Self-stabilisation of cellular automata on tilings
SSelf-stabilisation of cellular automata on tilings
Nazim Fat`es ∗ Ir`ene Marcovici † Siamak Taati ‡§ Abstract
Given a finite set of local constraints, we seek a cellular automaton (i.e., a local and parallelalgorithm) that self-stabilises on the configurations that satisfy these constraints. More precisely,starting from a finite perturbation of a valid configuration, the cellular automaton must eventu-ally fall back to the space of valid configurations where it remains still. We allow the cellularautomaton to use extra symbols, but in that case, the extra symbols can also appear in the initialfinite perturbation. For several classes of local constraints (e.g., k -colourings with k (cid:54) = 3, andNorth-East deterministic constraints), we provide efficient self-stabilising cellular automata withor without additional symbols that wash out finite perturbations in linear or quadratic time, butalso show that there are examples of local constraints for which the self-stabilisation problem isinherently hard. We also consider probabilistic cellular automata rules and show that in somecases, the use of randomness simplifies the problem. In the deterministic case, we show that iffinite perturbations are corrected in linear time, then the cellular automaton self-stabilises evenstarting from a random perturbation of a valid configuration, that is, when errors in the initialconfiguration occur independently with a sufficiently low density. Keywords: tilings, shifts of finite type, cellular automata, self-stabilisation, noise, fault-tolerance,reliable computing, symbolic dynamics.
Introduction 21 Terminology and notations 3 (cid:96) -fillable tiling spaces (with (cid:96) ≥
2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Deterministic SFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∗ Universit´e de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France. † Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France. ‡ Department of Mathematics, American University of Beirut, Beirut, Lebanon. § The work of ST was partially supported by NWO grant 612.001.409. a r X i v : . [ n li n . C G ] J a n Discussion and open problems 35
Acknowledgments 39References 40
Introduction
While all living organisms posses some ability to stabilise or repair themselves when subjected to per-turbations or attacks, artificial systems rarely have such an ability. In particular, in systems designedin engineering and computer science, a small local perturbation (e.g., due to noise or tampering byan adversary) can often propagate throughout the system leading to a total or partial devastation ofthe behaviour of the system. The inevitability of such perturbations has lead to the study of systemswhich, in addition to their normal functionality, have the self-stabilisation property. A self-stabilisingsystem has the capacity to re-enter a set of “legal” or “desirable” states once the system has beentaken out of its normal behaviour by an external perturbation.The concept of self-stabilisation in computational processes was first introduced in 1970s by Dijk-stra, who presented examples of networks of finite-state automata with a non-trivial self-stabilisationproperty [8]. Since then, self-stabilisation has been widely studied in the context of distributed com-puting (see e.g. [9, 2]). In the current paper, we explore the question of self-stabilisation in the contextof cellular automata.In a cellular automaton (CA), the components of the system, the cells , are arranged regularly onan infinite d -dimensional lattice. The cells are identical finite-state automata that interact locally andchange their states synchronously. The overall state of the cells is referred to as a configuration ofthe CA. We will assume that the set of legal configurations of the system is specified with a finitenumber of local constraints. As a prototypical example, one may consider the colouring constraints:each cell can have any of a finite number of colours, and the legal configurations are those in whichevery two adjacent cells have different colours. More generally, we think of the legal states as tilings of the lattice with a finite number of tile types (identified with the states of the cells) satisfying localmatching constraints. In the language of symbolic dynamics, the set of legal configurations is simplya shift space of finite type (SFT). We will clarify the terminology further in the following section.We require our CA to have the following form of self-stabilisation:(1) Starting from a configuration that deviates from a legal configuration only on a finite region, theCA must evolve back to a legal configuration in a finite number of steps.(2) Starting from a legal configuration, the CA must remain unchanged.We note that in our definition, the CA has no extra functionality other than to keep the constraintssatisfied. Depending on the local constraints, even this simplified notion of self-stabilisation can bequite challenging to achieve. The difficulty is that the cells are indistinguishable and the informationavailable to each cell is limited to the state of its close neighbours. Since the cells are not aware of theirown absolute position or the position and extent of the error region, it is thus for example not possibleto correct the error region by starting from the upper-left corner and then proceeding sequentially.Self-stabilisation can be understood as a weak form of fault-tolerance, in which the perturbationsoccur only at the beginning. Stronger forms of fault-tolerance have been studied in the setting ofcellular automata, although aside from a few strong proof-of-concept constructions, the field remainswide open. Around the same time as Dijkstra, Toom found a class of CA which self-stabilise even inpresence of sufficiently weak temporal noise [34, 35] (see Examples 3.1 and 6.2 below). G´acs and Reifexploited Toom’s simplest example (i.e., the NEC-majority rule) to construct a three-dimensional CAwhich, in presence of noise, can perform universal computation reliably [20]. Subsequently, G´acs wasable to construct a sophisticated one-dimensional CA capable of reliable universal computation [17, 18].In our setting, Toom’s NEC-majority CA solves the self-stabilisation problem for the constraint thatadjacent cells must have the same colour.Various other problems studied in the setting of cellular automata can be related to self-stabilisation.For instance, the density classification problem [6, 29] can be formulated as a problem of self-stabilisation2here the system needs to return to a homogeneous configuration (all-zero or all-one) with the ad-ditional requirement that the colour which appears less frequently in the initial configuration is theone which has to be wiped off. It is known that Toom’s NEC-majority CA solves this problem, atleast when the system starts from a biased Bernoulli random configuration [6]. Another example isthe global synchronisation problem, which can again be understood as a self-stabilisation problem withthe homogeneous configurations as the legal states, with the main difference is the requirement that,in its legal state, the system must oscillate rather than remain unchanged [31, 12].This article has grown out of a conference paper in which some of our results were presented [14].The scope of the current paper is however more general and contains various new results not presentin the earlier paper. The structure of the paper is as follows: • In Section 1, we introduce the terminology and notation. • In Section 2, we present a general construction for self-stabilising one-dimensional SFTs. • As is the case for many other problems regarding cellular automata and tilings, the self-stabilisationproblem in two and higher dimensions is significantly more complex than in one dimension.Section 3 is dedicated to the two-dimensional case, where we present several constructions ofself-stabilising CA depending on the type of the constraints. Here, the example of k -colouringsserves as a running example, as it has different levels of difficulty depending on the parameter k .While the cases of k = 2 and k ≥ k = 4 self-stabilises in quadratic time, and we could not find any solutionwhatsoever for the case k = 3. We also provide a linear-time solution for the case of determinis-tic SFTs. Deterministic SFTs encompass a relatively rich family of constraints, including somewhich admit only non-periodic legal configurations. • In Section 4, we investigate the self-stabilisation of probabilistic CA, and show that, in some cases,access to randomness simplifies the self-stabilisation problem. For instance, our probabilisticsolution for k -colourings with k ≥ • After having explored the “algorithmic” aspects of self-stabilisation, we turn to the questionof “complexity” in Section 5. We show that for some choices of the legal constraints, the self-stabilisation problem is inherently hard (i.e., requires super-polynomial stabilisation time, unless P = NP ). We also show that “isomorphic” constraints (i.e., isomorphic SFTs) admit solutionswith roughly the same stabilisation times. • Section 6 is about a different notion of self-stabilisation in which the initial perturbations arerandom rather than finite. We show that if a (deterministic) CA self-stabilises from finite per-turbations in linear time, then it also self-stabilises from sufficiently weak Bernoulli randomperturbations. The more interesting question of self-stabilisation in presence of temporal noise(as in the case of Toom’s CA) is left open. • The article ends with some remarks and open questions in Section 7.
Configurations and patterns.
Let Σ be an alphabet , that is, a finite set of symbols , and let d ≥ x : Z d → Σ is called a configuration of the lattice Z d . The symbol x i is the state (or colour ) of cell i . A configuration is said to be homogeneous if all the cells are in the same state. Given c ∈ Σ, we denote by c the homogeneous configuration in which all cell have symbol c .The restriction of the configuration x ∈ Σ Z d to a set A ⊆ Z d is denoted by x A . A pattern is anassignment p : A → Σ with finite shape A ⊆ Z d , i.e., a partial configuration with finite domain. Wedenote by Σ the set of all patterns.A sequence x (1) , x (2) , . . . of configurations is said to converge to another configuration x if the stateof each cell in x ( n ) eventually fixates at the value of the same cell in x , that is, if for every i ∈ Z d n i such that x ( n ) i = x i for all n ≥ n i . This is the notion of convergence in the producttopology on Σ Z d . The space Σ Z d with the product topology is compact and metrizable.The shift by k ∈ Z d is the map σ k : Σ Z d → Σ Z d defined by ∀ i ∈ Z d , σ k ( x ) i (cid:44) x k + i . Every shift iscontinuous in the product topology. Shift spaces of finite type. A shift space of finite type (SFT) is a set X ⊆ Σ Z d of configurationsidentified by a finite number of local constraints. More specifically, let F ⊆ Σ be a finite set of finitepatterns, which we refer to as the forbidden patterns . The set of configurations x ∈ Σ Z d that avoid thepatterns in F (i.e. σ k ( x ) A / ∈ F for every pattern p : A → Σ of F and every k ∈ Z d ) is called an SFTand is denoted by X F . Every SFT is closed (hence compact) in the product topology, and is invariantunder every shift.Observe that the choice of the defining forbidden sets F is not unique, and in our discussion, weoccasionally need to consider distinct collections defining the same SFT. The smallest integer m forwhich X can be identified by a collection of forbidden patterns with shape S m (cid:44) { , , . . . , m − } d isreferred to as the interaction range of X .A pattern (or partial configuration) p : A → Σ is said to be globally admissible in X if p = x A for some x ∈ X , and is said to be locally admissible with respect to F if it has no occurrence of thepatterns from F , that is, if σ k ( p ) B / ∈ F for all k ∈ Z d and finite k + B ⊆ A . Note that in general, alocally admissible pattern need not be globally admissible. Tiling spaces (or nearest-neighbour SFTs). A tiling space (or nearest-neighbour SFT ) is anSFT defined by a collection of nearest-neighbour forbidden patterns, that is to say, patterns whoseshapes consist of exactly two adjacent cells. Formally, let e , e , . . . , e d denote the standard basisvectors in R d . A nonempty set X ⊆ Σ Z d is a ( d -dimensional) tiling space if there exist functions v , v , . . . , v d : Σ → { , } such that X = (cid:110) x ∈ Σ Z d : ∀ c ∈ Z d , ∀ i ∈ { , , . . . , d } , v i ( x c , x c + e i ) = 1 (cid:111) . Example 1.1 (Homogeneous space) . We denote by H = { , } the d -dimensional SFT containingonly the two homogeneous configurations , ∈ { , } Z d . This can be seen as the tiling space definedby the functions v (cid:44) · · · (cid:44) v d (cid:44) v where v ( a, b ) (cid:44) a = b , and 0 if a (cid:54) = b . (cid:35) Example 1.2 ( k -colourings) . A k -colouring of the lattice Z d is an assignment of colours from Σ (cid:44) { , , . . . , k − } to each position in such a way that the adjacent positions have different colours. Theset of all k -colourings C k is a tiling space identified by the functions v (cid:44) · · · (cid:44) v d (cid:44) v where v ( a, b ) (cid:44) a (cid:54) = b , and 0 if a = b . (cid:35) Example 1.3 (Hard-core) . The d -dimensional hardcore tiling space on the set of symbols Σ = { , } is defined by the function v (cid:44) · · · (cid:44) v d (cid:44) v where v ( a, b ) (cid:44) a, b ) (cid:54) = ( , ). (cid:35) Example 1.4 (Wang tiles) . A general family of tiling spaces are those defined by Wang tiles. A
Wang tile is a unit square with coloured edges (see Figure 14). The colours indicate the matchingrules for tiling: two tiles placed next to each other must have the same colour on their touching edges.A finite collection Θ of Wang tiles identifies a two-dimensional tiling space X ⊆ Θ Z , consisting of all valid tilings (i.e., configurations that respect the matching rule). In other words, X is defined by thefunctions v , v : Θ → { , } where v ( a, b ) (cid:44) a has the same colouras the left edge of b , and v ( a, b ) (cid:44) a has the same colour as the bottomedge of b . (cid:35) Finite perturbations.
For two configurations x, y ∈ Σ Z d , we denote by ∆( x, y ) (cid:44) { i ∈ Z d : x i (cid:54) = y i } the set of cells at which x and y disagree. A finite perturbation of a configuration x ∈ Σ Z d in Σ Z d is aconfiguration ˜ x ∈ Σ Z d such that ∆( x, ˜ x ) is finite.The diameter of a finite set A ⊆ Z d , denoted by diam( A ), is the smallest m ∈ N such that A fitsin a hypercube of size m , that is, A ⊆ i + [0 , m ) d for some i ∈ Z d . For two configurations x, y ∈ Σ Z d ,the diameter of ∆( x, y ) is denoted by δ ( x, y ). 4iven an SFT X ⊆ Σ Z d , we denote by ˜ X (cid:104) Σ (cid:105) the set of finite perturbations of the elements of X in Σ Z d , that is ˜ X (cid:104) Σ (cid:105) (cid:44) (cid:8) y ∈ Σ Z d : ∃ x ∈ X, δ ( x, y ) < ∞ (cid:9) .Let us stress that the set ˜ X (cid:104) Σ (cid:105) depends on the choice of the alphabet Σ. A larger alphabet Σ (cid:48) ⊇ Σwould lead to a larger set ˜ X (cid:104) Σ (cid:48) (cid:105) of finite perturbations. When the choice of the alphabet is clear fromthe context, we will simply use the notation ˜ X as a shortcut. Case of tiling spaces.
When considering an element x ∈ ˜ X , we will often examine the set of cellswhere the constraints of the SFT are not respected. We will say that such cells are defects . In thespecific case of tiling spaces (or nearest-neighbour SFT), we introduce different notions of defects.For a configuration x ∈ Σ Z d , a cell c ∈ Z d is said to have a defect in direction e i (with respect to v i )if v i ( x c , x c + e i ) = 0. It has a defect in direction − e i if v i ( x c − e i , x c ) = 0. In the two-dimensional case,we will also use the terminology E-defect, W-defect, N-defect, S-defect instead of respectively defectin direction e , − e , e − e . The set of cells having a defect is then defined by D ( x ) (cid:44) { c ∈ Z d : ∃ e ∈ {± e , . . . , ± e d } , c has a defect in direction e } . A cell c ∈ Z d is said to be defect-free if it does not belong to D ( x ), meaning that it obeys the localconstraints in the 2 d directions.Note that in somes cases, even if a configuration contains only very few defects, it is necessary tomodify a much larger set of cells in order to reach a valid configuration. More precisely, for some tilingspaces X , neither the cardinality nor the diameter of D (˜ x ) gives much information about δ (˜ x, X ). Example 1.5 (3-colourings) . Let C be the set of two-dimensional 3-colourings. For any integer n ≥ x ∈ ˜ C such that D ( x ) contains only two adjacent cells, and δ (˜ x, C ) ≥ n .In other words, in order to correct a single defect, one may have to modify the state of cells that arearbitrarily far. The construction of such a configuration is illustrated in Figure 1, using the connectionbetween the set of 3-colourings and the six-vertex model. This connection and further discussion of3-colourings will be presented in Section 7.1. (cid:35) C with only two defects, but at anarbitrary distance from C .A symbol α ∈ Σ is called a safe symbol for an SFT X ⊆ Σ Z d if for every x ∈ X and each k ∈ Z d ,the configuration ˜ x obtained from x by replacing x k with α is again in X . In case of a tiling space, thismeans that v i ( α, σ ) = v i ( σ, α ) = 1 for all σ ∈ Σ and i ∈ { , , . . . , d } . For example, for the hardcore5iling space of Example 1.3, the symbol is a safe symbol. If a tiling space has a safe symbol, thenthe phenomenon discussed above for 3-colourings cannot occur, since one can always update the cellshaving a defect with a safe symbol in order to recover a valid configuration. Cellular automata. A cellular automaton (CA) is a dynamical system on Σ Z d obtained by repeatedparallel updating of the symbols on the lattice using a local rule. More specifically, given a finite set N ⊆ Z d and a map f : Σ N → Σ, we can define a mapping F : Σ Z d → Σ Z d by F ( x ) i (cid:44) f (cid:0) σ i ( x ) N (cid:1) . Thisis the global map of the CA defined with local rule f and neighbourhood N .If the neighbourhood can be chosen to be N = {− r, . . . , r } d , we say that the CA has neighbour-hood radius r . The neighbourhood M (cid:44) {− , , } d is called the Moore neighbourhood, and theneighbourhood N = { , ± e , . . . , ± e d } is referred to as the von Neumann neighbourhood.For a neighbourhood N and a set A ⊆ Z d , we write N ( A ) and A + N interchangeably. For aninteger t ≥
1, we also introduce the set N t = N + . . . + N (cid:124) (cid:123)(cid:122) (cid:125) t times . If the CA F has neighbourhood N , then F t is a CA of neighbourhood N t . Self-stabilisation.
