# Università degli Studi di Roma La Sapienza. Dottorato di Ricerca in Matematica XII ciclo

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1 Università degli Studi di Roma La Sapienza Dottorato di Ricerca in Matematica XII ciclo tesi Cellular Automata and Finitely Generated Groups di Francesca Fiorenzi Dipartimento di Matematica Università di Roma La Sapienza Piazzale Aldo Moro Roma e mail

2 Contents Introduction 3 1. Cellular Automata Cayley Graphs of Finitely Generated Groups Shift Spaces and Cellular Automata Shifts of Finite Type Edge Shifts Sofic Shifts Minimal Deterministic Presentations of a Sofic Shift Decision Problems A Decision Procedure for Surjectivity A Decision Procedure for Injectivity Entropy Surjunctivity and Density of Periodic Configurations Surjunctivity Periodic Configurations of a Shift Residually Finite Groups Density of Periodic Configurations Periodic Configurations of Euclidean Shifts Group Shifts Decision Problems for Group Shifts The Moore Myhill Property The Garden of Eden Theorem and the Moore Myhill Property Amenable Groups The Moore Myhill Property for Subshifts of A Z A counterexample to Moore property for a sofic subshift of A Z Gromov s Theorem Strongly Irreducible Shifts Semi Strongly Irreducible Shifts Bibliography 69 2

3 Introduction The notion of a cellular automaton has been introduced by Ulam [U] and von Neumann [vn]. In this classical setting, the universe is the lattice of integers Z n of Euclidean space R n. The set of states is a finite set A (also called the alphabet) and a configuration is a function c : Z n A. Time t goes on in discrete steps and represents a transition function τ : A Zn A Zn (if c is the configuration at time t, then τ(c) is the configuration at time t + 1), which is deterministic and local. Locality means that the new state at a point γ Z n at time t + 1 only depends on the states of certain fixed points in the neighborhood of γ at time t. More precisely, if c is the configuration reached from the automaton at time t then τ(c) γ = δ(c Nγ ), where δ : A N A is a function defined on the configurations with support the finite set N (the neighborhood of the point 0 Z n ), and N γ = γ + N is the neighborhood of γ obtained from N by translation. For these structures, Moore [Moo] has given a sufficient condition for the existence of the so called Garden of Eden (GOE) patterns, that is those configurations with finite support that cannot be reached at time t from another configuration starting at time t 1 and hence can only appear at time t = 0. Moore s condition (i.e. the existence of mutually erasable patterns a sort of non injectivity of the transition function on the finite configurations) was also proved to be necessary by Myhill [My]. This equivalence between local injectivity and local surjectivity of the transition function is the classical well known Garden of Eden theorem. The purpose of this work is to consider this kind of problems in the more general framework of symbolic dynamic theory, with particular regard to surjunctivity theorems (and, in this case, to the density of the periodic configurations) and to GOE like theorems restricted to the subshifts of the space A Γ (where Γ is a finitely generated group and A is a finite alphabet). Indeed for these sets is still possible to define a structure of a cellular automaton. More precisely, given a finitely generated group Γ, one can consider as universe the Cayley graph of Γ. A configuration is an element of the space A Γ (the so called full A shift), that is a function defined on Γ with values in a finite alphabet A. The space A Γ is naturally endowed with a metric and hence with an induced topology, this latter being equivalent to the usual product topology, 3

4 where the topology in A is the discrete one. A subset X of A Γ which is Γ invariant and topologically closed is called subshift, shift space or simply shift. In this setting a cellular automaton (CA) on a shift space X is given by specifying a transition function τ : X X which is local (that is the value of τ(c), where c X is a configuration, at a point γ Γ only depends on the values of c at the points of a fixed neighborhood of γ). In Chapter 1 we formally define all these notions, also proving that many basic results for the subshifts of A Z given in the book of Lind and Marcus [LinMar], can be generalized to the subshifts of A Γ. For example, we prove that the topological definition of shift is equivalent to the combinatorial one of set of configurations avoiding some forbidden blocks. Moreover we prove that a function between shift spaces is local if and only if it is continuous and commutes with the Γ action. This fact, together with the compactness of the shift spaces (notice that in A Γ closeness and compactness are equivalent), implies that the inverse of an invertible cellular automaton is also a cellular automaton and allows us to call conjugacy each invertible local function between two shifts. The invariants are those properties which are invariant under conjugacy. A fundamental notion we give is that of irreducibility for a shift; in the one dimensional case, this means that given any pair of words u, v in the language of the shift (i.e. the set of all finite words appearing in some bi infinite configuration), there is a word w such that the concatenation uwv still belongs to the language. We generalize this definition to a generic shift, using the patterns and their supports rather than the words. Then, in terms of forbidden blocks the notion of shift of finite type is given and we prove that, as in the one dimensional case, such a shift has a useful overlapping property that will be necessary in later chapters. Moreover we recall from [LinMar] the notions of edge shift and of sofic shift, and restate many of the basic properties of these one dimensional shifts strictly connected with the one dimensional shifts of finite type. As we have noticed, cellular automata have mainly been investigated in the Euclidean case and for the full shifts. The difference between the one dimensional cellular automata and the higher dimensional ones, is very deep. For example, Amoroso and Patt have shown in [AP] that there are algorithms to decide surjectivity and injectivity of local maps for one dimensional cellular automata. On the other hand Kari has shown in [K1] and [K2] that both the injectivity and the surjectivity problems are undecidable for Euclidean cellular automata of dimension at least two. Here we extend the Amoroso Patt s results to the one dimensional cellular automata over shifts of finite type. Finally, the notion of entropy for a generic shift as defined by Gromov in [G] is also given; we see that this definition involves the existence of a suitable sequence of sets and we prove that, in the case of non exponential growth of the group, these sets can be taken as balls centered at 1 and with increasing radius. Moreover, we see that the entropy is an invariant of the shifts. Then, following Lind and Marcus [LinMar], we recall its basic properties in the one dimensional case, also stating the principal result of the Perron Frobenius theory to compute 4

