Francesca Fiorenzi- Cellular Automata and Finitely Generated Groups

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    Universita 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

    Universita di Roma La Sapienza

    Piazzale Aldo Moro 2 - 00185 Roma

    email [email protected]

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    Contents

    Introduction 3

    1. Cellular Automata 71.1 Cayley Graphs of Finitely Generated Groups . . . . . . . . . . . 81.2 Shift Spaces and Cellular Automata . . . . . . . . . . . . . . . . 101.3 Shifts of Finite Type . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Edge Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    1.5 Sofic Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.1 Minimal Deterministic Presentations of a Sofic Shift . . . 19

    1.6 Decision Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6.1 A Decision Procedure for Surjectivity . . . . . . . . . . . 201.6.2 A Decision Procedure for Injectivity . . . . . . . . . . . . 21

    1.7 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2. Surjunctivity and Density of Periodic Configurations 252.1 Surjunctivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Periodic Configurations of a Shift . . . . . . . . . . . . . . . . . . 262.3 Residually Finite Groups . . . . . . . . . . . . . . . . . . . . . . 282.4 Density of Periodic Configurations . . . . . . . . . . . . . . . . . 292.5 Periodic Configurations of Euclidean Shifts . . . . . . . . . . . . 31

    2.6 Group Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.1 Decision Problems for Group Shifts . . . . . . . . . . . . . 34

    3. The MooreMyhill Property 363.1 The Garden of Eden Theorem and the MooreMyhill Property . 383.2 Amenable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 The MooreMyhill Property for Subshifts ofAZ . . . . . . . . . 43

    3.3.1 A counterexample to Mooreproperty for a sofic subshiftofAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    3.4 Gromovs Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Strongly Irreducible Shifts . . . . . . . . . . . . . . . . . . . . . . 543.6 SemiStrongly Irreducible Shifts . . . . . . . . . . . . . . . . . . 62

    Bibliography 69

    2

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    Introduction

    The notion of a cellular automaton has been introduced by Ulam [U] and vonNeumann [vN]. In this classical setting, the universe is the lattice of integersZn of Euclidean space Rn. The set of states is a finite set A (also called thealphabet) and a configuration is a function c : Zn A. Time t goes on indiscrete steps and represents a transition function : AZ

    n AZn (if c is theconfiguration at time t, then (c) is the configuration at time t + 1), whichis deterministic and local. Locality means that the new state at a point

    Zn at time t + 1 only depends on the states of certain fixed points in theneighborhood of at time t. More precisely, if c is the configuration reachedfrom the automaton at time t then (c)| = (c|N ), where : A

    N A isa function defined on the configurations with support the finite set N (theneighborhood of the point 0 Zn), and N = + N is the neighborhoodof obtained from N by translation. For these structures, Moore [Moo] hasgiven a sufficient condition for the existence of the socalled Garden of Eden(GOE) patterns, that is those configurations with finite support that cannot bereached at time t from another configuration starting at time t 1 and hencecan only appear at time t = 0. Moores condition (i.e. the existence of mutuallyerasable patterns a sort of noninjectivity of the transition function on thefinite configurations) was also proved to be necessary by Myhill [My]. This

    equivalence between local injectivity and local surjectivity of the transitionfunction is the classical wellknown 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 sur-junctivity theorems (and, in this case, to the density of the periodic configu-rations) and to GOElike theorems restricted to the subshifts of the space A

    (where is a finitely generated group and A is a finite alphabet). Indeed forthese sets is still possible to define a structure of a cellular automaton.

    More precisely, given a finitely generated group , one can consider as uni-verse the Cayley graphof . A configuration is an element of the space A (thesocalled full Ashift), that is a function defined on with values in a finitealphabet A. The space A is naturally endowed with a metric and hence with

    an induced topology, this latter being equivalent to the usual product topology,

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    where the topology in A is the discrete one. A subset X of A which is 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 speci-fying a transition function : X X which is local (that is the value of (c),where c X is a configuration, at a point only depends on the values ofc at the points of a fixed neighborhood of ).

    In Chapter 1 we formally define all these notions, also proving that manybasic results for the subshifts of AZ given in the book of Lind and Marcus[LinMar], can be generalized to the subshifts of A. For example, we prove thatthe topological definition of shift is equivalent to the combinatorial one of setof configurations avoiding some forbidden blocks.

    Moreover we prove that a function between shift spaces is local if and only ifit is continuous and commutes with the action. This fact, together with thecompactness of the shift spaces (notice that in A closeness and compactness areequivalent), implies that the inverse of an invertible cellular automaton is also acellular automaton and allows us to call conjugacy each invertible local functionbetween two shifts. The invariants are those properties which are invariantunder 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 languageof the shift (i.e. the set of all finite words appearing in some biinfinite con-figuration), there is a word w such that the concatenation uwv still belongs tothe language. We generalize this definition to a generic shift, using the patternsand their supports rather than the words.

    Then, in terms of forbidden blocks the notion of shift of finite type is givenand we prove that, as in the onedimensional case, such a shift has a usefuloverlapping property that will be necessary in later chapters. Moreover werecall from [LinMar] the notions ofedge shiftand ofsofic shift, and restate manyof the basic properties of these onedimensional shifts strictly connected withthe onedimensional shifts of finite type.

    As we have noticed, cellular automata have mainly been investigated inthe Euclidean case and for the full shifts. The difference between the onedimensional cellular automata and the higher dimensional ones, is very deep.For example, Amoroso and Patt have shown in [AP] that there are algorithmsto decide surjectivity and injectivity of local maps for onedimensional cellularautomata. On the other hand Kari has shown in [K1] and [K2] that both theinjectivity and the surjectivity problems are undecidable for Euclidean cellularautomata of dimension at least two. Here we extend the AmorosoPatts resultsto the onedimensional 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 suitablesequence of sets and we prove that, in the case of nonexponential growth of thegroup, 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 onedimensionalcase, also stating the principal result of the PerronFrobenius theory to compute

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    it.

    A selfmapping : X X on a set X is surjunctive if it is either noninjectiveor surjective. In other words a function is surjunctive if the implication injective surjective holds. Using the GOE theorem and the compactness of the spaceAZ

    n

    , Richardson has proved in [R] that a transition function : AZn

    AZ

    n

    is surjunctive. In Chapter 2, we consider the surjunctivity of the transitionfunction in a general cellular automaton over a group .

    A configuration of a shift is periodic if its orbit is finite; after proving somegeneralities about periodic configurations, we recall the class of residually finitegroups, proving that a group is residually finite if and only if the pe

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