We say that a CA F : Σ Z d → Σ Z d stabilises an SFT X ⊆ Σ Z d from finite pertur-bations ifi) ( consistency ) the configurations of X are fixed points, that is, F ( x ) = x for every x ∈ X ,ii) ( attraction ) finite perturbations of the elements of X evolve to X in finitely many steps, that is,for every ˜ x ∈ ˜ X (cid:104) Σ (cid:105) , there exists a time t ∈ N such that F t (˜ x ) ∈ X .The first such t is called the stabilisation time (or the recovery time ) starting from ˜ x . We say that F stabilises X from finite perturbations in time τ ( n ) if for each n ∈ N , the largest stabilisation timeamong all the configurations ˜ x with δ (˜ x, X ) = n is τ ( n ).Note that the above definition does not rule out the possibility that Σ has extra symbols in additionto those appearing in the elements of X . In other words, it is possible that X ⊆ Γ Z d for some Γ (cid:40) Σ.Let us emphasize that, according to the above definition, in order for F to stabilise X , it is necessarythat all finite perturbations of X in Σ Z d (not just those in Γ Z d ) evolve to X .Following the observation made earlier, note that if a tiling space X ⊆ Σ Z d has a safe symbol,then one can easily construct a CA that stabilises X from finite perturbations, without extra symbols.Indeed, suppose that α is a safe symbol of X . Then, the map F : Σ Z d → Σ Z d defined by ∀ c ∈ Z d , F ( x ) c (cid:44) (cid:40) α if c ∈ D ( x ), x c otherwise. (1)is a CA with neighbourhood N = { , ± e i } , and it stabilises X in one step.The aim of the current article is to present self-stabilising CAs for other families of SFTs, for whichfinding a CA achieving the stabilisation from finite perturbations is a non-trivial problem. In this section, we focus on the problem of self-stabilisation for a one-dimensional SFT X ⊆ Σ Z . As aspecific case, let us present a well-known example of a CA stabilising the one-dimensional tiling space H = { , } in linear time, without extra symbols. Example 2.1 (GKL) . The G´acs-Kurdyumov-Levin (GKL) cellular automaton is the CA
GKL : { , } Z →{ , } Z with neighbourhood N = {− , − , , , } defined for any x ∈ { , } Z and k ∈ Z by GKL ( x ) k (cid:44) (cid:40) maj( x k , x k +1 , x k +3 ) if x k = 1,maj( x k , x k − , x k − ) if x k = 0.It is known that this CA is both a -eroder and a -eroder , which precisely means that it stabilises H from finite perturbations [19]. Furthermore, the stabilisation occurs in linear time [21]. Anotherslightly simpler CA having the same property was proposed by Kari and Le Gloannec, under the nameof modified traffic [24]. (cid:35) X ⊆ Σ Z be a one-dimensional SFT. Then, there exists an integer k ≥ X can bedescribed by a set F of forbidden words of length k + 1. In this case, we say that X is a k -step SFT.Indeed, the constraints can be represented by a transition matrix A indexed by Σ k × Σ k such that fortwo words v = v · · · v k and w = w · · · w k +1 of length k , A ( v · · · v k , w · · · w k +1 ) (cid:44) (cid:40) v = w , . . . , v k = w k and v v · · · v k w k +1 (cid:54)∈ F ,0 otherwise.Note that if k = 1, then X is a tiling space, that is, a nearest-neighbour SFT.For an integer n ≥
1, we denote by L n ( X ) the set of words of length n occuring in X , that is, L n ( X ) (cid:44) (cid:8) w ∈ Σ n : ∃ x ∈ X, x · · · x n = w · · · w n (cid:9) . The set L ( X ) (cid:44) (cid:83) n ∈ N L n ( X ) is called the language of X . Remark 2.2.
In the specific case when X is a mixing one-dimensional SFT (meaning that there exists n ≥ u, v ∈ L ( X ) and every n ≥ n , there exists a word w ∈ L n ( X )such that uwv ∈ L ( X )) and has at least one homogeneous configuration, then a result of Maass impliesthat the self-stabilisation problem is trivial for X [27, Theorem 3.2]. Indeed, the result of Maass statesthat there exists a CA F : Σ Z → Σ Z (i.e., without extra symbols) such that1. F is the identity map on X ,2. F (Σ Z ) = X .In particular, we achieve self-stabilisation in one step. (cid:51) For u, v ∈ L k ( X ), we write u ∼ v if there is a word w ∈ Σ ∗ such that uwv ∈ L ( X ). We say that X is non-wandering if ∼ is an equivalence relation, meaning that the transition graph defined by theadjacency matrix A consists of strongly connected components, with no connections between them. Example 2.3 (Wandering vs. non-wandering) . The SFT { Z , Z } ⊆ { , } Z is non-wandering, but theSFT consisting in Z , Z and the translations of · · · · · · is not. (cid:35) Example 2.4 (A non-wandering SFT) . The 1-step SFT on the alphabet Σ = { , , , , } with thetransition matrix and graph illustrated in Figure 2 is non-wandering. We will use this example as ared thread to illustrate the constructions of this section. (cid:35) A = Figure 2: An example of a non-wandering SFT. (left) transition matrix (right) transition graph.
Theorem 2.5 (Self-stabilisation in one dimension) . For every non-wandering one-dimensional SFT X ⊆ Σ Z , there exists a CA F : Σ (cid:48) Z → Σ (cid:48) Z with Σ (cid:48) ⊇ Σ that stabilises X from finite perturbations inlinear time. Remark 2.6.
Ilkka T¨orm¨a has communicated with us an alternative construction showing that every one-dimensional SFT is stabilised by a CA with no extra symbol [37]. His construction uses longmarkers made of forbidden patterns, allowing to avoid the introduction of extra symbols. (cid:51) X is a k -step non-wandering SFT.We denote by m the smallest non-negative integer such that for every u, v ∈ L k ( X ) with u ∼ v , wehave uwv ∈ L ( X ) for some word w of length at most m .In the case of the tiling space of Example 2.4, recall that k = 1, and one can check that m = 2.Indeed, for u = v = 1 or u = v = 2, one needs a word w of length at least 2 in order to have uwv ∈ L ( X ), while in all the other cases, a word of length 0 or 1 is sufficient.For a configuration y ∈ Σ Z , we denote by D ( y ) (cid:44) { i ∈ Z : y [ i − k,i + k ] / ∈ L k +1 ( X ) } the set of cells at which a defect occurs. Note that if y is a (finite) perturbation of a configuration x ∈ X , then D ( y ) ⊆ ∆( x, y ) + {− k, − k + 1 , . . . , k } , (i.e., every defect on y is within distance k froman element of ∆( x, y )), but D ( y ) could be much smaller than ∆( x, y ).Let us consider again Example 2.4. We have L k +1 ( X ) = { , , , , , , , } .Let y ∈ Σ Z be the configuration below. The set D ( y ) contains four elements, which correspond tothe positions marked by a cross. But in order to recover a valid configuration, we need to change thevalues of at least seven cells (the ones taking values 3 and 4). · · · · · ·× × × × A sequential correction process.
We first describe a particular sequential procedure for correctingfinite “islands” of defects on X . Namely, the procedure applied on a configuration y involves updatingthe symbols on y one by one, from left to right, starting from a cell on the left of the leftmost elementof D ( y ). Each update is performed according to the same local rule, which we call the patching rule .If y is in X , then the procedure does not modify any symbol in y . If y is a finite perturbation of aconfiguration x ∈ X , then the procedure eventually turns y into a configuration z ∈ X , before reachingfew cells to the right of the rightmost element of D ( y ). The existence of an appropriate patching rulerelies on the fact that X is non-wandering.More specifically, the patching rule will be a function g : Σ k + m → Σ. In order to update thesymbol at position i on y , we replace it with g ( y [ i − k,i + m + k ) ). The sequential updating of y from acell i to a cell j ≥ i proceeds by first updating the symbol at cell i , then updating the symbol atcell i + 1, and so forth until we update the symbol at cell j .The patching rule g is constructed as follows. • For u ∈ L k ( X ) and q ∈ Σ m + k , we let r ∈ [0 , m ] be the smallest index (if exists) such that uwq [ r +1 ,r + k ] ∈ L ( X ) for some w ∈ Σ r . If no such r exists, we choose an arbitrary a ∈ Σ suchthat ua ∈ L ( X ) and set g ( uq ) (cid:44) a . If r exists, we choose a corresponding w ∈ Σ r and set g ( uq ) (cid:44) w if r > g ( uq ) (cid:44) q if r = 0. • For u / ∈ L k ( X ) and q ∈ Σ m + k , we simply set g ( uq ) (cid:44) q .Let us see on some examples how we can construct a patching rule in the context of the tiling spaceof Example 2.4. If u ∈ { , , } and q ∈ { , } , then, there is no index r ∈ [0 ,
2] as above. So, wecan simply set g ( uq q q ) (cid:44) ( u + 1) mod 3. Consider now for example g ( ). Then r = 2, and for w = 01, we have ∈ L ( X ). So, we can set g ( ) = , which is in fact the only possible choicehere. For g ( ), then r = 1, and we have also no other choice than to set g ( ) = .Intuitively, when applied at the leftmost defect i ∈ D ( y ), the patching rule updates y i based on the(roughly) shortest patch that would remove the defects on y . The following simple lemma formulatesthe main property of the patching rule. Lemma 2.7 (Sequential process) . Let y be a finite perturbation of a configuration x ∈ X . Let [ a, b ] be an interval containing D ( y ) . Then, the sequential updating of y from a to b + m using the above-constructed patching rule g ends with an element of X .Proof. For i ∈ [ a, b + m ], let y ( i ) denote the configuration obtained during the sequential updatingprocedure right after updating cell i , and set y ( a − (cid:44) y for consistency.Observe that y (cid:48) (cid:44) y ( b − has the following property: every finite word in either y (cid:48) ( −∞ ,b − or y (cid:48) [ b, ∞ ) is in L ( X ). Since y (cid:48) [ b − k,b − and y (cid:48) [ i,i + k ] for i ≥ b are in the same transitive component8f X (here we are using the non-wandering property), there must be an integer r ∈ [0 , m ] such that y (cid:48) [ b − k,b − wy (cid:48) [ b + r,b + r + k ] ∈ L ( X ) for some w ∈ Σ r . Now, the construction of g ensures that y ( b + r − isin X .Let us observe on Figure 3 how the patching rule described for Example 2.4 corrects the configu-ration y represented above, when applied from left to right. The smallest interval [ a, b ] containing thedefects is marked on the first configuration. In order to update the symbol at position i , we replaceit by g ( y i − y i y i +1 y i +2 ). The successive cells that are updated are represented in bold, and the lastconfiguration that is shown belongs to the SFT, so that afterwards, the patching rule does not induceany new change. · · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · · Figure 3: An example of evolution of the sequential correcting process in the context of the SFT ofExample 2.4. Time goes downwards.
From the sequential rule to the stabilising CA.
We now construct a CA T : Σ (cid:48) Z → Σ (cid:48) Z with anextended alphabet Σ (cid:48) ⊇ Σ that corrects finite islands of defects on X in linear time. The CA uses thepatching rule g to correct defects sequentially from left to right. Since the CA cannot a priori identifythe leftmost defect, it instead applies the patching rule simultaneously everywhere that locally lookslike the leftmost defect. Therefore, if there are several far apart defect regions, there will be a correctiontrail initiated from the left of each of them. These correction trails need not be consistent with oneanother. We use suitable signals to make sure that the leftmost trail of correction is “dominant”, andthe other ones do not continue forever to the right, corrupting the original configuration. To this end,each correction trail leaves a trace (using extra symbols) that is slowly faded away on its own. If acorrection trail coming from the left meets a trace in front of it, it sends a fast signal ahead (again usingextra symbols) to stop the correction trail that has left that trace. So, the leftmost trail eventuallystops all the trails in front of it and goes on to correct the entire island of defects. We must of coursemake sure that this scenario works even if the defects in the initial configuration involve symbols fromthe extended alphabet.The extended alphabet will be Σ (cid:48) (cid:44) Σ × { ◦ , • , ⊗} , in which { ◦ } × Σ is identified with Σ. Thesymbol • represents the trace and ⊗ signifies the stop signal. For convenience, we identify the configu-rations with alphabet Σ (cid:48) with pairs ( y, α ) where y ∈ Σ Z and α ∈ { ◦ , • , ⊗} Z . The configuration ( y, α )has a defect at cell i if either y has a defect at i , or i contains a signal symbol • or ⊗ . Extending thenotation D ( y ), we denote the set of defects on ( y, α ) by D ( y, α ) (cid:44) { i ∈ Z : i ∈ D ( y ) or α i (cid:54) = ◦ } . We also define the set D ( y, α ) (cid:44) (cid:8) i ∈ D ( y, α ) : y [ i − k − ,i + k − ∈ L k +1 ( X ) (cid:9) g can only affect the cells in D ( y, α ).The CA T will be constructed as a composition T ( T g T T ) of four CA maps T g , T , T , T : Σ (cid:48) Z → Σ (cid:48) Z . This composition will ensure that the fading of the traces is half as slow as the correction speed,while the stop signals propagate twice as fast as the correction speed. Patching . The map T g is defined by T g ( y, α ) (cid:44) ( y (cid:48) , α (cid:48) ), where( y (cid:48) i , α (cid:48) i ) (cid:44) (cid:40)(cid:0) g ( y [ i − k,i + m + k ] ) , • (cid:1) if α i − (cid:54) = ⊗ and i ∈ D ( y, α ),( y i , α i ) otherwise.It simply applies the patching rule on the elements of D ( y, α ) (if no stop signal on the left) andleaves a trace behind. It also erases any stop symbol which sees no stop symbol on its left neighbour,replacing it with a trace symbol. Generation of stop signals . The map T is responsible for generating stop signals, and is defined by T ( y, α ) (cid:44) ( y, α (cid:48) ) (i.e., no change on y ), where α (cid:48) i (cid:44) (cid:40) ⊗ if α i = • and i ∈ D ( y ), α i otherwise. Propagation of stop signals . The propagation of the stop signals is governed by the map T , which isdefined by T ( y, α ) (cid:44) ( y, α (cid:48) ) (i.e., no change on y ), where α (cid:48) i (cid:44) (cid:40) ⊗ if α i − = ⊗ and α i = • , α i otherwise. Fading of the traces . Finally, the map T handles the fading of the traces and is defined by T ( y, α ) (cid:44) ( y, α (cid:48) ) (i.e., no change on y ), where α (cid:48) i (cid:44) (cid:40) ◦ if α i − = ◦ and α i = • , α i otherwise.In Figure 4, we illustrate the operation of the cellular automaton in the context of Example 2.4.Two consecutive configurations correspond to one application of T g T T (in fact, just of T g here, sinceno stop symbol occurs in the configurations that are represented). Every two steps, the map T isalso applied, erasing one trace symbol on the left of each correction region. Figure 5 illustrates theevolution of the CA in longer time span. When the front of a correction trail meets the fading trace ofanother correction trail in front of it, a stop signal is created. This stop signal travels faster than thecorrection trails, and hence quickly catches up with all the correction trails in front of it. As a result,every correction trail which is not initiated by the leftmost defect is eventually stopped. Proof of Theorem 2.5.
