A Graph Theory Approach for Regional Controllability of Boolean Cellular Automata
Sara Dridi, Samira El Yacoubi, Franco Bagnoli, Allyx Fontaine
RRESEARCH ARTICLE
A Graph Theory Approach for Regional Controllability of BooleanCellular Automata.
S. Dridi a,b , S. El Yacoubi a , F. Bagnoli b , A. Fontaine c a Team Project IMAGES ESPACE-Dev, University of Perpignan, France; b Department ofPhysics and Astronomy and CSDC, University of Florence, Italy; also INFN, sez. Firenze c Universit´e de Guyane - UMR Espace-Dev, Cayenne, France
ARTICLE HISTORY
Compiled April 17, 2019
ABSTRACT
Controllability is one of the central concepts of modern control theory that allowsa good understanding of a system’s behaviour. It consists in constraining a systemto reach the desired state from an initial state within a given time interval. Whenthe desired objective affects only a sub-region of the domain, the control is saidto be regional. The purpose of this paper is to study a particular case of regionalcontrol using cellular automata models since they are spatially extended systemswhere spatial properties can be easily defined thanks to their intrinsic locality. Weinvestigate the case of boundary controls on the target region using an originalapproach based on graph theory. Necessary and sufficient conditions are given basedon the Hamiltonian Circuit and strongly connected component. The controls areobtained using a preimage approach.
KEYWORDS
Regional Controllability; Deterministic Cellular Automata; Graph Theory;Hamiltonian Circuit; Strongly connected component.
1. Introduction
Control theory is a branch of mathematics that deals with the behaviour of dynam-ical systems studied in terms of inputs and outputs. With the recent developmentsin computing, communications, and sensing technologies, the scope of control theoryis rapidly evolving to encompass the increasing complexity of real-life phenomena.Controllability and observability are two major concepts of control theory that havebeen extensively developed during the last two centuries. The concept of controlla-bility refers to the ability of designing control inputs so as to steer the state of thesystem to desired values within an interval time [0 , T ] while the observability describeswhether the internal state variables of the system can be externally measured. Theseconcepts are being increasingly useful in a wide range of applications such as: biology,biochemistry, biomedical engineering, ecology, economics etc. [1,2]. Controllable andobservable systems have been characterized so far using the Kalman condition in thelinear case. The aim of this paper is to find a general way to give a necessary and
Corresponding author: Samira El Yacoubi. Email: [email protected] a r X i v : . [ n li n . C G ] A p r ufficient condition for controllability of complex systems via cellular automata mod-els. We concentrate in this work on regional controllability via boundary actions onthe target region ω that consists in achieving an objective only in a subdomain of thelattice when some specific actions are exerted on the target region boundaries.The concept of controllability has been widely studied for both finite [3,4] andinfinite dimensional systems [5]. As in many practical problems one is interested inachieving some objectives only on a restricted given sub-region, the notion of control-lability has been extended to the so-called regional controllability concept that hasbeen introduced by El Jai and Zerrik in [6] and well studied in several works [7–9].For distributed parameter systems, the term regional has been used to refer to controlproblems in which the desired state is only defined and may be reachable on a portionof the domain Ω.As for controllability issue, one can consider controls applied on the boundary of thedomain or, in the case of regional controllability, on the boundaries of the consideredsubregion. The controls will steer the system from an initial state to a desired targeton a subregion ω during a fixed time T .Boundary regional controllability problems for distributed parameter systems havebeen mainly described so far, by partial differential equations and considered for linearor nonlinear, continuous or discrete systems. In this paper, we propose to investigatethese problems by using cellular automata as they have been often considered as agood alternative to partial differential equations. [10,11].Cellular automata (CA for short) are discrete dynamical systems considered as thesimplest models of spatially extended systems. They are widely used for studyingthe mathematical properties of discrete systems and for modelling physical systems,[10,12]. However, control of systems described by CA remains very difficult. The tech-niques for controlling discrete systems are quite different from those used in continuousones, since discrete systems are in general strongly nonlinear and the usual linear ap-proximation cannot be directly applied. We restrict our study to the case of some CArules.A