5 it. A selfmapping τ : X X on a set X is surjunctive if it is either non injective or surjective. In other words a function is surjunctive if the implication injective surjective holds. Using the GOE theorem and the compactness of the space A Zn, Richardson has proved in [R] that a transition function τ : A Zn A Zn is surjunctive. In Chapter 2, we consider the surjunctivity of the transition function in a general cellular automaton over a group Γ. A configuration of a shift is periodic if its Γ orbit is finite; after proving some generalities about periodic configurations, we recall the class of residually finite groups, proving that a group Γ is residually finite if and only if the periodic configurations are dense in A Γ. Hence if Γ is a residually finite group, a transition function τ of a cellular automaton on A Γ is surjunctive. In fact, we prove more generally that if the periodic configurations of a subshift X A Γ are dense, then a transition function on X is surjunctive. We also prove that the density of the periodic configurations is an invariant of the shifts, as is the number of the periodic configuration with a fixed period. The remaining part of the chapter is devoted to establish for which class of shifts the periodic configurations are dense. We prove the density of the periodic configurations for an irreducible subshift of finite type of A Z and hence, a sofic shift being the image under a local map of a shift of finite type, the density of the periodic configurations for an irreducible sofic subshift of A Z. We see that these results cannot be generalized to higher dimensions. If the alphabet A is a finite group G and Γ is abelian, the periodic configurations of a subshift which is also a subgroup of G Γ (namely a group shift), are dense (this result is a consequence of a more general theorem in [KitS2]). Moreover, this further hypothesis is not still included in the result about irreducible shifts of finite type; indeed a group shift is of finite type but there are examples of group shifts which are not irreducible. Finally, we list some other well known decision problems for Euclidean shifts proving that in the special case of a one dimensional shift they can be solved; more generally they can be solved for the class of group shifts using some results due to Wang [Wa] and Kitchens and Schmidt [KitS1]. In Chapter 3, we consider generalizations of the Moore s property and Myhill s property to a generic shift. In details, the GOE theorem has been proved by Machì and Mignosi [MaMi] more generally for cellular automata in which the space of configurations is the whole A shift A Γ and the group Γ has non exponential growth; more recently it has been proved by Ceccherini Silberstein, Machì and Scarabotti [CeMaSca] for the wider class of the amenable groups. Instead of the non existence of mutually erasable patterns, we deal with the notion of pre injectivity (a function τ : X A Γ A Γ is pre injective if whenever two configurations c, c X differ only on a finite non empty subset of Γ, then τ(c) τ( c)); this notion has been introduced by Gromov in [G]. In fact, we prove that these two properties are equivalent for local functions defined on 5

6 the full shift, but in the case of proper subshifts the former may be meaningless. On the other hand, we give a proof of the fact that the non existence of GOE patterns is equivalent to the non existence of GOE configurations, that is to the surjectivity of the transition function. Hence, in this language, the GOE theorem states that τ is surjective if and only if it is pre injective. We call Moore s property the implication surjective pre injective and Myhill s property the inverse one. Concerning one dimensional shifts, from the works of Hedlund [H] and Coven and Paul [CovP] we prove that the Moore Myhill property (MM property) holds for irreducible shifts of finite type of A Z. Moreover, using this result we prove that Myhill s property holds for irreducible sofic shifts of A Z. On the other hand, we give a counterexample of an irreducible sofic shift X A Z but not of finite type for which Moore s property does not hold. Concerning general cellular automata over amenable groups, from a result of Gromov [G] in a more general framework, it follows that the MM property holds for shifts of bounded propagation contained in A Γ. We generalize this result showing that the MM property holds for strongly irreducible shifts of finite type of A Γ (and we also show that strong irreducibility together with the finite type condition is strictly weaker that the bounded propagation property). The main difference between irreducibility and strong irreducibility is easily seen in the one dimensional case. Here the former property states that given any two words u, v in the language of a shift, there exists a third word w such that the concatenation uwv is still in the language. Strong irreducibility says that we can arbitrarily fix the length of this word (but it must be greater than a fixed constant only depending on the shift). The same properties for a generic shift refers to the way in which two different patterns in the language of the shift may appear simultaneously in a configuration. For irreducibility we have that two patterns always appear simultaneously in some configuration if we translate their supports. Strong irreducibility states that if the supports of the pattern are far enough, than it is not necessary to translate them in order to find a configuration in which both the patterns appear. These two irreducibility conditions are not equivalent, not even in the one dimensional case. Hence our general results about strongly irreducible shifts of finite type are strictly weaker than the one dimensional ones. In the attempt of using weaker hypotheses in the latter case, we give a new notion of irreducibility, the semi strong irreducibility. In the one dimensional case, this property means that the above word w may almost be of the length we want (provided that it is long enough): we must allow it to be a little longer or a little shorter; the length of this difference is bounded and only depends on the shift. In general, semi strong irreducibility states that if the supports of the patterns are far enough from each other, than translating them a little we find a configuration in which both the patterns appear. The reason for this choice lies in the fact that using the Pumping Lemma we can prove that a sofic subshift of A Z is irreducible if and only if is semi strongly irreducible. Moreover Myhill s property holds for semi strongly irreducible shifts of finite type of A Γ if Γ has non exponential growth. 6

