Experiments on Recognition of Infinite Grid Graph Labelling

Abstract

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.