Non-closure property of space-bounded two-dimensional alternating turing machines

  • Authors:

  • Tokio Okazaki;Atsuyuki Inoue;Katsushi Inoue;Akira Ito;Yue Wang

  • Affiliations:

  • Department of Electronics and Computer Science, Faculty of Science and Engineering, Science University of Tokyo in Yamaguchi, Onoda, Yamaguchi 756-0884, Japan;School of Information Science, Japan Advanced Institute of Science and Technology, Tatsukuchi-chou, Nobi-gun, Ishikawa 923-1292, Japan;Department of Computer Science and Systems Engineering, Faculty of Engineering, Yamaguchi University, Ube, Yamaguchi 755-8611, Japan;Department of Computer Science and Systems Engineering, Faculty of Engineering, Yamaguchi University, Ube, Yamaguchi 755-8611, Japan;Department of Computer Science and Systems Engineering, Faculty of Engineering, Yamaguchi University, Ube, Yamaguchi 755-8611, Japan

  • Venue:
  • Information Sciences—Applications: An International Journal
  • Year:
  • 2002

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Abstract

This paper investigates non-closure properties of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines and three-way two-dimensional alternating Turing machines. Let 2-ATM(L(m, n)) (resp., TR2-ATM(L(m, n))) be the class of sets accepted by L(m, n) space-bounded two-dimensional alternating Turing machines (resp., L(m, n) space-bounded three-way two-dimensional alternating Turing machines), where L(m, n): N2 → N ∪ {0} (N denotes the set of all the positive integers) is a function with two variables m ( = the number of rows of input tapes) and n (= the number of columns of input tapes). We show that (i) for any function g(n) = o(logn) (resp., g(n) = o(logn/log log n)) and any monotonic non-decreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, 2-ATM(L(m,n)) and TR2-ATM(L(m,n)) are not closed under column catenation, column +, and projection, and (ii) for any function f(m) = o(logm) (resp., f(m) = o(logm/log log m)) and any monotonic non-decreasing function g(n)which can be constructed by some one-dimensional deterministic Turing machine, 2-ATM(L(m, n)) and TR2-ATM(L(m, n)) are not closed under row catenation, row +, and projection, where L(m, n) = f(m) + g(n) (resp., L(m, n) = f(m) × g(n)).