Regional Control of Probabilistic Cellular Automata
Franco Bagnoli, Sara Dridi, Samira El Yacoubi, Raul Rechtman
RRegional Control of Probabilistic Cellular Automata
Franco Bagnoli (1) , Sara Dridi (2) , Samira El Yacoubi (2) , Raúl Rechtman (3)
1) Department of Physics and Astronomy and CSDC,University of Florence,via G. Sansone 1, 50019 Sesto Fiorentino. Italy.Also INFN, sez. Firenze. [email protected]
2) Team Project IMAGES_ESPACE-Dev,UMR 228 Espace-Dev IRD UA UM UG UR,University of Perpignan Via Domitia 52, Avenue Paul Alduy.66860-Perpignan cedex. France. [email protected]
3) Instituto de Energías Renovables,Universidad Nacional Autónoma de México,Apartado Postal 34, 62580 Temixco, Morelos, Mexico. [email protected]
May 16, 2018
Abstract
Probabilistic Cellular Automata are extended stochastic systems, widely used for modellingphenomena in many disciplines. The possibility of controlling their behaviour is therefore animportant topic. We shall present here an approach to the problem of controlling such systems byacting only on the boundary of a target region.
Cellular Automata (CA) are widely used for studying the mathematical properties of discrete systemsand for modelling physical systems [1–6]. They come in two major "flavours": deterministic CA(DCA) [9–14] and probabilistic CA (PCA) [15, 16].DCA are the discrete equivalent of continuous dynamical systems (i.e., differential equations ormaps) but are intrinsically extended, constituted by many elements, so they are in principle the discreteequivalent of system modelled by partial differential equations. DCA are defined by graph, a discreteset of states at the nodes of the graph, and a local transition function that gives the future state of anode as a function of the present state of the node connected to it, its so-called neighbourhood. Thisevolution rule is applied in parallel to all nodes. PCA can be thought as an extension of DCA wherethe transition function gives the probability that the target node goes in a certain state. If all theseprobabilities are either zero or one, that the PCA reduces to a DCA. In both cases, the state of theCA is the collection of states at the nodes of the graph and this state changes in time according tofunctions defined in every node of the graph.In analogy with continuous dynamical systems, it is important to develop methods for controllingthe behaviour of DCA and PCA. In particular, the main control problems for extended systems arereachability and drivability. The first is related to the possibility of applying a suitable control ableto make the system reach a given state or a set of states. For instance, assuming that the systemunder investigation represents a population of pests, the control problem could be that of bringing thepopulation towards extinction at a given time or to keep the population under a certain threshold.1 a r X i v : . [ n li n . C G ] J u l he drivability problem is somehow complementary to the reachability one; once that the systemis driven to a desired state or collection of states, what kind of control may make it follow a giventrajectory? For instance, one may want to stabilize a fixed point, or make the system follow a cycle,and so on.As usual in control problems, one aims at achieving the desired goal with the optimal cost orsmallest effort, and we speak of an optimal control problem. One may be interested not in controllingthe whole space, but rather the state of a given region, for instance how to avoid that a pollutantreaches a certain area.The techniques for controlling discrete systems are quite different from those used in continuousones, since discrete systems are in general strongly non-linear and the usual linear approximationscannot be directly applied. What one can do is to change the state at a node or a set of chosen nodes.For Boolean CA the state is either 0 or 1, so a change is either 1 or 0. The “intensity” of the controltherefore can be only associated to the average number of changes, and cannot be made arbitrarysmall. We are interested in regional control of PCA, that is, how to achieve a certain goal in a set ofneighbouring nodes of a graph.This problem is related to the so-called regional controllability introduced in Ref. [17], as a specialcase of output controllability [18–20]. The regional control problem consists in achieving an objectiveonly in a subregion of the domain when some specific actions are exerted on the system, in its domaininterior or on its boundaries. This concept has been studied by means of partial differential equations.Some results on the action properties (number, location, space distribution) based on the rank conditionhave been obtained depending on the target region and its geometry, see for example Ref. [17] and thereferences therein.Regional controllability has also been studied using CA models. In Ref. [21], a numerical approachbased on genetic algorithms has been developed for a class of additive CA in in one and two dimensions.In Ref. [22], an interesting theoretical study has been carried out for one dimensional additive CAwhere the effect of control is given through an evolving neighbourhood and a very sophisticated statetransition function. However, these studies did not provide a real insight in the regional controllabilityproblem.Some results for control techniques applied to one dimensional DCA can be found in Refs. [23–27].For DCA, once the states in the neighbouring nodes are known, the future state at the node underconsideration is fixed and for PCA we have in general only the probability of reaching a certain state.One advantage of PCA vs. DCA is that their dynamics can be fine-tuned. PCA are summarized inSec. 