A note on three-dimensional alternating Turing machines with space smaller than log m

Abstract This paper deals with the accepting powers of five-way three-dimensional alternating machines and six-way three-dimensional alternating machines whose input tapes are restricted to cubic ones. We first show that for space smaller than log m , five-way three-dimensional alternating Turing machines are less powerful than six-way three-dimensional alternating Turing machines. We then show that the set of all the three-dimensional connected tapes can be accepted by a six-way three-dimensional alternating finite automaton, but not accepted by any o (log m ) space-bounded five-way three-dimensional alternating Turing machine. Finally, we show that there exists a language accepted by a five-way three-dimensional alternating finite automaton, but not accepted by any o (log m ) space-bounded six-way three-dimensional nondeterministic Turing machine.