7 1. Cellular Automata In this chapter we give the notion of a cellular automaton over a finitely generated group. This notion generalizes the definition given by Ulam [U] and von Neumann [vn]. In that classical setting, the universe is the lattice of integers Z n of Euclidean space R n. The set of states is a finite set A (also called the alphabet) and a configuration is a function c : Z n A. Time t goes on in discrete steps and represents a transition function τ : A Zn A Zn (if c is the configuration at time t, then τ(c) is the configuration at time t + 1), which is deterministic and local. Locality means that the new state at a point γ Z n at time t + 1 only depends on the states of certain fixed points in the neighborhood of γ at time t. More precisely, if c is the configuration reached from the automaton at time t then τ(c) γ = δ(c Nγ ), where δ : A N A is a function defined on the configurations with support the finite set N (the neighborhood of the point 0 Z n ), and N γ = γ + N is the neighborhood of γ obtained from N by translation. Our purpose is, given a finitely generated group Γ, to consider as universe the Cayley graph of Γ. A configuration is an element of the space A Γ, that is a function defined on Γ with values in a finite alphabet A. The space A Γ is naturally endowed with a metric and hence with an induced topology, this latter being equivalent to the usual product topology, where the topology in A is the discrete one. A subset X of A Γ which is Γ invariant and topologically closed is called subshift, shift space or simply shift. In this setting a cellular automaton (CA) on a shift space X is given by specifying a transition function τ : X X which is local (that is the value of τ(c), where c X is a configuration, at a point γ Γ only depends on the values of c at the points of a fixed neighborhood of γ). Sections 1.1 and 1.2 of this chapter are dedicated to give a formal definition of all these notions, also proving that many basic results for the subshifts of A Z given in the book of Lind and Marcus [LinMar], can be generalized to the subshifts of A Γ. For example, we prove that the topological definition of shift 7

8 is equivalent to the combinatorial one of set of configurations avoiding some forbidden blocks. Moreover we prove that a function between shift spaces is local if and only if it is continuous and commutes with the Γ action. This fact, together with the compactness of the shift spaces (notice that in A Γ closeness and compactness are equivalent), implies that the inverse of an invertible cellular automaton is also a cellular automaton and allows us to call conjugacy each invertible local function between two shifts. The invariants are those properties which are invariant under conjugacy. Finally we extend to a general shift the notion of irreducibility and of being of finite type. In Sections 1.4 and 1.5, we recall from [LinMar] the notions of edge shift and of sofic shift, and restate many of the basic properties of these one dimensional shifts strictly connected with the one dimensional shifts of finite type. In Section 1.6, we generalize the Amoroso Patt s results about the decision problem of the surjectivity and the injectivity of local maps for one dimensional cellular automata on the full shift (see [AP] and [Mu]). Our results concern one dimensional cellular automata over shifts of finite type. Finally, in Section 1.7 the notion of entropy for a generic shift as defined by Gromov in [G] is given. We see that this definition involves the existence of a suitable sequence of sets and we prove that, in the case of non exponential growth of the group, these sets can be taken as balls centered at 1 and with increasing radius. Moreover, we see that the entropy is an invariant of the shifts. Then, following Lind and Marcus [LinMar, Chapter 4], we recall basic properties of the entropy in the one dimensional case. 1.1 Cayley Graphs of Finitely Generated Groups In this section, we give the basic notion of Cayley graph of a finitely generated group; as we have said, this graphs will constitute the universe of our class of cellular automata. Moreover, we recall the definition of growth of a finitely generated group Γ. Let Γ be a finitely generated group and X a fixed finite set of generators for Γ; then each γ Γ can be written as γ = x δ1 i 1 x δ2 i 2... x δn i n (1.1) where the x ij s are generators and δ j Z. We define the length of γ (with respect to X ) as the natural number γ X := min{ δ 1 + δ δ n γ is written as in (1.1)} so that Γ is naturally endowed with a metric space structure, with the distance given by dist X (α, β) := α 1 β X (1.2) and D X n := {γ Γ γ X n} 8

