Remarks on the recognizability of topological components by three-dimensional automata

It is conjectured that the three-dimensional pattern processing has its own difficulties not arising in two-dimensional case. One of these difficulties occurs in recognizing topological properties of three-dimensional patterns because the three-dimensional neighborhood is more complicated than two-dimensional case. Generally a property or relationship is topological only if it is preserved when an arbitrary "rubber-sheet" distortion is applied to the pictures. For example, adjacency and connectedness are topological; area, elongatedness, convexity, and straightness are not. In recent years, there have been many interesting papers on digital topological properties. For example, an interlocking component was defined as a new topological property in three-dimensional digital pictures, and it was proved that no one marker automaton can recognize interlocking components in a three-dimensional digital picture. In this paper, we deal with recognizability of topological components by three-dimensional Turing machines, and investigate some properties.