Some results concerning two-dimensional turing machines and finite automata
We show that emptiness is decidable for three-way two-dimensional nondeterministic finite automata as well as the universe problem for the corresponding class of deterministic automata. Emptiness is undecidable for three-way (and even two-way) two-dimensional alternating finite automata over a single-letter alphabet. Consequently inclusion, equivalence, and disjointness for these automata are undecidable properties.
We establish a hierarchy result for space bounded two-dimensional alternating Turing machines above logarithm where the languages witnessing the hierarchy are over single-letter alphabets. Below logarithm we prove that an infinite hierarchy of languages over larger alphabets exists.
The results rely mainly on a translational technique from one to two dimensions. Using this technique we can also show some connections between open problems of two-dimensional automata theory and one-dimensional complexity theory.
KeywordsTuring Machine Finite Automaton Input Tape Logarithmic Space Deterministic Automaton
Unable to display preview. Download preview PDF.
- J.Balcazár, J.Diáz, J.Gabárro: Structural Complexity II, Springer (1990).Google Scholar
- O.H.Ibarra, S.M.Kim, L.E.Rosier: Some characterizations of multihead finite automata, Inform. and Control 67 (1985) 114–125.Google Scholar
- K.Inoue, I.Takanami: A note on decision problems for three-way two-dimensional finite automata, IPL 10 (1980) 245–248.Google Scholar
- K.Inoue, I.Takanami: A survey of two-dimensional automata theory, Proc. Machines, Languages, and Complexity (J.Dassow, J.Kelemen eds.), Springer LNCS 381 (1989) 72–91.Google Scholar
- K.Inoue, I.Takanami: A survey of two-dimensional automata theory, Inf. Sci. 55 (1991) 99–121.Google Scholar
- T.Jiang, O.H.Ibarra, H.Wang: Some results concerning 2-d on-line tesselation acceptors and 2-d alternating finite automata, Theor. Comput. Sci. 125 (1994) 243–257.Google Scholar
- T.Jiang, O.H.Ibarra, H.Wang, Q.Zheng: A hierarchy result for 2-dimensional TM's operating in small space, Inf. Sci. 64 (1992) 49–56.Google Scholar
- K.N.King: Alternating multihead finite automata, Theor. Comput. Sci. 61 (1988) 149–174.Google Scholar
- B.Monien: The LBA-problem and the deterministic tape complexity of two-way one-counter languages over a one-letter alphabet, Acta Informatica 8 (1977) 371–382.Google Scholar
- H.Petersen: On space functions fully constructed by two-dimensional Turing machines, IPL 54 (1995) 9–10.Google Scholar
- H.Petersen: Alternation in simple devices, Proc. ICALP95.Google Scholar
- A.Rosenfeld: Picture Languages, Academic Press, New York (1979).Google Scholar
- J.C.Shepherdson: The reduction of two-way automata to one-way automata, IBM Journal of Res. and Dev. April (1959) 198–200.Google Scholar
- A.Szepietowski: Some remarks on two-dimensional finite automata, Inf. Sci. 63 (1992) 183–189.Google Scholar
- A.Szepietowski: Turing machines with sublogarithmic space, Springer LNCS 843 (1994).Google Scholar