Boolean cellular automaton is a collection of cells that can be in one of twostates, on and off, 1 or 0, S = { , } . The states of each cell varies in time dependingon the states of their neighbourhood and a local transition function that defines theconnections between the cells. This function can be deterministic or probabilistic,synchronous or asynchronous, linear or nonlinear.We focus in this study, on a particular case of deterministic and synchronous tran-sition rule that calculates the output of each cell, at each time step as a function ofthe current state of the cell and the states of its two immediate neighbours. A CAconfiguration or global state defines the image representing the states at time t , of thewhole lattice cells. The CA evolution is described by the succession of configurationsat different time steps.This evolution in the case of Boolean deterministic CA can be represented by anoriented graph where the vertices corresponds to the configurations obtained from thebinary representation converted to decimal. There is an arc between two vertices v and v if the configuration corresponding to v can be obtained from the configurationobtained from v where the local transition function is applied.Some regional control problems has been studied using CA, see for instance [13–15]. In [16], the control of 1D and 2D additive CA has been studied by exploring anumerical approach based on genetic algorithms. The regional control problem hasbeen studied on deterministic cellular automata in [13] and on probabilistic CA in[17]. In [18] an approach based on Markov Chain has been used to prove the regional2ontrollability of linear 1D and 2D cellular automata instead of using the well knownKalman criterion.In this paper, we pursue our investigation on regional control problems of deter-ministic CA by using a graph theory approach. The evolution of a controlled CA canbe represented by an oriented graph where the vertices represent the possible config-urations in the controlled region ω which are related to each other by arcs. A coupleof vertices ( v , v ) (CA configurations restricted to ω ) are related by arc if it existsa boundary control ( (cid:96), r ) such that v is reachable starting from v . In this paper wefocus on the problem of regional controllability by applying the controls on the bound-aries of the region ω . We prove the regional controllability by checking the existence ofHamiltonian Circuit which allows us to give a sufficient condition and necessary condi-tion. We address also the problem of decidability of regional controllability by lookingfor a strongly connected component in the graph related to the controlled CA. A nec-essary and sufficient condition for regional controllability is then obtained. Finally, thecontrols required on the boundaries of ω that ensure the regional controllability areobtained using a method for generating the preimages.The paper is organized as follows. Section 2 provides necessary definitions and sec-tion 3 presents the problem of regional controllability. Section 4 is devoted to theformulation of the problem using transition graphs and section 5 gives necessary andsufficient conditions for regional controllability. It first deals with the existence of aHamiltonian Circuit in the graph representing the Boolean CA global evolution andthen the decidability criterion of regional controllability by establishing a relation withstrongly connected component is given. According to this criterion, we give a classifi-cation of selected rules in the one-dimensional CA case. In section 6, we introduce amethod to trace the configurations where a regional control is possible using a methodbased on preimages. Finally, a conclusion will be given in section 7.
2. Basic definitionsDefinition 2.1. [14] A cellular automaton (CA for short) is defined by a tuple A = ( L , S, N , f ) where:(1) L is a cellular space which consists in a regular paving of domain Ω of R d , d = 1 , S is a finite set of possible states.(3) N is a function that defines the neighborhood of a cell c . We denote: N : L → L r c → N ( c ) = ( c i , c i , . . . , c i r )where c i j is a cell for j = 1 , . . . , r and r is the size of the neighborhood N ( c ) ofthe cell c .(4) f is the transition function that allows to compute the state of a cell at time t + 1 according to the state of its neighborhood at time t . It is defined as follow: f : S r → Ss t ( N ( c )) → f ( s t ( N ( c ))) = s t +1 ( c )3here s t ( c ) is the state of a cell c at time t and s t ( N ( c )) = { s t ( c (cid:48) ) , c (cid:48) ∈ N ( c ) } isthe state of the neighborhood of c . Definition 2.2. • The configuration of a CA at time t corresponds to the set { s t ( c ) , c ∈ L} . • The global dynamics of a CA is given by the function: F : S L → S L { s t ( c ) , c ∈ L} → { s t +1 ( c ) , c ∈ L} F maps a configuration of CA at time t a new configuration at time t + 1. • We denote by ω the region we want to control. We have: ω = { c , . . . , c n } . Definition 2.3.
