Terminating grid exploration with myopic luminous robots
TTerminating Grid Exploration with Myopic Luminous Robots
Shota Nagahama , Fukuhito Ooshita , and Michiko Inoue Nara Institute of Science and Technology, Japan
Abstract
We investigate the terminating grid exploration for autonomous myopic luminous robots.Myopic robots mean that they can observe nodes only within a certain fixed distance, andluminous robots mean that they have light devices that can emit colors. First, we prove that, inthe semi-synchronous and asynchronous models, three myopic robots are necessary to achievethe terminating grid exploration if the visible distance is one. Next, we give fourteen algorithmsfor the terminating grid exploration in various assumptions of synchrony (fully-synchronous,semi-synchronous, and asynchronous models), visible distance, the number of colors, and achirality. Six of them are optimal in terms of the number of robots.
Many studies about cooperation of autonomous mobile robots have been conducted in the field ofdistributed computing. These studies focus on the minimum capabilities of robots that permit toachieve a given task. To model operations of robots, the
Look-Compute-Move (LCM) model [21]is commonly used. In the LCM model, each robot repeats cycles of Look, Compute, and Movephases. In the Look phase, the robot observes positions of other robots. In the Compute phase,the robot executes its algorithm using the observation as its input, and decides whether it movessomewhere or stays idle. In the Move phase, it moves to a new position if the robot decidedto move in the Compute phase. To consider minimum capabilities, most studies assume thatrobots are identical ( i.e. , robots execute the same algorithm and have no identifier), oblivious ( i.e. , robots have no memory to record their past history), and silent ( i.e. , robots do not havecommunication capabilities). Furthermore, they have no global compass , i.e. , they do not agree onthe directions. Based on the LCM model, previous works clarified solvability of many tasks suchas exploration, gathering, and pattern formation in continuous environments (aka two- or three-dimensional Euclidean space) and discrete environments (aka graph networks) (see a survey [17]).In this paper, we focus on exploration in graph networks, which is one of the most centraltasks for mobile robots. Two variants of exploration tasks have been well studied: the perpetual exploration requires every robot to visit every node infinitely many times, and the terminating exploration requires robots to terminate after every node is visited by a robot at least once. Duringthe last decade, many works have considered the perpetual and terminating exploration on theassumption that each robot has unlimited visibility, i.e. , it observes all other robots in the network.The perpetual exploration has been studied for rings [1] and grids [2]. The terminating explorationhas been studied for lines [15], rings [13,16,18], trees [14], finite grids [10,11], tori [12], and arbitrarynetworks [6]. However, the capability of the unlimited visibility seems powerful and somewhat1 a r X i v : . [ c s . D C ] F e b ontradicts the principle of weak mobile robots. For this reason, some studies consider the morerealistic case of myopic robots [8, 9]. A myopic robot has limited visibility, i.e. , it can see nodes(and robots on them) only within a certain fixed distance φ . Datta et al. studied the terminatingexploration of rings for φ = 1 [8] and φ = 2 , luminous robots, have attracted a lot of attention. Each myopic luminous robot is equipped witha light device that can emit a constant number of colors to other robots, a single color at a time.The light color is persistent, i.e. , it is not automatically reset at the end of each cycle, and henceit can be used as a constant-space memory.Ooshita and Tixeuil [20] studied the perpetual and terminating exploration of rings for φ = 1in the synchronous (FSYNC), semi-synchronous (SSYNC), and asynchronous (ASYNC) models.They showed that the number of robots required to achieve the tasks can be reduced compared tonon-luminous robots. Nagahama et al. [19] studied the same problem in case of φ ≥ φ = 1.Bramas et al. studied the exploration of an infinite grid with myopic luminous and non-luminousrobots in the FSYNC model [3, 4]. Here they propose algorithms so that every node of an infinitegrid is visited by a robot at least once. In [3] robots agree on a common chirality , i.e. , robots agreeon common clockwise and counterclockwise directions. Bramas et al. [5] also studied the perpetualexploration of a (finite) grid with myopic luminous and non-luminous robots in the FSYNC modelon the assumption that robots agree on a common chirality. Algorithms proposed in [5] haveadditional nice properties: they work even if robots are opaque ( i.e. , a robot is able to see anotherrobot only if no other robot lies in the line segment joining them), and they are exclusive ( i.e. , notwo robots occupy a single node during the execution). This work also describes the way to extendtheir algorithms to acheive the terminating exploration and/or to work in the SSYNC and ASYNCmodels. More concretely, this gives three algorithms to achieve the terminating exploration of agrid in case of a common chirality: algorithms for two robots with φ = 1 and six colors in theFSYNC model, two robots with φ = 2 and five colors in the FSYNC model, and two robots with φ = 2 and six colors in the SSYNC and ASYNC models. However algorithms with a fewer numberof colors or no common chirality are not known yet. We focus on the terminating exploration of a (finite) grid with myopic luminous and non-luminousrobots, and clarify lower and upper bounds of the required number of robots in various assumptionsof synchrony, visible distance φ , the number of colors, and a chirality. Table 1 summarizes ourcontributions.First, we prove that, in the SSYNC and ASYNC models, three myopic robots are necessary toachieve the terminating exploration of a grid if φ = 1 holds. Note that this lower bound also holdsfor the perpetual exploration because we prove that robots cannot visit some nodes of a grid inthis case. Other lower bounds in Table 1 are given by Bramas et al. [5]. They are originally givenas impossibility results for the perpetual exploration, however they still hold for the terminatingexploration. This is because Bramas et al. prove that, if the number of robots is smaller in eachassumption, robots cannot visit some nodes.Second, we propose algorithms to achieve the terminating exploration of a grid in variousassumptions in Table 1. To the best of our knowledge, they are the first algorithms that achievethe terminating exploration of a grid by myopic robots with at most three colors and/or with no2able 1: Terminating grid exploration with myopic robots. Notation φ represents the visibledistance of a robot, (cid:96) represents the number of colors, and ∗ means the number of robots isminimum. Synchrony φ (cid:96) Common ∗ § § ∗ § § ∗ § § ∗ § § ∗ § § § § § ∗ § § § The system consists of k mobile robots and a simple connected graph G = ( V, E ), where V is a setof nodes and E is a set of edges. In this paper, we assume that G is a finite m × n grid (or a grid,for short) where m and n are two positive integers, i.e. , G satisfies the following conditions: • V = { v i,j | i ∈ { , , . . . , m − } , j ∈ { , , . . . , n − }} • E = { ( v i,j , v i (cid:48) ,j (cid:48) ) | v i,j , v i (cid:48) ,j (cid:48) ∈ V, | i − i (cid:48) | + | j − j (cid:48) | = 1 } The indices of nodes are used for notation purposes only and robots do not know them. Neithernodes nor edges have identifiers or labels, and consequently robots cannot distinguish nodes andcannot distinguish edges. Robots do not know m or n . Figure 1 shows global directions labeled byNorth, East, South, and West on a grid. Note that these directions are used only for explanations,and robots cannot access the global directions. Each robot is on a node of G at each instant. Whena robot r is on a node v , we say r occupies v and v hosts r . The distance between two nodes is thenumber of edges in a shortest path between the nodes. The distance between two robots r and r is the distance between two nodes occupied by r and r . Two robots r and r are neighbors ifthe distance between r and r is one.Robots we consider have the following characteristics and capabilities. Robots are identical , thatis, robots execute the same deterministic algorithm and do not have unique identifiers. Robots are3 ",$ 𝑣 ",$%& 𝑣 "’&,$ 𝑣 "%&,$ 𝑣 ",$’& 𝑣 "’&,$’& 𝑣 "’&,$%& 𝑣 ",%&$%& 𝑣 "%&,$’& North EastSouthWest
Figure 1: Global directions on a grid luminous , that is, each robot has a light (or state) that is visible to itself and other robots. Arobot can choose the color of its light from a discrete set
Col . When the set
Col is finite, (cid:96) denotesthe number of available colors ( i.e. , (cid:96) = | Col | ). Robots have no other persistent memory andcannot remember the history of past actions. Each robot can communicate by observing positionsand colors of other robots (for collecting information), and by changing its color and moving (forsending information). Robots are myopic , that is, each robot r can observe positions and colorsof robots within a fixed distance φ ( φ > φ (cid:54) = ∞ ) from its current position. Since robotsare identical, they share the same φ . Each robot distinguishes clockwise and counterclockwisedirections according to its own chirality . The robots agree on a common clockwise direction if andonly if they agree on a common chirality.Each robot executes an algorithm by repeating three-phase cycles: Look, Compute, and Movephases. During the Look phase, the robot takes a snapshot of positions and colors of robotswithin distance φ . During the Compute phase, the robot computes its next color and movementaccording to the observation in the Look phase. The robot may change its color at the end ofthe Compute phase. If the robot decides to move, it moves instantaneously to a neighboring nodeduring the
Move phase. To model asynchrony of executions, we introduce the notion of scheduler that decides when each robot executes phases. When the scheduler makes robot r execute somephase, we say the scheduler activates the phase of r or simply activates r . We consider three typesof synchronicity: the FSYNC (fully synchronous) model, the SSYNC (semi-synchronous) model,and the ASYNC (asynchronous) model. In all models, time is represented by an infinite sequenceof instants 0 , , , ... . No robot has access to this global time. In the FSYNC and SSYNC models,all the robots that are activated at an instant t execute a full cycle synchronously and concurrentlybetween t and t +1. In the FSYNC model, at every instant, the scheduler activates all robots. In theSSYNC model, at every instant, the scheduler selects a non-empty subset of robots and activatesthe selected robots. In the ASYNC model, the scheduler activates cycles of robots asynchronously:the time between Look, Compute, and Move phases is finite but unpredictable. Note that in theASYNC model, a robot r can move based on the outdated view obtained during the previous Lookphase. Throughout the paper we assume that the scheduler is fair , that is, each robot is activatedinfinitely often. 4 .2 Configuration, view, and algorithm Configuration.
