Closure properties of the classes of sets recognized by space-bounded two-dimensional probabilistic turing machines

• Authors: Tokio Okazaki;Katsushi Inoue;Akira Ito;Yue Wang
• Venue: Information Sciences: an International Journal
• Year: 1999

This paper investigates closure properties of the classes of sets recognized by spacebounded two-dimensional probabilistic Turing machines with error probability less than 12. Let 2-PTM(L(m,n)) be the class of sets recognized by L(m,n) space-bounded two-dimensional probabilistic Turing machines with error probability less than 12, where L(m, n): N^2 - N (N denotes the set of all the 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 first show that (i) for any function @?(m) = o(logm) (resp., @?(m) = o(log m/log log m)) and any monotonic nondecreasing function g(n) which can be constructed by some one-dimensional deterministic Turing machine, 2-PTM(L(m, n)) is not closed under row catenation, row closure, and projection, where L(m, n) = @?(m)+ g(n) (resp., L(m,n) = @?(m) x g(n)), and (ii) for any function g(n) = o(log n) (resp., g(n) = o(logn/loglogn)) and any monotonic nondecreasing function @?(m) which can be constructed by some one-dimensional deterministic Turing machine, 2-PTM(L(m, n)) is not closed under column catenation, column closure, and projection, where L(m,n) = @?(m)+g(n)(resp., L(m, n) = @?(m) x g(n)). We then show that 2-PTM^T(L(m, n)) is closed under union, intersection, and complementation for any L(m, n), where 2-PTM^T(L(m, n)) denotes the class of sets recognized by L(m, n) space-bounded two-dimensional probabilistic Turing machines with error probability less than 12 which always halt in the accepting or rejecting state for all the input tapes.