A converse to Moores theorem on cellular automata

A CONVERSE TO M OORE’S THEOREM ON CE LLULAR A UTOMA T A LAURENT BAR THOLDI Abstract. W e prov e a con ve rse to M oore’s “Garden-of-Eden” theorem: a group G i s amenable if and only if all cellular automat a l iving on G that admit mutually erasable patterns al so admi t gardens of Eden. It had already b een conjectured in [11; 1, Conject ure 6.2] tha t amenab ili t y could b e c haracterized by cellular automata. W e prov e the firs t part of that conjecture. 1. Introduction Definition 1.1. Let G b e a group. A finite c el lu lar aut omaton on G is a map θ : Q S → Q , where Q , the state set , is a finite set, and S is a finite subset o f G . Note that usually G is infinite; muc h of the theory holds trivially if G is finite. S could be ta ken to b e a gener ating set o f G , though this is not a necessity . A ce llular a utomaton should be thought of as a highly reg ular anima l, comp osed of many cells lab eled by G , each in a sta te ∈ Q . Each cell “ sees” its neig hbours a s defined by S , and “ evolv es” accor ding to its neighbours’ states. More for mally: a c onfigur ation is a map φ : G → Q . The evolution of the automaton θ : Q S → Q is the s elf-map Θ : Q G → Q G on configurations , defined b y Θ( φ )( x ) = θ ( s 7→ φ ( xs )) . Two pr op erties of cellular automata received sp ecia l atten tion. Le t us call p atch the res triction of a c onfiguration to a finite subset Y ⊆ G . On the o ne ha nd, there can exist patches that never app ear in the image of Θ. These are ca lle d Gar den of Eden (GOE ), the biblica l metapho r ex pressing the no tion of pa r adise los t forever. On the other hand, Θ can b e non-injective in a s trong sense: ther e can exist patches φ ′ 1 6 = φ ′ 2 ∈ Q Y such that, howev er one extends φ ′ 1 to a co nfiguration φ 1 , if one extends φ ′ 2 similarly (i.e. in s uc h a wa y that φ 1 and φ 2 hav e the same restriction to G \ Y ) then Θ( φ 1 ) = Θ( φ 2 ). These patches φ ′ 1 , φ ′ 2 are called Mut ual ly Er asable Patterns (MEP). Equiv alently 1 there are tw o configura tions φ 1 , φ 2 which differ o n a non- empt y finite s et, with Θ( φ 1 ) = Θ( φ 2 ). The absence of MEP is s ometimes called pr e-inje ctivity . Cellular automata were initially co nsidered on G = Z n . Celebrated theorems by Mo ore and Myhill [5 5, 5 6 6, 6] prove that, in this con text, a cellular a utomaton admits GOE if and only if it admits MEP . This result was generalized by Mach ` ı Date : t yp eset October 28, 20 18; last timestamp 20070 922. This w ork wa s partially supported by a CNRS visiting posi tion at Uni v ersit´ e de Prov ence, Marseille. 1 In the non-trivial direction, let φ 1 , φ 2 differ on a non-empt y finite se t F ; set Y = F ( S ∪ S − 1 ) and let φ ′ 1 , φ ′ 2 be the restriction of φ 1 , φ 2 to Y resp ectiv ely . 1 2 LAUR ENT BAR THOLDI and Mignosi [33, 3] to G of sub exp onential gr owth, a nd by Ceccherini, Mach ` ı and Scarab otti [1 1, 1] to G amenable. W e pro ve th at this last result is e ssentially optimal, and yields a characterization of amena ble g r oups: Theorem 1.2. L et G b e a gr oup. Then the fo l lowing ar e e quivalent: (1) the gro up G is amenab le; (2) al l c el lular automata on G that ad mit MEP a lso ad mit GOE. Sch upp had already asked in [77; 7, Ques tion 1] in which precise class o f gro ups the Mo ore-Myhill theo rem ho lds. Ceccherini et al. write in [1 1, 1] 2 : Conjecture 1.3 ([11; 1, Co njectur e 6.2]) . L et G b e a non-amenable finitely gen- er ate d gr oup. Then for any fin ite and