A configuration represents positions and colors of all robots. At instant t , let Q ( t )be the set of occupied nodes, and let M i,j ( t ) be the multiset of colors of robots on node v i,j ∈ Q ( t ).A configuration C ( t ) of the system at instant t is defined as C ( t ) = { ( v i,j , M i,j ( t )) | v i,j ∈ Q ( t ) } .If t is clear from the context, we simply write Q , M i,j and C instead of Q ( t ), M i,j ( t ), and C ( t ),respectively. View.
When a robot takes a snapshot of its environment, it gets a view up to distance φ . Considera robot r on node v i,j . Let c r be a color of r . We describe M i (cid:48) ,j (cid:48) = ⊥ if node v i (cid:48) ,j (cid:48) does not exist,that is, i (cid:48) / ∈ { , , . . . , m − } or j (cid:48) / ∈ { , , . . . , n − } holds. Since r does not know the globaldirection, it obtains one of the following four views in case of φ = 1 and a common chirality: • North view: V ,ν = ( c r , M i − ,j , M i,j − , M i,j , M i,j +1 , M i +1 ,j ) • East view: V ,e = ( c r , M i,j +1 , M i − ,j , M i,j , M i +1 ,j , M i,j − ) • South view: V ,s = ( c r , M i +1 ,j , M i,j +1 , M i,j , M i,j − , M i − ,j ) • West view: V ,w = ( c r , M i,j − , M i +1 ,j , M i,j , M i − ,j , M i,j +1 )In case of φ = 1 and no common chirality, r obtains one of eight views, which include the abovefour views and the mirror images of them: • Mirror image of V ,ν : V ,ν,µ = ( c r , M i − ,j , M i,j +1 , M i,j , M i,j − , M i +1 ,j ) • Mirror image of V ,e : V ,e,µ = ( c r , M i,j +1 , M i +1 ,j , M i,j , M i − ,j , M i,j − ) • Mirror image of V ,s : V ,s,µ = ( c r , M i +1 ,j , M i,j − , M i,j , M i,j +1 , M i − ,j ) • Mirror image of V ,w : V ,w,µ = ( c r , M i,j − , M i − ,j , M i,j , M i +1 ,j , M i,j +1 )When r obtains one of the views, it cannot recognize which view it obtains, however it can computeother views by rotating and/or flipping the view. Hence, we assume that, in case of a common chiral-ity, r obtains four views V ,ν , V ,e , V ,s , V ,w when it takes a snapshot. Note that r does not recognizewhich view corresponds to each of North, East, South, and West views. Similarly, we assume that,in case of no common chirality, r obtains eight views V ,ν , V ,e , V ,s , V ,w , V ,ν,µ , V ,e,µ , V ,s,µ , V ,w,µ when it takes a snapshot.Similarly, in case of φ = 2 and a common chirality, r obtains the following four views. • North view: V ,ν = ( c r , M i − ,j , M i − ,j − , M i − ,j , M i − ,j +1 , M i,j − , M i,j − , M i,j , M i,j +1 ,M i,j +2 , M i +1 ,j − , M i +1 ,j , M i +1 ,j +1 , M i +2 ,j ) • East view: V ,e = ( c r , M i,j +2 , M i − ,j +1 , M i,j +1 , M i +1 ,j +1 , M i − ,j , M i − ,j , M i,j , M i +1 ,j ,M i +2 ,j , M i − ,j − , M i,j − , M i +1 ,j − , M i,j − ) • South view: V ,s = ( c r , M i +2 ,j , M i +1 ,j +1 , M i +1 ,j , M i +1 ,j − , M i,j +2 , M i,j +1 , M i,j , M i,j − ,M i,j − , M i − ,j +1 , M i − ,j , M i − ,j − , M i − ,j ) 5 West view: V ,w = ( c r , M i,j − , M i +1 ,j − , M i,j − , M i − ,j − , M i +2 ,j , M i +1 ,j , M i,j , M i − ,j ,M i − ,j , M i +1 ,j +1 , M i,j +1 , M i − ,j +1 , M i,j +2 )In case of φ = 2 and no common chirality, r obtains eight views, which include the above four viewsand the mirror images of them: • Mirror image of V ,ν : V ,ν,µ = ( c r , M i − ,j , M i − ,j +1 , M i − ,j , M i − ,j − , M i,j +2 , M i,j +1 , M i,j ,M i,j − , M i,j − , M i +1 ,j +1 , M i +1 ,j , M i +1 ,j − , M i +2 ,j ) • Mirror image of V ,e : V ,e,µ = ( c r , M i,j +2 , M i +1 ,j +1 , M i,j +1 , M i − ,j +1 , M i +2 ,j , M i +1 ,j , M i,j ,M i − ,j , M i − ,j , M i +1 ,j − , M i,j − , M i − ,j − , M i,j − ) • Mirror image of V ,s : V ,s,µ = ( c r , M i +2 ,j , M i +1 ,j − , M i +1 ,j , M i +1 ,j +1 , M i,j − , M i,j − , M i,j ,M i,j +1 , M i,j +2 , M i − ,j − , M i − ,j , M i − ,j +1 , M i − ,j ) • Mirror image of V ,w : V ,w,µ = ( c r , M i,j − , M i − ,j − , M i,j − , M i +1 ,j − , M i − ,j , M i − ,j , M i,j ,M i +1 ,j , M i +2 ,j , M i − ,j +1 , M i,j +1 , M i +1 ,j +1 , M i,j +2 ) Algorithm.
An algorithm is described as a set of rules. Each rule is represented as a combinationof a label, a guard, and an action. The guard represents possible views obtained by a robot. Recallthat robot r obtains several views during the Look phase. If some view of robot r matches a guardin some rule, we say r is enabled. We also say the rule with the corresponding label is enabled.If r is enabled, r can execute the corresponding action ( i.e. , change its color and/or move to itsneighboring node) based on the directions of the matched view during Compute and Move phases.If several views of r match some guard or some view of r matches several guards, one combinationof a view and a rule is selected by the scheduler. Execution.
An execution from initial configuration C is a maximal sequence of configurations E = C , C , ..., C i , ... such that, for any j >
0, we have (i) C j − (cid:54) = C j , (ii) C j is obtained from C j − after some robots move or change their colors, and (iii) for every robot r that moves or changesits color between C j − and C j , there exists 0 ≤ j (cid:48) < j such that r takes its decision to move orchange its color according to its algorithm and its view in C j (cid:48) . The term “ maximal ” means thatthe execution is either infinite or ends in a terminal configuration , i.e. , a configuration in which norobot is enabled. Problem.
A problem P is defined as a set of executions: An execution E solves P if E ∈ P holds. An algorithm A solves problem P from initial configuration C if any execution from C solves P . We simply say an algorithm A solves problem P if there exists an initial configuration C such that A solves P from C . In this paper, we consider the terminating exploration problem. Definition 1 ( Terminating exploration problem ) . The terminating exploration is defined asa set of executions E such that 1) every node is visited by at least one robot in E and 2) there existsa suffix of E such that no robots are enabled. For simplicity, we describe a rule in an algorithm with a figure in Fig. 2. Figure 2(a) rep-resents a rule of an algorithm in case of φ = 1. Figure 2(b) represents a rule in case of6 !, 𝑀 !, 𝑀 !&%, 𝑀 !$%, 𝑀 !, 𝑐 ! 𝑀 !, 𝑀 !, 𝑀 !&%, 𝑀 !$%, 𝑀 !, 𝑐 ! 𝑀 !, 𝑀 !, 𝑀 !&%, 𝑀 !&%, 𝑀 !$’, 𝑀 !&’, 𝑀 !,$% 𝑀 !$%, 𝑅𝑢𝑙𝑒 ∶ 𝑐 " , 𝑀𝑜𝑣𝑒𝑚𝑒𝑛𝑡 𝑅𝑢𝑙𝑒 ∶ 𝑐 " , 𝑀𝑜𝑣𝑒𝑚𝑒𝑛𝑡 (a) (b)
Figure 2: Description of a rule in an algorithm φ = 2. Each graph in Fig. 2 represents a guard. The guard in Fig. 2(a) represents a view V = ( c r , M i − ,j , M i,j − , M i,j , M i,j +1 , M i +1 ,j ), and similarly the guard in Fig. 2(b) represents a view V . If M i (cid:48) ,j (cid:48) = ∅ holds, we paint the corresponding node white instead of writing ∅ . If M i (cid:48) ,j (cid:48) = ⊥ holds, we paint the corresponding node black instead of writing ⊥ . If both ∅ and ⊥ are acceptable,we paint the corresponding node gray. If some view of robot r with visible distance φ matches V φ , r is enabled. In this case, if the scheduler activates r , it executes an action represented by c new , M ovement . Notation c new represents a new color of the robot. Notation M ovement can be
Idle , ← , → , ↑ , ↓ and represents the movement: Idle implies a robot does not move, and ← (resp., → , ↑ , ↓ ) implies a robot moves toward the node corresponding to M i,j − (resp., M i,j +1 , M i − ,j , M i +1 ,j ) of the guard. In this section, we prove that, in the SSYNC model, two robots cannot achieve the terminatingexploration if φ = 1 holds. Since executions in the SSYNC model can happen in the ASYNCmodel, this impossibility also holds in the ASYNC model. This implies that, in case of φ = 1, atleast three robots are necessary to achieve the terminating exploration of grids in the SSYNC andASYNC models. In the following, we use terms of end nodes and inner nodes. We say node v isan end node if the degree of v is smaller than four. We say node v is an inner node if the distancefrom v to every end node is at least three. Theorem 1.