An elementary CA (ECA) is a one-dimensional cellular automatonconstituted by an array of cells that take their states in { , } and change it dependingonly on the states of their neighbours. One can say that the neighborhood of a cell isgiven by the cell itself and its right and left nearest neighbours.Since there are 2 = 8 possible binary states for a cell and its two immediateneighbours, there are a total of 2 = 256 ECA, each of which can be indexed with an8-bit binary number [19,20].
3. Problem Statement
Let us consider: • a 1D-cellular domain L of N cells, • a discrete time horizon I = { , , ..., T } , • a sub-domain ω that defines the controlled region where we want to drive theCA towards a given configuration.It will contain n cells denoted by c i , i = 1 , , · · · , n , n < N . • ω = { c , c , . . . , c n , c n +1 } = ω ∪ { c , c n +1 } where { c , c n +1 } are the boundarycells of ω where we apply control.We are interested in the problem of regional controllability via actions exerted onthe boundary of the target region ω which is a part of the cellular automaton space asillustrated in Figure 1 for n = 3. Our aim is to determine the values at the boundarycells in order to obtain a specific behaviour on ω . Definition 3.1. [16] Let s d ∈ S ω ( s d : ω → S ) be a desired profile to be reached on ω ⊂ L . The CA is said to be regionally controllable for ω at time T if there exists acontrol sequence u = ( u , . . . , u T − ) where u i = ( u i ( c ) , u i ( c n +1 )) , i = 0 , . . . , T − s T = s d on ω where s T is the final configuration at time T and s d is the desired configuration. Notation 1.
Let us introduce the following notations:4 igure 1.
Regional control of one-dimensional CA. (cid:96) t = u t ( c ) r t = u t ( c n +1 ) s ti = s t ( c i ) ∀ i. ≤ i ≤ n .( l · x · r ) is the concatenation operation describing the CA state on ω where x = s ( c ) , . . . , s ( c n ), (cid:96) = u ( c ) and r = u ( c n +1 ). Problem 1.
Starting from an initial condition and for a given desired configuration s d , the considered problem of regional controllability consists in finding the controlrequired on the boundaries { c , c n +1 } , in order to get at time T , the configuration s d in the controlled region { c , . . . , c n } , such that s d ( c i ) = s T ( c i ) ∀ i = 1 , . . . , n , for agiven time horizon T . Example 3.2.
Consider the Wolfram’s rule 90 for which the evolution can completelybe described by a table mapping the next state from all possible combinations of threeinputs ( s − , s , s +1 ) according to the sum modulo 2 of the state values of the cells toits left and to its right s − ⊕ s +1 : f : 111 (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ n = 6 (cf. Figure 2), if we assume starting at time 0 with aninitial configuration { s , s , s , s , s , s } = { } on ω = { c , · · · , c } and given adesired null state on ω , there exists a control u = ( u , u , u ) where u = ( (cid:96) , r ) =5000 011100 00000000 110110 00000000 110111 0000110101 0000 011100 10000001 110111 01000001 010101 0010000000 Figure 2.
The evolution of CA Wolfram rule 90 on the region ω = { c , · · · , c } starting with the same initialconfiguration; on the left without control and on the right with control. (0 , , u = ( (cid:96) , r ) = (1 , , u = ( (cid:96) , r ) = (1 ,
0) that are applied on cells c , c , suchthat the final CA configuration on ω obtained at time T = 3 from the evolution ofrule 90, is { s , s , s , s , s , s } = { } . Remark 1.
The same problem can be defined on two-dimensional CA. For example,we can apply the control on one side of the boundary or on the whole boundary cellsof the controlled region ω in order to get the desired state inside ω (see Figure 3). Figure 3.
Regional Control of two-dimensional CA
Example 3.3.
Consider the following local evolution rule of a two dimensional CAgiven by the function: s t +1 ( c i,j ) = s t ( c i − ,j ) ⊕ s t ( c i +1 ,j ) ⊕ s t ( c i,j − ) ⊕ s t ( c i,j +1 )We consider a controlled region given by the square ω = { c , , c , , c , , c , } .For a given initial configuration given by { s , , s , , s , , s , } = { , , , } on ω , wefirst let the system evolve without applying controls and get the final configuration { s , , s , , s , , s , } = { , , , } at time T = 1.6 T=0 − − − −− −− −− − − − T=1
T=0 − − − −− −− −− − − − T=1
Figure 4.
