Remarks on Recognizability of Four-Dimensional Topological Components

The study of four-dimensional automata as the computational model of four-dimensional pattern processing has been meaningful. However, it is conjectured that the three-dimensional pattern processing has its our difficulties not arising in two-or three-dimensional case. One of these difficulties occurs in recognizing topological properties of four-dimensional patterns because the four-dimensional neighborhood is more complicated than two-or three-dimensional case. Generally speaking, 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, straightness, etc. 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 multi-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 four-dimensional Turing machines, and investigate some properties.