On the directional movement of a collective of automata without a compass on a one-dimensional integer lattice

A collective of finite automata has to preserve unidirectional movement on one-dimensional integer lattice whose elements (vertices) are unlabelled. The automata does not distinguish between equally labelled vertices by their coordinates of direction (that means each automaton has no compass). We considered collectives consisting of an automaton and some pebbles, i.e. automata of the simplest form, whose positions are completely determined by automaton. We prove that a collective of automaton and a maximum of 2 pebbles cannot maintain movement direction on the one-dimensional integer lattice, but collective of automaton and 3 pebbles can.