We verify that the CA T constructed above corrects finite islands of defectson X in linear time.Let ( y, α ) : Z → Σ (cid:48) be a finite perturbation of a configuration x ∈ X (or more explicitly, a finiteperturbation of ( x, ◦ Z )). Let us call a cell i ∈ D ( y, α ) • active on ( y, α ) if i ∈ D ( y, α ) and α i − (cid:54) = ⊗ , • frozen if α i − = ⊗ , and • fading if α i = • .The tail of an active cell i is the longest (possibly empty) interval [ i − l, i −
1] of fading cells. Atail whose leftmost element is frozen is said to be freezing . An active cell together with its tail is acorrection trail . Observe that the leftmost element of D ( y, α ) is either active, or fading and non-frozen.The CA works intuitively as follows. Let [ a, b ] ⊆ Z be the smallest interval containing ∆(( x, ◦ Z ) , ( y, α )).The leftmost trail moves with speed at least at most t > · · · · ·• •· · · · · ·• • • •· · · · · ·• • • •· · · · · ·• • • • • •· · · · · ·• • • • • • T g T T T ( T g T T ) T g T T T ( T g T T ) Figure 4: An example of evolution of the celllular automaton in the context of the SFT of Example 2.4.Time goes downwards. • the leftmost active cell is on the right of a + 2 t , • the leftmost fading cell is on the right of a + t , • the rightmost active cell is on the left of b + 2 t , • the rightmost non-freezing tail is on the left of b + t .It follows that before time t (cid:44) b − a , the leftmost trail overpasses the rightmost non-freezing tail, andcauses it to freeze. From that moment on, it takes at most ( b − a ) / t (cid:44) t + (cid:98) ( b − a ) / (cid:99) < ( b − a ), the CA is then in a configuration( y (cid:48) , α (cid:48) ) with the following properties. • The configuration ( y (cid:48) , α (cid:48) ) is a finite perturbation of ( x, ◦ Z ) with ∆( x, y (cid:48) ) ⊆ [ a + 2 t , b + 2 t ] and∆(( x, ◦ Z ) , ( y (cid:48) , α (cid:48) )) ⊆ [ a + t , b + 2 t ]. • The only active cell on ( y (cid:48) , α (cid:48) ) is on the right of a + 2 t and every element of D ( y (cid:48) , α (cid:48) ) to the leftof it is fading and non-frozen.From time t onward, the CA essentially applies the patching rule, removing the defects on y in nomore than b − a + m steps. At time t (cid:44) t + b − a + m , the CA is in a configuration ( y (cid:48)(cid:48) , α (cid:48)(cid:48) ) in which • y (cid:48)(cid:48) has no defect, • α (cid:48)(cid:48) contains no stop symbol ⊗ and its trace symbols • are contained in region [ a + t , a + 2 t ].Eventually, in at most ( a + 2 t ) − ( a + t ) = t more steps, every trace symbol fades away, and wearrive at a configuration in X before time t (cid:44) t + t ≤ ( b − a ) + m . We now consider the problem of stabilising a two-dimensional SFT. In this section, we will focus onsome specific families of tiling spaces, for which different strategies can be impletemented in order toachieve self-stabilisation efficiently. The first cases we consider are inspired from the study of self-stabilisation for k -colourings [14]. Namely, Sections 3.1, 3.2, and 3.3 treat and extend respectively thecases of 2-colourings, k -colourings with k ≥
5, and 4-colourings. Finally, Section 3.4 is based on thesame construction as in Section 2, allowing us to handle the case of deterministic
SFTs. Most of theconstructions in this section can be easily generalized to higher dimensions, but for concreteness, wefocus on the two-dimensional case.As a first example, let us present a well-known CA stabilising the two-dimensional tiling space H = { , } in linear time and without extra symbols.11 ••••••••••••• ••••• ••• •••••••••• ••••••• ⊗⊗⊗⊗⊗⊗⊗⊗⊗ •••••⊗ Fading of the trace (speed 1) Front of the trace (speed 2) Propagation of the stop signal (speed 4)Figure 5: Illustration showing the behaviour of the different signals composing the cellular automaton.Time goes downwards.
Example 3.1 (Toom’s North-East-Center majority rule) . Toom’s (deterministic) majority cellularautomaton is the CA
NEC-Maj : { , } Z → { , } Z with neighbourhood N = { , e , e } defined, forany x ∈ { , } Z and k ∈ Z , by NEC-Maj ( x ) k (cid:44) maj( x k , x k + e , x k + e ) , where maj denotes the majority function, outputting the symbol which is in majority among the inputsymbols.It is known that this two-dimensional CA is both a -eroder and a -eroder , which is equivalentto the CA stabilising H = { , } from finite perturbations. Furthermore, the stabilisation occurs inlinear time [35, 36]. (cid:35) t = 0 t = 5 t = 10 t = 15Figure 6: Illustration of the evolution of Toom’s CA NEC-Maj .As we shall discuss in Section 5, it seems hopeless to be able to construct, for any two-dimensionalSFT, a CA that stabilises it from finite perturbations in linear time, or even in polynomial time.However, in the following of this section, we will present several classes of SFTs for which we are ableprovide CA that stabilise in linear or quadratic time. We first focus on colourings and similar tilingspaces, then we treat the case of two-dimensional deterministic tiling spaces.12 .1 Finite SFTs
In this section, we treat the case where the SFT contains only a finite number of configurations.Observe that the configurations of such a finite SFT are necessarily spatially periodic.
Example 3.2 (Finite SFTs) . The set H = { , } ⊆ { , } Z is a finite tiling space. The set C ofall 2-colourings of Z is also a finite tiling space, since it contains only two configurations, namely theodd and even chequerboard configurations. (cid:35) Let us consider an arbitrary finite SFT X ⊆ Σ Z . Then, for each configuration x ∈ X , there existintegers n , n ≥ horizontal and vertical periods of x ) such that σ n e ( x ) = σ n e ( x ) = x . Let N (respectively, N ) denote the least common multiple of the horizontal (resp., vertical) periods ofall the configurations in X . Since X is finite, N and N are finite. Then, for all x ∈ X , we have σ N e ( x ) = σ N e ( x ) = x . We define a CA F on Σ Z by F ( x ) k (cid:44) maj( x k , x k + N e , x k + N e ) , where the majority function maj( a, b, c ) assigns to three symbols a, b, c the symbol which is mostcommon among a, b, c , with the convention that when a, b, c are distinct, one can choose arbitrarilythe value of the function. Observe that F simply consists in applying Toom’s majority rule on eachsub-lattice generated by N e and N e . Proposition 3.3 (Self-stabilisation of finite SFTs) . Let X ⊆ Σ Z be a finite SFT. Then, the CA F : Σ Z → Σ Z defined above stabilises X from finite perturbations in linear time.Proof. It is clear from the definition that F ( x ) = x for every x ∈ X . For each n ∈ N , definethe triangular set T n = { k ∈ Z : k + k ≤ n, k , k ≥ } . Let x ∈ X and take y such that∆( x, y ) is finite. By translating x and y if needed, we can assume without loss of generality thatthe difference set ∆( x, y ) is included in the triangle T n for some n . It is then easy to verify that∆ (cid:0) x, F ( y ) (cid:1) ⊆ T n − . Indeed, for every cell outside T n , the local rule does not modify the state,whereas for the cells k ∈ Z which are inside T n and satisfy k + k = n , we have F ( y ) k = x k , since x k = x k + Ne = x k + Ne = y k + Ne = y k + Ne . Iterating F we obtain ∆ (cid:0) x, F t ( y ) (cid:1) ⊆ T n − t for each t ≥ t = n + 1,we obtain ∆ (cid:0) x, F n +1 ( y ) (cid:1) ⊆ T − = ∅ , hence F n +1 ( y ) = x ∈ X . This means that the configuration y has been corrected in at most n + 1 steps. Remark 3.4.
The above result clearly extends to dimensions d ≥
2. One can simply apply Toom’smajority rule on each (two-dimensional) sub-lattice generated by N e and N e , and in the proof,replace T n by the triangular prism T n = { k ∈ Z d : k + k ≤ n, k , k ≥ } . (cid:51) We say that a two-dimensional tiling space is single-cell fillable if there exists a map ψ : Σ → Σ suchthat, for any possible choice ( a, b, c, d ) ∈ Σ of symbols surrounding a cell (see Figure 7), assigningthe value ψ ( a, b, c, d ) to the central cell ensures that it is defect-free. More specifically, the value α = ψ ( a, b, c, d ) satisfies v ( a, α ) = v ( α, c ) = 1 and v ( α, b ) = v ( d, α ) = 1. One can observe that atiling space is single-cell fillable if and only if the functions v and v can be chosen in such a way thatevery locally admissible pattern is globally admissible [28]. αa b cd Figure 7: Illustration of the notion of single-cell fillabilityNote that any tiling space having a safe symbol is trivially single-cell fillable. The following is amore interesting family of single-cell fillable tiling spaces.13 xample 3.5 (Single-cell fillable) . For k ≥
5, the space C k of two-dimensional k -colourings is single-cell fillable.More generally, let G = ( V, E ) be a finite undirected graph (possibly with self-loops). Let X G ⊆ V Z d denote the set of all graph homomorphisms from Z d (with nearest-neigbour edges) to G . In other words, x ∈ X G if and only if x i x j ∈ E for every two adjacent cells i, j ∈ Z d . The set X G is clearly a tilingspace, defined by the functions v (cid:44) · · · (cid:44) v d (cid:44) v where v ( a, b ) (cid:44) ab ∈ E . Note thatif G is the complete graph on k vertices (without self-loops), then X G coincides (up to renaming ofthe vertices) with the space of k -colourings C k . It is easy to see that X G is single-cell fillable if andonly if G has the following property: for any subset A ⊆ V with | A | = 2 d , there exists a vertex b ∈ V such that ab ∈ E for every a ∈ A . In particular, X G is single-cell fillable if the minimum degree of thevertices in G is larger than (cid:0) − (2 d ) − (cid:1) | V | . See [5, 7] for more on the homomorphism spaces. (cid:35) Let X be a single-cell fillable tiling space. We introduce the following terminology for defects. Wesay that a cell ( i, j ) has a NE-defect if it has a N-defect or an E-defect (or both). For x ∈ Σ Z , wedenote by D NE ( x ) the set of cells having a NE-defect, that is: D NE ( x ) (cid:44) (cid:8) ( i, j ) ∈ Z : v ( x c , x c + e ) = 0 or v ( x c , x c + e ) = 0 (cid:9) . Let ψ : Σ → Σ be a function which assigns, to a choice ( a, b, c, d ) ∈ Σ of symbols surrounding acell, a symbol ψ ( a, b, c, d ) for the central cell ensuring that it is defect-free. We define a CA F on Σ Z by ∀ c ∈ Z , F ( x ) c (cid:44) (cid:40) ψ ( x c − e , x c − e , x c + e , x c + e ) if c ∈ D NE ( x ), x c otherwise. Proposition 3.6 (Self-stabilisation of single-cell fillable tiling spaces) . Let X ⊆ Σ Z be a single-cellfillable tiling space. Then, the CA F : Σ Z → Σ Z defined above stabilises X from finite perturbationsin linear time.Proof. It is clear from the definition that F ( x ) = x for every x ∈ X . Let us now take x ∈ ˜ X .First, observe that if c / ∈ D NE ( x ) , then c / ∈ D NE ( F ( x )), so that the set of NE-defects can onlydecrease under the action of F . Let us indeed take c / ∈ D NE ( x ). By definition of F , the value of cell c is not modified when applying F , and if the value of cell c + e (resp. c + e ) is modified, that is, if F ( x ) c + e (cid:54) = x c + e (resp. F ( x ) c + e (cid:54) = x c + e ), then the new value of c + e (resp. c + e ) is chosen insuch a way that v (cid:0) F ( x ) c , F ( x ) c + e (cid:1) = 1 (resp. v (cid:0) F ( x ) c , F ( x ) c + e (cid:1) = 1), so that c / ∈ D NE ( F ( x )).Second, if c ∈ D NE ( x ) is such that c + e , c + e / ∈ D NE ( x ), then c / ∈ D NE ( F ( x )), so that the set ofNE-defects is progressively eroded, from the NE to the SW. More formally, we can assume without lossof generality that there exists an integer n ≥ D NE ( x ) ⊆ T n , where T n = { ( i, j ) ∈ Z : i + j ≤ n, i, j ≥ } , and one can check that after t steps, we have D NE ( F t ( x )) ⊆ T n − t . Thus, after n + 1 steps,we have D NE ( F n +1 ( x )) = ∅ , meaning that the configuration is fully corrected: F n +1 ( x ) ∈ X . Remark 3.7.
