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Independent systems of automata in labyrinths
Abstract
We analyse the state of the art of a rather new field of automata theory – the study of behaviour of automata in labyrinths; more than a hundred publications devoted to this topic have been published. We consider key notions, problems, achievements, methods of problem solutions, and open problems in an important direction of this study, the behaviour of independent systems of automata in labyrinths. In a series of cases, we give base assertions in a more-strong form and give a more general presentation than the authors of the corresponding papers do. New results are also contained in this survey.
... Las celdas en las que un agente comienza y termina su recorrido deben ser libres, pero además deben estar conectadas por algún camino, i.e. debe haber una o más secuencias de celdas libres adyacentes. En particular, la topología del entorno debe ser tal que un entorno pueda tener dos o más caminos entre el comienzo y la meta que no sean redundantes; si éste no fuera el caso, existen estrategias que siempre tendrán éxito y el problema sería trivialmente resoluble (Kilibarda et al, 2003). El caso extremo donde un camino no redundante exista es el caso en el que existen celdas bloqueadas que no estén conectadas, i.e. en contacto, entre sí (como se ejemplifica en Figura 1). ...
... El tamaño del entorno se define como la cantidad de filas o columnas (la misma si son entornos cuadrados). La longitud del camino mínimo se define por las ubicaciones de las celdas de inicio y finalización, y dado que las celdas de un entorno pueden considerarse como los nodos de un grafo, la longitud de este camino puede relacionarse con el diámetro del grafo (Kilibarda et al., 2003). Por lo tanto, la probabilidad de éxito se tendrá en cuenta con respecto al tamaño de los entornos, y la mínima proporción del diámetro promedio del entorno (respecto a su tamaño) que podría necesitar un agente para alcanzar la meta. ...
Resumen Evaluar la navegación de agentes reactivos en entornos discretos rectangulares tiene una complejidad heterogénea, y el éxito de esta tarea depende tanto del diseño del agente como de la topología del entorno. Se explora el problema modelando agentes con máquinas de estados finitos con entradas y acciones disponibles mínimas. La navegación se parametriza respecto del tamaño del entorno y de la distancia entre el comienzo y la meta a alcanzar. Se caracterizó un generador aleatorio de entornos que fueron usados para testear agentes típicos. Como resultado, los modelos más exitosos siguen un comportamiento esperable, el de rodear obstáculos indefinidamente. PALABRAS CLAVE: SIMULACIONES-MÁQUINAS DE ESTADOS FINITOS-ENTORNOS DISCRETOS
Abstract The evaluation of the navigation of refex agents in rectangular discrete environments has a heterogeneous complexity, and the task success depends both on the agent design and the environment topology. The problem is explored by modeling agents as finite state machines with minimal available inputs and actions. The navigation is parameterized with respect to the environment size and the distance to the goal that has to be reached. A random environment generator was characterized, for testing typical agents. As a result, the most successful models follow expected behaviour, that is, they go around obstacles indefinitely.
... The cell where the agent starts its travel and the goal cell need to be free but also connected by some path. The environments topology should be such that an environment might have two or more non-redundant paths between two cells; if this were not the case, there are well-known strategies that will always succeed and the problem would be trivial [7]. The extreme case where a non-redundant path exists is the case where there are some unconnected blocked cells, as in Fig. 1. ...
... The environment size is defined by the quantity of rows and columns. The minimal path length is defined by the start and end cells, and an environment is taken as a graph whose nodes are the cells, this length is related to the diameter of the graph [7]. Hence, the probability will be taken into account with respect to the size of the environments, and the minimal proportion of the average environment diameter (regarding its size) that could need the FSM to reach the goal. ...
The success of a reflex agent facing the problem of traveling in a rectangular discrete environment from one place to another has an heterogeneous complexity. Their success not only depends on the design of the agent but also in the environment topology. Here this problem is explored for agents designed using Finite State Machines (FSM) as models, having a minimal set of perceptual inputs and a minimal set of available actions. Their success is considered regarding the size and the proportion of the environment diameter that corresponds to the minimal path length between the start and the end cells. A large pool of randomly generated environments was created to test the FSMs success and also to analyze the topological features of the environment. The FSMs designed to follow a wall on their side resulted very successful, overwhelming the other FSMs success rates. Moreover a linear relation between the environment average diameter and the size was found and a non linear relation between these and the success rate of the FSMs was also observed.
... Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Studies on the behavior of automata in finite and infinite labyrinths, which are embedded directed graphs of a specific form, have emerged and are intensively developing under this model [1,2,3]. Research in this area has various applications, such as image analysis [4,5] and mobile robotics navigation [6]. ...
We study the following problem: Can a collective of finite automata maintain directed movement on a two-dimensional integer lattice of width 2, where the elements (vertices) are anonymous? The automata do not distinguish between vertices based on their coordinates of direction (that means each automaton has no compass). We considered collectives consisting of an automaton and some pebbles, which are automata of the simplest form, whose positions are entirely determined by the automaton. We demonstrate that a collective of one automaton and a maximum of three pebbles cannot maintain a direction of movement on the lattice. However, a collective of one automaton and four pebbles can do so.
... The paper is self-contained, but one can find the basic notions and the basic results in the theory of automata in labyrinths in [3,4], where there is also a more or less complete list of literature on this theory. ...
It is shown that every automaton acceptable for rectangular labyrinths can be reduced to an automaton that behaves according to either the left-hand rule or the right-hand rule, or does not move at all, in every plane rectangular labyrinth without leaves. This enables us to approach certain fundamental problems of the theory of automata in labyrinths in a quite different way.
The author considers the problem of preserving the direction of movement by a collective of finite automata on a two-dimensional integer lattice of width two whose elements (vertices) are anonymous. The automata do not distinguish between equally labeled vertices by their direction coordinates (i.e., each automaton has no compass). The collectives consisting of a single automaton and several pebbles whose layout is completely determined by the automaton are considered. It is proved that a collective of an automaton and a maximum of three pebbles cannot maintain movement direction on the lattice, but a collective of an automaton and four pebbles can.
In recent years, topics related to automaton analysis of geometric environment have attracted widespread attention. The interaction of an automaton and an environment is often represented as a process of an automaton walks along a graph (labyrinth) of environment. Treating environment as vertex-labelled graph was suggested as one approach to addressing the problem of automaton analysis of environment properties. Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. This paper considers a four-way infinite lattice graph as a model of an environment for a graph-walking automaton. All vertices of this graph are labelled with labels from a known set but concrete labelling function is not a priori known. The automaton looking over neighbourhood of the current vertex and may travel to some neighbouring vertex selected by its label. The objective of the automaton is to determine two pairs of opposite directions on the aforementioned graph embedded onto integer lattice, i.e. recognizing the graph labelling. In previous work we have proposed a labelling thanks to which a finite automaton can walk on graph in any arbitrary direction. We have called this labelling deterministic. We have proved that minimal deterministic vertex labelling of infinite lattice graph uses labels of five different types. The paper demonstrates that there exists 240 different minimal deterministic labellings of the infinite lattice graph with a fixed set of labels. We prove that a single automaton cannot recognize minimal deterministic labelling of the infinite lattice graph, but automaton with one pebble can. The novelty associated with this paper is a new type of experiment with vertex labelled graphs designed to recognition their labelling. The method of constructing and performing recognition experiment for labelled infinite grid graph is proposed.
Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model a broad area of studies of the behaviour of automata in labyrinths arose and intensively developing last decades (a labyrinth is an embedded directed graph of special form). Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper deals with the problem of constructing square grid graph vertex labelling thanks to which a finite automaton without a compass can walk on graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may travel to some neighbouring vertex selected by its label. In this paper, we propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. In previous works we have proved that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. In this paper we prove that a collective of one automaton and three pebbles can construct this labelling on initially unlabelled infinite square grid graph. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective.
Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model broad area of studies of the behaviour of automata in finite and infinite labyrinths (a labyrinth is an embedded directed graph of special form) arose and intensively developing. Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper examines vertex labellings of infinite square grid graph thanks to these labellings a finite automaton without a compass can walk along graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may move to some neighbouring vertex selected by its label. We propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. It is shown that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. Minimal deterministic traversable labelling of subgraphs of infinite square grid graph whose degrees are less than four are developed. The key problem for automata and labyrinths is the problem of constructing a finite automaton that traverse a given class of labyrinths. We say that automaton traverse infinite graph if it visits any randomly selected vertex of this graph in a finite time. It is proved that a collective of one automaton and three pebbles can traverse infinite square grid graph with deterministic labelling and any collective with fewer pebbles cannot. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective. The results regarding to exploration of an infinite deterministic square grid graph coincide with the results of A.V. Andzhan (Andzans) regarding to traversal of an infinite mosaic labyrinth without holes. It follows from above that infinite grid graph after constructing a minimal traversable deterministic labelling on it and fixing two pairs of opposite directions on it becomes an analogue of an infinite mosaic labyrinth without holes.