9 is the disk of radius n centered at 1. Notice that D X 1 is the set X X 1 {1}. The asintotic properties of the group being independent on the choice of the set of generators X, from now on we fix a set X which is also symmetric (i.e. X 1 = X ) and we omit the index X in all the above definitions. For each γ Γ, the set D n provides, by left translation, a neighborhood of γ, that is the set γd n = D(γ, n). Indeed, if α γd n then α = γβ with β n. Hence dist(α, γ) = α 1 γ = β 1 n. Conversely, if α D(γ, n) then γ 1 α n (that is γ 1 α D n ), and α = γ γ 1 α. Now we define, for each E Γ and for each n N, the following subsets of Γ: the n closure of E E +n := D(α, n); the n interior of E the n boundary of E the n external boundary of E α E E n := {α E D(α, n) E}; n E := E +n \E n ; + n E := E+n \E and, finally, the n internal boundary of E n E := E\E n. For all these sets, we will omit the index n if n = 1. The Cayley graph of Γ, is the graph in which Γ is the set of vertices and there is an edge from γ to γ if there exists a generator x X such that γ = γx. Obviously this graph depends on the presentation of Γ. For example, we may look at the classical cellular decomposition of Euclidean space R n as the Cayley graph of the group Z n with the presentation a 1,..., a n a i a j = a j a i. If G = (V, E) is a graph with set of vertices V and set of edges E, the graph distance (or geodetic distance) between two vertices v 1, v 2 V is the minimal length of a path from v 1 to v 2. Hence the distance defined in (1.2) coincides with the graph distance on the Cayley graph of Γ. We recall that the function g : N N defined by g(n) := D n which counts the elements of the disk D n, is called growth function of Γ (with respect to X ). One can prove that the limit λ := lim n g(n) 1 n 9

10 always exists; if λ > 1 then, for all sufficiently large n, g(n) λ n, and the group Γ has exponential growth. If λ = 1, we distinguish two cases. Either there exists a polynomial p(n) such that for all sufficiently large n g(n) p(n), in which case Γ has polynomial growth, or Γ has intermediate growth (i.e. g(n) grows faster than any polynomial in n and slower then any exponential function x n with x > 1). Moreover, it is possible to prove that the type of growth is a property of the group Γ (i.e. it does not depend on the choice of a set of generators); for this reason we deal with the growth of a group. This notion has been indipendently introduced by Milnor [Mil], Efremovič [E] and Švarc [Š]; it is very useful in the theory of cellular automata. 1.2 Shift Spaces and Cellular Automata Let A be a finite alphabet; in the classical theory of cellular automata, the universe is the Cayley graph of the free abelian group Z n and a configuration is an element of A Zn, that is a function c : Z n A assigning to each point of the graph a letter of A. We generalize this notion taking as universe a Cayley graph of a generic finitely generated group Γ and taking suitable subsets of configurations in A Γ. Let A be a finite set (with at least two elements) called alphabet; on the set A Γ (i.e. the set of all functions c : Γ A), we have a natural metric and hence a topology. This latter is equivalent to the usual product topology, where the topology in A is the discrete one. By Tychonoff s theorem, A Γ is also compact. An element of A Γ is called a configuration. If c 1, c 2 A Γ are two configurations, we define the distance dist(c 1, c 2 ) := 1 n + 1 where n is the least natural number such that c 1 c 2 in D n. If such an n does not exist, that is if c 1 = c 2, we set their distance equal to zero. Observe that the group Γ acts on A Γ on the right as follows: (c γ ) α := c γα for each c A Γ and each γ, α Γ (where c α is the value of c at α). Now we give a topological definition of a shift space (briefly shift); we will see in the sequel that this definition is equivalent (in the Euclidean case) to the classical combinatorial one. 10