The result extends naturally to d -dimensional single-cell fillable tiling spaces, with d ≥ (cid:51) (cid:96) -fillable tiling spaces (with (cid:96) ≥ ) We say that a tiling space is strongly (cid:96) -fillable if there exists a map ψ : Σ (cid:96) → Σ (cid:96) such that, for anypossible choice ( a , . . . , a (cid:96) ) ∈ Σ (cid:96) of symbols surrounding an (cid:96) -square (that is, an (cid:96) × (cid:96) block of cells),assigning the values ( b , . . . , b (cid:96) ) = ψ ( a , . . . , a (cid:96) ) to the inner cells of the (cid:96) -square ensures that eachcell of the (cid:96) -square is defect-free (see Figure 8). Note that here, we do not assume any further conditionon ( a , . . . , a (cid:96) ) ∈ Σ (cid:96) . We refer to [1] for a similar but weaker condition of (cid:96) -fillability. Example 3.8 (Strongly 2-fillable) . The set C of two-dimensional 4-colourings is not single-cell fil-lable. We claim that C is strongly 2-fillable. To prove this, we need to show that for any possiblechoice ( a, b, c, d, e, f, g, h ) ∈ Σ of symbols surrounding a 2-square (see Figure 9), there exists a choice( α, β, γ, δ ) ∈ Σ for the cells of the 2-square such that the four cells of the 2-square are defect-free.If { a, d, e, h } (cid:40) Σ, then we can choose a colour from Σ \ { a, d, e, h } and assign it to both α and γ .We are then sure that we can find suitable colours for the two remaining cells, since each of these twocells is surrounded by at most three different colours. In the same way, if { b, c, f, g } (cid:40) Σ, we can finda valid pattern. 14 b . . .. . . b (cid:96) a a ... a (cid:96) a (cid:96) +1 a (cid:96) +2 . . . a (cid:96) a (cid:96) +1 a (cid:96) +2 ... a (cid:96) a (cid:96) +1 a (cid:96) +2 . . .a (cid:96) Figure 8: Illustration of the notion of strong (cid:96) -fillabilityLet us now assume that { a, d, e, h } = { b, c, f, g } = Σ. Without loss of generality, we can assumethat a = 0 , h = 1 , d = 2 , e = 3. The set of allowed colours for α is then { , } , and the set of allowedcolours for γ is { , } . If the allowed colours for β and δ are { , } and { , } respectively, then avalid pattern is given by ( α, β, γ, δ ) = (2 , , , β and δ are { , } , { , } respectively, then a valid pattern is given by ( α, β, γ, δ ) = (2 , , , (cid:35) αβ δγab c d feh g Figure 9: Illustration of the notion of strong 2-fillability
Example 3.9 (Strongly 2-fillable) . Consider the set Θ of all Wang tiles whose edges are colouredeither black or white, except the one with four white edges. It is easy to see that the space of validtilings with tiles from Θ is strongly 2-fillable but not single-cell fillable. (cid:35)
Example 3.10 (Strongly 2-fillable) . Consider the set of Wang tiles depicted in Figure 10. The deco-rations symbolise the edge colours, hence there are three possible colours: “horizontal line”, “verticalline” and “none”. Let X denote the space of all valid tilings. It can be verified that X is strongly2-fillable but not single-cell fillable. (cid:35) For (cid:96) = 1, strong (cid:96) -fillability corresponds to single-cell fillability. Therefore, in what follows, we willassume that (cid:96) ≥
2. The construction that we present below is slightly different from the one of [14],allowing for a simpler proof. The proof in the conference article had some inaccuracies which are nowavoided.Let X be a strongly (cid:96) -fillable tiling space, and let us design a CA that corrects finite perturbationsof X . Let ψ : Σ (cid:96) → Σ (cid:96) be a function that maps some ( a , . . . , a (cid:96) ) ∈ Σ (cid:96) to an element of Σ (cid:96) , suchthat the pattern formed by these values is a defect-free pattern. The aim is to apply ψ on a collectionof non-overlapping, non-touching (cid:96) -squares containing defects so as to reduce the number of defects.More specifically, the CA must (locally) select some (cid:96) -squares containing defects in such a way thatthe following two conditions are satisfied: 15a) (b)Figure 10: Illustration of Example 3.10. (a) The tile set. (b) An example of a valid tiling.1. Every two selected (cid:96) -squares are at distance at least 1 (they do not overlap or touch each other).2. If the configuration contains a defect, then at least one (cid:96) -square containing a defect is selected.To this end, let us first identify a set of cells that will play the role of the top-right corners of theselected (cid:96) -squares. Given a configuration x ∈ Σ Z , let us denote again the set of cells having a NE-defect by D NE ( x ) = (cid:8) c ∈ Z : v ( x c , x c + e ) = 0 or v ( x c , x c + e ) = 0 (cid:9) . For the sake of clarity, we firstdefine a stabilising CA in the case (cid:96) = 2, and then treat the general case. Case (cid:96) = 2 . We say that a cell c ∈ Z is a NE-corner if c ∈ D NE ( x ) and c − e + e , c − e + 2 e (cid:54)∈ D NE ( x ) ,c + 2 e − e , c + e , c + e , c − e + 2 e (cid:54)∈ D NE ( x ) ,c + 2 e , c + e + e , c + 2 e (cid:54)∈ D NE ( x ) ,c + 2 e + e , c + e + 2 e (cid:54)∈ D NE ( x ) , See Figure 11 for an illustration of the definition. We denote by C NE ( x ) the set of NE-corners in aconfiguration x ∈ Σ Z .Figure 11: Illustration of the notion of NE-corner in the case (cid:96) = 2. The central cell is a NE-cornerif at least one of the red lines (North or East or both) contains a defect and all the green lines aredefect-free.We define a CA F by the following rule: if a cell c = ( i, j ) ∈ Z is a NE-corner, then apply ψ to the 2-square whose NE-corner is c , that is, replace the symbols of the cells ( i − , j − , ( i − , j ) , ( i, j ) , ( i, j − ψ ( a, b, . . . , h ), where a = x i − ,j − , b = x i − ,j , . . . , h = x i − ,j − (see Figure 9). Let us first observethat the CA F given by this rule is well-defined. Indeed, by definition of a NE-corner, one can checkthat there are no two adjacent NE-corners, vertically or horizontally, or in diagonal. Consequently, ateach time step, the 2-squares that are updated do not overlap. In addition, note that the definitionof NE-corners ensures that the 2-squares that are updated cannot share a common edge, so that theproperty 1 above is satisfied. 16 eneral case ( (cid:96) ≥ ). In the general case, we modify the notion of NE-corner as follows. We saythat a cell c ∈ Z is a NE-corner if c ∈ D NE ( x ) and for any − (cid:96) ≤ i, j ≤ (cid:96) ,if 1 ≤ i + j ≤ (cid:96) − i + j = 0 and j > i ], then c + ie + je (cid:54)∈ D NE ( x ).See Figure 12 for an illustration in the case (cid:96) = 3. We denote again by C NE ( x ) the set of NE-cornersin a configuration x ∈ Σ Z .Figure 12: Illustration of the notion of NE-corner in the case (cid:96) = 3. The central cell is a NE-cornerif at least one of the red lines (North or East or both) contains a defect and all the green lines aredefect-free.Again, we define a CA F by the following rule: if a cell c = ( i, j ) ∈ Z is a NE-corner, then apply ψ to the (cid:96) -square whose NE-corner is c . By a similar argument as for (cid:96) = 2, F is well-defined, and inaddition, the (cid:96) -squares that are updated cannot share a common edge, so that the property 1 above issatisfied.We are now able to state the result, whose proof now consists in proving that the property 2 aboveis also satisfied. Proposition 3.11 (Self-stabilisation of strongly fillable tiling spaces) . Let X ⊆ Σ Z be a strongly (cid:96) -fillable tiling space, with (cid:96) ≥ . Then, the CA F : Σ Z → Σ Z defined above stabilises X from finiteperturbations in quadratic time.Proof. Let x ∈ ˜ X . We prove that if D ( x ) (cid:54) = ∅ , then C NE ( x ) (cid:54) = ∅ . Let us indeed sweep the configu-ration x by NW-SE diagonals, from the NE to the SW. Since D ( x ) is finite, we can consider the firstdiagonal which contains a NE-defect, and on this diagonal, we consider the leftmost NE-defect (whichis also the uppermost). By definition of a NE-corner, this NE-defect is a NE-corner.In order to end the proof, it is then sufficient to observe that while D ( x ) (cid:54) = ∅ , the number of defectsdecreases strictly when applying F . Indeed, at least all the NE-corners become defect-free, and sincethe (cid:96) -squares that are updated do not share any common edge, no new defect is created. We now extend the construction proposed in Section 2 for one-dimensional SFTs to a specific class ofhigher-dimensional SFTs: the deterministic
SFTs. To begin with, let us define this notion. Again, wefocus on the two-dimensional case, but the ideas can be applied also in higher dimensions.We say that a two-dimensional SFT X ⊆ Σ Z is NE-deterministic if it possible to describe it by aset F of forbidden patterns in such a way that all the forbidden patterns of F have shape { , e , e } ,and F has the additional property that for every a, b ∈ Σ, there exists at most an element c ∈ Σ suchthat the pattern p : { , e , e } → Σ defined by p ( e ) = a , p ( e ) = b and p (0) = c does not belong to F .The constraints are therefore completely specified by the partial function f : Σ × Σ → Σ, that maps( a, b ) ∈ Σ to the unique symbol c that is allowed, if any. Example 3.12 (Ledrappier SFT) . The SFT X (cid:44) { x ∈ { , } Z : x k = x k + e + x k + e (mod 2) for all k ∈ Z } known as the Ledrappier
SFT, or as the three-dot system , is NE-deterministic. (cid:35) (cid:96) = 2. Defects are represented inred, and cells that are NE-corners are shaded. For the other NE-defects, the constraints representedin green on Figure 11 are not all satisfied. As represented, we know that by construction, there existsat least one NE-corner on the first NW-SE diagonal that contains a NE-defect.In case of SFTs identified by Wang tiles, NE-determinism means that for every two tiles a, b ∈ Σ,there is at most one tile c = f ( a, b ) ∈ Σ such that the right edge of c is compatible with the left edgeof a and the upper edge of c matches the lower edge of b . Example 3.13 (Aperiodic deterministic tile sets) . The SFTs in two and higher dimensions can besignificantly more complex than the one-dimensional SFTs. For instance, in one dimension, everynon-empty SFT contains periodic configurations, whereas in two dimensions, there are various exam-ples of non-empty SFTs that do not contain periodic configurations. Moreover, it is algorithmicallyundecidable whether the SFT identified by a given finite set of forbidden patterns is non-empty [4, 32].Figure 14 shows a set of 16 Wang tiles, discovered by R. Ammann, which is aperiodic , meaning thatit admits valid tilings but no periodic valid tilings [22, Chapter 11]. Remarkably, Ammann’s tile setis both NE-deterministic and SW-deterministic. The question of whether a given finite set of Wangtiles admits a valid tiling remains undecidable when restricted to NE-deterministic (or even four-waydeterministic) tile sets [23, 26]. (cid:35)
Figure 14: Ammann’s set of Wang tiles is NE-deterministic and SW-deterministic.As illustrated by the two examples above, the class of NE-deterministic tiling spaces is thus richand multi-faceted.
Theorem 3.14 (Self-stabilisation of NE-deterministic SFTs) . For every two-dimensional NE-deterministicSFT X ⊆ Σ Z , there exists a CA F : Σ (cid:48) Z → Σ (cid:48) Z with Σ (cid:48) ⊇ Σ that stabilises X from finite perturba-tions in linear time. We construct a CA T : Σ (cid:48) Z → Σ (cid:48) Z with an extended alphabet Σ (cid:48) ⊇ Σ that corrects finite islands ofdefects on X in linear time. The construction is similar to the one-dimensional case (Section 2), except18hat the patching rule is replaced by the map f stemming from the fact that X is a deterministic tilingspace.Let us thus consider a NE-deterministic SFT X ⊆ Σ Z , identified by a partial map f : Σ × Σ → Σ.For a configuration y ∈ Σ Z , we denote by D ( y ) (cid:44) { ( i, j ) : y i,j (cid:54) = f ( y i +1 ,j , y i,j +1 ) } the set of cells at which a defect occurs.As before, the extended alphabet will be Σ (cid:48) (cid:44) Σ × { ◦ , • , ⊗} , in which { ◦ } × Σ is identifiedwith Σ, and • and ⊗ represent, respectively, the trace and stop signals. Again, for convenience, weidentify the configurations with alphabet Σ (cid:48) with pairs ( y, α ) where y ∈ Σ Z and α ∈ { ◦ , • , ⊗} Z . Theconfiguration ( y, α ) has a defect at cell ( i, j ) if either y has a defect at ( i, j ), or ( i, j ) contains a signalsymbol • or ⊗ . Extending the notation D ( y ), we denote the set of defects on ( y, α ) by D ( y, α ) (cid:44) { ( i, j ) ∈ Z : ( i, j ) ∈ D ( y ) or α i,j (cid:54) = ◦ } . The CA T will be, as before, constructed as a composition T ( T g T T ) of four CA maps T g , T , T , T :Σ (cid:48) Z → Σ (cid:48) Z . Patching . The map T g is defined by T g ( y, α ) (cid:44) ( y (cid:48) , α (cid:48) ), where( y (cid:48) i,j , α (cid:48) i,j ) (cid:44) ( f ( y i +1 ,j , y i,j +1 ) , • ) if α i +1 ,j (cid:54) = ⊗ , α i,j +1 (cid:54) = ⊗ ,( i, j ) ∈ D ( y, α ) and f ( y i +1 ,j , y i,j +1 ) exists,( y i,j , α i,j ) otherwise.It simply replaces the symbol y i,j with the one prescribed by its West and South neighbours (if noneof them contains a stop mark) and leaves a trace mark behind. Generation of stop signals . The map T is responsible for generating stop signals, and is defined by T ( y, α ) (cid:44) ( y, α (cid:48) ) (i.e., no change on y ), where α (cid:48) i,j (cid:44) (cid:40) ⊗ if α i,j = • and ( i, j ) ∈ E ( y ), α i,j otherwise. Propagation of stop signals . The propagation of the stop signals is governed by the map T , which isdefined by T ( y, α ) (cid:44) ( y, α (cid:48) ) (i.e., no change on y ), where α (cid:48) i,j (cid:44) (cid:40) ⊗ if ( α i +1 ,j = ⊗ or α i,j +1 = ⊗ ) and α i,j = • , α i,j otherwise. Fading of the traces . Finally, the map T handles the fading of the traces and is defined by T ( y, α ) (cid:44) ( y, α (cid:48) ) (i.e., no change on y ), where α (cid:48) i,j (cid:44) (cid:40) ◦ if α i +1 ,j = α i,j +1 = ◦ and α i,j = • , α i,j otherwise.The composition T ( T g T T ) ensures that the fading of the traces is half as slow as the correctionspeed, while the stop signals propagate twice as fast as the correction speed. Proof of Theorem 3.14.
The proof is similar to the proof of Theorem 2.5.
All the cellular automata discussed above provide directional solutions: the cells need to distinguishbetween the four directions North, South, East, West. In general, finding self-stabilising CA thatrespect the symmetries of the tiling space appears to be a difficult problem, and not always possible.For instance, in case of the homogeneous space H , Pippenger has shown that self-stabilisation cannotbe achieved by a monotone, self-dual (i.e., respecting the ↔ symmetry), centrosymmetric rule [30].19n this section, we will examine to which extent the use of randomness in the evolution of thecellular automata may provide us with a means to design simpler solutions. More precisely, we nowstudy probabilistic CA that achieve self-stabilisation with nearest-neighbour isotropic rules, that is,rules with von Neumann neighbourhood which treat the neighbours “equally”. Our aim is to showthat the use of randomness can extend the range of possibilities for designing self-stabilising processes.Since this is a broad topic, we will restrict our scope to two examples: finite SFTs and single-cellfillable tiling spaces. In Section 7.2, we will discuss some further questions related to other families oftiling spaces. To begin with, let us recall the notion of probabilistic CA , and formulate the concept of self-stabilisationin this context.
Probabilistic cellular automata.
The specificity of probabilistic CA is that the outcome of thelocal rule is now a probability distribution on Σ, and the cells of the lattice are updated simultaneouslyand independently at each time step, according to the distributions prescribed by the local rule.Formally, the local rule in this case is thus given by a function ϕ : Σ N → P (Σ), where P (Σ)denotes the set of probability distributions on Σ, and where we still denote by N ⊆ Z d the (finite)neighbourhood of the rule.We describe the evolution of the system as a time-homogeneous Markov chain ( x t ) t ∈ N with valuesin Σ Z d , such that for any finite C ⊆ Z d , and for any x , . . . , x t ∈ Σ Z d , we have: P ( x t +1 C = x t +1 C | x = x , . . . , x t = x t ) = P ( x t +1 C = x t +1 C | x tC + N = x tC + N )= (cid:89) c ∈ C ϕ (( x tc + i ) i ∈N )( { x t +1 c } ) . Self-stabilisation.
We say that a probabilistic CA stabilises a tiling space X ⊆ Σ Z d from finiteperturbations ifi) ( consistency ) the configurations of X are absorbing states, that is, if x t ∈ X , then x t +1 = x t ,ii) ( attraction ) finite perturbations of the elements of X evolve almost surely to X in finitely manysteps, that is, if x ∈ ˜ X , then there exists almost surely a time t ∈ N such that x t ∈ X .The first such t is called the stabilisation time (or the recovery time ) starting from x . Note that t is now a random variable. We say that F stabilises X from finite perturbations in time τ ( n ) if foreach n ∈ N , the maximum of the expected stabilisation time E [ t ] among all possible (deterministic)initial configurations ˜ x with δ (˜ x, X ) = n is τ ( n ). Let us first examine the case where the tiling space is the set H = { , } ⊆ { , } Z of Example 1.1.Toom’s North-East-Center majority rule (see Example 3.1) is a deterministic CA that stabilises H .We now present an isotropic self-stabilising probabilistic CA for this tiling space.Let Σ (cid:44) { , } , and consider the probabilistic CA Maj-Random-If-Equal on Σ Z , defined on thevon Neumann neighbourhood N (cid:44) { , e , − e , e , − e } by the local rule ϕ : Σ N → P (Σ) where ϕ (( x i ) i ∈N ) (cid:44) δ if x e + x e + x − e + x − e > δ if x e + x e + x − e + x − e < B (1 /
2) otherwise,where δ and δ are the Dirac distributions on and , respectively, and B (1 /
2) denotes the Bernoullirandom variable with parameter 1 /
2. In words, at every step, the state of each cell is changed to thestate which is in majority among its four adjacent cells, and in case of a tie, the tie is broken witha flip a of fair coin, independently of the other cells. A few sample snapshots from the evolution of
Maj-Random-If-Equal are shown in Figure 15. The continuous-time version of
Maj-Random-If-Equal was studied by Fontes, Schonmann and Sidoravicious [15] (see Examples 4.2 and 6.4 below).20 = 0 t = 100 t = 150 t = 200Figure 15: Snapshots from the evolution of the Maj-Random-If-Equal rule.
Proposition 4.1 (Isotropic self-stabilisation of H ) . The probabilistic CA defined above stabilises H from finite perturbations in at most cubic time.Proof. Let x ∈ ˜ H , and let us assume that the defects of x are initially included in a rectangle R . Bysymmetry, we can consider that the defects are s and that the system needs to return to the all- configuration. Observe that over time, the defects always stay within R .We first determine an upper bound for the average time it takes for the s of the upper row of R to disappear. Let us number from left to right by 1 , . . . , k the cells of the upper row of R . We alsoconsider the cell 0 which is on the left of cell 1 and the cell k + 1 which is on the right of cell k .We bound the evolution of the cells 1 , . . . , k by a new process, designed by imagining that thesecells evolve in an environment where for each cell i (1 ≤ i ≤ k ), its North neighbour is in state andits South neighbour in state (see Figure 16). Because of the monotonicity of the local rule ϕ , thenew process can be coupled with the original process in such a way that the state of the cells 1 , . . . , k in the original process remain dominated by their states in the new process (i.e., wherever the formerhas a , so does the latter). 0...00 0 0 . . . . . .