3.The control problem of PCA is more subtle than of DCA. In general, it is impossible to exactlydrive these systems towards a given configuration, but it is possible to increase the probability that thesystem will reach a target state in a collection of nodes, or, alternatively, to lower as much as possiblethe probability of the appearance of a given configuration, for instance the extinction of a species insidea given region.The evolution of a PCA can be seen as a Markov chain, where the elements of the transition matrixare given by the product of the local transition probabilities (Sec. 3). In particular we shall study herea particular PCA (BBR model) with two absorbing states in Sec. 4.A Markov chain is said to be ergodic if there is the possibility of going to any state in the graphto any other state in a finite number of steps. If this goal can be achieved for all pairs of statesat a given time, the Markov chain is said to be regular. This consideration allows us to define thereachability problem in terms of the probability, once summed over all possible realizations of thecontrol, of connecting any two sites. And since DCA can be considered as the extreme limit of PCA,this technique can be applied to them too, see Sec. 5.Finally, one should remark that the problem of controllability (in particular that of drivability) isstrictly related to that of synchronization (see Ref. [25] for instance). In this same issue the regionalsynchronization problem for the BBR model is addressed [28]. t+1 ! " ! ! $ ! % ! " ! % ! "& ! ! $& ! %& t-2t-1 t Target region Boundary region x i w j Figure 1: Left: The space-time lattice of 1D CA with periodic boundary conditions. Right: CAboundary-value problem.
Cellular Automata are defined on graph composed by N nodes identified by an index i = 1 , . . . , N ,by an adjacency matrix a ij that establishes the neighbourhood of each node with a ij = 1 ( a ij = 0 ) ifnode j is (is not) in node i ’s neighbourhood, and by a transition function f i that gives the new stateat node i given the states in its neighbourhood. The connectivity of node i is k i = (cid:80) j a ij . We shalldeal here with graphs having fixed connectivity k i = k and use the same transition function in all thenodes, f i = f .A lattice is a graph invariant by translation and the nodes are called sites. For a one dimensionallattice with N sites with connectivity k = 2 r + 1 , r = 1 , , . . . and r the range, the neighbourhoodof site i is the set { i − r, . . . , i + r } . We impose Periodic boundary conditions are generally imposed.The state at site i at time t , x i ( t ) , is chosen from a finite set of values, for Boolean CA, x i ( t ) ∈ { , } .Then x i ( t + 1) = f ( x i − r ( t ) , . . . , x i + r ( t )) On each node i of the graph there is one dynamical variable x i = x i ( t ) that for Boolean CA onlytakes values 0 and 1. We shall indicate with x (cid:48) i = x i ( t + 1) its value at the following time step.An ordered set of Boolean values like x , x , . . . , x N can be read as a Boolean vector or as base-twonumber and we shall indicate it as x , ≤ x < N . We shall also indicate with v i the state of allconnected neighbours. The state of x (cid:48) i depends on the state of the neighbourhood v i , and on somerandom number r i ( t ) for stochastic CA. In formulas (neglecting to indicate the random numbers) wehave x (cid:48) i = f ( v i ) . The function f is applied in parallel to all sites. Therefore, we can define a vector function F suchthat x (cid:48) = F ( x ) . The sequence of states { x ( t ) } t =0 ,... is a trajectory of the system with x (0) as the initial condition.When f depends symmetrically on the states of neighbours, it can be shown that f actually dependson the sum s i = (cid:80) j a ij x j . In this case we say that the cellular automaton is totalistic and write x i ( t + 1) = f T ( s i ( t )) , (1)with f T : { , . . . , k } → { , } . Totalistic cellular automata are generic, since they exhibit the wholevariety of behaviour of general rules [12]. It is possible to visualize the evolution of the automata ashappening on a space-time oriented graph or lattice, Fig. 1-left. p p p q p p p q Figure 2: Phase diagram of the BBR model. Left: Density phase diagram. Right: Damage phasediagram.