Evolution of the CA rule example 3.3 on ω in the autonomous and controlled cases on the left andright matrices respectively. We look for controls applied on the boundary cells of ω in order to obtain a desiredconfiguration consisting of all 1s on ω , this can be obtained by controls illustrated inred in Figure 4. Remark 2.
We can define asymmetric controls i.e, we keep a part of the boundariesfixed ( for instance at 0) and act on a subset of the boundary of the controlled region(red cells) in order to get the desired state (see Figure 5).
Figure 5.
Regional control of two dimensional CA with asymmetric controls
The above examples of CA in one and two dimensional cases show that it is possibleto steer a system from an initial state to a desired target on a subregion of the domainby acting on its boundary. The aim of this paper is to generalize this results and findnecessary and sufficient conditions for the regional controllability of some Boolean CArules. The proposed method in the following will be based on transition graphs.7 . Transition graph approach and regional controllability problem
In this section, we describe the main tool on which this paper is based: the transitiongraph Υ. It was inspired by the one introduced in ref [18] where the authors havebuilt a transformation matrix based on all possible state combinations of the CAto show the transition steps. Let us describe here the construction of Υ and thetransformation matrix C that is the associate adjacency matrix of Υ.Recall that the evolution of controlled CA for one step can be represented by adirected graph where the vertices represent the configurations and the arcs representthe transition from a configuration to another one in one step i.e. by applying theglobal transition function F . Consider an Elementary CA where the controlled region ω is of size | ω | and controls are applied on its two boundary cells { c , c n +1 } . Whenconsidering the restriction of F on S | ω | , there exists a bijection between S | ω | and theset of integers [0 : 2 | ω | −
1] that represents CA configurations on ω as | ω | -bit binarynumbers. Let λ be a vertex labelling such that for every vertex v , λ ( v ) is the Booleanconversion of vertex v .We define the transition graph Υ = ( V, A ) as follow where the vertices V correspondsto each possible configuration of the region ω and A is the set of arcs. Let v and v be two vertices in V , there is an arc from the vertex v to the vertex v if thereexists a control u = ( (cid:96), r ) ∈ { (0 , , , , } such that λ ( v ) is equal to F | S ω ( (cid:96) · λ ( v ) · r ), where the λ ( v ) denotes the configuration in the controlled regionat time t and λ ( v ) denotes the configuration in the controlled region at time t + 1.We denote by C the transition matrix which is the associate adjacency matrix ofthe graph Υ. The transition matrix is built as a Boolean matrix of size 2 | ω | × | ω | .There is a 1 at position ( i, j ), the i th row and j th column, if there is an arc betweenvertices i and j for all i, j in [0 : 2 | ω | − v , we compute the four configurations (representedby u , u , u , u ) obtained by the application of the global transition function F | S ω to the four possible configurations obtained by the concatenation of the controls((0 , , , , v . Then we add an arc from v to eachof the four u i . In total, the time complexity to build Υ is O ( | V | ) i.e. O (2 | ω | ) where | ω | is the size of the controlled region ω in the CA. The space complexity is the size ofthe Boolean matrix C : O ( | V | × | V | ) = O (2 | ω | × | ω | ). Note that the number of arcs isat most 4 × | V | . Remark 3.
Note that we have taken the binary representation for the controlledregion in a reverse order (the least significant bit is the first one). For instance, for acontrolled region of size 3, we note: λ (100) = 1, λ (110) = 3 and λ (001) = 4 Example 4.1.
For instance, consider the rule 30 where the controlled region is of size | ω | = 2 for more simplicity. The corresponding graph is represented in Figure 6.The corresponding table for rule 30 is: 8 : 111 (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ F | S ω (0010) = F | S ω (0011) = 11 and F | S ω (1010) = F | S ω (1011) = 01. Therefore thereare two arcs from 2: (2 ,
2) and (2 ,
3) as the binary conversion of 3 is 11.01 3 2
Figure 6.
Transition graph Υ for the CA rule 30 where the region to be controlled is of size 2 and λ (0) = 00, λ (1) = 10, λ (2) = 01, λ (3) = 11. Example 4.2. Rule when the controlled region is of size | ω | = 3.Another example is given in Figure 7. We have considered the rule 90 where thecontrolled region is of size 3. C = , Figure 7.