001 1 . . . x x . . . x k Figure 16: Study of the evolution of the upper row of a rectange of defects under the
Maj-Random-If-Equal probabilistic CA (see proof of Proposition 4.1).Since the two North and South neighbours have their state fixed, the evolution of the cells 1 , . . . , k can be modelled as a one-dimensional probabilistic CA ( y t ) t ∈ N with neighbourhood radius 1 and fixedboundary conditions y t = y tk +1 = .Let us analyse the evolution of this one-dimensional probabilistic CA, whose behaviour can beobserved in Figure 17. The local rule of this one-dimensional CA is given in the following table:Value of the neighbourhood
000 001 010 011 100 101 110 111
Probability of symbol / / / / x, y, z ), the new state of the central cell is equal21o x if x = z and to a random value with distribution B (1 /
2) otherwise. This observation allows us todescribe the evolution of this CA as the combination of two independent processes, one on the evenspace-time positions and the other on the odd space-time positions.Consider first the process on the odd space-time positions. In this process, the position of theleft-most cell in state is bounded from below by a symmetric random walk that is reflected on theleft boundary (cell 0) and vanishes when it reaches the right boundary (cell k + 1). One can showthat the expected time this random walk needs to reach k + 1 is of order k . Indeed, by a standardargument, if T i denotes the expected time needed to reach k + 1 from cell i , then we have the recursion T i = 1 + ( T i − + T i +1 ) /
2, with the boundary conditions T = 1 + T and T k +1 = 0. It follows that T is quadratic in k .The same result holds for the process on the even space-time positions. Since the time neededfor the one-dimensional CA to reach the all- configuration is the maximum of the times of the twoprocesses on the odd an even space-time positions, the former time is also quadratic. In particular, thetime it takes for the original two-dimensional CA to wipe out the first row of defects from rectangle R is quadratic in the diameter of the rectangle R . Finally, since we have a linear number of lines to wipeout, the stabilisation time is at most cubic.Figure 17: Space-time diagrams showing evolutions of the one-dimensional probabilistic CA appearingin the proof of Proposition 4.1, with fixed boundary conditions. Time goes from top to bottom. Blueand white squares respectively represent states and . (left) evolution from the all-one configura-tion; (middle) evolution showing how the central homogeneous zone disappears; (right) evolution withvarious appearances and reappearances of the central homogeneous zone.Experimental evidence suggests that the stabilisation is faster than cubic. Note that in the aboveargument, we did not use the fact that the different rows of the rectangle evolve simultaneously. Weconjecture that the true stabilisation time is in fact quadratic. Our intuition is further supported bya known result regarding the analogous model in continuous-time. Example 4.2 (Continuous-time
Maj-Random-If-Equal ) . Fontes, Schonmann and Sidoravicious stud-ied the continuous-time variant of
Maj-Random-If-Equal in which the cells are updated asynchronously,triggered by independent Poisson clocks with rate 1 [15]. Their Theorem 1.3 states that, in dimen-sion d ≥
2, the system stabilises H from finite perturbations, and that the stabilisation occurs intime O ( n d ), in the sense that there exist constants c, γ > t startingfrom a configuration ˜ x ∈ { , } Z d with δ (˜ x, H ) = n satisfies P ( t > cn d ) ≤ e − γn . (cid:35) Extension to finite SFTs.
We can extend the above isotropic probabilistic rule to stabilise anyfinite SFT in at most cubic time. Indeed, if we define N and N as in Section 3.1, then the followingrule is suitable: if a state appears strictly more than twice among x a + N ,b , x a,b + N , x a − N ,b , x a,b − N ,then this state becomes the new value for x a,b ; otherwise, choose one of the latter values at random In fact, Fontes et al. considered a more general family of local rules in which, in case of a neighbourhood tie, thecurrent state of the cell is flipped with probability 0 < α ≤
1. The
Maj-Random-If-Equal rule corresponds to α = 1 / The second case we examine is the one of single-cell fillable tiling spaces, such as k -colourings for k ≥ X ⊆ Σ Z be a single-cell fillable tiling space, and let ψ : Σ → Σ be a function as in Section 3.2,assigning, for any possible choice ( a, b, c, d ) ∈ Σ of symbols surrounding a cell, a consistent value ψ ( a, b, c, d ). For x ∈ Σ Z , recall that we denote by D ( x ) the set of cells having a defect, that is, D ( x ) (cid:44) { k ∈ Z : ∃ e ∈ {± e , ± e } , k has a defect in direction e } . Given α ∈ (0 , X which leaves the state of cell k unchanged if k / ∈ D ( x ) and changes it to ψ ( x k − e , x k − e , x k + e , x k + e ) with probability α if k ∈ D ( x ). In otherwords, the CA has von Neumann neighbourhood N (cid:44) { , e , − e , e , − e } and local rule ϕ (( x i ) i ∈N ) (cid:44) (cid:40) αδ ψ ( x − e ,x − e ,x + e ,x + e ) + (1 − α ) δ x if ( x i ) i ∈N contains a defect, δ x otherwise.Note that if ψ is isotropic, then so is ϕ .To estimate the stabilisation time of this CA, we will use the following standard lemma. Lemma 4.3 (Expectation of maximum of i.i.d. geometric random variables) . Let ε ∈ (0 , . Thereexist constants a, b > such that, whenever r , r , . . . , r n are independent geometric random variableswith parameter ε , we have E [max { r , r , . . . , r n } ] ≤ a log n + b .Proof sketch. This can be shown via a simple comparison with the maximum of n independent expo-nential random variables, for which the expected value can be calculated explicitly (see e.g. [11]). Proposition 4.4 (Isotropic self-stabilisation of single-cell fillable tiling spaces) . Let X ⊆ Σ Z be asingle-cell fillable tiling space. Then, for every α ∈ (0 , , the probabilistic CA defined above stabilises X from finite perturbations in at most logarithmic time.Proof. Let x ∈ ˜ X be an initial configuration. Let x , x , . . . denote the Markov process describedby the probabilistic CA starting from x = x . Note that if a cell k is non-defective at time t (i.e., c / ∈ D ( x t )), then it will remain non-defective at time t + 1, because according to the local rule, thestate of k will not change, and by the property of ψ , changes in the state of the neighbours of k cannotcreate a defect at k . Therefore, for every t we have D ( x t +1 ) ⊆ D ( x t ). On the other hand, at everystep, every defective cell has probability at least α (1 − α ) of becoming non-defective. Indeed, considera cell k ∈ D ( x t ). If at time t + 1, cell k is updated according to ψ and none of its four neighboursare updated according to ψ , then k becomes non-defective, and this event occurs with probability atleast α (1 − α ) . It follows that inside D ( x ), the probabilistic CA behaves like an absorbing finite-stateMarkov chain that eventually reaches a configuration with no defects.To analyse the stabilisation time, let us imagine that the process is constructed using a collection( z tk ) c ∈ Z ,t ∈ N of independent Bernoulli random variables with parameter α , representing the randomchoices taken at every time step and each cell. Namely, for t = 0 , , , . . . and k ∈ Z , we have x t +1 k (cid:44) (cid:40) ψ ( x tk − e , x tk − e , x tk + e , x tk + e ) if z t +1 k = and k ∈ D ( x t ), x tk otherwise.For each k ∈ Z , let t k (cid:44) inf (cid:8) s > z tk = and z tk + e = z tk − e = z tk + e = z tk − e = (cid:9) denote the first time that k is updated according to ψ and none of its neighbours are updated accordingto ψ . This is a geometric random variable with parameter ε (cid:44) α (1 − α ) . As we observed above, k / ∈ D ( x t ) for all t ≥ t k . In particular, defining t A (cid:44) max { t k : k ∈ A } for A ⊆ Z , we have x t ∈ X forall t ≥ t D ( x ) . We show that E [ t D ( x ) ] is logarithmic in |D ( x ) | , and hence also in δ (˜ x, X ).23he random variables t k (for k ∈ Z ) are not independent. However, if k , k , . . . are cells thatdo not share neighbours, then t k , t k , . . . are independent. Note that Z can be partitioned into fiveparts Q , . . . , Q in such a way that the cells in each part do not share neighbours with one another(see Figure 18). Therefore, setting D i ( x ) (cid:44) D ( x ) ∩ Q i , we have t D ( x ) = max { t D i ( x ) : i = 0 , , . . . , } ≤ (cid:88) i =0 t D i ( x ) and hence E [ t D ( x ) ] ≤ (cid:80) i =0 E [ t D i ( x ) ]. By Lemma 4.3, E [ t D i ( x ) ] = O (log |D i ( x ) | ). The claim follows.An alternative argument for the logarithmic stabilisation time can be given by considering anappropriate martingale as in [13, Lemma 6].0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 30 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3 0 124 3Figure 18: A partitioning of Z into 5 parts in such a way that the cells in each part do not shareneighbours with one another In this section, we consider the complexity of self-stabilisation as a computational task. One canconsider at least three different measures of complexity: the speed of stabilisation, the number ofextra symbols, the size of the neighbourhood (Section 5.1). Our focus in this article is on the speedof stabilisation, with preference for having no extra symbols. In Section 5.3, we show that there is anSFT whose self-stabilisation problem is inherently hard, in the sense that it requires super-polynomialtime (unless P = NP ). Interestingly, the optimal speed of stabilisation turns out to be a topologicalinvariant: if two SFTs are topologically isomorphic, then their stabilisation requires roughly the sameamount of time (Section 5.2). The problem of designing a cellular automaton that stabilises an SFT X is an algorithmic problem,albeit a parallel one with extra requirements. The examples discussed so far suggest that the complexityof this algorithmic problem could drastically vary with X . The efficiency of a CA F in stabilising X can be judged based on the resources it uses: Speed of stabilisation τ F ( n )How fast does τ F ( n ) grow with n ? Recall that τ F ( n ) denotes the maximum time it takes for F to correct a finite perturbation ˜ x with δ (˜ x, X ) = n . Number of extra symbols κ F How many extra states per cell does F have compared to the alphabet of X ? Neighbourhood radius r F How far does the local rule of F need to look in order to update the state of one cell?The complexity of self-stabilisation for X can be measured by the optimal values of τ F ( n ), κ F and r F .We let κ ∗ (cid:44) min F κ F and r ∗ (cid:44) min F r F , where the minimums are over all CA F that stabilise X .24e also allegorically use τ ∗ ( n ) to indicate the optimal (in order of magnitude) speed of stabilisationamong all CA F that stabilise X . Let us make a couple of remarks about these measures:1. The stabilisation can be linearly sped up at the cost of increasing the neighbourhood radius.Namely, if F has neighbourhood radius r F and stabilises X in time τ F ( n ), then F k has neigh-bourhood radius kr F and stabilises X in time (cid:100) k τ F ( n ) (cid:101) . In particular, the value of τ ∗ ( n ) ismeaningful only up to a multiplicative constant.2. As we will see in Proposition 5.3, the speed of stabilisation τ ∗ ( n ) is (almost) invariant undertopological isomorphisms. Namely, if two SFTs X and Y are topologically isomorphic, thentheir minimum stabilisation speeds are roughly the same. We suspect that no such invarianceholds for the minimum number of extra symbols κ ∗ or for the minimum neighbourhood radius r ∗ :for every SFT X , it should be possible to find an isomorphic SFT Y for which κ ∗ = 0 and r ∗ = 1.In this paper, we have focused on the speed of stabilisation τ ( n ) with preference towards havingno extra symbols. In this section, we show that the optimal stabilisation time τ ∗ ( n ) for an SFT is an isomorphisminvariant, meaning that topologically isomorphic SFTs have roughly the same optimal stabilisationtimes.To prove this claim, let us first recall some terminology from symbolic dynamics (see [25] for moredetails).The space Σ Z d of all d -dimensional configurations with symbols from a finite alphabet Σ is acompact metrisable space with the product topology. The shift maps σ k : Σ Z d → Σ Z d (for k ∈ Z d ) areall continuous. A closed subset X ⊆ Σ Z d is called a shift space if it is invariant under all shifts, thatis, σ k x ∈ X for all x ∈ X and k ∈ Z d . Clearly, every SFT is a shift space. Given a shift space X anda finite set M ⊆ Z d , we define L M ( X ) (cid:44) { x M : x ∈ X } as the set of all patterns with shape M thatappear in X .A homomorphism between two shift spaces X ⊆ Σ Z d and Y ⊆ Γ Z d is a continuous map Φ : X → Y that commutes with the shifts, that is, Φ ◦ σ k = σ k ◦ Φ for every k ∈ Z d . It can be verified that amap Φ : X → Y is a homomorphism if and only if it is realized by a local rule, that is, if and onlyif there exists a finite set N ⊆ Z d and a map ϕ : L N ( X ) → Γ such that Φ( x ) i = ϕ (cid:0) σ i ( x ) N (cid:1) for each x ∈ X and i ∈ Z d . A map Φ : X → Y is an isomorphism if it is bijective and both Φ and Φ − arehomomorphisms. Since every shift space is compact and Hausdorff, every bijective homomorphismis in fact an isomorphism. Two shift spaces X and Y are topologically isomorphic (or topologicallyconjugate ) if there is an isomorphism between them. Example 5.1 (Two isomorphic SFTs) . Let X ⊆ { , } Z and Y ⊆ { , , } Z be the 1-step SFTswhose transition graphs are depicted in Figure 19. It is easy to verify that the map Φ : Y → X givenby Φ( y ) i (cid:44) (cid:40) if y i ∈ { , } , if y i = ,is an isomorphism. Observe that is a safe symbol for X (see Example 1.3), whereas Y does not havea safe symbol. While the CA F defined in (1) stabilises X in one step, it is not immediately clear if F can be “translated” into a CA that stabilises Y . Indeed, Y has more symbols than X , and as a result,a configuration in Y can have many more finite perturbations than the corresponding configurationin X . Nevertheless, having the correspondence between X and Y in mind, one can re-implement themechanism of stabilisation by F (i.e., replacing each defect with the safe symbol) to obtain a CA thatstabilises Y . Namely, the CA G : { , , } Z → { , , } Z defined by G ( y ) i (cid:44) if y i − y i = or y i y i +1 ∈ { , , } or y i y i +1 y i +2 = , if y i y i +1 y i +2 ∈ { , } , y i otherwise, Note that this is not a rigorous notation. For instance, it might be that for each ε >
0, there exists an F with τ F ( n ) = O ( n ε ) but no F with τ F ( n ) = O ( n ), in which case τ ∗ ( n ) is not well defined. Y (in one step), and is in some sense the “translation” of F . (cid:35) X Y
Figure 19: Graphs of allowed transitions for two isomorphic one-dimensional 1-step SFTs.
