Constant leaf-size hierarchy of two-dimensional alternating turing machines

Abstract

“Leaf-size” (or “branching”) is the minimum number of leaves of some accepting computation tree 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-bonnded two-dimensional alternating Turing machines which have only universal states are equivalent to the same space bounded determinisitic 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 n ^k+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.