11 Definition A subset X of A Γ is called a shift if it is topologically closed and Γ invariant (i.e. X Γ = X). For every X A Γ and E Γ, we set X E := {c E c X}; a pattern of X is an element of X E where E is a non empty finite subset of Γ. The set E is called the support of the pattern; a block of X is a pattern of X with support a disk. The language of X is the set L(X) of all the blocks of X. If X is a subshift of A Z, a configuration is a bi infinite word and a block of X is a finite word appearing in some configuration of X. Hence a pattern with support E is a function p : E A. If γ Γ, we have that the function p : γe A defined as p γα = p α (for each α E), is the pattern obtained copying p on the translated support γe. Moreover, if X is a shift, we have that p X γe if and only if p X E. For this reason, in the sequel we do not make distinction between p and p (when the context makes it possible). For example, a word a 1... a n is simply a finite sequence of symbols for which we do not specify (if it is not necessary), if the support is the interval [1, n] or the interval [ n, 1]. Definition Let X be a subshift of A Γ ; a function τ : X A Γ is M local if there exists δ : X DM A such that for every c X and γ Γ where D M = {α 1,..., α m }. (τ(c)) γ = δ((c γ ) DM ) = δ(c γα1, c γα2,..., c γαm ), In this definition, we have assumed that the alphabet of the shift X is the same as the alphabet of its image τ(x). In this assumption there is no loss of generality because if τ : X A Γ B Γ, one can always consider X as a shift over the alphabet A B. Definition Let Γ be a finitely generated group with a fixed symmetric set of generators X, let A be a finite alphabet with at least two element and let D M the disk in Γ centered at 1 and with radius M. A cellular automaton is a triple (X, D M, τ) where X is a subshift of the compact space A Γ, D M is the neighborhood of 1 and τ : X X is an M local function. Let τ : X A Γ be a M local function; if c is a configuration of X and E is a subset of Γ, τ(c) E only depends on c E +M. Thus we have a family of functions (τ E +M : X E +M τ(x) E ) E Γ. The following characterization of local functions is, in the one dimensional case, known as the Curtis Lyndon Hedlund theorem. Here we generalize it to a general local function. Proposition A function τ : X A Γ is local if and only if it is continuous and commutes with the Γ action (i.e. for each c X and each γ Γ, one has τ(c γ ) = τ(c) γ ). 11

12 Proof Suppose that τ is M local and that D M = {α 1,..., α m }; then for γ Γ and c X, (τ(c γ )) α = δ((c γ ) αα1, (c γ ) αα2,..., (c γ ) ααm ) = δ(c γαα1,..., c γααm ). On the other hand ((τ(c)) γ ) α = (τ(c)) γα = δ(c γαα1,..., c γααm ). Hence τ commutes with the Γ action. We prove that τ is continuous. A generic element of a sub basis of A Γ is ξ := {c A Γ c α = a} with α Γ and a A. It suffices to prove that the set is open in X. Actually, if c ξ and ξ := τ 1 (ξ) = {c X τ(c) α = a} r := max( αα 1,..., αα m ), (1.3) 1 we claim that the ball B X (c, r+1 ) is contained in ξ. 1 Indeed, if c B X (c, then dist(c, c) < 1 r + 1 r+1 ) and on D r we have c Dr = c Dr ; since, from (1.3), αα i D r, we have τ( c) α = δ( c αα1,..., c ααm ) = δ(c αα1,..., c ααm ) = τ(c) α = a. Conversely, suppose that τ is continuous and commutes with the action of Γ. Since X is compact, τ is uniformly continuous; fix M in N such that for every c, c X, dist(c, c) < 1 dist(τ(c), τ( c)) < 1. M + 1 Hence, if c and c agree on D M, then τ(c) and τ( c) agree at 1, that is, for every c X, (τ(c)) 1 = δ(c α1,..., c αm ) where D M = {α 1,..., α m }. In general, since τ commutes with the action of Γ, we have (τ(c)) α = (τ(c)) α1 = ((τ (c)) α ) 1 = (τ(c α )) 1 = δ(c αα1,..., c ααm ) showing that τ is M local. From the previous theorem, it is clear that the composition of two local functions is still local. In any case, this can be easily seen by a direct proof that follows Definition

13 Now, fix γ Γ and consider the function X A Γ that associates with each c X its translated configuration c γ. In general, this function does not commute with the Γ action (and therefore it is not local). Indeed, if Γ is not abelian and γα αγ, then (c γ ) α (c α ) γ. However, this function is continuous. In order to see this, if n 0, fix a number m 0 such that γd n D m ; if dist(c, c) 1 m+1, then c and c agree on D m and hence on γd n. If α D n, we have c γα = c γα and then c γ α = c γ α that is c γ and c γ agree on D n so that dist(c γ, c γ ) < 1 n+1. This result will be necessary in the proof of Theorem 1.2.6, below. Observe that if X is a subshift of A Γ and τ : X A Γ is a local function, then, by Proposition 1.2.4, the image Y := τ(x) is still a subshift of A Γ. Indeed Y is closed (or, equivalently, compact) and Γ invariant: Y Γ = (τ(x)) Γ = τ(x Γ ) = τ(x) = Y. Moreover, if τ is injective then τ : X Y is a homeomorphism; if c Y then c = τ( c) for a unique c X and we have τ 1 (c γ ) = τ 1 (τ( c) γ ) = τ 1 (τ( c γ )) = c γ = (τ 1 (c)) γ that is, τ 1 commutes with the Γ action. By Proposition 1.2.4, τ 1 is local. Hence the well known theorem (see [R]), stating that the inverse of an invertible Euclidean cellular automaton is a cellular automaton, holds also in this more general setting. In the one dimensional case, Lind and Marcus [LinMar, Theorem ] give a direct proof of this fact. This result leads us to give the following definition. Definition Two subshifts X, Y A Γ are conjugate if there exists a local bijective function between them (namely a conjugacy). The invariants are the properties of a shift invariant under conjugacy. Now we prove that the topological definition of a shift space is equivalent to a combinatorial one involving the avoidance of certain forbidden blocks: this fact is well known in the Euclidean case. Theorem A subset X A Γ is a shift if and only if there exists a subset F n N such that X = X ADn F, where X F := {c A Γ c α D n / F for every α Γ, n N}. In this case, F is a set of forbidden blocks of X. Proof Suppose that X is a shift. X being closed, for each c / X there exists an integer n c > 0 such that the ball B A Γ(c, 1 ) X. n c 13