In case of φ = 1 and k = 2 , no algorithm solves the terminating exploration of gridsin the SSYNC model. This holds regardless of the number of colors and a common chirality.Proof. For contradiction, we assume that such an algorithm A exists. Consider an execution E = C , C , ... of A in a m × n grid G that satisfies m ≥ n ≥
9. Let i be the minimum indexsuch that some robot occupies an inner node at C i . Let r be a robot that occupies an inner nodeat C i and r be another robot. Let d be the distance between r and r at C i . We consider twocases: (1) d ≥ d ≤ d ≥ v and v be nodes that host r and r , respectively,at C i . We further consider two sub-cases: (1-1) v is not an end node, and (1-2) v is an end node.7irst assume that v is not an end node (Case 1-1). In this case, we can define nodes v (cid:48) and v (cid:48) suchthat v (cid:48) is a neighbor of v , v (cid:48) is a neighbor of v , v (cid:48) is not an end node, the distance between nodes w and w is at least two for any w ∈ { v , v (cid:48) } and any w ∈ { v , v (cid:48) } . Then we can prove thatthe scheduler makes r and r stay on nodes in { v , v (cid:48) } and { v , v (cid:48) } , respectively, forever after C i .Consider configuration C such that r and r stay on nodes in { v , v (cid:48) } and { v , v (cid:48) } , respectively.Since r and r cannot observe each other and they are not on end nodes, r x ( x ∈ { , } ) cannotdistinguish directions, that is, r x obtains four identical views when it takes a snapshot. This impliesthat, when r x moves, the scheduler can decide which direction r x moves toward. Hence, if r moves,the scheduler can move r to another node in { v , v (cid:48) } . Similarly, if r moves, the scheduler canmove r to another node in { v , v (cid:48) } . This implies that, at the configuration after C , r and r stay on nodes in { v , v (cid:48) } and { v , v (cid:48) } , respectively. Hence, inductively, after C i , robots r and r continue to stay on nodes in { v , v (cid:48) } and { v , v (cid:48) } , respectively. This means that robots can visit atmost two inner nodes until C i and visit at most two other inner nodes after C i . Since the number ofinner nodes in G is at least nine, robots cannot achieve the terminating exploration. Next assumethat v is an end node (Case 1-2). Let v (cid:48) be an inner node that is a neighbor of v . Similarly toCase 1-1, we can prove that, if r never observes r , r continues to stay on nodes in { v , v (cid:48) } . Thisimplies that, to achieve the terminating exploration, r moves toward r or visits the remainingnodes by itself. In any case, r leaves from end nodes, which reduces to Case 1-1.Consider Case 2, that is, d ≤ v be a node that hosts r . Let v be a node thathosts r if d = 1, and a neighbor of v if d = 0. We can prove that, as long as each robot movestoward another robot or stays on its current node, robots continue to stay on nodes in { v , v } :if two robots stay on different nodes, they can only move toward another node, and if two robotsstay on a single node v or v , the scheduler can move them to another node in { v , v } . Hence,eventually a robot moves to another node, say v , when the distance between two robots is one. Inthis moment, the scheduler activates only this robot. After the movement, the distance between r and r is two. Similarly to Case 1, after the configuration, robots can visit only two other innernodes. This implies that robots can visit at most two inner nodes ( v and v ) until C i and visit atmost three other inner nodes ( v and two other inner nodes) after C i . Since the number of innernodes in G is at least nine, they cannot achieve the terminating exploration.This is a contradiction.Note that this impossibility result also holds for the perpetual exploration because the proof ofTheorem 1 shows that robots cannot visit some nodes in this case. In this subsection, we give the overview of our algorithms. All of our algorithms make robotsexplore the grid according to the arrow in Fig. 3. In other words, robots start exploration from thenorthwest corner and repeat the following behaviors:1. Proceed east: Robots go straight to the east end of the grid.2. Turn west: They go one step south and turn west.3. Proceed west: Robots go straight to the west end of the grid.4. Turn east: They go one step south and turn east.8 "," 𝑣 ",$%& 𝑣 ’%&," 𝑣 ’%&,$%& Figure 3: Route of grid exploration with our algorithmIn each algorithm, we implement the behaviors of proceeding and turning. While proceeding,robots recognize their forward direction by their form. In the FSYNC model, since all robots areactivated at every instant, they move forward at every instant and keep their initial form. Therobots repeat this behavior until they reach the end of the grid. On the other hand, in the SSYNCand ASYNC models, not all robots are activated at the same time. For this reason, we propose theway to make robots move forward by moving a single robot at every instant.The difficult part is to implement the behaviors of turning. Since robots do not know globaldirections, they must understand the south direction from the local information. We realize this intwo different approaches. The first approach is to keep robots in two rows when proceeding eastor west. By making different forms in north and south rows, robots distinguish the two directions.Mainly we use this approach in the case of no common chirality. The second approach is usedonly in the case of a common chirality. In this approach, robots change their form of proceedingdepending on the directions. That is, robots distinguish the east and west directions by their form.In the case of a common chirality, robots can go south by turning right (resp. left) when theyproceed east (resp. west). In the second approach, robots do not have to keep themselves in tworows when proceeding. This is the main reason why we can reduce the number of robots in thecase of a common chirality.In the following subsections, we give terminating grid exploration algorithms in various assump-tions. We explain a set of rules and an execution from an initial configuration with figures. In theexplanations, we mention rules that can be applied in each configuration. We omit explanationswhy other rules cannot be applied, but readers can easily check it by comparing the configurationand the set of rules.
In this subsection, we give terminating grid exploration algorithms for the FSYNC model.9 lgorithm 1
Fully Synchronous Terminating Exploration for φ = 2 , (cid:96) = 2 , k = 2 with a CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) } Rules {W}{G} W
𝑅1: W , → {G} {W}G 𝑅2: G , → {G} {W}G 𝑅3: G , ↓ {W}W{G} 𝑅4: W , ↓ {G}G {W} 𝑅5: G , ← {G}G {W} 𝑅6: G , ← {W}W{G} 𝑅7: W , ← {G}G {W} 𝑅8: G , ↓ {W}W{G} 𝑅9: W , ↓ {G}G {W} 𝑅10: G , → φ = 2 , (cid:96) = 2 , a common chirality, and k = 2We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 2, (cid:96) = 2, a common chirality, and k = 2. A set of colors is Col = { G , W } . The algorithm is given inAlgorithm 1. Proceeding east.
From the initial configuration, robots with color G and W can execute rules R R
2, respectively. Hence, they proceed east while keeping the form.
Turning west.
The process of turning west is shown in Fig. 4. After robots proceed east, theyreach the east end of the grid (Fig. 4(a)). From this configuration, the robot with color G movessouth by rule R
3, and hence the configuration becomes one in Fig. 4(b). From this configuration,the robot with color W moves south by rule R
4. At the same time, the robot with color G moveswest by rule R
5. Hence, the configuration becomes one in Fig. 4(c).10
G} {W} {W}{G} {G}
𝑅5𝑅3 (a) (c)(b)
𝑅4 𝑅6 {W} 𝑅7 Figure 4: Turning west in an execution of Algorithm 1 {W}{G} {G} {W}{W} (a) (c)(b) {G}
𝑅8 𝑅7 𝑅9 𝑅2 𝑅1
Figure 5: Turning east in an execution of Algorithm 1
Proceeding west.
From the configuration in Fig. 4(c), the robot with color G and the robot withcolor W can execute rules R R
7, respectively. Hence, they proceed west while keeping theform.
Turning east.
The process of turning east is shown in Fig. 5. After robots proceed west, theyreach the west end of the grid (Fig. 5(a)). From this configuration, the robot with color G movessouth by rule R
8. At the same time, the robot with color W moves by rule R
7. Hence, theconfiguration becomes one in Fig. 5(b). From this configuration, the robot with color W movessouth by rule R
9, and hence the configuration becomes one in Fig. 5(c). From this configuration,two robots can proceed east again.
End of exploration.
After robots visit all nodes and reach a south corner of the grid, theconfiguration becomes terminal. In case that m is odd, two robots visit the south end nodes whileproceeding east, and hence they reach the southeast corner. Immediately after node v m − ,n − isvisited, the configuration is { ( v m − ,n − , { G } ) , ( v m − ,n − , { W } ) } . At this configuration, no robotsare enabled. In case that m is even, two robots visit the south end nodes while proceeding west, andhence they reach the southwest corner. Immediately after node v m − , is visited, the configurationis { ( v m − , , { G } ) , ( v m − , , { W } ) } . From this configuration, robots with colors G and W move byrules R
10 and R
7, respectively. Hence, the configuration becomes { ( v m − , , { G , W } ) } . At thisconfiguration, no robots are enabled. φ = 2 , (cid:96) = 2 , no common chirality, and k = 3We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 2, (cid:96) = 2, no common chirality, and k = 3. A set of colors is Col = { G , W } . The algorithm is given inAlgorithm 2. 11 lgorithm 2 Fully Synchronous Terminating Exploration for φ = 2 , (cid:96) = 2 , k = 3 WithoutCommon Chirality Initial configuration { ( v , , { G } ) , ( v , , { G } ) , ( v , , { W } ) } Rules {G}{G} G{W}
𝑅1: G , → {G} {G}{W}G 𝑅2: G , → {W}{G}W {G} 𝑅3: W , → {G} {G}{W}G 𝑅4: G , ↓ {W}{G}W {G} 𝑅5: W , ↓ {G}G{G} 𝑅6: G , ↓ {W}{G}W 𝑅7: W , → {G}{G} G{W} 𝑅8: G , ↓ Proceeding east.
