Collectives of automata on infinite grid graph with deterministic vertex labeling

Abstract

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.