14 Let F be the set F := {c Dnc c / X}; we prove that X = X F. If c / X F, there exists c / X such that c α D = c n c D n c for some α Γ. Then dist(c α, c) < 1 and hence c α B n c A Γ( c, 1 ) X which n c implies c / X, by the Γ invariance of X. This proves that X X F. For the other inclusion, if c / X then, by definition of F, c Dnc F and hence c / X F. Now, for the converse, we have to prove that a set of type X F is a shift. Observe that X F = p F X {p} and, if supp(p) = D n = {α 1,..., α N }, X {p} = {c A Γ c α D n p} = ( ) {c A Γ c α α i p αi }. α Γ α Γ α i D n Thus, in order to prove that X F is closed, it suffices to prove that for any i = 1,..., N the set {c A Γ c α α i p αi } (1.4) is closed. We have ({c A Γ c α α i p αi }) α = {c A Γ c αi p αi } and then the set in (1.4) is closed being the pre image of a closed set under a continuous function. Finally we have to prove that X F is Γ invariant. If γ Γ and c X F, for every α Γ and every n N we have c γα D n / F that is (c γ ) α D n / F and hence c γ X F Now we give the first, fundamental notion of irreducibility for a one dimensional shift and we see how to generalize this notion to a generic shift. Definition A shift X A Z is irreducible if for each pair of words u, v L(X), there exists a word w L(X) such that the concatenated word uwv L(X). The natural generalization of this property to any group Γ is the following. Definition A shift X A Γ is irreducible if for each pair of patterns p 1 X E and p 2 X F, there exists an element γ Γ such that E γf = and a configuration c X such that c E = p 1 and c γf = p 2. In other words, a shift is irreducible if whenever we have p 1, p 2 L(X), there exists a configuration c X in which these two blocks appear simultaneously on disjoint supports. This definition could seem weaker than Definition 1.2.7, in fact in this latter we establish that each word u L(X) must always appear in a configuration on the left of each other word of the language. In order to prove that the two definitions agree, suppose that X A Z is an irreducible 14

15 shift according with Definition If u, v are words in L(X), there exists a configuration c X such that c E = u and c F = v where E and F are finite and disjoint intervals. If max E < min F then there exists a word w such that uwv L(X) (where w = c I and I is the interval [max E +1, min F 1]). If, otherwise, max F < min E there exists a word w such that vwu L(X); consider the word vwu two times, there exists another word x such that vwu x vwu L(X) and hence uxv L(X). 1.3 Shifts of Finite Type Now we give the fundamental notion of shift of finite type. The basic definition is in terms of forbidden words; in a sense we may say that a shift is of finite type if we can decide whether or not a configuration belongs to the shift only checking its words of a fixed (and only depending on the shift) length. This fact implies an useful characterization of the one dimensional shifts of finite type, a sort of overlapping property for the words of the language. We prove that this overlapping property still holds for a generic shift of finite type. Definition A shift is of finite type if it admits a finite set of forbidden blocks. If X is a shift of finite type, since a finite set F of forbidden blocks of X has a maximal support, we can always suppose that each block of F has the disk D M as support (indeed each block that contains a forbidden block is forbidden). In this case the shift X is called M step and the number M is called the memory of X. If X is a subshift of A Z, we define the memory of X as the number M, where M + 1 is the maximal length of a forbidden word. For the shifts of finite type in A Z, we have the following useful property. Proposition [LinMar, Theorem 2.1.8] A shift X A Z is an M step shift of finite type if and only if whenever uv, vw L(X) and v M, then uvw L(X). Now we prove that this overlapping property holds more generally for subshifts of finite type of A Γ. Proposition Let X be an M step shift of finite type and let E be a subset of Γ. If c 1, c 2 X are two configurations that agree on + 2M E, then the configuration c A Γ that agrees with c 1 on E and with c 2 on E is still in X. Proof Let c 1, c 2 and c be three configurations as in the statement; if F is a finite set of forbidden blocks for X, each having D M = {α 1,..., α m } as support, we have to prove that for each γ Γ, c γ D M / F. Observe that either γd M E +2M or γd M E; 15