Transition graph and its adjacency matrix for the CA rule 90 where the controlled region is of size3. . Characterizing regional controllability for Boolean deterministic CA In this section we prove the regional controllability for one-dimensional and two-dimensional CA using a method based on the existence of a Hamiltonian circuit. TheCA is regionally controllable if all the states are reachable in the target region (start-ing from each vertex we can achieve another vertex in finite number of steps ). Theexistence of a Hamiltonian circuit ensures that all vertices (configurations) are visitedonce and ensures that it exists a time T such as all the configurations are reachable. Definition 5.1. [21] A Hamiltonian circuit of a graph G = ( V, A ) is a simple directedpath of G that includes every vertex exactly once. Notation 2.
We introduce the notation a (cid:32) b . This means that b is reachable from a i.e. there is a directed path starting from a to b . In other words, there exist vertices v , v , . . . , v i such that ( a, v ) , ( v , v ) , . . . , ( v i − , v i ) , ( v i , b ) are arcs in A . Theorem 5.2.
A Cellular Automaton is regionally controllable iff there exists a t such that the graph associated to the transformation matrix C t contains a Hamiltoniancircuit. Proof.
Let us start with the first implication. Let Υ = (
V, A ) be the transition graphbuilt in Section 4 for a CA with a controlled region of size | ω | and V = { v , . . . , v | ω | } .The graph Υ will be represented by an adjacency matrix C . Let G be the transitiongraph associated to the matrix C t . The proof is based on the following property ingraph theory: the ( i, j )th entry of the matrix C t corresponds to the number of pathsof length t from vertex i to j .Assume that the CA is regionally controllable at time T ≥ t . Then some configura-tions can be reached in less than T steps from any other one. That means that eachpair of vertices are linked by a directed path of length at most equal to T . Therefore, C T will be strictly positive as reported in the theorem in [18] which states that the CAis regionally controllable if there exists a power T such that C T >
0. The associatedgraph G T to the matrix C T is therefore a complete graph and it is trivial to find aHamiltonian circuit in a complete graph which implies that there exists t ≤ T suchthat the graph related to the matrix C t contains a Hamiltonian Circuit and the directimplication holds.To prove the converse one, let us assume that G contains a Hamiltonian cir-cuit. This means that there is a directed path that goes through all the verticesonce. Therefore there exists an order i , i , . . . , i | ω | such that: ( v i , v i ), ( v i , v i ), . . . ,( v i | ω |− , v i | ω | ), ( v i | ω | , v i ) are arcs in A . And then we have: v i (cid:32) v j ∀ i, j ∈ { , . . . , | ω | } and i (cid:54) = j Thus, ∃ T ≥ t such that all the vertices (configurations) are reachable. And the theoremholds.As the problem of proving the existence of a Hamiltonian circuit in a graph is NP-complete, the time complexity can be exponential in the number of vertices of thetransformation graph. We improve this criterion in the next section with a solution inpolynomial time that gives a necessary and sufficient condition.10 .2. Necessary and Sufficient Condition: Strongly connected component Definition 5.3. [21] A strongly connected component (SCC for short) of a directedgraph G is a maximal set of vertices C ⊂ V such that for every pair of vertices v and v in C , there is a directed path from v to v and a directed path from v to v . Theorem 5.4.
A CA is regionally controllable for a given rule iff the transition graph Υ associated to the rule has only one SCC. Proof.
Let Υ = (
V, A ) be the transition graph built in Section 4 from a controlledregion of size | ω | .Assume that the graph contains only one SCC. There exists a directed path whichrelates each pair of vertices of the graph. Hence there is a sequence of controls thatpermits to go from every configuration to any other one. The CA is then controllableon ω and the direct implication holds.Assume now that the graph Υ contains more than one SCC, let say it contains two.Then, the set of vertices can be divided in two sets related to each SCC such as: V = { v , . . . , v k } V = { v k +1 , . . . , v | ω | } and there is no arc between V and V . Therefore, there is no control that allows toobtain a configuration represented in V from a configuration in V according to theconstruction of Υ. It is impossible since the CA is regionaly controllable and so theconverse implication holds. Time complexity
To find the SCCs, we have used Tarjan’s algorithm [22] whichhas a linear time complexity: O ( | V | + | A | ) on the graph Υ = ( V, A ). If we considera controlled region ω of size | ω | and since in that case | A | ≤ | V | , then the timecomplexity is O ( | V | ) = O (2 | ω | ). Remark 4.
