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

Abstract

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-ptms's (resp. tr2-ptms's) 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-ptm's 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)[Formula: see text]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.