Tokio Okazaki

Space hierarchies of two-dimensional alternating turing machines. pushdown automata and counter automata

This paper investigates the space hierarchies of the language classes for two-dimensional Turing machines (2-TMs), two-dimensional pushdown automata (2-PDAs) and two-dimensional counter automata (2-CAs) with small space. We show that (1) if L(n) is space constructible by a 2-TM, L(n) ≤ log n and L′(n) = o(L(n)), then strong 2-DSPACE(L(n)) – weak 2-ASPACE(L′(n)) ≠ ∅, (2) if L(n) is space constructible by a 2-PDA, L(n) ≤ log n and L′(n) = o(L(n)), then strong 2-DPDA(L(n)) – weak 2-ASPACE(L′(n)) ≠ ∅, and (3) if L(n) is space-constructible by a 2-CA, L(n) ≤ n and L′(n) = o(L(n)), then strong 2-DCA(L(n)) – weak 2-ACA(L′(n)) ≠ ∅, (4) where strong 2-DSPACE(L(n)) (strong 2-DPDA(L(n)), strong 2-DCA(L(n))) denotes the class of sets accepted by strongly L(n) space-bounded deterministic 2-TMs (2-PDAs, 2-CAs), and weak 2-ASPACE(L′(n)) (weak 2-ACA(L′(n))) denotes the class of sets accepted by weakly L′(n) space-bounded alternating 2-TMs (2-CAs). We also investigate the closure property of space-bounded alternating 2-PDAs and 2-CAs under complementation, and show that (1) if L(n) = o(log log n), then the class of sets accepted by L(n) space-bounded alternating 2-PDAs is not closed under complementation, and (2) if L(n) is space-constructible by a 2-CA, L(n) ≤ n and , then the class of sets accepted by L′(n) space-bounded alternating 2-CAs is not closed under complementation.

A note on one-pebble two-dimensional Turing machines

This paper investigates a relationship among the accepting powers of deterministic, nondeterministic, and alternating one-pebble two-dimensional Turing machines, and shows that 1. nondeterminism are more powerful than determinism for o(log n) space-bounded one-pebble two-dimensional Turing machines whose input tapes are restricted to square ones, and 2. alternation are more powerful than nondeterminism for f(m) + g(n) (resp., f(m) × g(n)) space-bounded one-pebble two-dimensional Turing machines, where f : N → N is an arbitrary monotonic nondecreasing function space-constructible by a deterministic one-pebble two-dimensional Turing machine, and g : N → N is an arbitrary function such that g(n) = o(log n) (resp., g(n) = o(log n/log logn)).

CLOSURE PROPERTY OF SPACE-BOUNDED TWO-DIMENSIONAL ALTERNATING TURING MACHINES, PUSHDOWN AUTOMATA, AND COUNTER AUTOMATA

This paper investigates closure property of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines (2-atms) and space-bounded two-dimensional alternating pushdown automata (2-apdas), and space-bounded two-dimensional alternating counter automata (2-acas). 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-atms (2-apdas) 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-acas 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)).

A NOTE ON THREE-WAY TWO-DIMENSIONAL PROBABILISTIC TURING MACHINES

This paper introduces a three-way two-dimensional probabilistic Turing machine (tr2-ptm), and investigates several properties of the machine. The tr2-ptm is a two-dimensional probabilistic Turing machine (2-ptm) whose input head can only move left, right, or down, but not up. Let 2-ptms (resp. tr2-ptms) denote a 2-ptm (resp. tr2-ptm) whose input tape is restricted to square ones, and let 2-PTMs(S(n)) (resp. TR2-PTMs(S(n))) denote the class of sets recognized by S(n) space-bounded 2-ptmss (resp. tr2-ptmss) with error probability less than ½, where S(n): N→N is a function of one variable n (= the side-length of input tapes). Let TR2-PTM(L(m,n)) denote the class of sets recognized by L(m,n) space-bounded tr2-ptms with error probability less than ½, where L(m,n): N2→N is a function of two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). The main results of this paper are: (1) 2-NFAs - TR2-PTMs(S(n))≠ϕ for any S(n)=o(log n), where 2-NFAs denotes the class of sets of square tapes accepted by two-dimensional nondeterministic finite automata, (2) TR2-PTMsS(n)2-PTMs(S(n)) for any S(n)=o(log n), and (3) for any function g(n)=o(log n) (resp. g(n)=o(log n/log log n)) and any monotonic nondecreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, TR2-PTM(f(m)+g(n)) (resp. TR2-PTM(f(m)×g(n))) is not closed under column catenation, column closure, and projection. Additionally, we show that two-dimensional nondeterministic finite automata are equivalent to two-dimensional probabilistic finite automata with one-sided error in accepting power.