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

Abstract

This paper introduces a simple, natural complexity measure for space bounded two-dimensional alternating Turing machines, called “leaf-size”, and provides a hierarchy of complexity classes based on leaf-size bounded computations. Specifically, we show that for any positive integer k⩾1 and for any two functions L: N→ N and L′: N→ N such that (1) L is a two-dimensionally space-constructible function such that L( m) k+1 ⩽ m ( m⩾1), (2) lim m→∞L(m)L′(m) k log m=0 and (3) lim m→∞L′(m) L(m)=0 , L( m) space bounded and L( m>) k leaf-size bounded two-dimensional alternating Turing machines are more powerful than L( m) space bounded and L′( m) k leaf-size bounded two-dimensional alternating Turing machines.