Example 5.2 (Higher block presentation) . Let X ⊆ Σ Z d be an SFT and M ⊆ Z d be a finite set withat least two elements. Then, the map Φ : X → (Σ M ) Z d defined by Φ( x ) i (cid:44) x M is an isomorphismbetween X and Y (cid:44) Φ( X ). As in the previous example, the isomorphism between X and Y does notextend to a correspondence between the finite perturbations of X and the finite perturbations of Y ,and it is not immediately clear how a CA stabilising X from finite perturbations can be translatedinto a CA stabilising Y from finite perturbations. (cid:35) The following proposition shows that a CA stabilising an SFT X from finite perturbations canalways be translated into a CA stabilising an isomorphic SFT Y from finite perturbations (if weappropriately extend the alphabet of Y ) while keeping the stabilisation time roughly unchanged. Proposition 5.3 (Stabilisation time for isomorphic SFTs) . Let X and Y be two topologically isomor-phic SFTs. If there exists a CA that stabilises X from finite perturbations in time τ ( n ) , then therealso exists a CA that stabilises Y from finite perturbations in time τ (cid:0) n + O (1) (cid:1) .Proof. Let Σ and Γ be two finite alphabets such that X ⊆ Σ Z d and Y ⊆ Γ Z d . Let Φ : X → Y bethe isomorphism between X and Y , and denote its inverse by Ψ. Let F : Σ Z d → Σ Z d be a CA thatstabilises X from finite perturbations in time τ ( n ).Let us first assume that Φ can be extended to a one-to-one shift-invariant continuous map ˆΦ : Σ Z d → ˆΓ Z d , where ˆΓ is a finite alphabet including Γ. Let ˆΨ : ˆΦ(Σ Z d ) → Σ Z d denote the inverse of ˆΦ, andobserve that ˆΨ is an extension of Ψ, that is, ˆΨ | Y = Ψ. Let ˜Ψ : ˆΓ Z d → Σ Z d be a shift-invariantcontinuous extension of ˆΨ. Such an extension can be constructed by (arbitrarily) completing the localrule of ˆΨ. Let us now define G : ˆΓ Z d → ˆΓ Z d by Gy (cid:44) ˆΦ F ˜Ψ y . To see that G stabilises Y from finiteperturbations, first note that for every y ∈ ˆΓ Z d , the diagram x F x F x . . .y Gy G y . . .F F FG G G ˜Ψ ˜Ψ ˜ΨˆΦ ˆΦcommutes. If y ∈ Y , then clearly Gy = y . Suppose that y ∈ ˜ Y (cid:104) ˆΓ (cid:105) . Then, x ∈ ˜ X (cid:104) Σ (cid:105) . More specifically,let ¯ y ∈ Y be such that ∆(¯ y, y ) is finite with diameter n . Then, ¯ x (cid:44) ˜Ψ¯ y = Ψ¯ y is in X and the diameterof ∆(¯ x, x ) is at most n + C , where C is the diameter of the neighbourhood of the local rule of ˜Ψ.Following the above diagram, the iterations of G on y correspond to the iterations of F on x . Since F stabilises X , we have F t x ∈ X for some t ≤ τ ( n + C ). But as soon as F t x ∈ X , we also obtain G t y ∈ Y . It follows that G stabilises Y from finite perturbations in time τ ( n + C ).It remains to construct a one-to-one shift-invariant continuous extension ˆΦ : Σ Z d → ˆΓ Z d of Φ. Let ϕ : L N ( X ) → Σ and ψ : L M ( Y ) → Γ be the local rules of Φ and Ψ respectively. Without loss ofgenerality, we assume that Γ and Σ are disjoint. Let ˆΓ (cid:44) Γ ∪ Σ ∪ (Γ × Σ). Define ˆΦ with the local ruleˆ ϕ : Σ M + N → ˆΓ, given byˆ ϕ ( p ) (cid:44) ψ ( p N ) if p ∈ L M + N ( X ),( ψ ( p N ) , p ) if p / ∈ L M + N ( X ) but p N ∈ L N ( X ), p otherwise.26he first case in the definition ensures that ˆΦ | X = Φ. Furthermore, p can always be recoveredfrom ˆ ϕ ( p ), either directly, or by applying ψ on (cid:0) ψ ( σ i ( p ) N ) (cid:1) i ∈ M . This means that ˆΦ is indeed aone-to-one extension of Φ as required. In this section, we prove the following theorem:
Theorem 5.4 (SFT with slow stabilisation) . Let d ≥ . Unless P = NP , there exists a d -dimensionalSFT X which is not stabilised from finite perturbations by any CA in polynomial time. For concreteness, we present the proof for the two-dimensional case. The proof of the higher-dimensional case carries through similarly.The square tiling problem of a finite set of Wang tiles Θ is the decision problem of whether a squareof size n can be tiled admissibly using Θ in such a way as to achieve a prescribed colouring of itsboundary. The square tiling problem of every tile set is clearly in class NP . The existence of a set ofWang tiles for which the square tiling problem is NP -complete is folklore.The idea of the proof of Theorem 5.4 is that a CA that stabilises the SFT X associated to a setof Wang tiles Θ can be used to solve a variant of the square tiling problem for Θ. Namely, supposethat F is a CA that stabilises X in time τ ( n ), and let r be the neighbourhood radius of F . Then,given a configuration x ∈ ˜ X and a finite set A ⊆ Z such that x Z \ A is globally admissible, the cellularautomaton is able to “patch” the defects of x in τ (diam( A )) steps and turn it into a valid tiling by onlychanging the states of the cells that are no farther than rτ ( n ) from A . Note that only the states of thecells within distance rτ ( n ) from A can possibly change during this computation, and only the states ofthe cells within distance 2 rτ (diam( A )) from A are relevant for such changes. Thus, the relevant partsof the computation can be simulated by a Turing machine. As we will see, such a Turing machine canthen be used to solve the standard square tiling problem.To be specific, let us state the latter problem more explicitly. Let M (cid:44) {− , , } denote the Mooreneighbourhood. The (Moore) boundary of a set A ⊆ Z is the set ∂ M ( A ) (cid:44) M ( A ) \ A . For each n ∈ N ,let S n (cid:44) { , , . . . , n − } . Thus, for k ≥ M k ( S n ) represents the square {− k, − k + 1 . . . , n + k − } . Algorithmic Problem 5.1 (Global tiling patching) . Parameters:
A finite set of Wang tiles Θ and two functions α, β : N → N . Input:
A globally admissible tiling q of M α ( n )+ β ( n )+1 ( S n ) \ S n . Task:
Find a globally admissible pattern ˜ q on M α ( n )+1 ( S n ) that agrees with q on the band ∂ M ( M α ( n ) ( S n )).See Figure 20 for an illustration. Proposition 5.5 (Serial simulation) . Let X be the SFT of the valid tilings of a finite set of Wangtiles Θ . Suppose X (cid:54) = ∅ and that there exists a CA with neighbourhood radius r that stabilises X intime τ ( n ) . Then, there exists a Turing machine with two-dimensional tape that solves the global tilingpatching problem for Θ and α ( n ) (cid:44) β ( n ) (cid:44) rτ ( n ) in time O (cid:0) [ n + 4 rτ ( n )] τ ( n ) (cid:1) .Proof. The Turing machine simulates the CA on its two-dimensional tape. Once τ ( n ) steps of the CAsimulation have been carried out, the Turing machine stops and returns the current configuration ofthe CA.Serial simulation of the CA can be done in a standard fashion. The tape has two layers (numbered 0and 1) for storing the configurations at even and odd time steps. The input is initially provided onlayer 0. In an initialization stage, the input pattern is extended by writing an arbitrary symbolfrom Θ at every position in S n . When simulating time step t of the CA, the Turing machine readsthe configuration at time t − t −
1) mod 2 and writes the result on layer t mod 2. Atpositions in which not enough information is available (i.e., outside M rτ ( n )+1 − rt ( S n ) \ M rt ( S n )) theblank symbol is written instead.In order to keep of track of the number of simulated steps, the cells in M rτ ( n )+1 ( S n ) \ S n are, inthe initialization stage, marked with (cid:63) . At every stage of the simulation, the marks are also updated:a mark is kept if all its (2 r + 1) neighbours are marked and is erased otherwise. After τ ( n ) stages ofthe simulation, only the cells in the band M rτ ( n )+1 ( S n ) \ M rτ ( n ) ( S n ) are marked and thus the Turingmachine easily recognizes that it has to enter its final stage of computation. In the final stage, the27 n β ( n ) α ( n ) n β ( n ) 1 α ( n ) n (cid:55)→ S n α ( n ) n α ( n ) n (a) (b)Figure 20: Illustration of the global tiling patching problem. (a) The input is a globally admissiblepattern on the shaded region. (b) The output is a globally admissible pattern on the shaded regionwhich agrees with the input on the region which is shaded in blue.marks and the distinction between the two layers are erased so that only the configuration at time τ ( n ) is left on the tape.The simulation of each time step of the CA takes O (cid:0) [ n +4 rτ ( n )] (cid:1) time steps of the Turing machine.Likewise, the initialization stage and the final stage take only O (cid:0) [ n + 4 rτ ( n )] (cid:1) time steps each. Thus,in overall, the Turing machine accepts or rejects its input in O (cid:0) [ n + 4 rτ ( n )] τ ( n ) (cid:1) steps. Proposition 5.6 ( NP -hardness) . There exists a finite set of Wang tiles Θ such that for every twofunctions α, β : N → N with polynomial growth, the global tiling patching problem associated to Θ , α and β is NP -hard.Proof. Let Θ be a finite set of Wang tiles for which the square tiling problem is NP -complete, andlet C denote the set of colours that appear in the tiles of Θ . Let Θ be the set of Wang tiles depictedin Figure 21 and their four-fold rotations. Without loss of generality, we assume C (cid:44) {(cid:5) , , , , , , , , } ∪ { ( c, • ) : c ∈ C } is disjoint from C . Let Θ = Θ ∪ Θ .Let α, β : N → N be any two functions with polynomial growth. We show that the square tilingproblem for Θ can be reduced in polynomial time to the global tiling patching problem for Θ, α and β .It will then follow that the latter problem is NP -hard.The reduction is illustrated in Figure 22. An instance of the square tiling problem for Θ is givenby prescribing colours from C for the boundary of the square S n for some n as in Figure 22a. This istransformed into a locally admissible tiling of the region M α ( n )+ β ( n )+1 ( S n ) \ S n with tiles from Θ asdepicted in Figure 22b. Let us call the latter tiling q . Observe that the restriction of q to the band ∂ M ( M α ( n ) ( S n )) (the region shaded blue) enforces the tiling of the band ∂ M ( S n ) (the region shadedred), which in turn fixes the colouring of the boundary of the square S n . Observe also that if ˜ q is alocally admissible extension of q to M α ( n )+ β ( n )+1 ( S n ), then ˜ q is in fact globally admissible (can beextended to the entire plane).Let G ( q ) be an answer to the global tiling patching problem for Θ with the tiling of Figure 22b asinput.First, suppose that the there is a locally admissible tiling of S n using the tiles from Θ that achievesthe prescribed colouring in Figure 22a. It is easy to see that in this case, the partial tiling given inFigure 22b is globally admissible. Thus, G ( q ) will be globally admissible and will agree with q on ∂ M ( M α ( n ) ( S n )). In particular, by the above remark, G ( q ) S n will be a locally admissible tiling of S n with Θ compatible with the prescribed colouring of Figure 22a. Conversely, assume that theprescribed colouring of Figure 22a cannot be achieved by a locally admissible tiling of S n using Θ .Then, either G ( q ) is not locally admissible, or G ( q ) and q disagree on the band ∂ M ( S n ).28e see that, in either case, the answer to the square tiling problem for Θ on Figure 22a can easily(in polynomial time) be extracted from G ( q ). ( c, • ) ( c, • ) ( c, • ) • c Figure 21: The extra tiles used in the proof of Proposition 5.6. The tile set Θ consists of all thesetiles (for each c ∈ C ) and their rotations. The unlabeled edges are coloured with the blank symbol (cid:5) . a a a n c c c n b b b n d d d n (cid:55)→ a a a n c c c n b b b n d d d n β ( n ) α ( n ) β ( n ) α ( n ) (a) (b)Figure 22: The reduction in the proof of Proposition 5.6. (a) An instance of the square tiling problem,where a i , b i , c i , d i are colours from C . (b) An instance of the global tiling patching problem. Thecolour of the edges marked with • (not depicted) match accordingly.The proof of Theorem 5.4 is obtained by putting together Propositions 5.5 and 5.6. Proof of Theorem 5.4.
Let X (cid:54) = ∅ be the SFT of the valid tilings of the tile set Θ in Proposition 5.6.Suppose there exists a CA F that stabilises X in time τ ( n ) = O ( n k ) for some k ∈ N . By Proposition 5.5,this implies that there exists a Turing machine solving the global tiling patching problem associatedto Θ and α ( n ) (cid:44) β ( n ) (cid:44) rτ ( n ) (where r is the neighbourhood radius of F ) in polynomial time. Unless P = NP , this is a contradiction. Remark 5.7.
In principle, Proposition 5.5 may also lead to other hardness results. For instance, ifwe have a tile set for which solving the global tiling patching problem with a Turing machine (withtwo-dimensional tape) requires more than O ( n ) time steps, then it follows from Proposition 5.5 thatthere is no CA that stabilises the valid tilings of that tile set in linear time. (cid:51) Stability against random noise
A true fault-tolerant system should involve efficient error-correction mechanisms so as to maintain itsstructure and functionality in presence of random noise. In this scenario, noise is present everywhereand throughout the evolution of the system. Such level of stability is achieved by Toom’s
NEC-Maj
CAand its variants [35] (Examples 3.1 and 6.2) and G´acs’s (very sophisticated) reliable CA [17, 18], butappears very challenging in general.As a modest intermediate step, in this section we consider the problem of self-stabilisation startingfrom tilings that are perturbed by random noise. (Hence, noise is present only in the initial config-uration but not at every time step.) We show that if a CA stabilises from finite perturbations inlinear time, then it also stabilises from Bernoulli random perturbations with a sufficiently low densityof errors. The argument is based on the idea of sparseness due to G´acs [20, 17, 18] and Durand,Romashchenko and Shen [10]. In this section, we restrict ourselves to the case of stabilisation bydeterministic CA and leave the corresponding problem for probabilistic CA open.
Let us start by extending the notion of self-stabilisation to the case where the tiling is perturbed withrandom noise. This is not as straightforward as one might hope for, and there are indeed severalvariants depending on the requirements and the type of noise. Here we use one such possible notion.Consider a configuration x : Z d → Σ. Let ε ≥
0. An ε -perturbation of x in Σ Z d is a randomconfiguration ˜ x in Σ Z d with the property that for each finite set I ⊆ Z d , we have P (cid:0) ∆( x, ˜ x ) ⊇ I (cid:1) = P (˜ x i (cid:54) = x i for each i ∈ I ) ≤ ε | I | . We think of ˜ x as a “noisy version” of x where random errors have occurred leading to change in thestate of some cells. A special type of an ε -perturbation is a Bernoulli ε -perturbation for which the set∆( x, ˜ x ) is a Bernoulli random set with parameter ε , that is, a random subset of Z d in which each cell k ∈ Z d is included with probability ε independently of the other cells. The notion of ε -perturbationis however much more general. It turns out that the usual arguments regarding noise-resilience incomputational models often work equally well with this more general notion of noise. This was firstobserved by Toom [35].Let ε, δ ≥
0. We say that a CA F : Σ Z d → Σ Z d δ -stabilises an SFT X ⊆ Σ Z d from ε -perturbations if(i) ( consistency ) the configurations of X are fixed points, that is, F ( x ) = x for every x ∈ X ,(ii) for every x ∈ X and every ε -perturbation ˜ x of x , there exists a random configuration y ∈ X (in the same probability space) such that(ii.a) ( attraction ) lim t →∞ F t (˜ x ) = y almost surely in the product topology,(ii.b) ( stability ) P ( y k (cid:54) = x k ) ≤ δ for each k ∈ Z d .We say that F stabilises X from random perturbations if for every δ >
0, there exists an ε >
F δ -stabilises X from ε -perturbations. The stability condition may sound unnecessary at firstsight. The following example illustrates why it is a desired property. Example 6.1 (Unstable attraction) . Let F : { , } Z → { , } Z be the one-dimensional CA with neigh-bourhood N = {− , , } defined by F ( x ) k (cid:44) (cid:40) if x k − = x k = x k +1 = , otherwise.As usual, let H (cid:44) { , } be the finite tiling space consisting only of the two homogeneous configurationsin { , } Z . Then, F satisfies the conditions of consistency and attraction in the definition of stabilisationfrom random perturbations. However, note that if x is a non-trivial Bernoulli perturbation of , then F t ( x ) converges almost surely to rather than to . In other words, the system recovers from theinitial noise but the distinction between the elements of X is completely lost. (cid:35) Before stating our result, let us mention some examples.30 xample 6.2 (Toom’s CA; Continuation of Example 3.1) . Toom’s CA has strong forms of stabilityagainst noise.Buˇsi´c, Fat`es, Mairesse and Marcovici have shown that Toom’s CA classifies the Bernoulli randomconfigurations according to their density [6]. Namely, if we start from a Bernoulli random configu-ration x with parameter p (i.e., the state of different cells are chosen independently at random withprobability p of choosing and probability 1 − p of choosing ), then NEC-Maj t ( x ) → if p < / , NEC-Maj t ( x ) → if p > / .In particular, for δ = 0 and every ε < / , the CA δ -stabilises the homogeneous tiling space H (cid:44) { , } from Bernoulli ε -perturbations.Moreover, the two homogeneous configurations and remain stable under NEC-Maj even if thereis a small independent noise at every time step. In his original paper [35], Toom showed that for every δ >
0, there exists an ε > (cid:0) x ( t ) k (cid:1) k ∈ Z ,t ∈ N is an ε -perturbed trajectory of NEC-Maj startingfrom in the sense that for every finite set I ⊆ Z × N of space-time positions, we have P (cid:8) x ( s ) i deviates from the local rule of NEC-Maj at every ( i, s ) ∈ I (cid:9) ≤ ε | I | , then this trajectory remains δ -close to its initial configuration , that is, P (cid:0) x ( t ) k (cid:54) = (cid:1) ≤ δ for each ( k, t ) ∈ Z × N .In fact, Toom proved the same kind of stability for random perturbations of all monotonic eroders.Note that for NEC-Maj , a similar stability property holds by symmetry for the trajectories startingfrom . (cid:35) Example 6.3 (GKL and modified traffic; Continuation of Example 2.1) . Using the sparseness resultof Durand, Romashchenko and Shen [10], Taati showed that
GKL and modified traffic stabilise thehomogeneous tiling space H (cid:44) { , } from random perturbations [33]. Below, we use the samesparseness result to obtain a more general stabilisation result. (cid:35) Example 6.4 (Continuous-time
Maj-Random-If-Equal ; Continuation of Example 4.2) . The notionof stabilisation from random perturbations extends naturally to probabilistic CA and to continuous-time models. Fontes, Schonmann and Sidoravicius [15] proved that, in dimension two and higher,the continuous-time
Maj-Random-If-Equal of Example 4.2 stabilises the homogeneous tiling space H (cid:44) { , } from random perturbations. We conjecture that a similar result holds for the discrete-time version discussed in Section 4.2. (cid:35) Our objective is to prove the following result.
