Makoto Sakamoto

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.

Simulation of three-dimensional one-marker automata by five-way Turing machines

This paper investigates the simulation of three-dimensional one-marker automata by five-way Turing machines. We show that the necessary and sufficient space for fiveway three-dimensional deterministic Turing machines to simulate three-dimensional deterministic one-marker automata (three-dimensional nondeterministic one-marker automata) is 2Θ(lm log(lm)) (2Θ(l2m2) and the necessary and sufficient space for five-way three-dimensional nondeterministic Turing machines to simulate three-dimensional deterministic one-marker automata (three-dimensional nondeterministic one-marker automata) is lm log(lm)(l2m2), where l(m) is the number of rows (columns) on each plane of three-dimensional rectangular input tapes.

Three-dimensional alternating Turing machines with only universal states

Abstract We investigate several properties of three-dimensional alternating Turing machines whose input tapes are restricted to cubic ones. We first investigate the relationship between the classes of sets accepted by space-bounded and finitely leaf-size bounded five-way three-dimensional alternating Turing machines and the classes of sets which are finite intersections of sets accepted by space-bounded five-way three-dimensional nondeterministic Turing machines. Then, we investigate the accepting powers of three-dimensional alternating Turing machines with only universal states.