Necessary spaces for seven-way four-dimensional Turing machines to simulate four-dimensional one-marker automata

We think that recently, due to the advances in many application areas such as motion image processing, computer animation


Introduction
An improvement of picture recognizability of the finite automaton is the reason why the marker automaton was introduced. That is, a two-dimensional one-marker automaton can recognize connected pictures. This automaton has been widely investigated in the two-or three-dimensional case [2]. A multi-marker automaton is a finite automaton which keeps marks as 'pebbles' in the finite control, and cannot rewrite any input symbols but can make marks on its input with the restriction that only a bounded number of these marks can exist at any given time [1]. As is well known among the researchers of automata theory, one-dimensional one-marker automata are equivalent to ordinary finite state automata. In other words, there is no need of working space usage for oneway Turing machines to simulate one-marker automata, as well as finite state automata.
In the two-dimensional case, the following facts are known : the necessary and sufficient space for three-way two-dimensional deterministic Turing machines TR2-DTM's to simulate two-dimensional deterministic (nondeterministic) finite automata 2-DFA's (2-NFA's) is mlogm (m 2 ) and the corresponding space for three-way two-dimensional nondeterministic Turing machines TR2-NTM's is m (m), whereas the necessary and sufficient space for three-way two-dimensional deterministic Turing machines TR2-DTM's to simulate two-dimensional deterministic (nondeterministic) onemarker automata 2-DMA1's (2-NMA1's) is 2 mlogm (2 2 ) and the corresponding space for TR2-NTM's is mlogm (m 2 ), where m is the number of columns of twodimensional rectangular input tapes.
In the three-dimensional case, the following facts are known : the necessary and sufficient space for five-way three-dimensional deterministic Turing machines FV3-DTM's to simulate three-dimensional deterministic (nondeterministic) finite automata 3-DFA's (3-NFA's) is m 2 logm (m 3 ) and the corresponding space for five-way three-dimensional nondeterministic Turing machines FV3-NTM's is m 2 (m 2 ), whereas the necessary and sufficient space for five-way three-dimensional deterministic Turing machines FV3-DTM's to simulate three-dimensional deterministic (nondeterministic) onemarker automata 3-DMA1's (3-NMA1's) is 2 lmloglm (2 2 2 ) and the corresponding space for FV3-NTM's is lmloglm (l 2 m 2 ), where l(m) is the number of rows (columns) on each plane of three-dimensional rectangular input tapes.
In the four-dimensional case, we showed the sufficient spaces for four-dimensional Turing machines to simulate four-dimensional one-marker automata [3]. In this paper, we continue the investigations, and deal with the necessary spaces for four-dimensional Turing machines to simulate four-dimensional one-marker automata.

Preliminaries
An ordinary finite automaton cannot rewrite any symbols on input tape, but a marker automaton can make a mark on the input tape. We can think of the mark as a 'pebble' that M puts down in a specified position. If M has already put down the mark, and wants to put it down elsewhere, M must first go to the position of the mark and pick it up. Formally, we define it as follows.

Necessary spaces
In this section, we investigate the necessary spaces (i.e., lower bounds) for seven-way Turing machines to simulate one-marker automata.

Conclusion
In this paper, we showed the necessary spaces for fourdimensional Turing machines to simulate fourdimensional one-marker automata. It will be interesting to investigate how much space is necessary and sufficient for seven-way four-dimensional deterministic or nondeterministic Turing machines to simulate fourdimensional 'alternating' one-marker automata.