16 indeed, if γ E +M then by definition γd M (E +M ) +M = E +2M. If γ / E +M, suppose that there exists γ γd M E. Then, for some i, γ = γα i and hence γ = γα 1 i γd M E +M which is excluded; then γd M E. Now, if γ is such that γd M E +2M, we have c γ D M = (c γ 1 ) DM / F. If γ is such that γd M E, we have c γ D M = (c γ 2 ) DM / F. Corollary Let X be an M step shift of finite type and let E be a finite subset of Γ; if p 1, p 2 X E +2M are two patterns that agree on + 2M E, than there exist two extensions c 1, c 2 X of p 1 and p 2, respectively, that agree on E. Proof Indeed, if c 1 and c 2 are two configurations extending p 1 and p 2 respectively, then they agree on + 2M E. The configuration c of Proposition and the configuration c 2, coincide on E and they are extensions of p 1 and p 2, respectively. 1.4 Edge Shifts A relevant class of one dimensional subshifts of finite type, is that of edge shifts. This class is strictly tied up the one of finite graphs. This relation allows us to study the properties of an edge shift (possible quite complex) studying the properties of its graph. On the other hand, each one dimensional shift of finite type is conjugate to an edge shift and hence they have the same invariants; thus also in this case the properties of the shift depend on the structure of the accepting graph. Definition Let G be a finite directed graph with edge set E. The edge shift X G is the subshift of E Z defined by X G := {(e z ) z Z t(e z ) = i(e z+1 ) for all z Z} where, for e E, i(e) denotes the initial vertex of e and t(e) the terminal one. The structure of a finite graph (and hence of an edge shift) can be easily described by a matrix, the so called adjacency matrix of G. More precisely, let G be a graph with vertex set V = {1,..., n}; the adjacency matrix of G is the matrix A such that A ij is the number of edges in G with initial state i and terminal state j. It is easily seen that the (i, j) entry of the matrix A m (the product of A with itself m times), is the number of paths in G with length m from i to j. The edge shift X G is sometimes denoted as X A, where A is the adjacency matrix of G. The fundamental role of the adjacency matrix will be clarified in the sequel. Concerning the edge shifts, it is easy to see that every edge shift is a 1 step shift of finite type with set of forbidden blocks {ef t(e) i(f)}. On the other hand, each shift of finite type is conjugate to an edge shift accepted by a suitable graph G. Now we give an effective procedure to construct it. 16

17 Let X be a shift of finite type X with memory M, the vertices of G are the words of L(X) of length M and the edges are the words of L(X) of length M +1. There is an edge named a 1... a M a M+1 L(X) from a 1... a M to a 2... a M+1. a 1... a M a M+1 a 1... a M a 2... a M+1 The edge shift accepted by this graph is the (M + 1)th higher block shift of X and is denoted by X [M+1]. Consider the function τ : X [M+1] X defined setting, for each c X [M+1] and each z Z, τ(c) z equal to the first letter of the word c z ; τ is a bijective local function and hence a conjugacy.... a 1 a 0... a M a 0 a 1... a M+1 a 1 a 2... a M a 1 a 0 a 1... The above table points out the behavior of the function τ. 1.5 Sofic Shifts The class of sofic shifts has been introduced by Weiss in [Wei1] as the smallest class of shifts containing the shifts of finite type and closed under factorization (i.e. the image under a local map of a sofic shift is sofic). Equivalently, one can see that a shift is sofic if it is the set of all labels of the bi infinite paths in a finite labeled graph (or finite state automaton). In automata theory, this corresponds to the notion of regular language. Indeed a language (i.e. a set of finite words over a finite alphabet) is regular if it is the set of all labels of finite paths in a labeled graph. Definition Let A be a finite alphabet; a labeled graph G is a pair (G, L), where G is a finite graph with edge set E, and the labeling L : E A assigns to each edge e of G a label L(e) from A. We recall that a finite state automaton is a triple (A, Q, ) where A is a finite alphabet, Q is a finite set whose elements are called states and : Q A P(Q) is a function defined on the Cartesian product Q A with values in the family of all subsets of Q. Suppose that Q (Q, a); this means that when the automaton is in the state Q and reads the letter a, then it can pass to the state Q. The automaton is deterministic when : Q A Q { }, that is 17

18 the state Q above (if it exists) is determined from Q and a; the automaton is complete when : Q A P(Q)\{ }, that is for each state Q and each letter a there is at least a state Q such that Q (Q, a). Clearly each general finite state automaton determines a labeled graph setting V := Q and drawing an edge labeled a from Q to Q if and only if Q (Q, a). Also the converse holds: given a labeled graph G with set of vertices V, we can set Q := V and (i, a) is the set of all vertices j of the graph for which there is an edge e such that i(e) = i, t(e) = j and L(e) = a. In the sequel we will not distinguish between a labeled graph and the related finite state automaton. If ξ =... e 1 e 0 e 1... is a configuration of the edge shift X G, define the label of the path ξ to be L(ξ) =... L(e 1 )L(e 0 )L(e 1 ) A Z. The set of labels of all configurations in X G is denoted by X G := {c A Z c = L(ξ) for some ξ X G } = L(X G ). Definition A shift X A Z is sofic if X = X G for some labeled graph G. A presentation of a sofic shift X is a labeled graph G for which X G = X. Obviously each edge shift is sofic. Indeed if G is a graph with edge set E, we can consider the identity function on E as a labeling L : E E so that X G = X (G,idE). Moreover, each shift of finite type is sofic. Consider the graph G introduced in the previous section that accepts the edge shift X [M+1] ; labeling the edges of G setting L(a 1... a M a M+1 ) = a 1 (that is L is the previous function τ : X [M+1] X), it can be easily seen that the labeled graph G so obtained is such that X = X G. a 1 a 1... a M a 2... a M+1 a 3... a M+2 a 2 Notice that in a graph G (or in the labeled graph G having G as support), there can be a vertex from which no edges start or at which no edges end. Such a vertex is called stranded. Clearly no bi infinite paths in X G (or in X G ), involve a stranded vertex, hence the stranded vertices and the edges starting or ending at them are inessential for the edge shift X G or for the sofic shift X G. 18