At the initial configuration, the robot on v , can execute rule R
1, the roboton v , can execute rule R
2, and the robot on v , can execute rule R
3. By repeatedly executingthose rules, robots proceed east while keeping the form.
Turning west.
The process of turning west is shown in Fig. 6. After robots proceed east, theyreach the east end of the grid (Fig. 6(a)). From this configuration, two robots on west nodesmove south by rules R R
5. Hence, the configuration becomes one in Fig. 6(b). From thisconfiguration, the robot with color G at the east end of the grid moves south by rule R W moves east by rule R
7. Consequently, the configuration becomes one inFig. 6(c).
Proceeding west and turning east.
The form of robots in Fig. 6(c) is a mirror image of theone that robots make to proceed east. Hence, robots proceed west and turn east with the samerules as proceeding east and turning west, respectively.
End of exploration.
In case that m is odd, robots visit the south end nodes while proceedingwest. Eventually, the configuration becomes { ( v m − , , { G } ) , ( v m − , , { G } ) , ( v m − , , { W } ) } . Node v m − , has not been visited yet. From this configuration, the robot on v m − , moves to v m − , by rule R
8, and hence the configuration becomes { ( v m − , , { G } ) , ( v m − , , { G } ) , ( v m − , , { W } ) } . Atthis configuration, no robots are enabled. In case that m is even, robots terminate the algorithmsimilarly to the odd case. 12 a) (c)(b) {G} {G}{W} {G}{G}{W} {G} {G}{W} 𝑅4 𝑅5 𝑅7 𝑅6 𝑅1 𝑅2𝑅3
Figure 6: Turning west in an execution of Algorithm 2 {G} {W} {G}{G} {B} {G}
𝑅4𝑅3 (a) (c)(b)
𝑅5 𝑅6 𝑅7𝑅2
Figure 7: Turning west in an execution of Algorithm 3 φ = 2 , (cid:96) = 1 , a common chirality, and k = 3In executions of Algorithm 1, robots do not change their colors and robots with different colors donot occupy a single node. Therefore, by representing the robot of color W in Algorithm 1 with tworobots of color G , we can construct a terminating exploration algorithm in case of φ = 2, (cid:96) = 1, acommon chirality, and k = 3. φ = 2 , (cid:96) = 1 , no common chirality, and k = 4In executions of Algorithm 2, robots do not change their colors and robots with different colors donot occupy a single node. Therefore, by representing the robot of color W in Algorithm 2 with tworobots of color G , we can construct a terminating exploration algorithm in case of φ = 2, (cid:96) = 1, nocommon chirality, and k = 4. φ = 1 , (cid:96) = 3 , a common chirality, and k = 2We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 1, (cid:96) = 3, a common chirality, and k = 2. A set of colors is Col = { G , W , B } . The algorithm is givenin Algorithm 3. Proceeding east.
From the initial configuration, robots with colors W and G can execute rules R R
2, respectively. Hence, they proceed east while keeping the form.
Turning west.
The process of turning west is shown in Fig. 7. After robots proceed east, theyreach the east end of the grid (Fig. 7(a)). From this configuration, the robot with color W moves13 lgorithm 3 Fully Synchronous Terminating Exploration for φ = 1 , (cid:96) = 3 , k = 2 with CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) } Rules {W}{G} W
𝑅1 ∶ W , → {G} {W}G {W}{G} W {G}{G}G {G}{G}G{G}{B} G {B} {G}B {B}{G}B {G}{B}G{B} {G}B 𝑅10 ∶ G , ↓𝑅9 ∶ W , →𝑅8 ∶ B , ↓𝑅6 ∶ B , ← 𝑅4 ∶ B , ←𝑅3 ∶ G , ↓𝑅2 ∶ G , → 𝑅5 ∶ G , ↓𝑅7 ∶ G , ← {B} {G} {G}{B} (b)(a) {G} {W} (c) 𝑅7 𝑅10 𝑅2 𝑅1𝑅8 𝑅9
Figure 8: Turning east in an execution of Algorithm 3south by rule R
3. At the same time, the robot with color G moves east by rule R
2. Hence, theconfiguration becomes one in Fig. 7(b). From this configuration, the robot on a south node changesits color to B and moves west by rule R
4. At the same time, the robot on a north node movessouth by rule R
5. Consequently, the configuration becomes one in Fig. 7(c).
Proceeding west.
From the configuration in Fig. 7(c), the robot with color B and the robot withcolor G can execute rules R R
7, respectively. Hence, they proceed west while keeping theform.
Turning east.
The process of turning east is shown in Fig. 8. After robots proceed west, theyreach the west end of the grid (Fig. 8(a)). From this configuration, the robot with color B movessouth by rule R
8. At the same time, the robot with color G moves west by rule R
7. Hence, theconfiguration becomes one in Fig. 8(b). From this configuration, the robot with color B changesits color to W and moves east by rule R
9. At the same time, the robot with color G moves south14 lgorithm 4 Fully Synchronous Terminating Exploration for φ = 1 , (cid:96) = 3 , k = 4 WithoutCommon Chirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) , ( v , , { B } ) , ( v , , { W } ) } Rules {W}{W}{G} W
𝑅1 ∶ W , → {G} {W}{B}G {W}{W}{B} W {B} {W}{G}B {W}{W}{G} W{W,B}{G}{W}W {W}{W,B}W {W,B}{G}{W}B {G}{W,B}G{W}{W}{B} W 𝑅10 ∶ G , ↓𝑅9 ∶ B , ↓𝑅8 ∶ W , ←𝑅6 ∶ W , ↓ 𝑅4 ∶ B , →𝑅3 ∶ W , →𝑅2 ∶ G , → 𝑅5 ∶ W , ↓𝑅7 ∶ W , ← by rule R
10, and hence the configuration becomes one in Fig. 8(c). From this configuration, tworobots can proceed east again.
End of exploration.
In case that m is odd, two robots visit the south end nodes while proceedingeast, and hence they reach the southeast corner. Immediately after node v m − ,n − is visited, theconfiguration is { ( v m − ,n − , { G } ) , ( v m − ,n − , { W } ) } . From this configuration, the robot with color G moves, and hence the configuration becomes { ( v m − ,n − , { G , W } ) } . At this configuration, norobots are enabled. In case that m is even, two robots visit the south end nodes while proceedingwest, and hence they reach the southwest corner. Immediately after node v m − , is visited, theconfiguration is { ( v m − , , { B } ) , ( v m − , , { G } ) } . From this configuration, the robot with color G moves by rule R
7, and hence the configuration becomes { ( v m − , , { G , B } ) } . At this configuration,no robots are enabled. φ = 1 , (cid:96) = 3 , no common chirality, and k = 4We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 1, (cid:96) = 3,no common chirality, and k = 4. A set of colors is Col = { G , W , B } . The algorithm is given inAlgorithm 4. Proceeding east.
At the initial configuration, the robot on v , can execute rule R
1, the roboton v , can execute rule R
2, the robot on v , can execute rule R
3, and the robot on v , can executerule R
4. By repeatedly executing those rules, robots proceed east while keeping the form.
Turning west.
The process of turning west is shown in Fig. 9. After robots proceed east, theyreach the east end of the grid (Fig. 9(a)). From this configuration, two robots on east nodes move15 (a) (c)(b) 𝑅8 {G} {W}{B} {W} {G}{W,B}{W} {W} {G}{W} {B} 𝑅2 𝑅7 𝑅10 𝑅1 𝑅2𝑅3𝑅4 𝑅6𝑅5 𝑅9
Figure 9: Turning west in an execution of Algorithm 4south by rules R R
6. At the same time, the other robots move east by rules R R W move west by rules R R
8. At the same time, robots with color B and G move south byrules R R
10, respectively. Consequently, the configuration becomes one in Fig. 9(c).
Proceeding west and turning east.
The form of robots in Fig. 9(c) is a mirror image of theone that robots make to proceed east. Hence, robots proceed west and turn east with the samerules as proceeding east and turning west, respectively.
End of exploration.
In case that m is odd, robots visit the south end nodes while proceed-ing west, and hence they reach the southwest corner. Immediately after node v m − , is visited,the configuration is { ( v m − , , { W } ) , ( v m − , , { G } ) , ( v m − , , { W } ) , ( v m − , , { B } ) } . From this con-figuration, the robot on v m − , moves to v m − , by rule R
5. At the same time, robots withcolors G and B move west by rules R R
4, respectively. Hence, the configuration becomes { ( v m − , , { G } ) , ( v m − , , { W , W , B } ) } . At this configuration, no robots are enabled. In case that m is even, robots terminate the algorithm similarly to the odd case. φ = 1 , (cid:96) = 2 , a common chirality, and k = 3We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 1, (cid:96) = 2, a common chirality, and k = 3. A set of colors is Col = { G , W , B } . The algorithm is givenin Algorithm 5. Proceeding east.
At the initial configuration, the robot on v , can execute rule R
1, the roboton v , can execute rule R
2, and the robot on v , can execute rule R
3. By repeatedly executingthose rules, robots proceed east while keeping the form.
Turning west.
