CONSTANT LEAF-SIZE HIERARCHY OF TWO-DIMENSIONAL ALTERNATING TURING MACHINES

Author(s):  
AKIRA ITO ◽  
KATSUSHI INOUE ◽  
ITSUO TAKANAMI ◽  
YASUYOSHI INAGAKI
1989 ◽  
Vol 67 (1) ◽  
pp. 99-110 ◽  
Author(s):  
Katsushi Inoue ◽  
Itsuo Takanami ◽  
Juraj Hromkovič

Author(s):  
AKIRA ITO ◽  
KATSUSHI INOUE ◽  
ITSUO TAKANAMI ◽  
YASUYOSHI INAGAKI

“Leaf-size” (or “branching”) is the minimum number of leaves of some accepting computation trees of alternating devices. For example, one leaf corresponds to nondeterministic computation. In this paper, we investigate the effect of constant leaves of two-dimensional alternating Turing machines, and show the following facts: (1) For any function L(m, n), k leaf- and L(m, n) space-bounded two-dimensional alternating Turing machines which have only universal states are equivalent to the same space bounded deterministic Turing machines for any integer k≥1, where m (n) is the number of rows (columns) of the rectangular input tapes. (2) For square input tapes, k+1 leaf- and o(log m) space-bounded two-dimensional alternating Turing machines are more powerful than k leaf-bounded ones for each k≥1. (3) The necessary and sufficient space for three-way deterministic Turing machines to simulate k leaf-bounded two-dimensional alternating finite automata is nk+1, where we restrict the space function of three-way deterministic Turing machines to depend only on the number of columns of the given input tapes.


Author(s):  
KATSUSHI INOUE ◽  
ITSUO SAKURAMOTO ◽  
MAKOTO SAKAMOTO ◽  
ITSUO TAKANAMI

This paper deals with two topics concerning two-dimensional automata operating in parallel. We first investigate a relationship between the accepting powers of two-dimensional alternating finite automata (2-AFAs) and nondeterministic bottom-up pyramid cellular acceptors (NUPCAs), and show that Ω ( diameter × log diameter ) time is necessary for NUPCAs to simulate 2-AFAs. We then investigate space complexity of two-dimensional alternating Turing machines (2-ATMs) operating in small space, and show that if L (n) is a two-dimensionally space-constructible function such that lim n → ∞ L (n)/ loglog n > 1 and L (n) ≤ log n, and L′ (n) is a function satisfying L′ (n) =o (L(n)), then there exists a set accepted by some strongly L (n) space-bounded two-dimensional deterministic Turing machine, but not accepted by any weakly L′ (n) space-bounded 2-ATM, and thus there exists a rich space hierarchy for weakly S (n) space-bounded 2-ATMs with loglog n ≤ S (n) ≤ log n.


Author(s):  
Serge Miguet ◽  
Annick Montanvert ◽  
P. S. P. Wang

Several nonclosure properties of each class of sets accepted by two-dimensional alternating one-marker automata, alternating one-marker automata with only universal states, nondeterministic one-marker automata, deterministic one-marker automata, alternating finite automata, and alternating finite automata with only universal states are shown. To do this, we first establish the upper bounds of the working space used by "three-way" alternating Turing machines with only universal states to simulate those "four-way" non-storage machines. These bounds provide us a simplified and unified proof method for the whole variants of one-marker and/or alternating finite state machine, without directly analyzing the complex behavior of the individual four-way machine on two-dimensional rectangular input tapes. We also summarize the known closure properties including Boolean closures for all the variants of two-dimensional alternating one-marker automata.


Author(s):  
TOKIO OKAZAKI ◽  
KATSUSHI INOUE ◽  
AKIRA ITO ◽  
YUE WANG

This paper investigates closure property of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines (2-atm's) and space-bounded two-dimensional alternating pushdown automata (2-apda's), and space-bounded two-dimensional alternating counter automata (2-aca's). Let L(m, n): N2 → N (N denotes the set of all positive integers) be 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 f(m) = o( log m) (resp. f(m) = o( log m/ log log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional Turing machine (2-Tm) (resp. two-dimensional pushdown automaton (2-pda)), the class of sets accepted by L(m,n) space-bounded 2-atm's (2-apda's) is not closed under row catenation, row + or projection, and (ii) for any function f(m) = o(m/ log ) (resp. for any function f(m) such that log f(m) = o( log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional counter automaton (2-ca), the class of sets accepted by L(m, n) space-bounded 2-aca's is not closed under row catenation, row + or projection, where L(m, n) = f(m) + g(n) (resp. L(m, n) = f(m) × g(n)).


1988 ◽  
Vol 45 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Akira Ito ◽  
Katsushi Inoue ◽  
Itsuo Takanami

1983 ◽  
Vol 27 (1-2) ◽  
pp. 61-83 ◽  
Author(s):  
Katsushi Inoue ◽  
Itsuo Takanami ◽  
Hiroshi Taniguchi

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