symmet ric gener ating set S for G ther e exist c el lular a utomata θ 1 , θ 2 with that S such t hat • In θ 1 ther e ar e MEP bu t no GOE; • In θ 2 ther e ar e GOE bu t no MEP. As a first s tep, w e will pro ve Theorem 1.2, in whic h w e allo w ourselves to c ho ose an appropr iate subs et S of G . Next, we extend a little the co nstruction to answer the first pa rt o f Co njecture 1 .3: Theorem 1 .4. L et G = h S i b e a finitely gener ate d, non-amenable gr oup. Then ther e exists a c el lular automaton θ : Q S → Q that has M EP but no GOE. W e conclude that the pr op erty of “satisfying Moor e’s theor em” is independent of the ge ner ating set, a fact which was not obvious a prio ri . 2. Proof of Theorem 1.2 The implication (1) ⇒ (2) ha s b e en prov en b y Ceccherini et a l.; see also [22; 2, § 8 ] for a slick er pr o of. W e prove the conv erse. Let us there fore b e g iven a non-amenable group G . Let us also, as a first step, b e given a large eno ugh finite subset S of G . Then there exists a “bo unded propagation 2 : 1 compressing vector field” on G : a map f : G → G such that f ( x ) − 1 x ∈ S and # f − 1 ( x ) = 2 for all x ∈ G . W e constr uct the following a uto maton θ . Its stateset is Q = S × { 0 , 1 } × S. Order S in an arbitrary manner, and cho ose an a r bitrary q 0 ∈ Q . Define θ : Q S → Q as follows: (2.1) θ ( φ ) =      ( p, α, q ) for the minima l pa ir s < t in S with ( φ ( s ) = ( s, α, p ) , φ ( t ) = ( t, β , q ) , q 0 if no such s, t exis t . 2 I changed sligh tly their w ording to matc h this pap er’s A CONVERSE TO MOORE’S THEOREM ON CELLULAR AUTOMA T A 3 2.1. Θ is surjective. Namely , θ do es not admit GOE. Let indeed φ be any con- figuration. W e construct a configur a tion ψ with Θ( ψ ) = φ . Consider in turn all x ∈ G ; wr ite φ ( x ) = ( p, α, q ), a nd f − 1 ( x ) = { xs, xt } for some s, t ∈ S ordered a s s < t . Set then (2.2) ψ ( xs ) = ( s, α, p ) , ψ ( xt ) = ( t, 0 , q ) . Note that ψ ( z ) = ( f − 1 ( z ) z , ∗ , ∗ ) for a ll z ∈ G . Since # f − 1 ( z ) = 2 for all z ∈ G , it is clear that, for ev ery x ∈ G , there are exactly tw o s ∈ S such that ψ ( xs ) = ( s, ∗ , ∗ ); call them s, t , ordered such that ψ ( xs ) = ( s, α, p ) and ψ ( xt ) = ( t, 0 , q ). Then Θ( ψ )( x ) = ( p, α, q ), so Θ( ψ ) = φ . 2.2. Θ i s not pre- injectiv e. Namely , θ admits MEP . Let indee d φ : G → Q be an y configuratio n; then construct ψ following (2.2), and define ψ ′ as follows. C ho ose any y ∈ G , wr ite φ ( y ) = ( p, α, q ), and write f − 1 ( y ) = { y s, y t } for some s, t ∈ S , ordered as s < t . Define ψ ′ : G → Q by ψ ′ ( x ) = ( ψ ( x ) if x 6 = yt, ( t, 1 , q ) if x = yt. Then ψ a nd ψ ′ differ only at y t ; and Θ( ψ ) = Θ( ψ ′ ) b ecause the v alue of β is un used in (2.1 ). W e conclude tha t θ has MEP . 3. Proof of Theorem 1.4 W e b egin by a new formulation of amenability for finitely generated gro ups: Lemma 3.1. L et G b e a finitely gener ate d gr oup. The fol lowing ar e e quivalent: (1) the gro up G is not amenable; (2) for every gener ating set S of G , ther e exist m > n ∈ N and a “ m : n c ompr essing c orr esp ondenc e on G with pr op agation S ”; i.e. a function f : G × G → N such that ∀ y ∈ G : X x ∈ G f ( x, y ) = m, (3.1) ∀ x ∈ G : X y ∈ G f ( x, y ) = n, (3.2) ∀ x, y ∈ G : f ( x, y ) 6 = 0 ⇒ y ∈ xS. (3.3) Note that this definition gener alizes the notion of “2 : 1 co mpressing v ector field” int ro duced ab ov e. Pr o of. F o r the forward dir ection, assuming that G is non-a menable, there exists a rational m/n > 1 such that every finite