2DFAs and Regular Sets

Abstract

In this lecture we show that 2DFAs are no more powerful than ordinary DFAs. Here is the idea. Consider a long input string broken up in an ar-bitrary place into two substrings xz. How much information about x can the machine carry across the boundary from x into z? Since the machine is two-way; it can cross the boundary between x and z several times. Each time it crosses the boundary moving from right to left, that is, from z into x, it does so in some state q. When it crosses the boundary again moving from left to right (if ever), it comes out of x in some state, say p. Now if it ever goes into x in the future in state q again, it will emerge again in state p. because its future action is completely determined by its current configuration (state and head position). Moreover, the state p depends only on q and x. We will write Tx (q) = p to denote this relationship. We can keep track of all such information by means of a finite table

$${T_x}:(Q \cup \{ \bullet \} ) \to (Q \cup \{ \bot \} ), $$

where Q is the set of states of the 2DFA M, and • and ⊥ are two other objects not in Q whose purpose is described below.