The process of turning west is shown in Fig. 10. After robots proceed east, theyreach the east end of the grid (Fig. 10(a)). From this configuration, the robot at the east end movessouth by rule R
4. At the same time, the other robots move east by rules R R
3. Hence,the configuration becomes one in Fig. 10(b). From this configuration, the robot with color G on asouth node moves south by rule R
5. At the same time, the robot with color W moves west by rule R
6, and the robot on a north node changes its color to W and moves south. Consequently, theconfiguration becomes one in Fig. 10(c). 16 lgorithm 5 Fully Synchronous Terminating Exploration for φ = 1 , (cid:96) = 2 , k = 3 with CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { G } ) , ( v , , { W } ) } Rules {G}{G} G
𝑅1 ∶ G , → {G} {G}{W}G {W}{G}W {G}{G} G {G,W}{G}G{G}{G,W}G {W} {W}W {W}{G}{W} W {G}{W}G{G,W}{G}W 𝑅10 ∶ G , ←𝑅9 ∶ W , ←𝑅8 ∶ W , ←𝑅6 ∶ W , ← 𝑅4 ∶ G , ↓𝑅3 ∶ W , →𝑅2 ∶ G , → 𝑅5 ∶ G , ↓𝑅7 ∶ W , ↓ {W} {W}W 𝑅11 ∶ W , ↓ {G,W}{W}W {G,W}{W}G {W}{G,W}W 𝑅14 ∶ G , ↓𝑅13 ∶ G , →𝑅12 ∶ W , ↓ (a) (c)(b) {G} {G}{W} {G}{G,W} {W} {W}{G} 𝑅2 𝑅4 𝑅5 𝑅7 𝑅8 𝑅9𝑅10𝑅3 𝑅6
Figure 10: Turning west in an execution of Algorithm 517
W}{G,W} {G} {G}{W} {W}{G} (a) (c)(b)
𝑅10 𝑅9 𝑅14 {W}
𝑅11 𝑅13 𝑅2 𝑅1𝑅3𝑅12
Figure 11: Turning east in an execution of Algorithm 5
Proceeding west.
At the configuration in Fig. 10(c), the robot on a west node can execute rule R
8, the robot with color W on a east node can execute rule R
9, and the robot with color G canexecute rule R
10. Hence, they proceed west while keeping the form.
Turning east.
The process of turning east is shown in Fig. 11. After robots proceed west, theyreach the west end of the grid (Fig. 11(a)). From this configuration, the robot on a west node movessouth by rule R
11. At the same time, the other robots move west by rule R R
10. Hence,the configuration becomes one in Fig. 11(b). From this configuration, the robot with color W on asouth node moves south by rule R
12. At the same time, the robot with color G moves east rule R
13, and the robot on a north node changes its color to G and moves south by rule R
14. Hence,the configuration becomes one in Fig. 11(c). From this configuration, three robots can proceed eastagain.
End of exploration.
In case that m is odd, robots visit the south end nodes while proceed-ing west. Eventually, the configuration becomes { ( v m − , , { W } ) , ( v m − , , { W } ) , ( v m − , , { G } ) } .Node v m − , has not been visited yet. From this configuration, the robot on v m − , moves to v m − , by rule R
11. At the same time, the other robots move west by rules R R { ( v m − , , { W } ) , ( v m − , , { G , W } ) } . From this configura-tion, the robot on v m − , moves to v m − , by rule R
14, and hence the configuration becomes { ( v m − , , { G , G , W } ) } . At this configuration, no robots are enabled. In case that m is even,robots visit the south end nodes while proceeding east. Eventually, the configuration becomes { ( v m − ,n − , { G } ) , ( v m − ,n − , { G } ) , ( v m − ,n − , { W } ) } . Node v m − ,n − has not been visited yet.From this configuration, the robot on v m − ,n − moves to v m − ,n − by rule R
4. At the sametime, the other robots move east by rules R R
3, and hence the configuration becomes { ( v m − ,n − , { G } ) , ( v m − ,n − , { G , W } ) } . From this configuration, the robot on v m − ,n − moves to v m − ,n − by rule R
7, and hence the configuration becomes { ( v m − ,n − , { G , W , W } ) } . At this con-figuration, no robots are enabled. φ = 1 , (cid:96) = 2 , no common chirality, and k = 5In executions of Algorithm 4, robots do not change their colors and robots with colors G and B donot occupy a single node. Therefore, by representing the robot of color B in Algorithm 4 with tworobots of color G , we can construct a terminating exploration algorithm in case of φ = 1, (cid:96) = 2, nocommon chirality, and k = 5. 18 lgorithm 6 Asynchronous Terminating Exploration for φ = 2 , (cid:96) = 3 , k = 2 with CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) } Rules {W}{G} W
𝑅1: W , → {G}G {W} 𝑅2: G , → {W}{G} W 𝑅3: G , ↓ {G}G {W} 𝑅4: B , ↓ {B} {W}B 𝑅5: B , ← {W}W{B} 𝑅6: W , ← {B} {W}B 𝑅7: B , ↓ {B}B {W} 𝑅8: G , 𝐼𝑑𝑙𝑒 {W}W{G} 𝑅9: W , ↓ In this subsection, we give terminating exploration algorithms for the ASYNC model. Clearlyrobots can achieve terminating exploration with those algorithms also in the SSYNC and FSYNCmodels. φ = 2 , (cid:96) = 3 , a common chirality, and k = 2We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 2, (cid:96) = 3, a common chirality, and k = 2. A set of colors is Col = { G , W , B } . The algorithm is givenin Algorithm 6. Proceeding east.
From the initial configuration, the robot with color W moves east by rule R { ( v , , { G } ) , ( v , , { W } ) } . From this configuration, the robot19 G} {W} {G} {W} {B} {W} 𝑅3 (a) (d)(b) 𝑅4 𝑅5 {B} {W} (c) 𝑅4 Figure 12: Turning west in an execution of Algorithm 6 {B} {W} {W}{G} (c)(a) {G} {W} (d)
𝑅7 𝑅9 𝑅1 {W}{B} (b) 𝑅8 Figure 13: Turning east in an execution of Algorithm 6with color G moves east by rule R
2, and hence the configuration becomes { ( v , , { G } ) , ( v , , { W } ) } .After that, robots proceed east while keeping the form by repeatedly executing those rules. Turning west.
The process of turning west is shown in Fig. 12. After robots proceed east,they reach the east end of the grid (Fig. 12(a)). From this configuration, the robot with color W moves south by rule R
3, and hence the configuration becomes one in Fig. 12(b). From thisconfiguration, the robot with color G changes its color to B and moves south by rule R
4. In theASYNC model, after the robot with color G changes its color, the other robot may observe theintermediate configuration (Fig. 12(c)). However, there are no rules that the other robot can executein the intermediate configuration. Consequently, the configuration becomes one in Fig. 12(d). Proceeding west.
From the configuration in Fig. 12(d), the robot with color B moves west byrule R
5. Next, the robot with color W moves west by rule R
6. After that, robots proceed westwhile keeping the form by repeatedly executing those rules.
Turning east.
The process of turning east is shown in Fig. 13. After robots proceed west, theyreach the west end of the grid (Fig. 13(a)). From this configuration, the robot with color B movessouth by rule R
7, and hence the configuration becomes one in Fig. 13(b). From this configuration,the robot with color B changes its color to G by rule R
8, and hence the configuration becomes onein Fig. 13(c). From this configuration, the robot with color W moves south by rule R
9, and hencethe configuration becomes one in Fig. 13(d). From this configuration, two robots can proceed eastagain.
End of exploration.
In case that m is odd, two robots visit the south end nodes while proceedingeast, and hence they reach the southeast corner. Immediately after node v m − ,n − is visited, the20 lgorithm 7 Asynchronous Terminating Exploration for φ = 2 , (cid:96) = 3 , k = 3 Without CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) , ( v , , { B } ) } Rules {B}{G}B {W}
𝑅1: B , → {W}{B}{G} W 𝑅2: W , → {G}G {W}{B} 𝑅3: G , → {B}{G}B {W} 𝑅4: B , ↓ {G} {W}G {B} 𝑅5: W , ↓ {B}{W}B 𝑅6: B , → {W}W {B}{W} 𝑅7: G , ↓ {W} {G}W {B} 𝑅8: W , ↓ configuration is { ( v m − ,n − , { G } ) , ( v m − ,n − , { W } ) } . At this configuration, no robots are enabled.In case that m is even, two robots visit the south end nodes while proceeding west, and hencethey reach the southwest corner. Immediately after node v m − , is visited, the configuration is { ( v m − , , { B } ) , ( v m − , , { W } ) } . At this configuration, no robots are enabled. φ = 2 , (cid:96) = 3 , no common chirality, and k = 3We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 2, (cid:96) = 3, a common chirality, and k = 2. A set of colors is Col = { G , W , B } . The algorithm is givenin Algorithm 7. Proceeding east.
From the initial configuration, the robot with color B moves by rule R
1, and hence the configuration becomes { ( v , , { G } ) , ( v , , { W } ) , ( v , , { B } ) } . From thisconfiguration, the robot with color W by rule R
2, and hence the configuration becomes { ( v , , { G } ) , ( v , , { W } ) , ( v , , { B } ) } . From this configuration, the robot with color G by rule R { ( v , , { G } ) , ( v , , { W } ) , ( v , , { B } ) } . After that, robots pro-ceed east while keeping the form by repeatedly executing those rules. Turning west.