19 Following Lind and Marcus [LinMar, Definition 2.2.9], a graph is essential if no vertex is stranded. Removing step by step the stranded vertices of G, we get an essential graph Ḡ that gives rise to the same edge shift. This procedure is effective, because G has a finite number of vertices. Moreover, this essential form of G is unique. This property of the edge shift X G holds true for the sofic shift X G : the labeled graph Ḡ = (Ḡ, L E(Ḡ) ) is such that X G = X Ḡ (indeed X G = L(X G ) = L(X Ḡ ) = X Ḡ ). If we deal only with essential graphs, it is easy to see that the language of a sofic shift is regular. Among the subshifts of A Z, the sofic shifts are the images under a local function (briefly factors) of a shift of finite type. Indeed each labeling is a local function defined on an edge shift and, on the other hand, the factor of a shift of finite type is also the image under a local function of a suitable edge shift (each shift of finite type being conjugate to an edge shift). This characterization allows us to call sofic a shift X A Γ which is factor of a shift of finite type Minimal Deterministic Presentations of a Sofic Shift In automata theory, a finite state automaton is deterministic if, given a state Q and a letter a, there is at most one successive state Q (determined by Q and a). This corresponds, in the finite graph representing the automata, to the fact that from a vertex i (the state) there is at most one edge carrying the label a. Although this restrictive condition, one can prove that for each regular language there is a deterministic automaton accepting it. This property holds true for sofic shifts: each sofic shift admits a deterministic presentation. A minimal deterministic presentation of a sofic shift is a deterministic presentation with the least possible number of vertices. We will see that the irreducibility of the sofic shift implies the existence of only one (up to labeled graphs isomorphism) minimal deterministic presentation of it. Definition A labeled graph G = (G, L) is deterministic if, for each vertex i of G, the edges starting at i carry different labels. Proposition [LinMar, Theorem 3.3.2] Every sofic shift has a deterministic presentation. Definition A minimal deterministic presentation of a sofic shift X is a deterministic presentation of X having the least number of vertices among all deterministic presentations of X. One can prove that any two minimal deterministic presentations of an irreducible sofic shift are isomorphic as labeled graphs (see [LinMar, Theorem ]), so that one can speak of the minimal deterministic presentation of an irreducible sofic shift. In the following propositions, we clarify the relation between the irreducibility of a sofic (or edge) shift and the strong connectedness of its presentations. 19

20 On the other hand, it can be easily seen that the strong connectedness of G is equivalent to the following property of the n n adjacency matrix A of G: for each i, j {1,..., n}, there exists an m N such that the (i, j) entry of A m is not zero. Proposition [LinMar, Lemma ] Let X be an irreducible sofic shift and G = (G, L) the minimal deterministic presentation of X. Then G is a strongly connected graph. Proposition [LinMar, Proposition ] If G is a strongly connected graph, then the edge shift X G is irreducible. As a consequence of this two facts, we have the following corollary that will be useful in the sequel. Corollary Let X be an irreducible sofic shift and G = (G, L) the minimal deterministic presentation of X. Then the edge shift X G is irreducible. As we have seen, a shift is sofic if and only if it is the image under a local function of a shift of finite type. From the previous result it follows a stronger property. Corollary A shift is an irreducible sofic shift if and only if it is the image under a local function of an irreducible shift of finite type. Notice that, in general, the image under a local function of an irreducible shift is also irreducible. 1.6 Decision Problems Two natural decision problems arising in the theory of cellular automata concern the existence of effective procedures to establish the surjectivity and the injectivity of the transition function. For Euclidean cellular automata over full shifts, Amoroso and Patt have shown in [AP] that there are algorithms to decide surjectivity and injectivity of local maps for one dimensional cellular automata (see also [Mu]). On the other hand Kari has shown in [K1] and [K2] that both the injectivity and the surjectivity problems are undecidable for Euclidean cellular automata of dimension at least two. In this section we extend the problem to cellular automata over subshifts of finite type of A Z, giving in both cases a positive answer to the existence of decision procedures A Decision Procedure for Surjectivity If X is a shift of finite type, the problem of deciding whether or not a function τ : X X given in terms of local map is surjective, is decidable. In fact we can decide if a local function τ : X A Z is such that τ(x) X. 20

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