On the recognition of topological invariants by 4-way finite automata

It is the purpose of this paper to show that 4-way nondeterministic finite automata (4NFAs) cannot recognize simple topological properties of twodimensional patterns, whereas 4NFAs with a finite number of markers (pebbles) available can recognize many such properties. Multidimensional finite automata with markers have been studied by several authors, among them Blum and Hewitt [1968] who were concerned with 4-way deterministic, halting FAs that were presented with two-dimensional rectangular patterns; Shank [1971] who dealt with graph-walking automata computing properties of arbitrary directed graphs with all edges leaving each node ordered; and Milgram and Rosenfeld [1971] whose work involves FAs that can write on the input pattern. One conclusion that can be drawn from this research is that it is very difficult to show just how powerful FAs are for recognizing simple properties of patterns. The patterns we are concerned with are finite connected subsets of the twodimensional square grid with their points labeled black (0) or white (1) and having a boundary (Fig. l(a)). ~r(0), or(l), ~r(0,1) will denote, respectively, the sets of all black, white, and black-or-white points (cells) in pattern ~r, while X will denote the set of all points on the two-dimensional square grid. The boundary of 7r consists of all points not in ~'(0,1) with a neighbor in ~r(0,1) along any one of the four directions or a diagonal. Such points will be labeled B. Every pattern will be assumed to have an initial point where any computation on the pattern will have to begin. A 4-way nondeterministic finite automaton (4NFA) A consists of a 5-tuple