Probabilistic CA constitute an extension of DCA. Let us introduce the transition probability τ (1 | v ) that, given a certain configuration v = v i of the neighbourhood of site i , gives the probability ofobserving x (cid:48) i = 1 at next time step. Clearly τ (0 | v ) = 1 − τ (1 | v ) . DCA are such that τ (1 | v ) is either0 or 1, while for PCA it can take any value in the middle. For a PCA with k inputs, there are k independent transition probabilities, and for totalistic PCA there are k + 1 independent probabilities.If one associate each transition probability to a different axis, the space of all possible PCA is an unithypercube, with corners corresponding to DCA.PCA can be also partially deterministic, i.e., the transition probability τ (1 | v ) can be zero or onefor certain v . This opens the possibility for the automata to have one or more absorbing state, i.e.,configurations that always originate the same configuration (or give origin to a cyclic behaviour). TheBBR model illustrated below has one or two absorbing states.The evolution of all possible configurations x of a PCA can be written as a Markov chain. Letus define the probability P ( x , t ) , i.e., the probability of observing the configuration x at time t . Itsevolution is given by P ( x , t + 1) = (cid:88) y M ( x | y ) P ( y , t ) , (2)where the matrix M is such that M ( x | y ) = N (cid:89) i =1 τ ( x i | v i ( y )) . (3)For a CA on a 1D lattice and k = 3 we have M ( x | y ) = N (cid:89) i =1 τ ( x i | y i − , y i , y i ) . (4)Phase transitions for PCA can be described as degeneration of eigenvalues in the limit N → ∞ and (subsequently) T → ∞ [29].Notice that since DCA are limit cases of PCA, they also can be seen as particular Markov chains.A Markov chain such that, for some t , ( M t ) ij > for all i, j is said to be regular, and this impliesthat any configuration can be reached by any configuration in time t . A weaker condition (ergodicity)igure 3: Damage spreading; time runs downwards. Left: CA Rule 150. Right: CA rule 126.says that t may depend on the pair i, j (for instance, one may have an oscillating behaviour suchthat certain pairs can be connected only for even or odd values of t ). Also for ergodic systems allconfigurations are connected. We shall use as a testbed model the one presented in Ref. [30], which is an extension of the Domany-Kinzel CA [15]. We shall refer to it as the BBR model from the name of its authors. It is a totalisticPCA defined on a one-dimensional lattice, with connectivity k = 3 . The transition probabilities of themodel are τ (1 |
0) = 0; τ (1 |
1) = p ; τ (1 |
2) = q ; τ (1 |
3) = w. (5)This model has one absorbing state, corresponding to configuration = (0 , , , . . . ) , For w = 1 alsothe configuration = (1 , , , . . . ) is an absorbing state. This is the version studied in Ref. [30].Notice that for p = 1 , q = 1 , w = 0 we have DCA rule 126 while for p = 1 , q = 0 , w = 1 we haveDCA rule 150. In the following we shall use w = 1 .The implementation of a stochastic model makes use of one of more random numbers. For instance,the BBR model can be implemented using the function x (cid:48) i = f ( x i − , x i , x i +1 ; r i ) = [ r i < p ]( x i − ⊕ x i ⊕ x i +1 ⊕ x i − x i x i +1 ) ⊕ [ r i < q ]( x i − x i ⊕ x i − x i +1 ⊕ x i x i +1 ⊕ x i − x i x i +1 ) ⊕ x i − x i x i +1 , (6)where [ · ] is the truth function which takes value one if · is true and zero otherwise, and ⊕ is the summodulo two. The r i = r i ( t ) random numbers have to be extracted for each site and for each time.One can think of extracting them once and for all at the beginning of the simulation, i.e., running thesimulation on a space-time lattice on which a random field r i ( t ) , i = 1 , . . . , N ; t = 0 , . . . is defined.Notice that in this way one has a deterministic CA over a quenched random field.The phase diagram of the BBR model is reported in Fig. 2-left. One can see three regions. The onemarked in white, for p < . , is where the only asymptotically stable configuration is the absorbingstate formed by all