Theorem 6.5 (Self-stabilisation from random perturbations) . Let F : Σ Z d → Σ Z d be a CA and X ⊆ Σ Z d an SFT. If F stabilises X from finite perturbations in linear time, then F also stabilises X fromrandom perturbations. A proof of Theorem 6.5 will be provided in the next subsection. Here we discuss the idea and themain ingredient, namely the notion of sparseness. Let us remark that a recent result of G´acs [16] wouldallow us to improve the above result by replacing the “linear time” condition with the “sub-quadratictime” condition. See Remark 6.8 below for more details.A basic idea in the proof of Theorem 6.5 is that the speed of the propagation of information in acellular automaton is bounded.
Observation 6.6 (Speed of light) . Let F : Σ Z d → Σ Z d be a CA with neighbourhood N (cid:44) {− r, . . . , r } d .Then, for every A ⊆ Z d and t ≥ , and every two configurations x, y ∈ Σ Z d , we have (a) If x N t ( A ) = y N t ( A ) , then F t ( x ) A = F t ( y ) A . (b) If ∆( x, y ) ⊆ A , then ∆ (cid:0) F t ( x ) , F t ( y ) (cid:1) ⊆ N t ( A ) . F : Σ Z d → Σ Z d stabilises an SFT X ⊆ Σ Z d from finite perturbations. Sinceinformation propagates at bounded speed, this immediately implies that F stabilises starting from anyperturbation in which the errors are “sufficiently sparse”. Indeed, if the set of errors in a perturbation ˜ x can be decomposed into well-separated finite islands, then under the iterations of F , each island willdisappear before sensing or affecting the other islands (see Figure 23).(a) τ ( (cid:96) ) (cid:96)rτ ( (cid:96) ) rτ ( (cid:96) ) rτ ( (cid:96) ) rτ ( (cid:96) ) t i m e (b) t i m e (c) k t i m e Figure 23: Stabilisation from a sparse set of errors. (a) An isolated island of errors. (b) Well-separatedislands of errors disappear before sensing or affecting one another. (c) The disappearance of all islandsof error is not sufficient for stabilisation.More specifically, suppose that F stabilises X in time τ ( n ). Suppose that F has a neighbourhoodradius r and X has an interaction range m . Let ˜ x ∈ Σ Z d be a (possibly infinite) perturbation of aconfiguration x ∈ X , and let S (cid:44) ∆( x, ˜ x ) denote the set of positions at which an error has occurred.Let us call a subset A ⊆ S with diameter (cid:96) an isolated island if S \ A is at distance more than2 rτ ( (cid:96) ) + m from S \ A , where r denotes the neighbourhood radius of F (Figure 23a). Note that underthe iterations of F on ˜ x , every isolated island of S is corrected before interacting with the rest of S .Let S (cid:48) ⊆ S be the subset of S obtained by removing all its isolated islands. Then, again under theiterations of F on ˜ x , every isolated island of S (cid:48) is corrected before interacting with the rest of S (cid:48) . Thisreasoning can be repeated to identify an infinite hierarchy of islands in S , each of which is correctedwithout interacting with the rest of S . If every element of S is included in one of these islands, thenthat means that every error on ˜ x is eventually patched (Figure 23b). However, note that this alonedoes not guarantee attraction towards an element of X , for it is possible that a cell k ∈ Z d is withinthe interaction range of infinitely many islands of S (Figure 23c) and hence never stabilises.To make this idea precise, we need to introduce the notion of sparseness.Let ρ : N → N be an arbitrary function, and let M (cid:44) {− , , } d be the Moore neighbourhood. Wedefine the ρ -territory of a finite set A ⊆ Z d as the set M ρ ( A ) (cid:44) M ρ (diam( A )) of all cells k that arewithin ρ (diam( A )) from A . (We use the Moore neighbourhood and the (cid:96) ∞ distance on Z d , but thisis an arbitrary choice.) We say that a set S ⊆ Z d is ρ -sparse if there is a partitioning C ( S ) of S intofinite sets, called the ρ -islands of S , such that(i) (separation) every two distinct islands A, B ∈ C ( S ) are well separated from each other, that is,either A ∩ M ρ ( B ) = ∅ or M ρ ( A ) ∩ B = ∅ .32ii) (thinness) every cell a ∈ Z d is in the territory of no more than finitely many islands, that is, { C ∈ C ( S ) : M ρ ( C ) (cid:51) a } is finite.The significance of the concept of sparseness in the context of fault-tolerant computation wasnoticed by G´acs [20, 17, 18], who used more sophisticated variants of it in the scenario in which errorsdue to noise occur at every time step. The above definition of sparseness is equivalent to the oneintroduced by Durand, Romashchenko and Shen [10], who used it in a context very similar to (butdifferent from) ours.Let ε >
0. By an ε -random subset of Z d , we shall mean a random set S ⊆ Z d with the propertythat for each finite I ⊆ Z d , P ( S ⊇ I ) ≤ ε | I | . Durand, Romashchenko and Shen [10, Section 9.2] proved the following.
Theorem 6.7 (Linear sparseness of ε -random sets) . Let ρ : N → N be such that ρ ( (cid:96) ) = O ( (cid:96) ) as (cid:96) → ∞ .For every δ > , there exists an ε > such that (i) every ε -random set S ⊆ Z d is almost surely ρ -sparse, (ii) for every ε -random set S ⊆ Z d and each k ∈ Z d , the probability that cell k is in the ρ -territoryof some ρ -island in S is less than δ . Let us clarify that Durand, Romashchenko and Shen proved their result for Bernoulli ε -randomsets (for all sufficiently small ε ). However, their proof goes through without modification for arbitrary ε -random sets. Remark 6.8.
The proof of Theorem 6.5 relies on the linear sparseness of ε -random sets (Theorem 6.7).Recently, G´acs has strengthened the latter sparseness result by proving the sub-quadratic sparseness of ε -random sets [16]. More precisely, he has shown that if ρ : N → N is any sub-quadratic function (i.e., ρ ( (cid:96) ) = O ( (cid:96) β ) for some β < ε > ε -random set is almost surely ρ -sparse. Applying the result of G´acs in place of Theorem 6.7, we immediately obtain that F stabilises X from random perturbations as long as it stabilises X from finite perturbations in sub-quadratic time. (cid:51) In this section, we prove Theorem 6.5Suppose that F stabilises X from finite perturbations in time τ ( (cid:96) ). There is no loss of generalityto assume that τ ( (cid:96) ) is an increasing function with τ ( (cid:96) ) → ∞ as (cid:96) → ∞ . If not, we can replace τ ( (cid:96) )with τ (cid:48) ( (cid:96) ) (cid:44) max { (cid:96), τ ( k ) : k ≤ (cid:96) } in the argument below.Suppose that F has neighbourhood radius r so that M r = {− r, . . . , r } d is a neighbourhood forthe local rule of F . Suppose that X has interaction range m so that it can be identified by a set F of forbidden patterns with shape S m (cid:44) { , , . . . , m − } d . Define ρ ( (cid:96) ) (cid:44) rτ ( (cid:96) ) + m . (We use thischoice rather than ρ ( (cid:96) ) (cid:44) rτ ( (cid:96) ) + m used in the intuitive explanation above to simplify the proof.)By assumption, τ ( (cid:96) ) = O ( (cid:96) ), hence ρ ( (cid:96) ) = O ( (cid:96) ). Let δ >
0, and choose ε >
F δ -stabilises X from ε -perturbations.The consistency property of F is satisfied by assumption. Let x ∈ X , and let ˜ x be an ε -perturbationof x . Let Q (cid:44) ∆( x, ˜ x ) denote the set of positions where an error has occurred. Note that Q is an ε -random set. Thus, by the choice of ε , we know that Q is almost surely ρ -sparse. Let C ( Q ) be apartition of Q into ρ -islands, witnessing the ρ -sparseness of Q . For each (cid:96) ≥
0, let us define C (cid:96) ( Q ) (cid:44) (cid:8) C ∈ C ( Q ) : diam( C ) = (cid:96) (cid:9) , C >(cid:96) ( Q ) (cid:44) (cid:8) C ∈ C ( Q ) : diam( C ) > (cid:96) (cid:9) = (cid:91) k>(cid:96) C k ( Q ) , C ≥ (cid:96) ( Q ) (cid:44) (cid:8) C ∈ C ( Q ) : diam( C ) ≥ (cid:96) (cid:9) = C (cid:96) ( Q ) ∪ C >(cid:96) ( Q ) . Note that by the separation property of Q , each C ∈ C (cid:96) ( S ) is at distance more than ρ ( (cid:96) ) = 3 rτ ( (cid:96) ) + m from every C (cid:48) ∈ C ≥ (cid:96) ( Q ) distinct from C . Lemma 6.9 (Recursive correction) . For all (cid:96) ≥ , there exist random configurations ˜ x ( (cid:96) ) ∈ Σ Z d and y ( (cid:96) ) ∈ X (in the same probability space) such that the following conditions are satisfied: F t (˜ x ) = F t (cid:0) ˜ x ( (cid:96) ) (cid:1) for all t ≥ τ ( (cid:96) ) , that is, the distinction between ˜ x and ˜ x ( (cid:96) ) isforgotten in τ ( (cid:96) ) time steps. (b) (patching) y (0) = x and ∆ (cid:0) y ( (cid:96) − , y ( (cid:96) ) (cid:1) ⊆ (cid:83) C ∈ C (cid:96) ( Q ) M rτ ( (cid:96) ) ( C ) ⊆ (cid:83) C ∈ C (cid:96) ( Q ) M ρ ( C ) for (cid:96) ≥ ,that is, y ( (cid:96) − and y ( (cid:96) ) differ only in the territory of the ρ -islands with diameter (cid:96) in C ( Q ) . (c) (correction) ∆ (cid:0) y ( (cid:96) ) , ˜ x ( (cid:96) ) (cid:1) ⊆ (cid:83) C ∈ C >(cid:96) ( Q ) C , that is, the ρ -islands of errors on ˜ x ( (cid:96) ) are simply the ρ -islands of diameter larger than (cid:96) in C ( Q ) .Proof. The configurations ˜ x ( (cid:96) ) and y ( (cid:96) ) are constructed recursively. We start with ˜ x (0) (cid:44) ˜ x and y (0) (cid:44) x .In step (cid:96) ≥
1, assuming we have constructed ˜ x ( (cid:96) − and y ( (cid:96) − , we construct ˜ x ( (cid:96) ) as follows. Let C , C , . . . be an enumeration of the elements of C (cid:96) ( Q ). Let z ( n ) be the configuration that agreeswith ˜ x ( (cid:96) − on C n and with y ( (cid:96) − outside C n . Note that z ( n ) is a finite perturbation of y ( (cid:96) − with∆ (cid:0) y ( (cid:96) − , z ( n ) (cid:1) ⊆ C n . Since F stabilises X from finite perturbations in time τ ( · ), we know that F τ ( (cid:96) ) ( z ( n ) ) ∈ X . By Observation 6.6, the two configurations y ( (cid:96) − and F τ ( (cid:96) ) ( z ( n ) ) agree everywhereexcept possibly in M rτ ( (cid:96) ) ( C n ), that is the set of positions within distance rτ ( (cid:96) ) from C n . Define˜ x ( (cid:96) ) k (cid:44) (cid:40) F τ ( (cid:96) ) ( z ( n ) ) k if k ∈ M rτ ( (cid:96) ) ( C n ) for some n ∈ N ,˜ x ( (cid:96) − k otherwise. y ( (cid:96) ) k (cid:44) (cid:40) F τ ( (cid:96) ) ( z ( n ) ) k if k ∈ M rτ ( (cid:96) ) ( C n ) for some n ∈ N , y ( (cid:96) − k otherwise.We use induction to show that y ( (cid:96) ) ∈ X and conditions (a)–(c) are satisfied. The case (cid:96) = 0 istrivial. Assuming that these conditions are satisfied in the first (cid:96) − (cid:96) .(–) Let us first verify that y ( (cid:96) ) ∈ X . To see this, note that for each n ∈ N , the configuration y ( (cid:96) ) agreeswith F τ ( (cid:96) ) ( z ( n ) ) not only on M rτ ( (cid:96) ) ( C n ) but on M rτ ( (cid:96) )+ m ( C n ). Namely, by Observation 6.6, F τ ( (cid:96) ) ( z ( n ) ) agrees with y ( (cid:96) − on M rτ ( (cid:96) )+ m ( C n ) \ M rτ ( (cid:96) ) ( C n ), and on the same set, y ( (cid:96) − agreeswith y ( (cid:96) ) by construction. Now, for every i ∈ Z d , the set i + S m is either entirely in M rτ ( (cid:96) )+ m ( C n )for some n or entirely in Z d \ (cid:83) ∞ n =1 M rτ ( (cid:96) ) ( C n ). In the first case, y ( (cid:96) ) i + S m = F τ ( (cid:96) ) ( z ( n ) ) i + S m / ∈ F and in the second case y ( (cid:96) ) i + S m = y ( (cid:96) − i + S m / ∈ F . Thus, y ( (cid:96) ) ∈ X as claimed.(a) By the induction hypothesis, F t (˜ x ) = F t (cid:0) ˜ x ( (cid:96) − (cid:1) for t ≥ τ ( (cid:96) − τ ( (cid:96) − < τ ( (cid:96) ), it isenough to show that F t (cid:0) ˜ x ( (cid:96) ) (cid:1) = F t (cid:0) ˜ x ( (cid:96) − (cid:1) for t ≥ τ ( (cid:96) ).Note that for each n ∈ N , the configuration x ( (cid:96) ) agrees with F τ ( (cid:96) ) ( z ( n ) ) not only on M rτ ( (cid:96) ) ( C n )but on M rτ ( (cid:96) ) ( C n ). Namely, by construction, Observation 6.6, and the fact that the ele-ments of X are fixed points, F τ ( (cid:96) ) ( z ( n ) ) agrees with y ( (cid:96) − outside M rτ ( (cid:96) ) ( C n ). Furthermore,by the induction hypothesis, ∆ (cid:0) y ( (cid:96) − , ˜ x ( (cid:96) − (cid:1) ⊆ (cid:83) C ∈ C ≥ (cid:96) ( Q ) C . Since C n has distance morethan ρ ( (cid:96) ) = 3 rτ ( (cid:96) ) + m from the rest of C ≥ (cid:96) ( Q ), we find that y ( (cid:96) − agrees with ˜ x ( (cid:96) − on M rτ ( (cid:96) ) ( C n ) \ M rτ ( (cid:96) ) ( C n ). Lastly, again by construction and the fact that C n has distance morethan ρ ( (cid:96) ) = 3 rτ ( (cid:96) ) + m from the rest of C (cid:96) ( Q ), the two configurations ˜ x ( (cid:96) − and ˜ x ( (cid:96) ) also agreeon M rτ ( (cid:96) ) ( C n ) \ M rτ ( (cid:96) ) ( C n ).Now, Observation 6.6 and the fact that F τ ( (cid:96) ) ( z ( n ) ) ∈ X is a fixed point imply that for each n ∈ N ,the configurations F τ ( (cid:96) ) (˜ x ( (cid:96) − ), F τ ( (cid:96) ) ( z ( n ) ) and F τ ( (cid:96) ) (˜ x ( (cid:96) ) ) agree on M rτ ( (cid:96) ) ( C n ). On the otherhand, by construction, ∆ (cid:0) ˜ x ( (cid:96) − , ˜ x ( (cid:96) ) (cid:1) ⊆ (cid:83) ∞ n =1 M rτ ( (cid:96) ) ( C n ). Therefore, according to Observa-tion 6.6, the two configurations F τ ( (cid:96) ) (˜ x ( (cid:96) − ) and F τ ( (cid:96) ) (˜ x ( (cid:96) ) ) agree outside (cid:83) ∞ n =1 M rτ ( (cid:96) ) ( C n ).We conclude that F τ ( (cid:96) ) (˜ x ( (cid:96) − ) and F τ ( (cid:96) ) (˜ x ( (cid:96) ) ) agree everywhere. That F t (˜ x ( (cid:96) − ) = F t (˜ x ( (cid:96) ) )for all t ≥ τ ( (cid:96) ) follows immediately.(b) That ∆ (cid:0) y ( (cid:96) − , y ( (cid:96) ) (cid:1) ⊆ (cid:83) C ∈ C (cid:96) ( Q ) M rτ ( (cid:96) ) ( C ) ⊆ (cid:83) C ∈ C (cid:96) ( Q ) M ρ ( C ) is immediate from the con-struction.(c) By the induction hypothesis,∆ (cid:0) y ( (cid:96) − , ˜ x ( (cid:96) − (cid:1) ⊆ (cid:91) C ∈ C ≥ (cid:96) ( Q ) C , ∆ (cid:0) y ( (cid:96) − , y ( (cid:96) ) (cid:1) ⊆ (cid:91) C ∈ C (cid:96) ( Q ) M rτ ( (cid:96) ) ( C ) .