F ⊆ G satisfies #( F S ) ≥ m/n # F . Construct the following bipartite oriented graph: its vertex set is G × { 1 , . . . , m } ⊔ G × {− 1 , . . . , − n } . There is an edge from ( g , i ) to ( g s, − j ) for a ll s ∈ S and all i ∈ { 1 , . . . , m } , j ∈ { 1 , . . . , m } . By hypothes is, this gr a ph satisfies : every finite F ⊆ G × { 1 , . . . , m } has at least # F neighbo ur s. Since m > n and m ultiplication by a generator is a bijection, ev ery finit e F ⊆ G × {− 1 , . . . , − n } also has at least # F neighbour s. W e now inv oke the Hall-Rado theorem [4 4, 4]: if a bipa r tite graph is such that every subset of any of the parts has as many neig hbours as its cardinality , then 4 LAUR ENT BAR THOLDI there exists a “per fect matc hing” — a subset I o f the edge set of the graph suc h that every vertex is c o nt ained in precisely one edge in I . Set then f ( x, y ) = # { ( i , j ) ∈ { 1 , . . . , m }×{ 1 , . . . , n } : I contains the edg e from ( x, i ) to ( y , − j ) } . F or the bac kward dir ection: if G is amenable, then there exists an inv ariant mea- sure on G , hence on b ounded natura l-v alued functions on G . Let f be a b ounded- propaga tion m : n compres sing cor resp ondence; then m = X x ∈ G Z { x }× G f = X y ∈ G Z G ×{ y } f = n, contradicting m > n .  Let now G = h S i be a non-amenable group, and apply Lemma 3.1 to G = h S − 1 i , yielding m > n ∈ N and a con tracting m : n c orresp ondence f . Consider the following cellular automaton θ , with s tateset Q = ( S × { 0 , 1 } × S n ) n . Cho ose q 0 ∈ Q , and give a total ordering to S × { 1 , . . . , n } . Consider φ ∈ Q S . T o define θ ( φ ), le t ( s 1 , k 1 ) < · · · < ( s m , k m ) b e the lexic o- graphically minimal se quence in ( S × { 1 , . . . , n } ) m such that φ ( s j ) k j = ( s j , α j , t j, 1 , . . . , t j,n ) ∈ S × { 0 , 1 } × S n for j = 1 , . . . , m. If no such s 1 , k 1 , . . . , s m , k m exist, set θ ( φ ) = q 0 ; other wise, s e t (3.4) θ ( φ ) = (( t 1 , 1 , α 1 , t 2 , 1 , . . . , t n +1 , 1 ) , . . . , ( t 1 ,n , α n , t 2 ,n , . . . , t n +1 ,n )) ∈ Q. The sa me arguments a s b efor e apply . Given φ : G → Q , we constr uc t ψ : G → Q such that Θ( ψ ) = φ , a s follows. W e think of the co ¨ ordinates ψ ( x ) k of ψ ( x ) as n “slots”, initially all “free” . By definition, # f − 1 ( x ) = m for all x ∈ G , while # f ( x ) = n . Consider in turn all x ∈ G ; wr ite f − 1 ( x ) = { xs 1 , . . . , xs m } , and let k 1 , . . . , k m ∈ { 1 , . . . , n } b e “free” slots in ψ ( xs 1 ) , . . . , ψ ( xs m ) r esp ectively . By the definition of f , there always exist sufficiently many free slots. Mark now these slots as “oc c upied”. Reorder s 1 , k 1 , . . . , s m , k m in such a w ay that ( s 1 , k 1 , . . . , s m , k m ) is minimal a mo ng its m ! p ermutations. Set then ψ ( xs j ) k j = ( s j , α j , t j, 1 , . . . , t j,n ) for j = 1 , . . . , m, where α n +1 , . . . , α m are taken to b e a r bitrary v alues (say 0 for definiteness) and φ ( x ) = (( t 1 , 1 , α 1 , t 2 , 1 , . . . , t n +1 , 1 ) , . . . , ( t 1 ,n , α 1 , t 2 ,n , . . . , t n +1 ,n )) . Finally , define ψ arbitrar ily on slots that are still “ fr ee”. It is clea r that Θ ( ψ ) = φ , so θ does not hav e GOE. On the o ther hand, θ has MEP as b e fore, b ecause the v alues of α j in (3.4) are not used for j ∈ { n + 1 , . . . , m } . 4. Remarks 4.1. G -sets. A cellula r automaton could more generally b e defined on a right G -set X . Ther e is a natura l notion of amenability for G -sets, but it is not clea r exa ctly to which extent Theorem 1 .2 ca n b e gener alized to that setting. 