zeros, i.e., the asymptotic probability distribution of configurations P ( x ) is a deltaon zero. The symmetric region marked in black, for q > . is where the only stable configuration isformed by all ones. Actually, in a region near the diagonal q = 1 − p , for p < . the two absorbingstates are both stable, the transition line is fixed by the initial configuration, which in the figure isdrawn at random with the same probability of extracting a zero and a one. These regions are denotedwith the term “quiescent”. The region marked in shades of grey, for p > . and q < . is a regionwhere the two absorbing states are unstable, and the asymptotic probability distribution is distributedover many configurations, with average number of ones proportional to the shades of grey. In the insectit is reported the asymptotic average number of ones (the “density”) computed along the dashed lines.This region is denoted with the term “active”. .1 Damage spreading One possibility for controlling the evolution of a system with little efforts is offered by the sensitivedependence on initial conditions, i.e., when a small variation in the initial state propagates to thewhole system. Indeed, this is also the main ingredient of chaos, which in general prevents a carefulcontrol. But in discrete systems the situation is somehow different. These systems are not affectedby infinitesimal perturbations in the variables (assuming that they can be extended in the continuoussense), only to finite ones. The study of the propagation of a finite perturbation in CA goes under thename of “damage spreading”, indicating how an initial disturbance (a “defect” or “damage”) can spreadin the system. A CA where a damage typically spreads is said to be chaotic.Mathematically, one has two copies of the same system, say x and y , evolving with the same rulebut starting from different initial conditions. We shall indicate with z i = x i ⊕ y i the local difference atsite i . Typical patterns of the spreading of a damage (i.e., the evolution of z ) are reported in Fig. 3For PCA, the concept of damage spreading is meant “given the random field”. The phase diagramof the damage z for the BBR model is shown in Fig 2-right. We shall mainly deal here with the problem of regional control via boundary actions, i.e., boundaryreachability as illustrated in Fig.1-right, however the techniques of analysis can be extended to othercases.Let us now consider the problem of computing the probability M xy ( a, b ) = M ( x | y ; a, b ) whichis the probability of getting configuration x at time t + 1 given the configuration y at time t , andboundaries a and b (for simplicity we refer here only to one-dimensional cases). The Markov matrix M ( a, b ) is given by M xy ( a, b ) = τ ( x | a, y , y ) τ ( x | y , y , y ) . . . τ ( x n | y n − , y n , b ) , where n indicates the size of the target region.For a given control sequence a = a , . . . , a T and b = b , . . . , b T , the resulting Markov matrix fortime T is M ( a , b ) = T (cid:89) t =1 M ( a t , b t ) . We can define several control problems. A first one is about ergodicity: which is the best controlsequence a and b so that M xy ( a , b ) > for all pairs x , y and minimum time T ? Another is: given acertain time T and a pair x , y , which is the best control sequence a and b that maximises M xy ( a , b ) > ? Clearly, one can also be interested in avoiding certain configurations, for instance, if x i = 1 repre-sents the presence of some animal or plant in position i at time t , one could be interested in devisinga control that prevents the extinction of animals, i.e., avoid the state x = 0 .As we shall show in the following, so far we have not found algorithms for finding the best controlbut exhaustive search.Beyond finding the actual sequence that maximises the observable, one could be rather interestedin determining the existence of such a sequence, for a certain time interval T , or to find the minimumtime T for which an optima sequence exists.In particular this latter problem can be faced with less computer efforts than finding the actualsequence for the best control. If one considers the matrix C = 14 (cid:88) a,b M ( a, b ) = 14 (cid:0) M (0 ,