The regional controllability depends on the rule and the size of thecontrolled region. The size of the controlled region for the same rule has an impacton the number of SCCs. According to Theorem 5.4, by changing the size of CA, arule can be sometimes regionally controllable and sometimes not.In Table 1 the results of our simulations are highlighted.
Table 1.
Classification of some rules of one-dimensional CARules Decidability Criterion number of SCC0,255 not controllable 64 for | ω | = 6 and 16 for | ω | = 41 not controllable | ω | = 4, controllable | ω | = 2 8 for | ω | = 4 and 1 for | ω | = 260,90,102,150,170 controllable 1204 not controllable ∨ ∨ | ω | = 2, not controllable | ω | = 5 1 for | ω | = 2 and ∨ | ω | = 526 controllable 2 ≤ | ω | ≤
3, not controllable | ω | = 4 1 for 2 ≤ | ω | ≤ | ω | = 4233 not controllable ∨
13 controllable | ω | = 2, otherwise not controllable 1 for | ω | = 2 , otherwise ∨ To illustrate the obtained results, we shall give in the following section, some ex-amples in both one and two dimensional cellular automata.11 .3. Examples
Example 5.5.
Let us consider the two linear Wolfram rules 150 and 90 [23]. For acontrolled region of | ω | = 4, the graph of the matrix C associated to these two rules isillustrated in the figure Fig. 5.5. It contains one strongly connected component whichmeans that there exists a time T where each configuration is reachable. (a) One SCC associated to the graph of rule 150. (b) One SCC associated to the graph of rule 90. Figure 8.
Graphs of the matrices C and C respectively. Example 5.6.
Wolfram Rule 0 is not controllable neither its Boolean complementrule 255 as they converge to a fixed point. The graph of their matrix C contains morethen one strongly connected component and the previous theorem states that theserules are not regionally controllable for every region ω . (a) rule 0, | ω | = 6. (b) rule 255, | ω | = 4. Figure 9.
Graphs related to the matrices C and C respectively. Example 5.7.
Let us consider now a two-dimensional cellular automaton. Its local12volution is given by the transition function: s t +1 ( c i,j ) = s t ( c i − ,j ) ⊕ s t ( c i +1 ,j ) ⊕ s t ( c i,j − ) ⊕ s t ( c i,j +1 )that is also denoted by rule 170 using Wolfram’s formalism. We impose asymmetriccontrols by setting all cells on the boundaries to 0 except for the two red colored cellsillustrated in Fig. 5.The obtained graph of the matrix C for | ω | = 2 × , ω = { c , , c , , c , , c , } ,contains one strongly connected component and so the CA is regionally controllable. Figure 10.
Graph of the matrix C in two dimensional CA Example 5.8.
Finally, an example with rule 1 is given to show that the decidabilitycriterion for regional controllability may change for the same rule, according to thesize of ω . With a region of size | ω | = 6, the graph of the matrix C contains morethan one SCC while for | ω | = 2 it contains only one strongly connected component.Consequently, the CA is not regionally controllable in the first case and regionallycontrollable in the second one.
6. Pre-images of a regional controlled area
Let { s i , s i , . . . , s in } be the configuration at time i of the region to be controlled. Theidea is to find a boundary control given as a sequence ( (cid:96) , r ) , ( (cid:96) , r ) , . . . , ( (cid:96) T − , r T − )so as to obtain a desired configuration { s T , s T , . . . , s Tn } at time T from an initial one { s , s , . . . , s n } .Let us define in what follows some needed notions and present the data structurerequired to solve the problem. 13 a) rule 1, | ω | = 6. (b) rule 1, | ω | = 2. Figure 11.
