Nondeterminism and the size of two way finite automata

An important goal of the theory of computation is the classification of languages according to computational difficulty. Classes such as P, NP, and LOGSPACE provide a natural framework for this, though it is a fundamental open problem to demonstrate languages distinguishing them. The complete languages of Cook, Karp, and others [1-7] are candidates for such languages in the sense that, if the classes are in fact different, these languages witness the difference. We consider two questions on regular languages resembling these open problems. One of these questions concerns 2-way non-deterministic (2n) and 2-way deterministic (2d) finite automata: