On three-way two-dimensional turing machines

Abstract

This paper solves several open problems concerning closure properties of three-way tape-bounded Turing machines. It is shown that: (1) the class of sets of square tapes accepted by nondeterministic three-way L( m) tape-bounded Turing machines is closed under complementation if L( m)⩾ m 2 is constructible, (2) the class of sets of square tapes accepted by nondeterministic three-way L( m) tape-bounded Turing machines is closed neither under row nor column cyclic closure if L( m) < log m, (3) if L( m, n) = mg( n) or L( m, n) = g( m) n, then L( m, n)⩾ mn space is necessary for the class of sets of (general) tapes accepted by deterministic three-way L( m, n) tape-bounded Turing machines to be closed under row or column catenation, row or column closure or row or column cyclic closure.