4.2. Myhill’ s Theorem . It seems harder to produce counterexamples to My hill’s theorem (“ GOE imply MEP”) for ar bitrary non-amenable groups, a lthough there exists an e x ample o n C = C 2 ∗ C 2 ∗ C 2 , due to Muller 3 . Let us make our task even 3 Unive rsi ty of Illinois 1976 class notes A CONVERSE TO MOORE’S THEOREM ON CELLULAR AUTOMA T A 5 harder, a nd res trict ourselves to linea r a utomata over finite r ings (so we assume Q is a mo dule ov er a finite ring and the map θ : Q S → Q is linear). The follo wing approach seems pro mising. Conjecture 4.1 (F olklore? I learnt it from V. Guba) . L et G b e a gr oup. The fol lowing ar e e quivalent: (1) The gr oup G is amenable; (2) L et K b e a field. Then K G admits right c ommon multiples, i.e. for any α, β ∈ K G ther e exist γ , δ ∈ K G with αγ = β δ and ( γ , δ ) 6 = (0 , 0) . The implication (1) ⇒ (2) is easy , and follows from Følner’s criterion of amenabil- it y by linear algebra. Assume now the “ hard” dir e ction of the conjecture. Given G non-a menable, we may then find a finite field K , and α, β ∈ K G that do not hav e a common right m ultiple. Set Q = K 2 with basis ( e 1 , e 2 ), let S contain the in verses of the suppor ts of α and β , and define the cellular automaton θ : Q S → Q by θ ( φ ) = X x ∈ G  α ( x − 1 ) h φ ( x ) | e 1 i − β ( x − 1 ) h φ ( x ) | e 2 i , 0  . Then θ has GOE , indee d any configur ation not in ( K × 0) G is a GOE. On the other hand, if θ had MEP , then by linear ity w e migh t as w ell assume Θ( φ ) = 0 for some non-zero finitely- s uppo rted φ : G → Q . W rite φ = ( γ , δ ) in co¨ ordinates; then Θ( φ ) = 0 g ives αγ = β δ , showing that α, β actually did hav e a common right m ultiple. Muller’s example is in fact a s pec ia l ca se o f this constr uction, with G = h x, y, z | x 2 , y 2 , z 2 i , K = F 2 , and α = x , β = y + z . References [1] ]cite.cecc heri ni-m-s:ca1T. G. Cecc herini- Silb erstein, A. Mach ` ı, and F. Scarabotti, Am enable gr oups and c el lular auto mata , Ann. Inst. F ouri er (Grenoble) 4 9 (1 999), no. 2, 673–685 (English, with E ngli sh and F renc h summaries). MR 16973 76 (2000k: 43001) [2] ]cite.gromo v:endomorphisms2M. Gromov, Endom orphisms of symb olic algebr aic varieties , J. Eur. M ath. So c. (JEMS) 1 (19 99), no. 2, 109–197. MR 1 694588 (2 000f: 14003) [3] ]cite.mac hi-m:ca3Antonio Mach ` ı and Fil ippo Mignosi, Gar den of Eden c onfigur ations for c el- lular automata on Cayley gr aphs of gr oups , SIAM J. Discrete Math. 6 (1993), no. 1, 44–56. MR 1 201989 (95 a: 68084) [4] ]cite.mirsky:transversal4Leon id Mirsky, T r ansversal t he ory. An ac c ount of some asp e cts of c ombinatorial mathematics , Mathematics in Science and Engineering, V ol. 75, Academic Press, New Y ork, 197 1. MR 02828 53 (44 # 87) [5] ]cite.moore:ca5Edward F. Mo or e, Machine mo dels of self-r epr o duction , Essa ys on cellular au- tomata, 19 70, pp. 17–33. MR 029 9409 (45 #8457) [6] ]cite.m yhill: ca6John Myhill, The c onverse of Mo or e’s Gar den-of-Eden the or em , Pro c. Amer. Math. Soc. 14 (1963), 685–686. M R 0155764 (27 #5698) [7] ]cite.sc hupp :arr a ys7Paul E. Sc hupp , Arr ays, automata and gr oups—some inter c onne ctions , Automata net works (Ar gel ` es-V i llage, 1986), Lecture N otes in Comput. Sci., vol. 3 16, Springer, Berlin, 198 8, pp. 19–28. MR 9612 74 ´ Ecole Pol ytechnique F ´ ed ´ erale de Lausanne (EPFL), Institut de Ma th ´ ema tiques B (IMB), 1015 Lausanne, Switzerland E-mail addr ess : laure nt.bartholdi @gmail.com