This paper considers the problem of exploration of finite undirected graphs by a collective of agents. Two agents-researchers simultaneously traverse a graph, read and change labels of graph elements, and send necessary information to the agent-experimenter constructing a representation of the graph being explored. An exploration algorithm is proposed with a linear (with respect to the number of vertices) time complexity and a quadratic space complexity. An optimization procedure is developed that partitions a graph with a view to exploring its parts by different agents. Each of the agents traversing a graph needs two different colors (three colors are used in the aggregate) for exploring a graph. The algorithm is based on the depth-first traversal method.
In this paper we study mosaic labyrinths with the help of words generated by them in the alphabet of labels attached to arcs
and vertices of a labyrinth. We consider the problem of the characterization of words generated by a labyrinth. We propose
a constructive recognition criterion, it defines whether a word is generated by a labyrinth or not. We establish conditions
under which a word can be generated by a unique labyrinth, by a finite number of labyrinths, or by infinitely many labyrinths.
We study the process of pursuing several independent of one another automata (preys) by a system of automata (predators) on a plane. We show that there exists a finite collective of predators which catches any finite independent system of preys such that the prey velocity is less than the predator velocity and their field of vision is not greater that the field of vision of predators, under any initial disposition of the preys provided that the predators start from one point.
The behaviour of automata in labyrinths is a rather new field of automata theory, but more than one hundred papers devoted to this topic have been published. In this paper, we consider the key notions, problems, achievements, methods to solve problems, and open problems related to an important direction of this field, the behaviour of collectives of automata in labyrinths. In a series of cases, we give base assertions in a more strong form and give a more general presentation than the authors of the corresponding papers do. New results are also contained in this survey.
In this paper we consider rectangular and s-labyrinths. We investigate problems similar to classical ones in the automata theory, namely, the distinguishability of vertices
and the labyrinths equivalence. We prove that for the considered class of labyrinths these problems are solvable and estimate
the distinguishing word length. For rectangular labyrinths we prove that the isomorphism and equivalence relations coincide.
This article reviews over 80 works on the behaviour of automata systems in labyrinths which have appeared in the last two decades. Main concepts, problems, results, methods for solving problems and problems to be solved are selected. In a number of cases the main statements are given in a stronger form compared to their original formulations. This article also contains some new results on the problem of searching labyrinths by automata.
In 1975 L. Budach proved in a groundbreaking paper ([7]) that there is no finite automaton which is able to search (to master) all finite (cofinite) plane labyrinths. On the other side we have a result of M. Blum, D. Kozen (1978, [4]) saying that the search can be implemented with just two pebbles. The aim of our paper is to show that one pebble does not suffice, answering a question of Blum,Kozen. Furthermore we present a new construction for universal traps (see H.Antelmann, L.Budach, H.-A.Rollik 1979, [2]).
In the following will be considered the interaction of a finite automaton with a finite or infinite environment. The automaton will not be considered as a passive machine which obtains given information of its environment and it can only accept or process them but we assume the automaton to have the possibility to influence the information originated from its environment. This active behavior may consist in change of place or in modifications of the environment. As one sees in algorithm — theory the possibilities of this automaton grow dependent on the environment it is acting in. The interaction model “automaton — environment” can be applied in problem solving, artificial intelligence (design of robots) and in the analysis of real computing machinary. In the latter case processing part and memory constitute the environment, whereas the control part represents the finite automaton 0l. This interpretation already standart in computer science was thoroughly investigated by V.M. Glushkov, A.A. Letichevskij and others(see f. i. [3]) and applied to structural problems of computers. C.P. Schnorr [5] used a similar interaction model to develop and to treat the theory of recursive functions. The use of partially defined actions in an environment makes it possible to integrate labyrinths into this concept. As will be seen later these labyrinths are universal in that sense, that all environments can be encoded by certain labyrints.
The embedding problem for a class of graphs called rectilinear graphs is discussed. These graphs have applications in many VLSI Layout Problems. An interesting topological characterization of these graphs lead to efficient algorithms for recognizing and embedding rectilinear graphs which are embeddable on the plane.
Berlin, Akad. d. Wiss. d. DDR, Diss., 1983 (Nicht f.d. Austausch).
First Page of the Article
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