The process of turning west is shown in Fig. 14. After robots proceed east,they reach the east end of the grid (Fig. 14(a)). From this configuration, the robot with color B moves south by rule R
4, and hence the configuration becomes one in Fig. 14(b). From thisconfiguration, the robot with color G changes its color to W and moves south by rule R
5. Inthe ASYNC model, after the robot with color G changes its color, other robots may observe the21 a) (c)(b) {G} {W}{B} {G} {W}{B} {W} {W}{B} 𝑅5 (e) (g)(f) {W}{W} {B} {G}{W} {B} {W} {G}{B} 𝑅7 𝑅1 (d) {W}{W}{B}
𝑅6𝑅4 𝑅5𝑅7
Figure 14: Turning west in an execution of Algorithm 7intermediate configuration (Fig. 14(c)). However, there are no rules that the other robot can executein the intermediate configuration. Hence, the configuration becomes one in Fig. 14(d). From thisconfiguration, the robot with color B moves east by rule R
6, and hence the configuration becomesone in Fig. 14(e). From this configuration, the robot with color W changes its color to G and movessouth by rule R
7. In the ASYNC model, after the robot with color W changes its color, other robotsmay observe the intermediate configuration (Fig. 14(f)). However, there are no rules that the otherrobot can execute in the intermediate configuration. Consequently, the configuration becomes onein Fig. 14(g). Proceeding west and turning east.
The form of robots in Fig. 14(g) is a mirror image of theone that robots make to proceed east. Hence, robots proceed west and turn east with the samerules as proceeding east and turning west, respectively.
End of exploration.
In case that m is odd, robots visit the south end nodes while proceedingwest. Eventually, the configuration becomes { ( v m − , , { W } ) , ( v m − , , { G } ) , ( v m − , , { B } ) } . Node v m − , has not been visited yet. From this configuration, the robot with color W moves to v m − , by rule R
8, and hence the configuration becomes { ( v m − , , { G } ) , ( v m − , , { W } ) , ( v m − , , { B } ) } . Atthis configuration, no robots are enabled. In case that m is even, robots terminate the algorithmsimilarly to the odd case. φ = 2 , (cid:96) = 2 , a common chirality, and k = 3We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 2, (cid:96) = 2, a common chirality, and k = 3. A set of colors is Col = { G , W } . The algorithm is given inAlgorithm 8. Proceeding east.
From the initial configuration, the robot with color W moves east by rule R
1, and hence the configuration becomes { ( v , , { G } ) , ( v , , { W } ) , ( v , , { G } ) } . From this con-22 lgorithm 8 Asynchronous Terminating Exploration for φ = 2 , (cid:96) = 2 , k = 3 with CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) , ( v , , { G } ) } Rules {W}{G} W{G}
𝑅1: W , → {G}{G}G {W} 𝑅2: G , → {G}G {G} 𝑅3: G , → {W}{G} W{G} 𝑅4: W , ↓ {G} {W}{G}G 𝑅5: W , 𝐼𝑑𝑙𝑒 {G}{W}G {W} 𝑅6: G , → {W}{G}{W} W 𝑅7: W , ↓ {G}G {W}{W} 𝑅8: G , ↓ {W} {G}W {W} 𝑅9: W , ← {G}{W}G{W} 𝑅10: G , ← {W}W{G} 𝑅11: W , ← {W} {G}W {W} 𝑅12: W , ↓ {W} {W}W {G} 𝑅13: G , 𝐼𝑑𝑙𝑒 {G}{W}G{G} {G} {W}{G}G {G}G {G} {W} 𝑅14: G , ← 𝑅16: G , ↓𝑅15: G , ↓ a) (c)(b) {G} {W}{G} {G}{G} {W} {G}{W} {W} 𝑅6𝑅5𝑅4 (d) (f)(e) {G}{W} {W} {G}{W} {W} {W} {G}{W}
𝑅7 𝑅8 𝑅9
Figure 15: Turning west in an execution of Algorithm 8figuration, the robot on v , moves east by rule R
2, and hence the configuration becomes { ( v , , { G } ) , ( v , , { W } ) , ( v , , { G } ) } . From this configuration, the robot on v , moves east by rule R
3, and hence the configuration becomes { ( v , , { G } ) , ( v , , { W } ) , ( v , , { G } ) } . After that, robotsproceed east while keeping the form by repeatedly executing those rules. Turning west.
The process of turning west is shown in Fig. 15. After robots proceed east, theyreach the east end of the grid (Fig. 15(a)). From this configuration, the robot with color W movessouth by rule R
4, and hence the configuration becomes one in Fig. 15(b). From this configuration,the robot with color G on a south node changes its color to W by rule R
5, and hence the configurationbecomes one in Fig. 15(c). From this configuration, the robot with color G moves east by rule R W moves south by rule R
7, and hence the configuration becomes one in Fig. 15(e). From thisconfiguration, the robot with color G moves south by rule R
8, and hence the configuration becomesone in Fig. 15(f).
Proceeding west.
From the configuration in Fig. 15(f), the robot with color W on a west nodemoves west by rule R
9. Next, the robot with color G moves west by rule R
10. Then, the robot withcolor W on a east node moves west by rule R
11. After that, robots proceed west while keeping theform by repeatedly executing those rules.
Turning east.
The process of turning east in an execution of Algorithm 8 is shown in Fig. 16.After robots proceed west, they reach the west end of the grid (Fig. 16(a)). From this configuration,the robot with color W on a west node moves south by rule R
12, and hence the configurationbecomes one in Fig. 16(b). From this configuration, the robot with color W on a west node changesits color to G by rule R
13, and hence the configuration becomes one in Fig. 16(c). From thisconfiguration, the robot with color G on a north node moves west by rule R
14, and hence theconfiguration becomes one in Fig. 16(d). From this configuration, the robot with color G on a24 G}{W} {W} {G}{G} {W}{W} {G}{W} (a) (c)(b) {G} {W} {G} {W}{G}{G} {W} (d) (f)(e) {G} {G}
𝑅12 𝑅13 𝑅14𝑅15 𝑅16 𝑅1
Figure 16: Turning east in an execution of Algorithm 8south node moves south by rule R
15, and hence the configuration becomes one in Fig. 16(e). Fromthis configuration, the robot with color G on a north node moves south by rule R
16, and hencethe configuration becomes one in Fig. 16(f). From this configuration, two robots can proceed eastagain.
End of exploration.
In case that m is odd, robots visit the south end nodes while proceedingwest. Eventually, the configuration becomes { ( v m − , , { W } ) , ( v m − , , { G } ) , ( v m − , , { W } ) } . Node v m − , has not been visited yet. From this configuration, the robot on v m − , moves to v m − , by rule R
12, and hence the configuration becomes { ( v m − , , { G } ) , ( v m − , , { W } ) , ( v m − , , { W } ) } .At this configuration, no robots are enabled. In case that m is even, robots visitthe south end nodes while proceeding east. Eventually, the configuration becomes { ( v m − ,n − , { G } ) , ( v m − ,n − , { W } ) , ( v m − ,n − , { G } ) } . Node v m − ,n − has not been visited yet. Fromthis configuration, the robot on v m − ,n − moves to v m − ,n − by rule R
4, and hence the configura-tion becomes { ( v m − ,n − , { G } ) , ( v m − ,n − , { G } ) , ( v m − ,n − , { W } ) } . At this configuration, no robotsare enabled. φ = 2 , (cid:96) = 2 , no common chirality, and k = 4We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 2, (cid:96) = 2, no common chirality, and k = 4. A set of colors is Col = { G , W } . The algorithm is given inAlgorithm 9. Proceeding east.
The process of proceeding east is shown in Fig. 17. At the initial configurationor at a configuration immediately after turning east, robots make the form in Fig. 17(a). From thisconfiguration, the robot with color W on a south node moves east by rule R
1, and hence theconfiguration becomes one in Fig. 17(b). From this configuration, the robot with color W on aneast node moves east by rule R
2, and hence the configuration becomes one in Fig. 17(c). Fromthis configuration, the robot with color W neighboring to the robot with color G moves east by25 lgorithm 9 Asynchronous Terminating Exploration for φ = 2 , (cid:96) = 2 , k = 4 Without CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) , ( v , , { W } ) , ( v , , { W } ) } Rules {W}{G}W {W}
𝑅1: W , → {W}{W} W{G} {W} 𝑅2: W , → {W}{W}{G} W {W} 𝑅3: W , → {G}G {W}{W} 𝑅4: G , → {W}{W} W{G} {W} 𝑅5: W , ↓ {W}{W}{G} W {W} 𝑅6: G , 𝐼𝑑𝑙𝑒 {G} {G}G {W} 𝑅7: G , ↓ {G}{W}G {W}{G} 𝑅8: G , → {G} {W}G {W} 𝑅9: W , 𝐼𝑑𝑙𝑒 {W}{G}{W} W{W} 𝑅10: W , ↓ W} {W} (a) {G}{W} 𝑅1 {W} {W}{W} (b) {G} 𝑅2 {W}{W} (c) {G} {W} 𝑅3 {W}{W} (d) {G} {W} 𝑅4 ・・・ Figure 17: Proceeding east in an execution of Algorithm 9rule R
3, and hence the configuration becomes one in Fig. 17(d). From this configuration, the robotwith color G moves east by rule R
4. After that, robots proceed east while keeping the form byrepeatedly executing those rules.
Turning west.