Graphs of the matrix C for two sizes of ω . We define the distance function∆ i : vertex (cid:55)→ list of verticesthat associates to each vertex v ∈ Υ, the list of vertices from which v can be reachedwithin a path of length exactly i .Where Υ is the transition graph introduced in Section 4.Therefore ∆ i ( v ) gives all initial configurations from which the desired configuration v can be reached in exactly i steps with the application of control.Let T be the time at which we want to reach a desired state. Representing ∆ as amap, we shall consecutively construct the functions ∆ i by searching the predecessorsfrom ∆ i − . We start by ∆ and go until ∆ T . The algorithm is given in the appendix.If δ is the maximum number of predecessors of a vertex, the time complexity is O ( T × δ × | V | ) = O ( T × δ × | ω | ). We address two problems in this section.
Problem 2.
Find one state configuration that can be driven to a desired state config-uration b f in k steps and the relative control sequence. To solve this problem, we construct the distance function ∆ i for 1 ≤ i ≤ k . Then weconsider the ancestors b i at distance k of b f (stored in ∆ k ( b f )). If there is no ancestor,that means that it is not possible to reach this state in k steps. Otherwise, we canfind the path of length k with end extremity b f . To do so, we pick one predecessor of b f , say b k − , that can be reached in k − i.e. one among the vertices in the list∆ k − ( b f ). Then we search the first one among the predecessors of b k − , say b k − , thatcan be reached in k − i.e. one among the vertices in the list ∆ k − ( b f ) andso forth until we find b . We obtain the path of configurations b , . . . , b k − , b f . It justremains to find the appropriate control by applying the rules to each configurationextended with the boundaries (0 , , (0 , , (1 , , (1 , O ( k ) time. Problem 3.
Find all the needed controls and the intermediate states of the controlledregion required to obtain a desired state b f in exactly k steps ( i.e. ) from a state con-figuration b . For this problem, instead of checking if the list of ancestors is not empty (and takingone among the vertices), we need to check if among the ancestors there is the initialconfiguration b .Therefore, the time complexity is O ( k × δ ).
7. Conclusion
We have studied in this paper the problem of regional controllability of Boolean cel-lular automata focusing on actions performed on the boundary of a target region.We established some necessary and sufficient conditions using graph theory tools. Weshowed that the existence of a Hamiltonian circuit or a strongly connected componentguarantees the regional controllability. To obtain the control that allows the system toreach the desired state during a given time horizon and starting from a given initialcondition, an efficient algorithm for generating preimages was used. Several examplesof Elementary cellular automata has been considered. The obtained control is notunique at this stage and the problem of optimality will be addressed afterward. A firstproblem of regional controllability in minimal time is currently under study.
Acknowledgement(s)
Sara DRIDI who was supported by the Algerian grant Averroes, would like to thankthe Algerian government for this funding.
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16e present in the appendix, the algorithms to construct the data structures usedin the paper.Construction of the transition graph Υ.
Algorithm transGraph( d ω , F ) d ← d ω d Υ ← d Υ ← [0] | d Υ × d Υ (zero matrix of size d Υ ) forall ≤ i < d Υ (for every configuration i ) λ ( v ) ← F (0 · ˘ i | d · λ ( v ) ← F (0 · ˘ i | d · λ ( v ) ← F (1 · ˘ i | d · (add the boundary controlsand apply the rule) λ ( v ) ← F (1 · ˘ i | d · forall ≤ j < d Υ (for every configuration j ) if λ ( v ) , λ ( v ) , λ ( v ) or λ ( v ) equal ˘ j | d (add an edge if thereis a boundary control) Υ( i, j ) ← for i that leads to j) return ΥConstruction of the distance function.
Algorithm distanceFunction( G Υ , k, v)Let d Υ be the number of vertices of G Υ Let ∆ be an empty matrix of size d Υ × k for each vertex v in G Υ ∆( v, ← predecessors( G Υ , v )for < (cid:96) ≤ k for each vertex v in G Υ for each vertex u in ∆( v, (cid:96) − add( ∆( v, (cid:96) ) , predecessors( G Υ , u ))return ∆Finding the control in k steps to reach the desired state. Algorithm pathControllability( G Υ , k , v init , v desired ) p ← [ v desired ]∆ ← distanceFunction ( G Υ , k, v desired ) if v init / ∈ ∆( v, k ) return p pred ← v desired for i from k to predList ← predecessors( G Υ , pred )pred ← find( x ∈ predList .v init ∈ ∆( x, i ) )add(pred, p )return pp