA is applied orretly . Similarly , w e pro v e that for all n ≥ N − 2 N r − k, 2 N r + k + N , the sequene ( F n ( x ) − k , . . . , F n ( x ) k ) is a onstan t sequene whi h do es not ev olv e. There- fore, there exists c ∈ Z su h that lim n →∞ d ( F n ( x ) , c ) = 0 . W e just pro v ed that A is nilp oten t, i.e. lim n →∞ d ( F n ( x ) , c ) = 0 for all x ∈ B , if and only if S is nilp oten t (b eause of the equiv alene of denitions giv en b y Corollary 4.2 ), so Nil is undeidable (Prop osition 4.3 ). 5 Conlusion In this artile w e ha v e on tin ued the study of sand automata, b y in tro duing a ompat top ology on the SA. In this new on text of study , the haraterization of SA funtions of [6 , 8 ℄ still holds. Moreo v er, a top ologial onjugay of an y SA with a suitable CA ating on a partiular subshift migh t failitate future studies ab out dynamial and top ologial prop erties of SA, as for the pro of of the equiv alene b et w een equion tin uit y and ultimate p erio diit y (Prop osition 3.6 ). Then, w e ha v e giv en a denition of nilp oteny . Although it diers from the standard one for CA, it aptures the in tuitiv e idea that a nilp oten t automaton destro ys ongurations. Ev en though nilp oten t SA ma y not ompletely de- stro y the initial onguration, they atten them progressiv ely . Finally , w e ha v e pro v ed that SA nilp oteny is undeidable (Theorem 4.5 ). This fat enhanes the idea that the b eha vior of a SA is hard to predit. W e also think that this result migh t b e used as a fundamen tal undeidabilit y result, whi h ould b e redued to other SA prop erties. 19 Among these, deiding dynamial b eha viors remains a ma jor problem. More- o v er, the study of global prop erties su h as injetivit y and surjetivit y and their orresp onding dimension-dep enden t deidabilit y problems ould help un- derstand if d -dimensional SA lo ok more lik e d -dimensional or d + 1 -dimensional CA. Still in that idea is the op en problem of the di hotom y b et w een sensitiv e SA and those with equion tin uous ongurations. A p oten tial oun ter-example w ould giv e a more preise idea of the dynamial b eha viors represen ted b y SA. Referenes [1℄ E. Akin. 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