Decision Problems for Restricted Variants of Two-Dimensional Automata

  • Taylor J. SmithEmail author
  • Kai SalomaaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11601)


A two-dimensional finite automaton has a read-only input head that moves in four directions on a finite array of cells labelled by symbols of the input alphabet. A three-way two-dimensional automaton is prohibited from making upward moves, while a two-way two-dimensional automaton can only move downward and rightward.

We show that the language emptiness problem for unary three-way nondeterministic two-dimensional automata is Open image in new window -complete, and is in Open image in new window for general-alphabet two-way nondeterministic two-dimensional automata. We show that the language equivalence problem for two-way deterministic two-dimensional automata is decidable. This is the first known positive decidability result for the equivalence problem on two-dimensional automata over a general alphabet. We show that there exists a unary three-way deterministic two-dimensional automaton with a nonregular column projection, and we show that the row projection of a unary three-way nondeterministic two-dimensional automaton is always regular.


Decision problem Language emptiness Language equivalence Three-way automata Two-dimensional automata Two-way automata 


  1. 1.
    Anselmo, M., Giammarresi, D., Madonia, M.: New operations and regular expressions for two-dimensional languages over one-letter alphabet. Theor. Comput. Sci. 340(2), 408–431 (2005). Scholar
  2. 2.
    Blum, M., Hewitt, C.: Automata on a 2-dimensional tape. In: Miller, R.E. (ed.) SWAT 1967, pp. 155–160 (1967).
  3. 3.
    Dong, J., Jin, W.: Comparison of two-way two-dimensional finite automata and three-way two-dimensional finite automata. In: Yang, X. (ed.) CSSS 2012, pp. 1904–1906 (2012).
  4. 4.
    Galil, Z.: Hierarchies of complete problems. Acta Inf. 6(1), 77–88 (1976). Scholar
  5. 5.
    Greibach, S.: Checking automata and one-way stack languages. J. Comput. Syst. Sci. 3(2), 196–217 (1969). Scholar
  6. 6.
    Hunt III, H.B.: On the time and tape complexity of languages I. In: Aho, A.V. (ed.) STOC 1973, pp. 10–19 (1973).
  7. 7.
    Ibarra, O.H., Dang, Z., Li, Q.: Accepting runs in a two-way finite automation. Inf. Comput. 260, 1–8 (2018). Scholar
  8. 8.
    Inoue, K., Takanami, I.: A note on decision problems for three-way two-dimensional finite automata. Inf. Process. Lett. 10(4–5), 245–248 (1980). Scholar
  9. 9.
    Inoue, K., Takanami, I.: A survey of two-dimensional automata theory. Inf. Sci. 55(1–3), 99–121 (1991). Scholar
  10. 10.
    Kari, J., Moore, C.: Rectangles and squares recognized by two-dimensional automata. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds.) Theory Is Forever. LNCS, vol. 3113, pp. 134–144. Springer, Heidelberg (2004). Scholar
  11. 11.
    Kari, J., Salo, V.: A survey on picture-walking automata. In: Kuich, W., Rahonis, G. (eds.) Algebraic Foundations in Computer Science. LNCS, vol. 7020, pp. 183–213. Springer, Heidelberg (2011). Scholar
  12. 12.
    Kinber, E.B.: Three-way automata on rectangular tapes over a one-letter alphabet. Inform. Sci. 35, 61–77 (1985). Scholar
  13. 13.
    Petersen, H.: Some results concerning two-dimensional Turing machines and finite automata. In: Reichel, H. (ed.) FCT 1995. LNCS, vol. 965, pp. 374–382. Springer, Heidelberg (1995). Scholar
  14. 14.
    Pighizzini, G.: Two-way finite automata: old and recent results. Fund. Inf. 126(2–3), 225–246 (2013). Scholar
  15. 15.
    Rosenfeld, A.: Picture Languages: Formal Models for Picture Recognition. Computer Science and Applied Mathematics. Academic Press, New York (1979)zbMATHGoogle Scholar
  16. 16.
    Smith, T.J.: Two-dimensional automata. Technical report 2019–637, Queen’s University, Kingston (2019)Google Scholar
  17. 17.
    Taniguchi, K., Kasami, T.: Some decision problems for two-dimensional nonwriting automata. Trans. Inst. Electron. Comm. Engrs. Jpn. 54–C(7), 578–585 (1971)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of ComputingQueen’s UniversityKingstonCanada

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