0) + M (0 ,
1) + M (1 ,
0) + M (1 , (cid:1) , and then computes its power C T , all possible control sequences of length T are contained in such apower. Therefore, the problem of the existence of a control sequence for a given time T reduces tochecking if ( W T ) xy > . One can also quantify the effective of the control by computing the ratio η igure 4: The ratio η = min( C ) / max( C ) for the BBR model with n = 5 for T = 3 (lower, blue curve)and T = 5 (upper, red curve). Left: q = 0 , Right: q = 1 − p between the minimum and maximum values of C . If this ratio is zero, it means that there are certainpairs of configurations that cannot be connected by any control sequence, while η = 1 means that allpairs of configurations can be connected with equal easiness.Let us illustrate some of these concepts for the BBR model, for p = q and for q = 0 . In Fig. 4 weshow the easiness parameter η in function of p for q = 0 and q = 1 − p , for n = 5 and different valuesof T . One can see that in the “quiescent” phase p < . the control is almost impossible, and that onthe line q = 1 − p , for p > . , the easiness of the control rises with T faster that on the line q = 0 .Indeed, referring to Fig. 2, one can see that this portion of the diagram corresponds to the “active”phase, where the BBR model is ergodic. One can also notice that the easiness of the control is notrelated to the damage spreading phase: considering for instance the line q = p , from Fig. 2-right onesees that the damage spreading phase starts for p > . , while from Fig. 4-right one sees that thecontrol is possible well before this threshold. The control properties are probably associated to the“chaoticity” of the associated deterministic CA over the random quenched field, a problem which willbe faced in the future (for “chaotic” CA and the associated Boolean derivatives, see Refs. [31–33]).Let us now turn to the problem of finding the best control. For compactness, let us consider thecase n = 3 , for which the minimum control time is T = 2 . The highest probability for each pair ofconfigurations x (row index in base two) and y (column index in base two) for q = 1 − p and p = 0 . is M = , orresponding to controls a and b (again in base two) a = b = . These results should be read in this way. Let us consider for instance the initial configuration y = 3 = 110 | (numbers are coded in reverse order) and final configuration x = 4 = 001 | . The bestcontrol is given by a sequence a = 0 = 00 | and b = 3 = 11 | , which is reasonable since one is tryingto force zeros on the left side of the configurations and ones on the right side.Notice however that the entries for a and b are not always either 0 or 3, meaning that the bestcontrol is not a uniform one for all pairs. For instance, for going from y = 3 = 110 | to x = 1 = 100 | one has to apply a = 2 = 01 | and b = 1 = 10 | , exploiting the fact that q = τ (1 |
3) = 1 − p = 0 . andtherefore for forcing a zero in the presence of a neighbourhood already containing a one, it is better toinsert another one than a zero. We have introduced the problem of controlling probabilistic cellular automata by an action performedon the boundary of a target region (boundary control or boundary reachability problem). We haveformulated the problem and presented the first results.The field of control of cellular automata and discrete systems is extremely recent and only a handfulof results are known [26, 27]. In particular, the control of probabilistic cellular automata is still to beexplored in depth, and more efficient algorithms for finding the best control sequence are needed if onewants to exert control on large regions, and in higher dimensions.A promising possibility is that of exploring the relationship between the control and the “chaotic”properties of the associated deterministic CA over a quenched random field.
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