The process of turning west is shown in Fig. 18. After robots proceed east,they reach the east end of the grid, and the configuration becomes one in Fig. 18(a). From thisconfiguration, the robot at the east end moves south by rule R
5, and hence the configurationbecomes one in Fig. 18(b). From this configuration, the robot with color W on a north nodechanges its color to G by rule R
6, and hence the configuration becomes one in Fig. 18(c). Fromthis configuration, the robot with color G on a west node moves south by rule R
7, and hence theconfiguration becomes one in Fig. 18(d). From this configuration, the robot with color G on a northnode moves east by rule R
8, and hence the configuration becomes one in Fig. 18(e). From thisconfiguration, the robot with color G on a west node changes its color to W by rule R
9, and hencethe configuration becomes one in Fig. 18(f). From this configuration, the robot with color W on aneast node moves south by rule R
10, and hence the configuration becomes one in Fig. 18(g). Fromthis configuration, the robot with color G moves south by rule R
4, and hence the configurationbecomes one in Fig. 18(h).
Proceeding west and turning east.
The form of robots in Fig. 18(h) is a mirror image of theone that robots make to proceed east. Hence, robots proceed west and turn east with the samerules as proceeding east and turning west, respectively.
End of exploration.
In case that m is odd, robots visit the southend nodes while proceeding west. Eventually, the configuration becomes { ( v m − , , { W } ) , ( v m − , , { W } ) , ( v m − , , { G } ) , ( v m − , , { W } ) } . Node v m − , has not been vis-ited yet. From this configuration, the robot on v m − , moves to v m − , by rule R
5, and hencethe configuration becomes { ( v m − , , { W } ) , ( v m − , , { G } ) , ( v m − , , { W } ) , ( v m − , , { W } ) } . At thisconfiguration, no robots are enabled. In case that m is even, robots terminate the algorithmsimilarly to the odd case. 27 a) (c)(b) {G} {W}{W} {G} {W}{W} {G} {G}{W} 𝑅7𝑅5 {W} {W} 𝑅6 {W} (d) (f)(e) {G}{G} {W} {G} {W} {W} {W} 𝑅8 {W} {G}{W} 𝑅9 {G}{W} 𝑅10 (g) (h) {W} {W} {W} {W} 𝑅4 {G}{W} {G}{W} 𝑅1 Figure 18: Turning west in an execution of Algorithm 928 lgorithm 10
Asynchronous Terminating Exploration for φ = 1 , (cid:96) = 3 , k = 3 with CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) , ( v , , { W } ) } Rules {G} {W}G
𝑅1 ∶ G , → {G,W} {W}W {G,W}{G} G {G,W}{G} G {G,W}{B}G{W}{B}W {W,B}{B} B {W,B} {W}W {W,B} {W}W{G,B}{W}G 𝑅10 ∶ G , ↓𝑅9 ∶ B , ←𝑅8 ∶ W , ←𝑅6 ∶ B , ← 𝑅4 ∶ B , ↓𝑅3 ∶ W , →𝑅2 ∶ G , → 𝑅5 ∶ G , ↓𝑅7 ∶ W , ↓ {W,B}{G}W 𝑅11 ∶ B , ↓ {G,B}{B}B 𝑅12 ∶ G , → {B}{G}B {G,B} {G}B {G,B}{G} B 𝑅14 ∶ B , →𝑅13 ∶ B , ↓ 𝑅15 ∶ W , 𝐼𝑑𝑙𝑒 φ = 1 , (cid:96) = 3 , a common chirality, and k = 3We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 1, (cid:96) = 3, a common chirality, and k = 3. A set of colors is Col = { G , W , B } . The algorithm is givenin Algorithm 10. Proceeding east.
The process of proceeding east is shown in Fig. 19. We use the same procedureas a ring exploration algorithm in [20]. At the initial configuration or at a configuration immediatelyafter turning east, robots make the form in Fig. 19(a). From this configuration, the robot withcolor G moves east by rule R
1, and hence the configuration becomes one in Fig. 19(b). From thisconfiguration, the robot with color W on a west node changes its color to G and moves east byrule R
2. In the ASYNC model, after it changes its color to G , other robots may observe theintermediate configuration (Fig. 19(c)). However, there are no rules that the other robots canexecute in the intermediate configuration. Hence, the configuration becomes one in Fig. 19(d).From this configuration, the robot with color G on an east node changes its color to W and moveseast by rule R
3. In the ASYNC model, after it changes its color to W , other robots may observethe intermediate configuration (Fig. 19(e)). However, there are no rules that the other robots canexecute in the intermediate configuration. Hence, the configuration becomes one in Fig. 19(f). After29 W} {W} (a) {G} {G,W} {W} (b) 𝑅2 {G,G} {W} (c) 𝑅2 {G} {G,W} (d) 𝑅3 ・・・ 𝑅1 {G} {W,W} (e) 𝑅3 {G} {W} (f) {W} 𝑅1 Figure 19: Proceeding east in an execution of Algorithm 10 {G} {G,W} {G} {B,W} {G,W}{B} 𝑅4 (a) (d)(b) {G} {W}{B} (c) 𝑅1𝑅4 𝑅5 {W}{G,B} {W}{B} {B} (g)(e) {W}{B,B} (f)
𝑅6 𝑅6 𝑅7 {B} {W,B} (h) 𝑅8 Figure 20: Turning west in an execution of Algorithm 10that, robots proceed east while keeping the form by repeatedly executing those rules.
Turning west.
The process of turning west is shown in Fig. 20. After robots proceed east,they reach the east end of the grid, and the configuration becomes one in Fig. 20(a). From thisconfiguration, the robot with color G on an east node changes its color to B and moves southby rule R
4. In the ASYNC model, after it changes its color to B , other robots may observe theintermediate configuration (Fig. 20(b)). However, there are no rules that the other robots canexecute in the intermediate configuration. Hence, the configuration becomes one in Fig. 20(c).From this configuration, the robot with color G moves east by rule R
1, and hence the configurationbecomes one in Fig. 20(d). From this configuration, the robot with color G moves south by rule R G changes its color to B and moves west by rule R
6. In the ASYNC model, after it changes its colorto B , other robots may observe the intermediate configuration (Fig. 20(f)). However, there are norules that the other robots can execute in the intermediate configuration. Hence, the configurationbecomes one in Fig. 20(g). From this configuration, the robot with color W moves south by rule R
7, and hence the configuration becomes one in Fig. 20(h).30
W,B} {W} {B} {W}{G} (c)(a) {W,B}{G} (d)
𝑅10 𝑅7 {G,B} {W} (b) {B}{G,B} (f) {B}{G,G} (g) {B,B}{G} (e) {B}{G} {G} (h)
𝑅10 𝑅11𝑅11 𝑅12 𝑅12 𝑅13 {G} {G,B} (j) {G} {G,W} (k) {G,B} {G} (i)
𝑅15𝑅14 𝑅3
Figure 21: Turning east in an execution of Algorithm 10
Proceeding west.
The process of proceeding west is similar to that of proceeding east. Robotswith colors W and B for proceeding west move in the same way as robots with colors G and W for proceeding east, respectively. The form in Fig. 20(h) corresponds to one in Fig. 19(b). Rules R R
8, and R R R
2, and R Turning east.
The process of turning east is shown in Fig. 21. After robots proceed west, theyreach the west end of the grid (Fig. 21(a)). From this configuration, the robot with color W on a westnode changes its color to G and moves south by rule R
10. In the ASYNC model, after it changesits color to W , other robots may observe the intermediate configuration (Fig. 21(b)). However,there are no rules that the other robots can execute in the intermediate configuration. Hence, theconfiguration becomes one in Fig. 21(c). From this configuration, the robot with color W moveswest by rule R
7, and hence the configuration becomes one in Fig. 21(d). From this configuration,the robot with color W changes its color to B and moves south by rule R
11. In the ASYNC model,after it changes its color to B , other robots may observe the intermediate configuration (Fig. 21(e)).However, there are no rules that the other robots can execute in the intermediate configuration.Hence, the configuration becomes one in Fig. 21(f). From this configuration, the robot with color B on a south node changes its color to G and moves east by rule R
12. In the ASYNC model,after it changes its color to G , other robots may observe the intermediate configuration (Fig. 21(g)).However, there are no rules that the other robots can execute in the intermediate configuration.Hence, the configuration becomes one in Fig. 21(h). From this configuration, the robot with color B moves south by rule R
13, and hence the configuration becomes one in Fig. 21(i). From thisconfiguration, the robot with color B moves east by rule R
14, and hence the configuration becomesone in Fig. 21(j). From this configuration, the robot with color B changes its color to W by rule R
15, and hence the configuration becomes one in Fig. 21(k). From this configuration, robots can31 lgorithm 11
Asynchronous Terminating Exploration for φ = 1 , (cid:96) = 3 , k = 6 Without CommonChirality Initial configuration { ( v , , { G } ) , ( v , , { W } ) , ( v , , { W } ) , ( v , , { W , B } ) , ( v , , { W } ) } Rules {G} {W}{W,B}G
𝑅1 ∶ G , → {W,B} {W}W {G,W} {W}{W,B}W {W,B}{G}{B} B {G,W}{W}{G} G{G,W}{W}{G} W {W,B} {W,B}{G}W {G} {G}{B}G {W,B}{G}{G,B} B{B} {W}B 𝑅10 ∶ B , ↓𝑅9 ∶ G , ↓𝑅8 ∶ W , ↓𝑅6 ∶ B , → 𝑅4 ∶ W , →𝑅3 ∶ G , →𝑅2 ∶ B , → 𝑅5 ∶ W , →𝑅7 ∶ B , ↓ {G}{W}G 𝑅11 ∶ G , ↓ {G,W}{B}{G,B} W 𝑅12 ∶ W , ↓ {G,B} {G}{W}B 𝑅13 ∶ W , 𝐼𝑑𝑙𝑒 proceed east again since their form is the same as one in Fig. 19(d). End of exploration.