34n the other hand, by construction, ˜ x ( (cid:96) ) and y ( (cid:96) ) agree on (cid:83) C ∈ C (cid:96) ( Q ) M rτ ( (cid:96) ) ( C ). It follows that∆ (cid:0) y ( (cid:96) ) , ˜ x ( (cid:96) ) (cid:1) ⊆ (cid:83) C ∈ C >(cid:96) ( Q ) C .This completes the proof of the lemma.Item (b) in Lemma 6.9 ensures that y ( (cid:96) ) → y for some y ∈ X . Indeed, let k ∈ Z d . By the thinnesscondition in the definition of ρ -sparseness, we have k ∈ M ρ ( C ) for at most finitely many C ∈ C ( Q ).It follows that y ( (cid:96) − k (cid:54) = y ( (cid:96) ) k for no more than finitely many values of (cid:96) . Define y k as the eventualvalue of y ( (cid:96) ) k . Then, y ( (cid:96) ) → y in the product topology. Furthermore, y ∈ X because X is closed. Lemma 6.10 (Attraction) . F t (˜ x ) → y as t → ∞ .Proof. Let t ≥
0, and choose (cid:96) such that τ ( (cid:96) ) ≤ t ≤ τ ( (cid:96) + 1). By item (a) in Lemma 6.9, F t (˜ x ) = F t (cid:0) ˜ x ( (cid:96) ) (cid:1) . By item (c) in Lemma 6.9, ∆ (cid:0) y ( (cid:96) ) , ˜ x ( (cid:96) ) (cid:1) ⊆ (cid:83) C ∈ C >(cid:96) ( Q ) C . This, together with Observa-tion 6.6, implies that ∆ (cid:16) F t (cid:0) y ( (cid:96) ) (cid:1) , F t (cid:0) ˜ x ( (cid:96) ) (cid:1)(cid:17) ⊆ (cid:83) C ∈ C >(cid:96) ( Q ) M rt ( C ). Recall that y ( (cid:96) ) is an element of X and thus F t (cid:0) y ( (cid:96) ) (cid:1) = y ( (cid:96) ) . Note further that for C ∈ C >(cid:96) ( Q ), we have 2 rt ≤ ρ ( (cid:96) + 1) ≤ ρ (cid:0) diam( C ) (cid:1) ,and thus M rt ( C ) ⊆ M ρ ( C ). It follows that∆ (cid:16) y ( (cid:96) ) , F t (cid:0) ˜ x (cid:1)(cid:17) = ∆ (cid:16) F t (cid:0) y ( (cid:96) ) (cid:1) , F t (cid:0) ˜ x (cid:1)(cid:17) = ∆ (cid:16) F t (cid:0) y ( (cid:96) ) (cid:1) , F t (cid:0) ˜ x ( (cid:96) ) (cid:1)(cid:17) ⊆ (cid:91) C ∈ C >(cid:96) ( Q ) M rt ( C ) ⊆ (cid:91) C ∈ C >(cid:96) ( Q ) M ρ ( C ) . From item (b) in Lemma 6.9, it also follows that ∆ (cid:0) y , y ( (cid:96) ) (cid:1) ⊆ (cid:83) C ∈ C >(cid:96) ( Q ) M ρ ( C ). Thus,∆ (cid:16) y , F t (cid:0) ˜ x (cid:1)(cid:17) ⊆ (cid:91) C ∈ C >(cid:96) ( Q ) M ρ ( C ) . Let k ∈ Z d . By the thinness condition in the definition of ρ -sparseness, we know that k ∈ M ρ ( C ) forno more than finitely many C ∈ C ( Q ). Thus, the value of F t (cid:0) ˜ x (cid:1) k will eventually stabilise at y k . Weconclude that F t (cid:0) ˜ x (cid:1) → y as t → ∞ . Lemma 6.11 (Stability) . P ( y k (cid:54) = x k ) ≤ δ for each k ∈ Z d .Proof. In order to have y k (cid:54) = x k , we must have y ( (cid:96) ) k (cid:54) = y ( (cid:96) − k for some k ∈ Z . By item (b) in Lemma 6.9,this means that k ∈ (cid:83) C ∈ C ( Q ) M ρ ( C ). However, by property (ii) in Theorem 6.7, P (cid:16) k ∈ (cid:91) C ∈ C ( Q ) M ρ ( C ) (cid:17) ≤ δ . The claim follows.This concludes the proof of Theorem 6.5. -colourings In Section 3, we presented self-stabilising CA for different families of two-dimensional tiling spaces, in-cluding k -colourings for k (cid:54) = 3. When k (cid:54) = 3, the k -colourings have the property that any configurationwith a finite island of defects can be corrected in a purely local manner, by modifying the configura-tion only in a bounded neighbourhood of the defects. In contrast, as remarked in Example 1.5, this isnot always possible for two-dimensional 3-colourings. This makes the self-stabilisation problem morechallenging. Question 7.1 (Self-stabilisation of 3-colourings) . Is there a deterministic CA that stabilises -colouringsfrom finite perturbations in polynomial time, ideally without additional symbols?
35e conjecture that it is not possible to stabilise 3-colourings in linear time, but that it could be possiblein quadratic time, at least if we allow additional symbols.In order to shed some light on this problem, let us describe a representation of the two-dimensional3-colourings based on the configurations of the so-called six-vertex model, and in the process, explainFigure 1. The correspondence between 3-colourings and the six-vertex model was first discovered byA. Lenard (see [3, Section 8.13]).
Connection with the six-vertex model.
Let us start with a valid 3-colouring in two dimensions.For each pair of neighbouring cells (horizontal or vertical), let us draw an arrow on the boundarybetween the two cells according to the following rule. Let q and q (cid:48) be the colours of the two neighbouringcells. Since q (cid:48) (cid:54) = q , we either have q (cid:48) = q + 1 (mod 3) or q (cid:48) = q − • The arrow on a vertical boundary is directed upwards if the colour on its right is one more thanthe colour on its left (modulo 3), and downwards otherwise. • The arrow on a horizontal boundary is directed towards the right if the colour below it is onemore than the colour above it (modulo 3), and towards the left otherwise.These conventions are depicted in Figure 24. q q + 1 q q − qq + 1 qq − A sequential correction procedure.
In order to tackle the problem of self-stabilisation for 3-colourings, it may be helpful to address the simpler question of how to correct a perturbed 3-colouringwith a conventional, sequential, non-local algorithm.Suppose we are given a 3-colouring with a finite number of defects and a square region S ofsize (cid:96) containing all the defective cells. Is there a simple (sequential, non-local) algorithm to decide36f the tiling can be corrected by modifying only the colour of the cells inside S ? The 3-colouringproblem on general graphs is NP -complete, and exhaustive search takes exponential amount of timein (cid:96) . Nevertheless, in this case, the representation in terms of the six-vertex configurations allowsus to solve the decision problem in linear time, and to find the actual correction (when it exists) inquadratic time.More precisely, consider the pattern of incoming and outgoing arrows on the boundary of S . Inorder to decide if the colouring of S can be corrected, we just need to know if it is possible to pairthe incoming and outgoing arrows on the boundary of S in an admissible way. This is easy to dosequentially in linear time. Starting from the NE-corner, let us enumerate the incoming arrows on theNorth and the West sides from 1 to n in counter-clockwise, and the outgoing arrows on the East andthe South sides from 1 to n out clockwise. The square can be coloured if we can match each incomingarrow number k with the outgoing arrow number k by a SE-path of arrows. (In particular, this wouldimply n in = n out .) In order to know if this can be done, we try to match successively the incomingand outgoing arrows from 1 to n in = n out with disjoint paths, by moving East if the edge has notalready been selected, and South otherwise. As an additional condition, we need to ensure that ateach step, the path does not go beyond the corresponding outgoing arrow or come across another path.This procedure succeeds if and only if there is at least one admissible matching. See Figure 25 for anillustration. 12345 12345Figure 25: Example of valid matching of the arrows of the contour, with the procedure that matchsuccessively the incoming and outgoing arrows, by moving East if possible, and South otherwise.At this point, we may want to try to turn the above sequential procedure into a self-stabilisingCA. The CA would start by marking squares encompassing the defects. On each such square, the CAwould then simulate a Turing machine that performs the above sequential procedure. If the procedureis successful, a signal is propagated throughout the square in order to erase the markings. If theprocedure is unsuccessful, another signal is sent to increase the size of the square, and to repeat thesimulation of the Turing machine on this larger square. If two squares collide, they merge and forma larger square. Such a CA can indeed be constructed. However, the main difficulty in turning sucha construction into a self-stabilising CA is that the initial perturbation could now involve the extrastates. The construction must ensure stabilisation starting from any such perturbation, not just theperturbations involving the three colours. In this paper, we have barely touched the topic of self-stabilisation with probabilistic rules. There areseveral questions left to be answered, and much more to be explored.37 n isotropic candidate for stabilising -colourings. In Section 4, we have presented isotropicprobabilistic cellular automata achieving self-stabilisation on two-dimensional k -colourings, with k = 2and k ≥
5. We now propose a rule which we believe does the job for k = 4, but for which we haveno formal proof of convergence. The idea is to modify the method used for the case k ≥
5, and makean exception when there is no colour available to directly correct a defective cell. More specifically,for α ∈ (0 , k is not defective, itsstate is kept unchanged. Otherwise, the state of k is changed, with probability α , to a colour whichis chosen at random from among all colours consistent with the current colours of its four neighbours.If no consistent colour exists, a colour is chosen at random from among all possible colours. As usual,different cells are updated independently. Experimentally, we have observed that this rule rapidlycorrects the defects. However, unlike the case k ≥
5, for k = 4, we cannot ensure with the aboverule that the defects stay in some bounded area. See Figure 26 for an example of stabilisation in thisprobabilistic CA. t = 0 t = 1 t = 3 t = 8Figure 26: Illustration of the evolution of the probabilistic CA proposed in Section 7.2 for stabilising4-colourings. The dots indicate the defective cells. Question 7.2 (Probabilistic self-stabilisation of 4-colourings) . Does the probabilistic CA defined abovestabilise -colourings from finite perturbations, for any or for some values of α ∈ (0 , ? We suspect that the answer is positive. To support this claim, one can try to look for configurationsfor which this rule could potentially fail to stabilise. Consider the configuration depicted in Figure 27.It has only two defective cells in the center. Furthermore, all cells (including the defective ones), seethe three other colours in their neighbourhoods. Consequently, if one of the defective cells changesits state alone, it will remain defective. For this specific configuration, some kind of coordination isthus necessary, which cannot here occur by a specific mechanism as in the deterministic case. This atfirst might suggest that the defects may propagate arbitrary far from their origin. However, we haveexperimentally observed that this is not the case: defects have a tendency to stay in the same area,and the correcting process is more rapid than the diffusion of the defects. Surprisingly enough, evenwhen the cells are updated successively at random (i.e., the fully asynchronous case), we also noticedthat the rule succeeds in correcting the defects. Indeed, when the defects propagate, they modify theconfiguration in such a way that the property of seeing three different colours in the neighbourhoodis lost, which finally enables a correction to take place. This observation supports the idea that whenwe use parallel updates, as we do in our rule, we can only increase the possibilities of correction.
No isotropic candidate for stabilising -colourings. In contrast with the previous cases, whenwe have three colours, we cannot use the method of taking an available colour or a random colourwhen no colour is available. We noticed experimentally that the errors diffuse and we could not findany rule that keeps them confined, even in statistical terms.
Question 7.3 (Probabilistic self-stabilisation of 3-colourings) . Is there a probabilistic CA that sta-bilises -colourings from finite perturbations, ideally rapidly and having the same symmetries (isotropy,symmetry of colours) as the tiling space? Self-stabilisation of the
Maj-Random-If-Equal rule.
In Section 4.2, we showed that the
Maj-Random-If-Equal probabilistic CA stabilises the homogeneous space H in no more than cubic time.38 0 03 3 32 2 21 1 12 2 21 1 10 0 03 3 33 3 32 20 01 1 13 3 32 2 21 1 10 0 01 1 10 0 03 3 32 2 2 Figure 27: A 3-colouring with a single defect (i.e., two defective cells). The configuration has theproperty that every cell sees the other three colours in its neighbourhood.
Question 7.4 (Speed of stabilisation in the
Maj-Random-If-Equal
CA) . What is the precise speed ofstabilisation in the
Maj-Random-If-Equal probabilistic CA?
We conjecture that the stabilisation occurs in quadratic time. See the discussion after the proof ofProposition 4.1.
Self-stabilisation from random perturbations.
Does the result of Theorem 6.5 extend in someform to probabilistic CA?
Question 7.5 (Stabilisation from random perturbations) . Suppose that a probabilistic CA stabilisesan SFT X from finite perturbations in linear (or sub-quadratic) time. Does the CA also stabilise X from random perturbations? As a special case, it would be interesting to know if the
Maj-Random-If-Equal rule stabilises fromrandom perturbations, as in the case of its continuous-time counterpart (see Example 6.4).
Stabilising any tiling space.
In the current paper, we have searched for efficient solutions to theself-stabilisation problem for specific classes of SFTs. However, even if we drop the efficiency ambition,it is not clear if one can always find a CA stabilising any given SFT.
Question 7.6 (General solution) . Is it true that for every SFT X , there exists a deterministic CAthat stabilises X from finite perturbations, possibly in exponential time in the number of errors? Whatif we require the solution to have no extra symbols compared to the alphabet of the SFT? We conjecture that an approach similar to the one suggested for 3-colourings in the last paragraph ofSection 7.1 should be possible for a general SFT. In particular, one should be able to come up with ageneral construction for translating a Turing machine that solves the functional version of the squaretiling problem for a finite set of Wang tiles (see Section 5.3) into a self-stabilising CA for the validtilings. As before, the main difficulty would be to handle the appearance of the extra symbols in theinitial perturbation.
Stabilisation in presence of temporal noise.
Recall from Example 6.2 that Toom’s
NEC-Maj
CAmaintains a form of stability on H even in presence of temporal noise. A comparison argument withToom’s CA can be used to show that the two CA discussed in Sections 3.1 and 3.2 have the samestability property. Question 7.7 (Self-stabilisation in presence of temporal noise) . Under what conditions does a CAstabilise an SFT in presence of temporal noise?
In particular, do the CA discussed in Sections 2 and 3.4 self-stabilise in presence of temporal noise?
Acknowledgments
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