In case that m is odd, robots visit the south end nodes while proceedingeast. Eventually, the configuration becomes { ( v m − ,n − , { G } ) , ( v m − ,n − , { G , W } ) } . At this con-figuration, no robots are enabled. In case that m is even, robots visit the south end nodes whileproceeding east. Eventually, the configuration becomes { ( v m − , , { W , B } ) , ( v m − , , { W } ) } . At thisconfiguration, no robots are enabled. φ = 1 , (cid:96) = 3 , no common chirality, and k = 6We give a terminating exploration algorithm for m × n grids ( m ≥ , n ≥
3) in case of φ = 1, (cid:96) = 3,no common chirality, and k = 6. A set of colors is Col = { G , W , B } . The algorithm is given inAlgorithm 11. Proceeding east.
The process of proceeding east is shown in Fig. 22 and Fig. 23. At theinitial configuration or at a configuration immediately after turning east, robots make the formin Fig. 22(a). From this configuration, the robot with color G moves east by rule R
1, and hencethe configuration becomes one in Fig. 22(b). From this configuration, the robot with color W W} {W}{W} (a) {G}{W,B} {G,W} {W}{W} (b) {W,B} 𝑅1 {G,W} {W}{W} (c) {B,B} {G,W} {W}{W,B} (d) {B}{G,G} {W}{W,B} (e) {B} 𝑅3 {G} {G,W}{W,B} (f) {B}{G} {G,W}{W,W} (g) {B} {G} {G,W}{W} {W} (h) {B} 𝑅5𝑅2𝑅2 𝑅3𝑅4𝑅4 𝑅6
Figure 22: Proceeding east in executions of Algorithm 11 (I)33
G} {G,W}{W} {W} (h) {B} {G} {G,W}{W,B} {W} (i) {G} {W,W}{W} {W} (j) {B} {G} {W,W}{W,B} {W} (k) {G} {W}{W} {W} (l) {B} {W} {G} {W}{W,B} {W} (m) {W}
𝑅5𝑅6 𝑅5𝑅5𝑅6𝑅6 𝑅5𝑅1
Figure 23: Proceeding east in executions of Algorithm 11 (II)34n a west node changes its color to B and moves east by rule R
2. In the ASYNC model, afterit changes its color to B , other robots may observe the intermediate configuration (Fig. 22(c)).However, there are no rules that the other robots can execute in the intermediate configuration.Hence, the configuration becomes one in Fig. 22(d). From this configuration, the robot with color W occupying the same node as the robot with color G changes its color to G and moves east byrule R
3. In the ASYNC model, after it changes its color to G , other robots may observe theintermediate configuration (Fig. 22(e)). However, there are no rules that the other robots canexecute in the intermediate configuration. Hence, the configuration becomes one in Fig. 22(f).From this configuration, the robot with color B occupying the same node as the robot with color W changes its color to W and moves east by rule R
4. In the ASYNC model, after it changes its colorto W , other robots may observe the intermediate configuration (Fig. 22(g)). However, there are norules that the other robots can execute in the intermediate configuration. Hence, the configurationbecomes one in Fig. 22(h).Fig. 23(h) denotes the same configuration as one in Fig. 22(h). We show that the configurationeventually becomes one in Fig. 23(m) regardless of the scheduler. At the configuration in Fig. 23(h),let r be the robot with color W on a northeast node and let r be the robot with color B . Then, r can execute rule R
5, and r can execute rule R
6. If r finishes R r finishes the computephase of R
5, the configuration becomes one in Fig. 23(i). If r finishes the compute phase of R r finishes R
6, the configuration becomes one in Fig. 23(j). If r finishes the compute phaseof R r finishes R R
5, and hence theconfiguration eventually becomes one in Fig. 23(m). At the configuration in Fig. 23(j), robots cannotexecute rules except R R
6. From this configuration, if r finishes R r finishes R
5, theconfiguration becomes one in Fig. 23(k). If r finishes R r finishes R
6, the configurationbecomes one in Fig. 23(l). If r finishes R r finishes R R
6, and hence the configuration eventually becomes one in Fig. 23(m). From the above discussion,the configuration eventually becomes one in Fig. 23(m) in any case. In this configuration, the formof robots is the same as in Fig. 22(a). Hence, robots proceed east while keeping their form byrepeatedly executing those rules.
Turning west.
The process of turning west is shown in Fig. 24 and Fig. 25. After robots proceedeast, they reach the east end of the grid, and the configuration becomes one in Fig. 24(a). At thisconfiguration, let r be the robot with color B , and let r be the robot with color G on a northeastnode. Then, r can execute rule R
6, and r can execute rule R
7. If r finishes the compute phaseof R r finishes R
6, the configuration becomes one in Fig. 24(b). If r finishes R r finishes the compute phase of R
7, the configuration becomes one in Fig. 24(d). If r finishes R r finishes the compute phase of R R
7, and hence the configuration eventually becomes one in Fig. 24(f). At the configuration inFig. 24(b), robots cannot execute rules except R R
7. From this configuration, if r finishes R r finishes R
6, the configuration becomes one in Fig. 24(c). If r finishes R r finishes R
7, the configuration becomes one in Fig. 24(e). If r finishes R r finishes R R
6, and hence the configuration eventually becomes one in Fig. 24(f).Fig. 25(f) denotes the same configuration as one in Fig. 24(f). From this configuration, the robotwith color W on a southwest node moves south by rule R
8, and hence the configuration becomes35 a) (e)(b) {G}{B} {W} {G}{B} {W}{G}{W,B} 𝑅7 {G,W}{W} {G,B}{W}{G,B}{W} (d) {G}{W,B} {G,W}{W} 𝑅6 𝑅7 𝑅6 (c) {G}{B} {W} {G}{W,B} (f) {G}{W,B} {G}{W,B}
𝑅8𝑅7 𝑅6𝑅7
Figure 24: Turning west in an execution of Algorithm 11 (I) (f) (h)(g) {G}{W,B} {G}{B}{W} {G,B}{W}{G}{W,B} {G}{W,B} {G}{W,B} (i) (k)(j) {G,B}{W} {G,B}{W} {G,B}{W}{G}{W}{B} {G,W}{B} {G}{W,B} (l) (m) {G,W}{W} {W,W}{W}{G}{W,B} {G}{W,B} (n) {W} {W}{W} {G}{W,B}
𝑅1𝑅9 𝑅10 𝑅13𝑅8 𝑅11 𝑅12𝑅5 𝑅5
Figure 25: Turning west in an execution of Algorithm 11 (II)36ne in Fig. 25(g). From this configuration, the robot with color G on a northwest node moves southby rule R
9, and hence the configuration becomes one in Fig. 25(h). From this configuration, therobot with color B on an east node moves south by rule R
10, and hence the configuration becomesone in Fig. 25(i). From this configuration, the robot with color G on an east node moves south byrule R
11, and hence the configuration becomes one in Fig. 25(j). From this configuration, the robotwith color W on an east node moves south by rule R
12, and hence the configuration becomes onein Fig. 25(k). From this configuration, the robot with color B on a west node changes its color to W by rule R
13, and hence the configuration becomes one in Fig. 25(l). From this configuration,the robot with color G on a northwest node changes its color to W and moves west by rule R W , other robots may observe the intermediateconfiguration (Fig. 25(m)). However, there are no rules that the other robots can execute in theintermediate configuration. Hence, the configuration becomes one in Fig. 25(n). Proceeding west and turning east.
The form of robots in Fig. 25(n) is a mirror image of theone that robots make to proceed east. Hence, robots proceed west and turn east with the samerules as proceeding east and turning west, respectively.
End of exploration.
In case that m is odd, robots visit the southend nodes while proceeding west. Eventually, the configuration becomes { ( v m − , , { G } ) , ( v m − , , { G } ) , ( v m − , , { W , B } ) , ( v m − , , { W , B } ) } . At this configuration, norobots are enabled. In case that m is even, robots terminate the algorithm similarly to the oddcase. In this paper, we have investigated terminating exploration algorithms for myopic robots in finitegrids. First, we have proved that, in the SSYNC and ASYNC models, three myopic robots arenecessary to achieve the terminating exploration of a grid if φ = 1 holds. Second, we have proposedfourteen algorithms to achieve the terminating exploration of a grid in various assumptions ofsynchrony, visible distance, the number of colors, and a chirality. To the best of our knowledge,they are the first algorithms that achieve the terminating exploration of a grid by myopic robotswith at most three colors and/or with no common chirality. In addition, six proposed algorithmsare optimal in terms of the number of robots.For the future work, it is interesting to close the gap between the lower and upper bounds ofthe number of required robots. It is also interesting to consider other tasks and topologies withmyopic luminous robots. References [1] L. Blin, A. Milani, M. Potop-Butucaru, and S. Tixeuil. Exclusive perpetual ring explorationwithout chirality. In , pages 312–327,2010.[2] F. Bonnet, A. Milani, M. Potop-Butucaru, and S. Tixeuil. Asynchronous exclusive perpetualgrid exploration without sense of direction. In , pages 251–265, 2011. 373] Q. Bramas, S. Devismes, and P. Lafourcade. Infinite grid exploration by disoriented robots.In , pages 129–145, 2020.[4] Q. Bramas, P. Lafourcade, and S. Devismes. Finding water on poleless using melomaniacmyopic chameleon robots. In , volume157, pages 6:1–6:19, 2020.[5] Q. Bramas, P. Lafourcade, and S. Devismes. Optimal exclusive perpetual grid explorationby luminous myopic opaque robots with common chirality. In
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